Large language models (LLMs) and their agentic frameworks are increasingly adopted to perform development tasks such as automated program repair (APR). While prior work has identified security risks in LLM-generated code, most have focused on synthetic, simplified, or isolated tasks that lack the complexity of real-world program repair. In this study, we present the first large-scale security analysis of LLM-generated patches using 20,000+ GitHub issues. We evaluate patches proposed by developers, a standalone LLM (Llama 3.3 Instruct-70B), and three top-performing agentic frameworks (OpenHands, AutoCodeRover, HoneyComb). Finally, we analyze a wide range of code, issue, and project-level factors to understand the conditions under which generating insecure patches is more likely. Our findings reveal that Llama introduces many new vulnerabilities, exhibiting unique patterns not found in developers' code. Agentic workflows also generate a number of vulnerabilities, particularly when given more autonomy. We find that vulnerabilities in LLM-generated patches are associated with distinctive code characteristics and are commonly observed in issues missing specific types of information. These results suggest that contextual factors play a critical role in the security of the generated patches and point toward the need for proactive risk assessment methods that account for both issue and code-level information.