Subgroups — C₃₁×#21148 #21477907

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1940	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1939	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#424	—	2	4	no	—
18	C₃₁×#424	—	2	4	no	—
19	C₃₁×#423	—	2	4	no	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
23	C₃₁×#405	—	2	4	no	—
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#389	C₄	1	2	yes	—
27	C₃₁×#389	C₄	1	2	yes	—
28	C₃₁×#385	C₄	1	4	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂×M₁₆×C₃₁	—	2	2	no	—
38	C₂×M₁₆×C₃₁	—	2	2	no	—
39	C₂×M₁₆×C₃₁	—	2	2	no	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	D₄	1	4	yes	—
56	C₃₁×#117	Q₈	1	4	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₃₁×C₄⋊C₈	—	2	8	no	—
71	C₃₁×C₄⋊C₈	—	2	8	no	—
72	C₃₁×C₄⋊C₈	—	2	8	no	—
73	C₃₁×#97	—	2	4	no	—
74	C₄×C₈×C₃₁	—	2	4	no	—
75	C₃₁×#95	C₂×C₄	1	4	yes	—
76	C₃₁×#95	C₂×C₄	1	4	yes	—
77	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₃₁	#41	1	2	yes	—
91	C₂²×C₄×C₃₁	#41	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₃₁	—	2	8	no	—
94	M₁₆×C₃₁	—	4	4	no	—
95	C₂×C₈×C₃₁	—	2	8	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
111	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	8	no	—
114	C₄⋊C₄×C₃₁	—	4	4	no	—
115	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
116	C₂²⋊C₄×C₃₁	—	2	4	no	—
117	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	C₂×D₄	1	4	yes	—
125	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
126	#21148	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	2	2	no	—
128	C₂³×C₃₁	—	2	2	no	—
129	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₃₁	C₄×D₄	1	2	yes	—
131	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
132	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	4	no	—
135	C₂×C₄×C₃₁	—	2	4	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	4	8	no	—
144	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
145	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₃₁	#121	1	4	yes	—
149	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
150	C₂×C₄×C₃₁	#125	1	4	yes	—
151	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
152	C₂×C₄×C₃₁	#131	1	4	yes	—
153	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
159	C₈×C₃₁	—	2	4	no	—
160	C₈×C₃₁	—	2	2	no	—
161	C₈×C₃₁	—	2	2	no	—
162	C₈×C₃₁	—	4	2	no	—
163	C₈×C₃₁	—	4	2	no	—
164	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
165	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
166	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
167	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
168	#2296	C₂×C₃₁	1	1	yes	characteristic
169	#1940	C₂×C₃₁	1	4	yes	complemented
170	#1939	C₂×C₃₁	1	4	yes	complemented
171	#1729	C₂×C₃₁	1	1	yes	characteristic
172	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
173	C₂²×C₃₁	—	2	4	no	—
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	#389	1	4	yes	—
178	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₃₁	C₂×#131	1	2	yes	—
182	C₂²×C₃₁	#568	1	2	yes	—
183	C₄×C₃₁	—	2	8	no	—
184	C₄×C₃₁	—	2	4	no	—
185	C₄×C₃₁	—	4	4	no	—
186	C₄×C₃₁	—	4	2	no	—
187	C₄×C₃₁	#389	1	4	yes	—
188	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
189	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
190	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
191	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
192	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
198	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
199	#424	—	2	4	no	—
200	#424	—	2	4	no	—
201	#423	—	2	4	no	—
202	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
203	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
204	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
205	#405	—	2	4	no	—
206	C₂×#97	C₂²×C₃₁	1	2	yes	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
208	#389	C₄×C₃₁	1	2	yes	—
209	#389	C₄×C₃₁	1	2	yes	—
210	#385	C₄×C₃₁	1	4	yes	—
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
212	#336	C₂²×C₃₁	1	2	yes	—
213	#336	C₂²×C₃₁	1	2	yes	—
214	#336	C₂²×C₃₁	1	2	yes	—
215	#336	C₂²×C₃₁	1	2	yes	—
216	C₂×C₃₁	—	2	4	no	—
217	C₂×C₃₁	C₂×#389	1	1	yes	characteristic
218	C₂×C₃₁	#2906	1	2	yes	—
219	C₂×C₃₁	#2933	1	2	yes	—
220	C₂×C₃₁	#2960	1	1	yes	characteristic
221	C₂×C₃₁	#2961	1	1	yes	characteristic
222	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
223	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
236	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
237	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
238	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
239	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	#117	D₄×C₃₁	1	4	yes	—
244	#117	Q₈×C₃₁	1	4	yes	—
245	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
248	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
249	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
250	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
257	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
258	C₄⋊C₈	—	2	8	no	—
259	C₄⋊C₈	—	2	8	no	—
260	C₄⋊C₈	—	2	8	no	—
261	#97	—	2	4	no	—
262	C₄×C₈	—	2	4	no	—
263	#95	C₂×C₄×C₃₁	1	4	yes	—
264	#95	C₂×C₄×C₃₁	1	4	yes	—
265	C₃₁	#21148	1	1	yes	characteristic, direct product factor, Sylow
266	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
272	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
273	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
279	C₂²×C₄	#41×C₃₁	1	2	yes	—
280	C₂²×C₄	#41×C₃₁	1	2	yes	—
281	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
282	M₁₆	—	2	8	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	2	8	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	4	no	—
298	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
299	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
300	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	8	no	—
303	C₄⋊C₄	—	4	4	no	—
304	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	C₂×D₄×C₃₁	1	4	yes	—
314	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
318	C₂³	C₄×D₄×C₃₁	1	2	yes	—
319	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
320	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	8	no	—
332	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
333	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
334	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
335	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
336	C₂×C₄	C₃₁×#121	1	4	yes	—
337	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
338	C₂×C₄	C₃₁×#125	1	4	yes	—
339	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
340	C₂×C₄	C₃₁×#131	1	4	yes	—
341	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
347	C₈	—	2	4	no	—
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	2	no	—
356	C₂²	C₃₁×#389	1	4	yes	—
357	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
358	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
359	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
360	C₂²	C₂×C₃₁×#131	1	2	yes	—
361	C₂²	C₃₁×#568	1	2	yes	—
362	C₄	—	2	8	no	—
363	C₄	—	2	4	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	4	no	—
366	C₄	C₃₁×#389	1	4	yes	—
367	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
368	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
369	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
370	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	C₂×C₃₁×#389	1	1	yes	characteristic
373	C₂	C₃₁×#2906	1	2	yes	—
374	C₂	C₃₁×#2933	1	2	yes	—
375	C₂	C₃₁×#2960	1	1	yes	characteristic
376	C₂	C₃₁×#2961	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. C₃₁×#1940	C₂	semidirect
7. C₃₁×#1939	C₂	semidirect
126. #21148	C₃₁	direct (N × H)
169. #1940	C₂×C₃₁	semidirect
170. #1939	C₂×C₃₁	semidirect
265. C₃₁	#21148	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#424	2
C₂×C₃₁×#423	2
C₂×C₃₁×#405	2
C₃₁×#2296	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1729	2
C₂×C₃₁×#336	2
#21148	31

Subgroups — C31×#21148 #21477907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21148 #21477907