Subgroups — C₃₁×#26278 #21483037

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2100	C₂	1	2	yes	maximal
4	C₃₁×#2084	C₂	1	2	yes	maximal
5	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1919	C₂	1	2	yes	maximal
10	C₃₁×#1919	C₂	1	2	yes	maximal
11	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#398	—	2	2	no	—
22	C₃₁×#391	—	2	1	no	—
23	C₃₁×#389	—	2	2	no	—
24	C₃₁×#389	—	2	1	no	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#384	C₂²	1	2	yes	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	—	2	1	no	—
42	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	—	4	4	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	2	2	no	—
51	C₄×Q₈×C₃₁	—	4	4	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#106	—	2	2	no	—
73	C₃₁×#106	—	2	4	no	—
74	C₃₁×#106	Q₈	1	4	yes	complemented
75	C₃₁×C₄⋊C₈	—	2	2	no	—
76	C₄×C₈×C₃₁	—	2	2	no	—
77	C₃₁×#95	—	2	1	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	4	2	no	—
85	C₂×Q₈×C₃₁	—	4	2	no	—
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	SD₁₆	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	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	1	no	—
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	4	no	—
122	C₄⋊C₄×C₃₁	—	4	4	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	—	8	1	no	—
125	C₄⋊C₄×C₃₁	—	8	2	no	—
126	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
127	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
128	C₄⋊C₄×C₃₁	#41	1	2	yes	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	4	4	no	—
132	C₄²×C₃₁	—	4	2	no	—
133	C₄²×C₃₁	—	4	2	no	—
134	C₄²×C₃₁	#41	1	2	yes	—
135	C₄²×C₃₁	#41	1	2	yes	—
136	#26278	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₃₁	—	2	8	no	—
139	Q₈×C₃₁	—	8	4	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	1	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	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	#123	1	2	yes	—
166	C₂×C₄×C₃₁	#123	1	2	yes	—
167	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
168	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
169	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
170	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
176	C₈×C₃₁	—	2	2	no	—
177	C₈×C₃₁	—	4	1	no	—
178	C₈×C₃₁	—	4	2	no	—
179	#2546	C₂×C₃₁	1	1	yes	characteristic
180	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
181	#2100	C₂×C₃₁	1	2	yes	—
182	#2084	C₂×C₃₁	1	2	yes	—
183	#1990	C₂×C₃₁	1	1	yes	characteristic
184	#1988	C₂×C₃₁	1	1	yes	characteristic
185	#1984	C₂×C₃₁	1	1	yes	characteristic
186	#1948	C₂×C₃₁	1	1	yes	characteristic
187	#1919	C₂×C₃₁	1	2	yes	—
188	#1919	C₂×C₃₁	1	2	yes	—
189	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
190	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
191	C₂²×C₃₁	#548	1	1	yes	characteristic
192	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
193	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
194	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂²×C₃₁	#577	1	1	yes	characteristic
196	C₂²×C₃₁	#578	1	1	yes	characteristic
197	C₄×C₃₁	—	2	4	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	—	8	2	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	—	8	1	no	—
206	C₄×C₃₁	#481	1	4	yes	—
207	C₄×C₃₁	#533	1	1	yes	characteristic
208	C₄×C₃₁	#541	1	1	yes	characteristic
209	C₄×C₃₁	#556	1	1	yes	characteristic
210	C₄×C₃₁	#557	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	—	2	1	no	—
212	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄×Q₈	—	2	1	no	—
214	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#106	C₂²×C₃₁	1	2	yes	—
217	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
220	#398	—	2	2	no	—
221	#391	—	2	1	no	—
222	#389	—	2	1	no	—
223	#389	—	2	2	no	—
224	#389	C₂²×C₃₁	1	2	yes	—
225	#389	C₂²×C₃₁	1	2	yes	—
226	#384	C₂²×C₃₁	1	2	yes	—
227	#384	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
231	#340	C₂²×C₃₁	1	2	yes	—
232	#340	C₂²×C₃₁	1	2	yes	—
233	#340	C₂²×C₃₁	1	1	yes	characteristic
234	#340	C₂²×C₃₁	1	2	yes	—
235	#340	C₂²×C₃₁	1	2	yes	—
236	#340	C₂²×C₃₁	1	2	yes	—
237	#340	C₂²×C₃₁	1	1	yes	characteristic
238	#336	C₂²×C₃₁	1	2	yes	—
239	#336	C₂²×C₃₁	1	2	yes	—
240	C₂×C₃₁	#2446	1	1	yes	characteristic
241	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
242	C₂×C₃₁	#3097	1	1	yes	characteristic
243	C₂×C₃₁	#3359	1	1	yes	characteristic
244	C₂×C₃₁	#3368	1	1	yes	characteristic
245	C₂×C₃₁	Q₈×SD₁₆	1	1	yes	characteristic
246	C₂×C₃₁	#3386	1	1	yes	characteristic
247	C₂²×Q₈	—	2	1	no	—
248	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₄⋊Q₈	—	2	2	no	—
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	—	4	4	no	—
255	C₄×Q₈	—	2	2	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₄×Q₈	—	4	4	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	—	2	1	no	—
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
277	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
278	#106	—	2	2	no	—
279	#106	—	2	4	no	—
280	#106	Q₈×C₃₁	1	4	yes	complemented
281	C₄⋊C₈	—	2	2	no	—
282	C₄×C₈	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	1	no	—
285	#95	—	2	1	no	—
286	#95	—	2	2	no	—
287	#95	—	2	2	no	—
288	C₃₁	#26278	1	1	yes	characteristic, direct product factor, Sylow
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	4	2	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	4	4	no	—
329	C₄⋊C₄	—	4	4	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	8	1	no	—
332	C₄⋊C₄	—	8	2	no	—
333	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
334	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
335	C₄⋊C₄	#41×C₃₁	1	2	yes	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	4	4	no	—
339	C₄²	—	4	2	no	—
340	C₄²	—	4	2	no	—
341	C₄²	#41×C₃₁	1	2	yes	—
342	C₄²	#41×C₃₁	1	2	yes	—
343	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
344	Q₈	—	2	8	no	—
345	Q₈	—	8	4	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	C₃₁×#123	1	2	yes	—
372	C₂×C₄	C₃₁×#123	1	2	yes	—
373	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
374	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
375	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
376	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
381	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	2	no	—
385	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
386	C₂²	C₃₁×#548	1	1	yes	characteristic
387	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
388	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
389	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₂²	C₃₁×#577	1	1	yes	characteristic
391	C₂²	C₃₁×#578	1	1	yes	characteristic
392	C₄	—	2	4	no	—
393	C₄	—	2	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	—	8	1	no	—
401	C₄	C₃₁×#481	1	4	yes	—
402	C₄	C₃₁×#533	1	1	yes	characteristic
403	C₄	C₃₁×#541	1	1	yes	characteristic
404	C₄	C₃₁×#556	1	1	yes	characteristic
405	C₄	C₃₁×#557	1	1	yes	characteristic
406	C₂	C₃₁×#2446	1	1	yes	characteristic
407	C₂	C₂×C₃₁×#481	1	1	yes	characteristic
408	C₂	C₃₁×#3097	1	1	yes	characteristic
409	C₂	C₃₁×#3359	1	1	yes	characteristic
410	C₂	C₃₁×#3368	1	1	yes	characteristic
411	C₂	Q₈×SD₁₆×C₃₁	1	1	yes	characteristic
412	C₂	C₃₁×#3386	1	1	yes	characteristic

	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 ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
74. C₃₁×#106	Q₈	semidirect
136. #26278	C₃₁	direct (N × H)
280. #106	Q₈×C₃₁	semidirect
288. C₃₁	#26278	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×#2546	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2100	2
C₃₁×#2084	2
C₃₁×#1990	2
C₃₁×#1988	2
C₃₁×#1984	2
C₃₁×#1948	2
C₃₁×#1919	2
C₄×C₃₁×#106	2
#26278	31

Subgroups — C31×#26278 #21483037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26278 #21483037