Subgroups — C₃₁×#25049 #21481808

Order 7936.

776 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2421	C₂	1	2	yes	maximal
4	C₃₁×#2074	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2036	C₂	1	2	yes	maximal
6	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#470	—	2	4	no	—
17	C₃₁×#441	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#400	—	2	2	no	—
29	C₃₁×#397	—	2	2	no	—
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#388	—	2	2	no	—
33	C₃₁×#387	—	2	2	no	—
34	C₃₁×#382	C₂²	1	2	yes	—
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	4	no	—
56	C₄×Q₈×C₃₁	—	4	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#107	—	2	2	no	—
70	C₃₁×#107	—	4	2	no	—
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×#106	—	2	2	no	—
73	C₃₁×#103	—	2	4	no	—
74	C₃₁×#103	—	2	4	no	—
75	C₃₁×#103	—	2	4	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×#97	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	2	yes	—
86	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	Q₁₆×C₃₁	—	2	8	no	—
108	M₁₆×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₃₁	#41	1	2	yes	—
119	C₂×C₈×C₃₁	#41	1	2	yes	—
120	C₂×C₈×C₃₁	#41	1	2	yes	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	—	4	4	no	—
129	C₄⋊C₄×C₃₁	—	4	4	no	—
130	C₂²⋊C₄×C₃₁	—	4	2	no	—
131	C₂²⋊C₄×C₃₁	—	4	4	no	—
132	C₂²⋊C₄×C₃₁	—	4	4	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	4	4	no	—
138	#25049	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	1	no	—
140	C₂³×C₃₁	—	2	1	no	—
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	4	no	—
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	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	1	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	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	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₈×C₃₁	#123	1	4	yes	—
184	C₂×#470	C₂×C₃₁	1	1	yes	characteristic
185	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#2421	C₂×C₃₁	1	2	yes	—
187	#2074	C₂×C₃₁	1	2	yes	complemented
188	#2036	C₂×C₃₁	1	2	yes	—
189	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1863	C₂×C₃₁	1	2	yes	complemented
192	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1809	C₂×C₃₁	1	1	yes	characteristic
194	#1753	C₂×C₃₁	1	1	yes	characteristic
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	—	4	1	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	#532	1	1	yes	characteristic
201	C₂²×C₃₁	D₄²	1	1	yes	characteristic
202	C₂²×C₃₁	#547	1	1	yes	characteristic
203	C₂²×C₃₁	#575	1	2	yes	—
204	C₂²×C₃₁	#578	1	2	yes	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	4	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	8	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
213	C₄×C₃₁	#542	1	1	yes	characteristic
214	C₄×C₃₁	#547	1	1	yes	characteristic
215	C₄×C₃₁	#553	1	1	yes	characteristic
216	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
217	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
219	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
220	#470	—	2	4	no	—
221	#441	—	2	4	no	—
222	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#107	C₂²×C₃₁	1	2	yes	—
224	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#103	C₂²×C₃₁	1	2	yes	—
227	C₂×#103	C₂²×C₃₁	1	2	yes	—
228	C₂×#103	C₂²×C₃₁	1	2	yes	—
229	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
232	#400	—	2	2	no	—
233	#397	—	2	2	no	—
234	#391	C₂²×C₃₁	1	2	yes	—
235	#389	C₂²×C₃₁	1	2	yes	—
236	#388	—	2	2	no	—
237	#387	—	2	2	no	—
238	#382	C₂²×C₃₁	1	2	yes	—
239	#380	C₂²×C₃₁	1	2	yes	—
240	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
241	#340	C₂²×C₃₁	1	2	yes	—
242	#340	C₂²×C₃₁	1	2	yes	—
243	#336	C₂²×C₃₁	1	1	yes	characteristic
244	#336	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₃₁	—	8	1	no	—
246	C₂×C₃₁	#2422	1	1	yes	characteristic
247	C₂×C₃₁	#3177	1	1	yes	characteristic
248	C₂×C₃₁	#3183	1	1	yes	characteristic
249	C₂×C₃₁	#3277	1	2	yes	—
250	C₂×C₃₁	#3349	1	2	yes	—
251	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
252	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×Q₁₆	—	2	2	no	—
255	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²⋊Q₈	—	2	4	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×Q₈	—	4	2	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	C₂³×C₃₁	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
276	C₂×C₄²	—	2	2	no	—
277	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
279	#107	—	2	2	no	—
280	#107	—	4	2	no	—
281	#106	—	2	2	no	—
282	#106	—	2	2	no	—
283	#103	—	2	4	no	—
284	#103	—	2	4	no	—
285	#103	—	2	4	no	—
286	C₂²⋊C₈	—	2	2	no	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	D₄×C₃₁	1	2	yes	—
296	C₃₁	#25049	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
317	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
318	Q₁₆	—	2	8	no	—
319	M₁₆	—	4	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
329	C₂×C₈	#41×C₃₁	1	2	yes	—
330	C₂×C₈	#41×C₃₁	1	2	yes	—
331	C₂×C₈	#41×C₃₁	1	2	yes	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	4	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	4	4	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	2	1	no	—
351	C₂³	—	2	1	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
355	Q₈	—	2	4	no	—
356	Q₈	—	2	4	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	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	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	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	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	C₃₁×#123	1	4	yes	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	2	no	—
399	C₂²	C₃₁×#532	1	1	yes	characteristic
400	C₂²	D₄²×C₃₁	1	1	yes	characteristic
401	C₂²	C₃₁×#547	1	1	yes	characteristic
402	C₂²	C₃₁×#575	1	2	yes	—
403	C₂²	C₃₁×#578	1	2	yes	—
404	C₄	—	4	2	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	4	no	—
407	C₄	—	4	2	no	—
408	C₄	—	8	2	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	2	no	—
411	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
412	C₄	C₃₁×#542	1	1	yes	characteristic
413	C₄	C₃₁×#547	1	1	yes	characteristic
414	C₄	C₃₁×#553	1	1	yes	characteristic
415	C₂	—	8	1	no	—
416	C₂	C₃₁×#2422	1	1	yes	characteristic
417	C₂	C₃₁×#3177	1	1	yes	characteristic
418	C₂	C₃₁×#3183	1	1	yes	characteristic
419	C₂	C₃₁×#3277	1	2	yes	—
420	C₂	C₃₁×#3349	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#441	C₂	semidirect
4. C₃₁×#2074	C₂	semidirect
6. C₃₁×#1969	C₂	semidirect
7. C₃₁×#1943	C₂	semidirect
8. C₃₁×#1863	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
138. #25049	C₃₁	direct (N × H)
185. C₂×#441	C₂×C₃₁	semidirect
187. #2074	C₂×C₃₁	semidirect
189. #1969	C₂×C₃₁	semidirect
190. #1943	C₂×C₃₁	semidirect
191. #1863	C₂×C₃₁	semidirect
192. #1846	C₂×C₃₁	semidirect
296. C₃₁	#25049	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×#470	2
C₂×C₃₁×#441	2
C₃₁×#2421	2
C₃₁×#2074	2
C₃₁×#2036	2
C₃₁×#1969	2
C₃₁×#1943	2
C₃₁×#1863	2
C₃₁×#1846	2
C₃₁×#1809	2
C₃₁×#1753	2
#25049	31

Subgroups — C31×#25049 #21481808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25049 #21481808