Subgroups — C₃₁×#21025 #21477784

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 396 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×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1917	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1916	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1728	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	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₃₁×#405	—	2	4	no	—
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#384	C₄	1	1	yes	characteristic
24	C₃₁×#384	C₄	1	1	yes	characteristic
25	C₃₁×#382	C₄	1	1	yes	characteristic
26	C₃₁×#382	C₄	1	1	yes	characteristic
27	C₃₁×#380	C₄	1	4	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	4	yes	—
31	C₃₁×#336	C₂²	1	4	yes	—
32	C₃₁×#336	C₂²	1	4	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	Q₈	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	D₄	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	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	—	2	2	no	—
56	C₃₁×#117	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₃₁×C₄⋊C₈	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	4	no	—
74	C₃₁×#97	—	2	4	no	—
75	C₄×C₈×C₃₁	—	2	4	no	—
76	C₃₁×#95	C₂×C₄	1	4	yes	—
77	C₃₁×#95	C₂×C₄	1	4	yes	—
78	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	4	no	—
81	C₂²×C₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₃₁	#41	1	2	yes	—
92	C₂²×C₄×C₃₁	#41	1	2	yes	—
93	C₂²×C₄×C₃₁	#41	1	2	yes	—
94	C₂²×C₄×C₃₁	#41	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
96	M₁₆×C₃₁	—	2	8	no	—
97	M₁₆×C₃₁	—	4	4	no	—
98	C₂×C₈×C₃₁	—	2	8	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	1	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	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	8	no	—
120	C₄⋊C₄×C₃₁	—	4	4	no	—
121	C₂²⋊C₄×C₃₁	—	2	4	no	—
122	C₂²⋊C₄×C₃₁	—	2	4	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	#21025	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	#117	1	2	yes	—
135	C₂³×C₃₁	C₄×D₄	1	2	yes	—
136	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
137	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	4	no	—
146	C₂×C₄×C₃₁	—	2	4	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	4	no	—
149	C₂×C₄×C₃₁	—	2	4	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	4	8	no	—
152	C₂×C₄×C₃₁	#117	1	2	yes	—
153	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
155	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₄×C₃₁	#123	1	4	yes	—
157	C₂×C₄×C₃₁	#125	1	4	yes	—
158	C₂×C₄×C₃₁	#126	1	4	yes	—
159	C₂×C₄×C₃₁	#131	1	4	yes	—
160	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₈×C₃₁	—	2	4	no	—
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	2	2	no	—
169	C₈×C₃₁	—	4	2	no	—
170	C₈×C₃₁	—	4	2	no	—
171	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
172	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
173	#2285	C₂×C₃₁	1	1	yes	characteristic
174	#1917	C₂×C₃₁	1	4	yes	complemented
175	#1916	C₂×C₃₁	1	4	yes	complemented
176	#1729	C₂×C₃₁	1	1	yes	characteristic
177	#1729	C₂×C₃₁	1	1	yes	characteristic
178	#1728	C₂×C₃₁	1	1	yes	characteristic
179	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
180	C₂²×C₃₁	—	2	4	no	—
181	C₂²×C₃₁	—	2	2	no	—
182	C₂²×C₃₁	—	2	2	no	—
183	C₂²×C₃₁	—	2	4	no	—
184	C₂²×C₃₁	#387	1	4	yes	—
185	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
186	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
188	C₂²×C₃₁	C₂×#131	1	2	yes	—
189	C₂²×C₃₁	#568	1	2	yes	—
190	C₄×C₃₁	—	2	4	no	—
191	C₄×C₃₁	—	4	2	no	—
192	C₄×C₃₁	—	4	2	no	—
193	C₄×C₃₁	—	4	2	no	—
194	C₄×C₃₁	—	4	2	no	—
195	C₄×C₃₁	—	4	2	no	—
196	C₄×C₃₁	#387	1	4	yes	—
197	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
198	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
199	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
200	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
201	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
207	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
208	#424	—	2	4	no	—
209	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
210	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
211	#405	—	2	4	no	—
212	C₂×#97	C₂²×C₃₁	1	2	yes	—
213	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
214	#384	C₄×C₃₁	1	1	yes	characteristic
215	#384	C₄×C₃₁	1	1	yes	characteristic
216	#382	C₄×C₃₁	1	1	yes	characteristic
217	#382	C₄×C₃₁	1	1	yes	characteristic
218	#380	C₄×C₃₁	1	4	yes	—
219	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
220	#336	C₂²×C₃₁	1	2	yes	—
221	#336	C₂²×C₃₁	1	4	yes	—
222	#336	C₂²×C₃₁	1	4	yes	—
223	#336	C₂²×C₃₁	1	4	yes	—
224	#336	C₂²×C₃₁	1	2	yes	—
225	C₂×C₃₁	—	2	4	no	—
226	C₂×C₃₁	C₂×#387	1	1	yes	characteristic
227	C₂×C₃₁	#2924	1	2	yes	—
228	C₂×C₃₁	#2931	1	1	yes	characteristic
229	C₂×C₃₁	#2932	1	1	yes	characteristic
230	C₂×C₃₁	#2962	1	2	yes	—
231	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
232	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	2	2	no	—
243	C₂²×C₈	—	2	2	no	—
244	C₂²×C₈	D₄×C₃₁	1	4	yes	—
245	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
246	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
247	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
250	#117	—	2	2	no	—
251	#117	—	2	2	no	—
252	#117	—	2	2	no	—
253	#117	—	2	2	no	—
254	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
260	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
261	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×C₄²	—	2	2	no	—
263	C₂×C₄²	—	2	2	no	—
264	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
267	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
269	C₄⋊C₈	—	2	4	no	—
270	C₄⋊C₈	—	2	4	no	—
271	#97	—	2	4	no	—
272	C₄×C₈	—	2	4	no	—
273	#95	C₂×C₄×C₃₁	1	4	yes	—
274	#95	C₂×C₄×C₃₁	1	4	yes	—
275	C₃₁	#21025	1	1	yes	characteristic, direct product factor, Sylow
276	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
281	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
284	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
288	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
289	C₂²×C₄	#41×C₃₁	1	2	yes	—
290	C₂²×C₄	#41×C₃₁	1	2	yes	—
291	C₂²×C₄	#41×C₃₁	1	2	yes	—
292	C₂²×C₄	#41×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
294	M₁₆	—	2	8	no	—
295	M₁₆	—	4	4	no	—
296	C₂×C₈	—	2	8	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	4	4	no	—
310	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
311	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
312	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	—	4	4	no	—
319	C₂²⋊C₄	—	2	4	no	—
320	C₂²⋊C₄	—	2	4	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	C₃₁×#117	1	2	yes	—
332	C₂³	C₄×D₄×C₃₁	1	2	yes	—
333	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
334	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	8	no	—
349	C₂×C₄	C₃₁×#117	1	2	yes	—
350	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
351	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
352	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
353	C₂×C₄	C₃₁×#123	1	4	yes	—
354	C₂×C₄	C₃₁×#125	1	4	yes	—
355	C₂×C₄	C₃₁×#126	1	4	yes	—
356	C₂×C₄	C₃₁×#131	1	4	yes	—
357	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
359	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
362	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
363	C₈	—	2	4	no	—
364	C₈	—	2	2	no	—
365	C₈	—	2	2	no	—
366	C₈	—	4	2	no	—
367	C₈	—	4	2	no	—
368	C₂²	—	2	4	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	4	no	—
372	C₂²	C₃₁×#387	1	4	yes	—
373	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
374	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
375	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
376	C₂²	C₂×C₃₁×#131	1	2	yes	—
377	C₂²	C₃₁×#568	1	2	yes	—
378	C₄	—	2	4	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	2	no	—
381	C₄	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	C₃₁×#387	1	4	yes	—
385	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
386	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
387	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
388	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
389	C₂	—	2	4	no	—
390	C₂	C₂×C₃₁×#387	1	1	yes	characteristic
391	C₂	C₃₁×#2924	1	2	yes	—
392	C₂	C₃₁×#2931	1	1	yes	characteristic
393	C₂	C₃₁×#2932	1	1	yes	characteristic
394	C₂	C₃₁×#2962	1	2	yes	—

	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
4. C₃₁×#1917	C₂	semidirect
5. C₃₁×#1916	C₂	semidirect
131. #21025	C₃₁	direct (N × H)
174. #1917	C₂×C₃₁	semidirect
175. #1916	C₂×C₃₁	semidirect
275. C₃₁	#21025	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×#405	2
C₃₁×#2285	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1729	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#21025	31

Subgroups — C31×#21025 #21477784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21025 #21477784