Subgroups — C₃₁×#24746 #21481505

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#462	C₂	1	2	yes	maximal
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2066	C₂	1	2	yes	maximal
7	C₃₁×#2043	C₂	1	2	yes	maximal
8	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1878	C₂	1	2	yes	maximal
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#462	—	2	8	no	—
20	C₃₁×#462	—	2	4	no	—
21	C₃₁×#441	—	2	4	no	—
22	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₃₁×#398	—	2	1	no	—
33	C₃₁×#398	—	2	1	no	—
34	C₃₁×#391	C₂²	1	2	yes	—
35	C₃₁×#389	—	2	2	no	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#382	—	2	1	no	—
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	4	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	—	4	2	no	—
65	C₄×Q₈×C₃₁	D₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	2	4	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#103	—	4	2	no	—
92	C₃₁×#97	—	2	2	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	D₄	1	1	yes	characteristic
99	C₃₁×#95	D₄	1	1	yes	characteristic
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
115	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₃₁	#41	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	Q₁₆×C₃₁	—	2	8	no	—
130	Q₁₆×C₃₁	—	4	8	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	2	4	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	4	no	—
148	C₄⋊C₄×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	—	4	4	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₄²×C₃₁	—	2	4	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	4	4	no	—
159	C₄²×C₃₁	—	4	2	no	—
160	C₄²×C₃₁	C₂×D₄	1	2	yes	—
161	C₄²×C₃₁	#41	1	2	yes	—
162	#24746	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₃₁	—	2	2	no	—
165	Q₈×C₃₁	—	2	2	no	—
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	2	2	no	—
169	Q₈×C₃₁	—	4	1	no	—
170	Q₈×C₃₁	—	4	1	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	4	2	no	—
175	Q₈×C₃₁	#121	1	4	yes	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	#121	1	2	yes	—
200	C₂×C₄×C₃₁	#121	1	2	yes	—
201	C₂×C₄×C₃₁	#137	1	4	yes	—
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈×C₃₁	—	2	2	no	—
209	C₈×C₃₁	—	4	2	no	—
210	C₈×C₃₁	—	4	1	no	—
211	C₂×#462	C₂×C₃₁	1	2	yes	—
212	C₂×#462	C₂×C₃₁	1	1	yes	characteristic
213	C₂×#441	C₂×C₃₁	1	1	yes	characteristic
214	#2546	C₂×C₃₁	1	1	yes	characteristic
215	#2351	C₂×C₃₁	1	1	yes	characteristic
216	#2066	C₂×C₃₁	1	2	yes	—
217	#2043	C₂×C₃₁	1	2	yes	—
218	#1960	C₂×C₃₁	1	1	yes	characteristic
219	#1940	C₂×C₃₁	1	1	yes	characteristic
220	#1878	C₂×C₃₁	1	2	yes	—
221	#1805	C₂×C₃₁	1	1	yes	characteristic
222	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
223	C₂²×C₃₁	D₄²	1	1	yes	characteristic
224	C₂²×C₃₁	#548	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×#137	1	2	yes	—
226	C₂²×C₃₁	#578	1	2	yes	—
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	—	8	1	no	—
237	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
238	C₄×C₃₁	#537	1	1	yes	characteristic
239	C₄×C₃₁	#547	1	1	yes	characteristic
240	C₄×C₃₁	#553	1	1	yes	characteristic
241	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
242	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	#462	—	2	8	no	—
249	#462	—	2	4	no	—
250	#441	—	2	4	no	—
251	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
254	C₂×#103	C₂²×C₃₁	1	2	yes	—
255	C₂×#103	C₂²×C₃₁	1	2	yes	—
256	C₂×#103	C₂²×C₃₁	1	2	yes	—
257	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#103	C₂²×C₃₁	1	2	yes	—
260	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
261	#398	—	2	1	no	—
262	#398	—	2	1	no	—
263	#391	C₂²×C₃₁	1	2	yes	—
264	#389	—	2	2	no	—
265	#389	—	2	2	no	—
266	#389	C₂²×C₃₁	1	2	yes	—
267	#382	—	2	1	no	—
268	#382	C₂²×C₃₁	1	1	yes	characteristic
269	C₄×C₄⋊C₄	—	2	1	no	—
270	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
271	#340	C₂²×C₃₁	1	2	yes	—
272	#336	C₂²×C₃₁	1	2	yes	—
273	C₂×C₃₁	#2416	1	1	yes	characteristic
274	C₂×C₃₁	#3039	1	1	yes	characteristic
275	C₂×C₃₁	#3044	1	1	yes	characteristic
276	C₂×C₃₁	#3270	1	2	yes	—
277	C₂×C₃₁	#3374	1	2	yes	—
278	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
279	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×Q₁₆	—	2	2	no	—
282	C₂×Q₁₆	—	2	2	no	—
283	C₂×Q₁₆	—	2	4	no	—
284	C₂×Q₁₆	—	2	4	no	—
285	C₂×Q₁₆	—	2	4	no	—
286	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
287	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄⋊Q₈	—	2	4	no	—
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	—	4	2	no	—
294	C₄×Q₈	—	2	2	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₄×Q₈	—	4	2	no	—
299	C₄×Q₈	D₄×C₃₁	1	4	yes	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
316	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	#106	—	2	2	no	—
318	C₄⋊C₈	—	2	4	no	—
319	C₄⋊C₈	—	2	2	no	—
320	#103	—	2	2	no	—
321	#103	—	2	4	no	—
322	#103	—	2	4	no	—
323	#103	—	2	2	no	—
324	#103	—	2	4	no	—
325	#103	—	4	2	no	—
326	#97	—	2	2	no	—
327	#95	—	2	1	no	—
328	#95	—	2	2	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	#95	D₄×C₃₁	1	1	yes	characteristic
333	#95	D₄×C₃₁	1	1	yes	characteristic
334	C₃₁	#24746	1	1	yes	characteristic, direct product factor, Sylow
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
350	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
351	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×Q₈	#41×C₃₁	1	2	yes	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
364	Q₁₆	—	2	8	no	—
365	Q₁₆	—	4	8	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	4	4	no	—
383	C₄⋊C₄	—	4	4	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₄²	—	2	4	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	4	4	no	—
394	C₄²	—	4	2	no	—
395	C₄²	C₂×D₄×C₃₁	1	2	yes	—
396	C₄²	#41×C₃₁	1	2	yes	—
397	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	Q₈	—	2	2	no	—
399	Q₈	—	2	2	no	—
400	Q₈	—	2	2	no	—
401	Q₈	—	2	2	no	—
402	Q₈	—	2	2	no	—
403	Q₈	—	4	1	no	—
404	Q₈	—	4	1	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	2	no	—
409	Q₈	C₃₁×#121	1	4	yes	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	C₃₁×#121	1	2	yes	—
434	C₂×C₄	C₃₁×#121	1	2	yes	—
435	C₂×C₄	C₃₁×#137	1	4	yes	—
436	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	4	2	no	—
444	C₈	—	4	1	no	—
445	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
446	C₂²	D₄²×C₃₁	1	1	yes	characteristic
447	C₂²	C₃₁×#548	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×#137	1	2	yes	—
449	C₂²	C₃₁×#578	1	2	yes	—
450	C₄	—	2	4	no	—
451	C₄	—	2	4	no	—
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	1	no	—
460	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
461	C₄	C₃₁×#537	1	1	yes	characteristic
462	C₄	C₃₁×#547	1	1	yes	characteristic
463	C₄	C₃₁×#553	1	1	yes	characteristic
464	C₂	C₃₁×#2416	1	1	yes	characteristic
465	C₂	C₃₁×#3039	1	1	yes	characteristic
466	C₂	C₃₁×#3044	1	1	yes	characteristic
467	C₂	C₃₁×#3270	1	2	yes	—
468	C₂	C₃₁×#3374	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
162. #24746	C₃₁	direct (N × H)
334. C₃₁	#24746	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×#462	2
C₂×C₃₁×#441	2
C₃₁×#2546	2
C₃₁×#2351	2
C₃₁×#2066	2
C₃₁×#2043	2
C₃₁×#1960	2
C₃₁×#1940	2
C₃₁×#1878	2
C₃₁×#1805	2
#24746	31

Subgroups — C31×#24746 #21481505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24746 #21481505