Subgroups — C₃₁×#17770 #21474529

Order 7936.

1248 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2990	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1555	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1551	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#331	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×H(5,𝔽₂)	C₄	1	2	yes	—
10	C₃₁×#550	—	2	2	no	—
11	D₄²×C₃₁	—	2	2	no	—
12	D₄²×C₃₁	—	2	4	no	—
13	C₃₁×#535	C₄	1	4	yes	—
14	C₃₁×#535	C₄	1	2	yes	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#127	C₂²	1	2	yes	—
19	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#446	—	2	2	no	—
21	C₃₁×#435	—	2	4	no	—
22	C₈×D₄×C₃₁	—	2	2	no	—
23	C₃₁×#432	C₂²	1	4	yes	complemented
24	C₃₁×#431	C₂²	1	2	yes	complemented
25	C₃₁×#423	C₂²	1	2	yes	complemented
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
27	C₃₁×#333	C₂²	1	2	yes	complemented
28	C₃₁×#333	C₂²	1	2	yes	complemented
29	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#331	C₂²	1	4	yes	complemented
32	C₃₁×#331	C₂²	1	4	yes	complemented
33	C₃₁×#331	C₂²	1	2	yes	complemented
34	C₃₁×H(5,𝔽₂)	—	4	8	no	—
35	C₂×#41×C₃₁	—	2	2	no	—
36	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	—	2	4	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	2	4	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	—	2	4	no	—
47	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	4	no	—
50	C₃₁×#127	C₂×C₄	1	2	yes	—
51	C₃₁×#127	C₂×C₄	1	4	yes	—
52	C₃₁×#127	C₂³	1	2	yes	—
53	C₃₁×#127	C₂³	1	1	yes	characteristic
54	C₃₁×#124	C₂×C₄	1	2	yes	—
55	C₃₁×#124	C₂×C₄	1	4	yes	—
56	C₃₁×#121	—	2	4	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	—	2	8	no	—
59	C₃₁×#121	—	2	4	no	—
60	C₃₁×#120	—	2	4	no	—
61	C₃₁×#120	—	2	8	no	—
62	C₄×Q₈×C₃₁	D₄	1	2	yes	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₄×D₄×C₃₁	D₄	1	2	yes	—
66	C₄×D₄×C₃₁	D₄	1	4	yes	—
67	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#117	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	—	2	4	no	—
78	C₃₁×#97	—	2	4	no	—
79	C₄×C₈×C₃₁	—	2	2	no	—
80	C₂⁴×C₃₁	—	2	2	no	—
81	C₂⁴×C₃₁	—	2	4	no	—
82	#41×C₃₁	—	2	4	no	—
83	#41×C₃₁	—	4	8	no	—
84	#41×C₃₁	—	4	4	no	—
85	#41×C₃₁	—	4	8	no	—
86	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	8	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	4	8	no	—
97	C₂×D₄×C₃₁	—	4	4	no	—
98	C₂×D₄×C₃₁	—	4	16	no	—
99	C₂×D₄×C₃₁	—	4	8	no	—
100	C₂×D₄×C₃₁	—	8	4	no	—
101	C₂×D₄×C₃₁	—	8	2	no	—
102	C₂×D₄×C₃₁	—	8	4	no	—
103	C₂×D₄×C₃₁	—	8	2	no	—
104	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
105	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	M₁₆×C₃₁	—	8	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	4	no	—
118	C₂×C₈×C₃₁	—	8	4	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
121	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
122	C₂²⋊C₄×C₃₁	—	2	4	no	—
123	C₂²⋊C₄×C₃₁	—	2	8	no	—
124	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
125	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
126	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂×D₄	1	2	yes	—
129	C₄²×C₃₁	C₂×D₄	1	4	yes	—
130	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
131	#17770	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	4	4	no	—
135	C₂³×C₃₁	—	4	8	no	—
136	C₂³×C₃₁	—	8	8	no	—
137	C₂³×C₃₁	—	8	4	no	—
138	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
140	Q₈×C₃₁	—	2	2	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	2	4	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	4	no	—
145	D₄×C₃₁	—	4	8	no	—
146	D₄×C₃₁	—	4	4	no	—
147	D₄×C₃₁	—	4	8	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	4	4	no	—
152	D₄×C₃₁	—	4	4	no	—
153	D₄×C₃₁	—	8	8	no	—
154	D₄×C₃₁	—	8	4	no	—
155	D₄×C₃₁	#100	1	4	yes	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	4	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
167	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
171	C₈×C₃₁	—	4	2	no	—
172	C₈×C₃₁	—	4	4	no	—
173	C₈×C₃₁	—	4	2	no	—
174	#3498	C₂×C₃₁	1	1	yes	characteristic
175	#2990	C₂×C₃₁	1	2	yes	complemented
176	#1560	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#1555	C₂×C₃₁	1	2	yes	complemented
178	#1551	C₂×C₃₁	1	4	yes	complemented
179	#1545	C₂×C₃₁	1	2	yes	complemented
180	#1543	C₂×C₃₁	1	1	yes	characteristic, complemented
181	C₂×#331	C₂×C₃₁	1	2	yes	complemented
182	C₂²×C₃₁	—	2	4	no	—
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	—	4	4	no	—
186	C₂²×C₃₁	—	4	2	no	—
187	C₂²×C₃₁	—	4	4	no	—
188	C₂²×C₃₁	—	4	8	no	—
189	C₂²×C₃₁	—	16	8	no	—
190	C₂²×C₃₁	—	16	4	no	—
191	C₂²×C₃₁	C₂×#100	1	4	yes	—
192	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	2	no	—
195	C₄×C₃₁	—	2	4	no	—
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	C₂×#100	1	2	yes	—
199	C₄×C₃₁	#576	1	2	yes	—
200	C₄×C₃₁	#577	1	2	yes	—
201	C₂×H(5,𝔽₂)	C₄×C₃₁	1	2	yes	—
202	#550	—	2	2	no	—
203	D₄²	—	2	2	no	—
204	D₄²	—	2	4	no	—
205	#535	C₄×C₃₁	1	4	yes	—
206	#535	C₄×C₃₁	1	2	yes	—
207	#532	C₂²×C₃₁	1	1	yes	characteristic
208	#532	C₂²×C₃₁	1	2	yes	—
209	C₂×#127	C₂²×C₃₁	1	2	yes	—
210	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
211	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
212	#446	—	2	2	no	—
213	#435	—	2	4	no	—
214	C₈×D₄	—	2	2	no	—
215	#432	C₂²×C₃₁	1	4	yes	complemented
216	#431	C₂²×C₃₁	1	2	yes	complemented
217	#423	C₂²×C₃₁	1	2	yes	complemented
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
219	#333	C₂²×C₃₁	1	2	yes	complemented
220	#333	C₂²×C₃₁	1	2	yes	complemented
221	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
222	#331	C₂²×C₃₁	1	4	yes	complemented
223	#331	C₂²×C₃₁	1	2	yes	complemented
224	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
225	#331	C₂²×C₃₁	1	4	yes	complemented
226	C₂×C₃₁	—	2	4	no	—
227	C₂×C₃₁	—	4	1	no	—
228	C₂×C₃₁	—	8	2	no	—
229	C₂×C₃₁	—	8	4	no	—
230	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
231	C₂×C₃₁	#2897	1	2	yes	—
232	H(5,𝔽₂)	—	4	8	no	—
233	C₂×#41	—	2	2	no	—
234	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
235	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×D₄	—	2	4	no	—
238	C₂²×D₄	—	2	2	no	—
239	C₂²×D₄	—	2	4	no	—
240	C₂²×D₄	—	2	2	no	—
241	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
242	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂²×C₈	—	2	4	no	—
245	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
246	#127	—	2	2	no	—
247	#127	—	2	4	no	—
248	#127	C₂×C₄×C₃₁	1	4	yes	—
249	#127	C₂×C₄×C₃₁	1	2	yes	—
250	#127	C₂³×C₃₁	1	1	yes	characteristic
251	#127	C₂³×C₃₁	1	2	yes	—
252	#124	C₂×C₄×C₃₁	1	2	yes	—
253	#124	C₂×C₄×C₃₁	1	4	yes	—
254	#121	—	2	4	no	—
255	#121	—	2	2	no	—
256	#121	—	2	8	no	—
257	#121	—	2	4	no	—
258	#120	—	2	4	no	—
259	#120	—	2	8	no	—
260	C₄×Q₈	D₄×C₃₁	1	2	yes	—
261	C₄×D₄	—	2	2	no	—
262	C₄×D₄	—	2	4	no	—
263	C₄×D₄	D₄×C₃₁	1	2	yes	—
264	C₄×D₄	D₄×C₃₁	1	4	yes	—
265	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
266	#117	C₂×C₄×C₃₁	1	2	yes	—
267	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
269	C₄⋊C₈	—	2	4	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
272	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
273	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	2	4	no	—
276	#97	—	2	4	no	—
277	C₄×C₈	—	2	2	no	—
278	C₃₁	#17770	1	1	yes	characteristic, direct product factor, Sylow
279	C₂⁴	—	2	2	no	—
280	C₂⁴	—	2	4	no	—
281	#41	—	2	4	no	—
282	#41	—	4	8	no	—
283	#41	—	4	4	no	—
284	#41	—	4	8	no	—
285	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
286	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	4	no	—
289	C₂×D₄	—	2	4	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	2	8	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	4	no	—
295	C₂×D₄	—	4	8	no	—
296	C₂×D₄	—	4	4	no	—
297	C₂×D₄	—	4	16	no	—
298	C₂×D₄	—	4	8	no	—
299	C₂×D₄	—	8	4	no	—
300	C₂×D₄	—	8	2	no	—
301	C₂×D₄	—	8	4	no	—
302	C₂×D₄	—	8	2	no	—
303	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
304	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
305	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
313	M₁₆	—	8	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	4	no	—
317	C₂×C₈	—	8	4	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
320	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₂²⋊C₄	—	2	8	no	—
323	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
324	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
325	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
326	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₃₁	1	2	yes	—
328	C₄²	C₂×D₄×C₃₁	1	4	yes	—
329	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
330	C₂³	—	2	4	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	4	8	no	—
334	C₂³	—	8	8	no	—
335	C₂³	—	8	4	no	—
336	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
337	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
338	Q₈	—	2	2	no	—
339	Q₈	—	4	2	no	—
340	D₄	—	2	4	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	4	no	—
343	D₄	—	4	8	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	8	no	—
346	D₄	—	4	4	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	4	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	4	no	—
351	D₄	—	8	8	no	—
352	D₄	—	8	4	no	—
353	D₄	C₃₁×#100	1	4	yes	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	8	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
364	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
365	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
366	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
369	C₈	—	4	2	no	—
370	C₈	—	4	4	no	—
371	C₈	—	4	2	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	4	no	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	8	no	—
379	C₂²	—	16	8	no	—
380	C₂²	—	16	4	no	—
381	C₂²	C₂×C₃₁×#100	1	4	yes	—
382	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	2	2	no	—
385	C₄	—	2	4	no	—
386	C₄	—	2	2	no	—
387	C₄	—	4	2	no	—
388	C₄	C₂×C₃₁×#100	1	2	yes	—
389	C₄	C₃₁×#576	1	2	yes	—
390	C₄	C₃₁×#577	1	2	yes	—
391	C₂	—	2	4	no	—
392	C₂	—	4	1	no	—
393	C₂	—	8	2	no	—
394	C₂	—	8	4	no	—
395	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
396	C₂	C₃₁×#2897	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2990	C₂	semidirect
3. C₃₁×#1560	C₂	semidirect
4. C₃₁×#1555	C₂	semidirect
5. C₃₁×#1551	C₂	semidirect
6. C₃₁×#1545	C₂	semidirect
7. C₃₁×#1543	C₂	semidirect
8. C₂×C₃₁×#331	C₂	semidirect
23. C₃₁×#432	C₂²	semidirect
24. C₃₁×#431	C₂²	semidirect
25. C₃₁×#423	C₂²	semidirect
26. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
27. C₃₁×#333	C₂²	semidirect
28. C₃₁×#333	C₂²	semidirect
29. C₃₁×#331	C₂²	semidirect
30. C₃₁×#331	C₂²	semidirect
31. C₃₁×#331	C₂²	semidirect
32. C₃₁×#331	C₂²	semidirect
33. C₃₁×#331	C₂²	semidirect
131. #17770	C₃₁	direct (N × H)
175. #2990	C₂×C₃₁	semidirect
176. #1560	C₂×C₃₁	semidirect
177. #1555	C₂×C₃₁	semidirect
178. #1551	C₂×C₃₁	semidirect
179. #1545	C₂×C₃₁	semidirect
180. #1543	C₂×C₃₁	semidirect
181. C₂×#331	C₂×C₃₁	semidirect
215. #432	C₂²×C₃₁	semidirect
216. #431	C₂²×C₃₁	semidirect
217. #423	C₂²×C₃₁	semidirect
218. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
219. #333	C₂²×C₃₁	semidirect
220. #333	C₂²×C₃₁	semidirect
221. #331	C₂²×C₃₁	semidirect
222. #331	C₂²×C₃₁	semidirect
223. #331	C₂²×C₃₁	semidirect
224. #331	C₂²×C₃₁	semidirect
225. #331	C₂²×C₃₁	semidirect
278. C₃₁	#17770	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3498	2
C₃₁×#2990	2
C₃₁×#1560	2
C₃₁×#1555	2
C₃₁×#1551	2
C₃₁×#1545	2
C₃₁×#1543	2
C₂×C₃₁×#331	2
#17770	31

Subgroups — C31×#17770 #21474529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17770 #21474529