Subgroups — C₃₁×#22197 #21478956

Order 7936.

822 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2323	C₂	1	2	yes	maximal
3	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1894	C₂	1	2	yes	maximal
6	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×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	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₃₁×#467	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₃₁×#391	C₂²	1	2	yes	—
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#388	—	2	2	no	—
28	C₃₁×#387	—	2	2	no	—
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×Q₁₆×C₃₁	—	4	1	no	—
39	C₂×Q₁₆×C₃₁	—	4	1	no	—
40	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	—	4	1	no	—
43	C₂×M₁₆×C₃₁	—	4	1	no	—
44	C₂²×C₈×C₃₁	—	4	1	no	—
45	C₂²×C₈×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×C₂²⋊Q₈	—	2	4	no	—
51	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
52	C₄×Q₈×C₃₁	—	4	2	no	—
53	C₄×Q₈×C₃₁	—	4	2	no	—
54	C₃₁×#117	—	4	1	no	—
55	C₃₁×#117	—	4	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₃₁×#107	—	4	1	no	—
71	C₃₁×#107	—	4	1	no	—
72	C₃₁×#107	C₂×C₄	1	4	yes	—
73	C₃₁×#103	—	4	2	no	—
74	C₃₁×#103	—	4	2	no	—
75	C₃₁×#103	C₂×C₄	1	8	yes	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	#41	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	Q₁₆×C₃₁	—	4	4	no	—
103	M₁₆×C₃₁	—	4	4	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	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	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
114	C₄⋊C₄×C₃₁	—	2	8	no	—
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	2	no	—
120	C₄⋊C₄×C₃₁	—	4	1	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	—	4	4	no	—
123	C₄⋊C₄×C₃₁	—	8	2	no	—
124	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
125	C₂²⋊C₄×C₃₁	—	2	8	no	—
126	C₂²⋊C₄×C₃₁	—	4	2	no	—
127	C₂²⋊C₄×C₃₁	—	8	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	4	4	no	—
131	C₄²×C₃₁	—	8	2	no	—
132	#22197	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	2	1	no	—
134	C₂³×C₃₁	—	2	1	no	—
135	C₂³×C₃₁	—	2	1	no	—
136	C₂³×C₃₁	—	2	1	no	—
137	C₂³×C₃₁	—	2	1	no	—
138	C₂³×C₃₁	—	2	1	no	—
139	C₂³×C₃₁	C₄×D₄	1	2	yes	—
140	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	2	4	no	—
142	Q₈×C₃₁	—	2	4	no	—
143	Q₈×C₃₁	—	4	2	no	—
144	Q₈×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
176	C₈×C₃₁	—	2	2	no	—
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	—	4	2	no	—
179	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
180	#2323	C₂×C₃₁	1	2	yes	—
181	#1928	C₂×C₃₁	1	2	yes	complemented
182	#1918	C₂×C₃₁	1	2	yes	complemented
183	#1894	C₂×C₃₁	1	2	yes	—
184	#1878	C₂×C₃₁	1	2	yes	complemented
185	#1863	C₂×C₃₁	1	2	yes	complemented
186	#1808	C₂×C₃₁	1	1	yes	characteristic
187	#1729	C₂×C₃₁	1	1	yes	characteristic
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	#534	1	1	yes	characteristic
197	C₂²×C₃₁	#537	1	1	yes	characteristic
198	C₂²×C₃₁	#576	1	1	yes	characteristic
199	C₂²×C₃₁	#578	1	1	yes	characteristic
200	C₂²×C₃₁	#578	1	1	yes	characteristic
201	C₂²×C₃₁	#578	1	1	yes	characteristic
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	4	4	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	4	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	—	8	2	no	—
209	C₄×C₃₁	#518	1	1	yes	characteristic
210	C₄×C₃₁	#520	1	1	yes	characteristic
211	C₄×C₃₁	#538	1	1	yes	characteristic
212	C₄×C₃₁	#539	1	1	yes	characteristic
213	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
214	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
215	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
218	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
219	#467	C₄×C₃₁	1	8	yes	complemented
220	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#103	C₂²×C₃₁	1	2	yes	—
224	C₂×#103	C₂²×C₃₁	1	2	yes	—
225	C₂×#103	C₂²×C₃₁	1	2	yes	—
226	#391	C₂²×C₃₁	1	2	yes	—
227	#389	—	2	2	no	—
228	#389	C₂²×C₃₁	1	2	yes	—
229	#388	—	2	2	no	—
230	#388	—	2	2	no	—
231	#387	—	2	2	no	—
232	#380	C₂²×C₃₁	1	2	yes	—
233	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
234	#340	C₂²×C₃₁	1	2	yes	—
235	#340	C₂²×C₃₁	1	2	yes	—
236	#340	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	2	yes	—
239	C₂×C₃₁	—	4	2	no	—
240	C₂×C₃₁	#2325	1	1	yes	characteristic
241	C₂×C₃₁	#2950	1	1	yes	characteristic
242	C₂×C₃₁	#2951	1	1	yes	characteristic
243	C₂×C₃₁	#3202	1	1	yes	characteristic
244	C₂×C₃₁	#3203	1	1	yes	characteristic
245	C₂×C₃₁	#3261	1	1	yes	characteristic
246	C₂×C₃₁	#3262	1	1	yes	characteristic
247	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
248	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×Q₁₆	—	4	1	no	—
250	C₂×Q₁₆	—	4	1	no	—
251	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	4	1	no	—
254	C₂×M₁₆	—	4	1	no	—
255	C₂²×C₈	—	4	1	no	—
256	C₂²×C₈	—	4	1	no	—
257	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
258	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
259	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₂²⋊Q₈	—	2	4	no	—
262	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
263	C₄×Q₈	—	4	2	no	—
264	C₄×Q₈	—	4	2	no	—
265	#117	—	4	1	no	—
266	#117	—	4	1	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
281	#107	—	4	1	no	—
282	#107	—	4	1	no	—
283	#107	C₂×C₄×C₃₁	1	4	yes	—
284	#103	—	4	2	no	—
285	#103	—	4	2	no	—
286	#103	C₂×C₄×C₃₁	1	8	yes	—
287	#95	—	2	2	no	—
288	#95	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	2	no	—
292	C₃₁	#22197	1	1	yes	characteristic, direct product factor, Sylow
293	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	#41×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
314	Q₁₆	—	4	4	no	—
315	M₁₆	—	4	4	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	4	4	no	—
324	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	4	no	—
335	C₄⋊C₄	—	8	2	no	—
336	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
337	C₂²⋊C₄	—	2	8	no	—
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	8	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	4	4	no	—
343	C₄²	—	8	2	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	1	no	—
347	C₂³	—	2	1	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	C₄×D₄×C₃₁	1	2	yes	—
351	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
352	Q₈	—	2	4	no	—
353	Q₈	—	2	4	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
381	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
384	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
385	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
386	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
387	C₈	—	2	2	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
398	C₂²	C₃₁×#534	1	1	yes	characteristic
399	C₂²	C₃₁×#537	1	1	yes	characteristic
400	C₂²	C₃₁×#576	1	1	yes	characteristic
401	C₂²	C₃₁×#578	1	1	yes	characteristic
402	C₂²	C₃₁×#578	1	1	yes	characteristic
403	C₂²	C₃₁×#578	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	4	4	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	4	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	2	no	—
411	C₄	C₃₁×#518	1	1	yes	characteristic
412	C₄	C₃₁×#520	1	1	yes	characteristic
413	C₄	C₃₁×#538	1	1	yes	characteristic
414	C₄	C₃₁×#539	1	1	yes	characteristic
415	C₂	—	4	2	no	—
416	C₂	C₃₁×#2325	1	1	yes	characteristic
417	C₂	C₃₁×#2950	1	1	yes	characteristic
418	C₂	C₃₁×#2951	1	1	yes	characteristic
419	C₂	C₃₁×#3202	1	1	yes	characteristic
420	C₂	C₃₁×#3203	1	1	yes	characteristic
421	C₂	C₃₁×#3261	1	1	yes	characteristic
422	C₂	C₃₁×#3262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1928	C₂	semidirect
4. C₃₁×#1918	C₂	semidirect
6. C₃₁×#1878	C₂	semidirect
7. C₃₁×#1863	C₂	semidirect
16. C₃₁×#467	C₄	semidirect
132. #22197	C₃₁	direct (N × H)
181. #1928	C₂×C₃₁	semidirect
182. #1918	C₂×C₃₁	semidirect
184. #1878	C₂×C₃₁	semidirect
185. #1863	C₂×C₃₁	semidirect
219. #467	C₄×C₃₁	semidirect
292. C₃₁	#22197	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×#467	2
C₃₁×#2323	2
C₃₁×#1928	2
C₃₁×#1918	2
C₃₁×#1894	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1808	2
C₃₁×#1729	2
#22197	31

Subgroups — C31×#22197 #21478956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22197 #21478956