Subgroups — C₃₁×#8565 #21465324

Order 7936.

688 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	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	—	2	1	no	—
10	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
17	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
18	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₄³×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#336	—	2	1	no	—
21	C₃₁×#336	—	2	1	no	—
22	C₃₁×#336	—	2	1	no	—
23	C₃₁×D₄⋊C₈	—	2	4	no	—
24	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	—	2	1	no	—
26	C₂³×C₄×C₃₁	—	2	1	no	—
27	C₂³×C₄×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₄×D₄×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	C₈	1	4	yes	complemented
36	C₄×D₄×C₃₁	D₄	1	2	yes	—
37	C₄×D₄×C₃₁	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	4	4	no	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊C₈	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₄×C₈×C₃₁	—	2	2	no	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₃₁×#95	—	2	1	no	—
61	C₃₁×#95	—	2	1	no	—
62	C₃₁×#95	—	2	1	no	—
63	C₂⁴×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	—	2	1	no	—
65	C₂×D₄×C₃₁	—	2	1	no	—
66	C₂×D₄×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₃₁	M₁₆	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	—	2	4	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	4	2	no	—
92	C₂×C₈×C₃₁	—	4	2	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	—	4	2	no	—
95	C₂×C₈×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
101	C₄⋊C₄×C₃₁	M₁₆	1	2	yes	—
102	C₂²⋊C₄×C₃₁	—	2	4	no	—
103	C₂²⋊C₄×C₃₁	—	2	4	no	—
104	C₂²⋊C₄×C₃₁	—	2	2	no	—
105	C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	4	1	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	C₂×C₈	1	2	yes	—
116	C₄²×C₃₁	M₁₆	1	2	yes	—
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₃₁	#41	1	1	yes	characteristic
120	C₄²×C₃₁	#41	1	1	yes	characteristic
121	#8565	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	2	2	no	—
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	4	1	no	—
126	C₂³×C₃₁	—	4	1	no	—
127	C₂³×C₃₁	—	4	1	no	—
128	C₂³×C₃₁	—	4	1	no	—
129	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	D₄×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	2	2	no	—
132	D₄×C₃₁	—	2	2	no	—
133	D₄×C₃₁	C₄⋊C₈	1	4	yes	complemented
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	4	8	no	—
149	C₂×C₄×C₃₁	—	4	8	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	4	no	—
157	C₂×C₄×C₃₁	—	4	4	no	—
158	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
159	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
160	C₂×C₄×C₃₁	#104	1	2	yes	—
161	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
163	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
164	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
171	C₂×C₄×C₃₁	#135	1	2	yes	—
172	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₈×C₃₁	—	4	4	no	—
174	C₈×C₃₁	—	8	2	no	—
175	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
176	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
178	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	—
179	#1279	C₂×C₃₁	1	2	yes	complemented
180	C₂²×C₃₁	—	2	4	no	—
181	C₂²×C₃₁	—	4	2	no	—
182	C₂²×C₃₁	—	4	2	no	—
183	C₂²×C₃₁	—	4	2	no	—
184	C₂²×C₃₁	—	4	2	no	—
185	C₂²×C₃₁	—	4	2	no	—
186	C₂²×C₃₁	—	4	2	no	—
187	C₂²×C₃₁	#380	1	1	yes	characteristic
188	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
189	C₂²×C₃₁	#407	1	1	yes	characteristic
190	C₂²×C₃₁	#416	1	1	yes	characteristic
191	C₂²×C₃₁	#418	1	1	yes	characteristic
192	C₂²×C₃₁	C₂×#104	1	1	yes	characteristic
193	C₂²×C₃₁	#421	1	1	yes	characteristic
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	4	1	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
204	C₄×C₃₁	#423	1	1	yes	characteristic
205	C₄×C₃₁	#431	1	1	yes	characteristic
206	C₄×C₃₁	#432	1	1	yes	characteristic
207	C₄×C₃₁	#463	1	2	yes	—
208	C₄×C₃₁	#478	1	2	yes	—
209	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²×C₄²	—	2	1	no	—
213	C₂×C₄⋊C₈	—	2	1	no	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
216	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
217	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
218	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
219	C₄×C₂²⋊C₄	—	2	1	no	—
220	C₄×C₂²⋊C₄	—	2	1	no	—
221	C₄×C₂²⋊C₄	—	2	1	no	—
222	C₄³	C₂²×C₃₁	1	1	yes	characteristic
223	#336	—	2	1	no	—
224	#336	—	2	1	no	—
225	#336	—	2	1	no	—
226	D₄⋊C₈	—	2	4	no	—
227	C₂×C₃₁	—	4	4	no	—
228	C₂×C₃₁	#1476	1	1	yes	characteristic
229	C₂×C₃₁	#1478	1	1	yes	characteristic
230	C₂×C₃₁	#1488	1	1	yes	characteristic
231	C₂×C₃₁	#1490	1	1	yes	characteristic
232	C₂×C₃₁	#1797	1	1	yes	characteristic
233	C₂×C₃₁	#1804	1	1	yes	characteristic
234	C₂×C₃₁	#1806	1	1	yes	characteristic
235	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
236	C₂³×C₄	—	2	1	no	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	2	1	no	—
246	C₄×D₄	C₈×C₃₁	1	4	yes	complemented
247	C₄×D₄	D₄×C₃₁	1	2	yes	—
248	C₄×D₄	Q₈×C₃₁	1	2	yes	—
249	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₂²⋊C₄	—	2	1	no	—
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₂²⋊C₄	—	2	1	no	—
255	C₂×C₄²	—	2	1	no	—
256	C₂×C₄²	—	2	1	no	—
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	—	4	4	no	—
259	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	#95	—	2	1	no	—
272	#95	—	2	1	no	—
273	#95	—	2	1	no	—
274	C₃₁	#8565	1	1	yes	characteristic, direct product factor, Sylow
275	C₂⁴	—	2	1	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
280	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
281	C₂×D₄	M₁₆×C₃₁	1	2	yes	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	4	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	4	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
312	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
313	C₄⋊C₄	M₁₆×C₃₁	1	2	yes	—
314	C₂²⋊C₄	—	2	4	no	—
315	C₂²⋊C₄	—	2	4	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	1	no	—
320	C₄²	—	2	1	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	4	2	no	—
325	C₄²	—	4	2	no	—
326	C₄²	—	4	2	no	—
327	C₄²	C₂×C₈×C₃₁	1	2	yes	—
328	C₄²	M₁₆×C₃₁	1	2	yes	—
329	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
331	C₄²	#41×C₃₁	1	1	yes	characteristic
332	C₄²	#41×C₃₁	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	2	2	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	4	1	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	1	no	—
340	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
341	D₄	—	2	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	C₃₁×C₄⋊C₈	1	4	yes	complemented
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	8	no	—
360	C₂×C₄	—	4	8	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
370	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
371	C₂×C₄	C₃₁×#104	1	2	yes	—
372	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
374	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
375	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
376	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
377	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₃₁×#135	1	2	yes	—
383	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
384	C₈	—	4	4	no	—
385	C₈	—	8	2	no	—
386	C₂²	—	2	4	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	C₃₁×#380	1	1	yes	characteristic
394	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
395	C₂²	C₃₁×#407	1	1	yes	characteristic
396	C₂²	C₃₁×#416	1	1	yes	characteristic
397	C₂²	C₃₁×#418	1	1	yes	characteristic
398	C₂²	C₂×C₃₁×#104	1	1	yes	characteristic
399	C₂²	C₃₁×#421	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	2	no	—
403	C₄	—	2	2	no	—
404	C₄	—	2	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	2	no	—
409	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
410	C₄	C₃₁×#423	1	1	yes	characteristic
411	C₄	C₃₁×#431	1	1	yes	characteristic
412	C₄	C₃₁×#432	1	1	yes	characteristic
413	C₄	C₃₁×#463	1	2	yes	—
414	C₄	C₃₁×#478	1	2	yes	—
415	C₂	—	4	4	no	—
416	C₂	C₃₁×#1476	1	1	yes	characteristic
417	C₂	C₃₁×#1478	1	1	yes	characteristic
418	C₂	C₃₁×#1488	1	1	yes	characteristic
419	C₂	C₃₁×#1490	1	1	yes	characteristic
420	C₂	C₃₁×#1797	1	1	yes	characteristic
421	C₂	C₃₁×#1804	1	1	yes	characteristic
422	C₂	C₃₁×#1806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1831	C₂	semidirect
3. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
35. C₄×D₄×C₃₁	C₈	semidirect
121. #8565	C₃₁	direct (N × H)
133. D₄×C₃₁	C₄⋊C₈	semidirect
176. #1831	C₂×C₃₁	semidirect
177. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
179. #1279	C₂×C₃₁	semidirect
246. C₄×D₄	C₈×C₃₁	semidirect
274. C₃₁	#8565	direct (N × H)
344. D₄	C₃₁×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₃₁	2
C₃₁×#1831	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×D₄⋊C₈	2
C₃₁×#1279	2
#8565	31

Subgroups — C31×#8565 #21465324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8565 #21465324