Subgroups — C₃₁×#17812 #21474571

Order 7936.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1574	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1573	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1572	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1571	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	—	2	1	no	—
14	C₈×Q₈×C₃₁	—	2	1	no	—
15	C₃₁×#436	—	2	1	no	—
16	C₃₁×#435	—	2	2	no	—
17	C₈×D₄×C₃₁	—	2	2	no	—
18	C₈×D₄×C₃₁	—	2	1	no	—
19	C₃₁×#433	C₄	1	4	yes	—
20	C₃₁×#432	C₂²	1	2	yes	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₃₁×#431	C₂²	1	2	yes	—
26	C₃₁×#424	C₄	1	2	yes	—
27	C₃₁×#423	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃₁×#408	C₄	1	2	yes	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#335	C₂²	1	2	yes	—
35	C₃₁×#335	C₂²	1	2	yes	—
36	C₃₁×#335	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	2	yes	—
39	C₃₁×#335	C₂²	1	2	yes	—
40	C₃₁×#334	C₂²	1	1	yes	characteristic
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×#334	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₃₁	—	2	1	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	1	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
52	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
53	C₄×D₄×C₃₁	D₄	1	2	yes	—
54	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
55	C₄×D₄×C₃₁	Q₈	1	2	yes	—
56	C₄×D₄×C₃₁	Q₈	1	1	yes	characteristic
57	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
58	C₃₁×#117	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	—	2	2	no	—
65	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
66	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
70	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₃₁×C₂²⋊C₈	—	2	1	no	—
74	C₃₁×C₂²⋊C₈	—	2	2	no	—
75	C₃₁×C₂²⋊C₈	—	2	1	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
78	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
79	C₃₁×#97	—	2	2	no	—
80	C₃₁×#97	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₃₁	—	2	1	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	1	no	—
84	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂³	1	2	yes	—
88	#41×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
98	M₁₆×C₃₁	—	2	4	no	—
99	M₁₆×C₃₁	—	8	2	no	—
100	C₂×C₈×C₃₁	—	2	4	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	8	1	no	—
108	C₂×C₈×C₃₁	—	8	2	no	—
109	C₂×C₈×C₃₁	—	8	1	no	—
110	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
120	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
122	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂×D₄	1	2	yes	—
124	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
125	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
127	#17812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	Q₈×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	2	4	no	—
132	D₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	4	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#131	1	4	yes	—
147	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₃₁	—	2	2	no	—
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	4	no	—
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	4	1	no	—
155	C₈×C₃₁	—	4	2	no	—
156	C₈×C₃₁	—	4	1	no	—
157	C₈×C₃₁	—	4	1	no	—
158	C₈×C₃₁	—	4	1	no	—
159	#2989	C₂×C₃₁	1	1	yes	characteristic
160	#2988	C₂×C₃₁	1	1	yes	characteristic
161	#2922	C₂×C₃₁	1	1	yes	characteristic
162	#1574	C₂×C₃₁	1	2	yes	complemented
163	#1573	C₂×C₃₁	1	2	yes	complemented
164	#1572	C₂×C₃₁	1	2	yes	complemented
165	#1571	C₂×C₃₁	1	2	yes	complemented
166	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	2	no	—
175	C₂²×C₃₁	#424	1	2	yes	—
176	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₃₁	—	2	2	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	1	no	—
181	C₄×C₃₁	—	2	1	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	#424	1	2	yes	—
184	C₄×C₃₁	C₂×#131	1	2	yes	—
185	C₄×C₃₁	#576	1	1	yes	characteristic
186	C₄×C₃₁	#578	1	1	yes	characteristic
187	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
188	#446	—	2	1	no	—
189	C₈×Q₈	—	2	1	no	—
190	#436	—	2	1	no	—
191	#435	—	2	2	no	—
192	C₈×D₄	—	2	2	no	—
193	C₈×D₄	—	2	1	no	—
194	#433	C₄×C₃₁	1	4	yes	—
195	#432	C₂²×C₃₁	1	2	yes	—
196	#431	C₂²×C₃₁	1	1	yes	characteristic
197	#431	C₂²×C₃₁	1	1	yes	characteristic
198	#431	C₂²×C₃₁	1	2	yes	—
199	#431	C₂²×C₃₁	1	2	yes	—
200	#431	C₂²×C₃₁	1	2	yes	—
201	#424	C₄×C₃₁	1	2	yes	—
202	#423	C₂²×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
206	#408	C₄×C₃₁	1	2	yes	—
207	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
209	#335	C₂²×C₃₁	1	2	yes	—
210	#335	C₂²×C₃₁	1	2	yes	—
211	#335	C₂²×C₃₁	1	1	yes	characteristic
212	#335	C₂²×C₃₁	1	1	yes	characteristic
213	#335	C₂²×C₃₁	1	2	yes	—
214	#335	C₂²×C₃₁	1	2	yes	—
215	#334	C₂²×C₃₁	1	1	yes	characteristic
216	#334	C₂²×C₃₁	1	1	yes	characteristic
217	#334	C₂²×C₃₁	1	2	yes	—
218	#334	C₂²×C₃₁	1	2	yes	—
219	C₂×C₃₁	—	2	2	no	—
220	C₂×C₃₁	—	4	2	no	—
221	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
222	C₂×C₃₁	#2913	1	1	yes	characteristic
223	C₂×C₃₁	#2916	1	1	yes	characteristic
224	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
231	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
232	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
233	C₄×D₄	D₄×C₃₁	1	2	yes	—
234	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
235	C₄×D₄	Q₈×C₃₁	1	2	yes	—
236	C₄×D₄	Q₈×C₃₁	1	1	yes	characteristic
237	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
238	#117	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
246	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
247	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
251	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
252	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₂²⋊C₈	—	2	1	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
258	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
259	#97	—	2	2	no	—
260	#97	C₂×C₄×C₃₁	1	2	yes	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
265	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₄×C₈	C₂³×C₃₁	1	2	yes	—
267	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₃₁	#17812	1	1	yes	characteristic, direct product factor, Sylow
269	#41	—	2	4	no	—
270	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
272	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	8	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	8	1	no	—
289	C₂×C₈	—	8	2	no	—
290	C₂×C₈	—	8	1	no	—
291	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
293	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
294	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
295	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
296	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
297	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
298	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
299	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
300	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
301	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
303	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₃₁	1	2	yes	—
305	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
306	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
307	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
309	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
322	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₃₁×#131	1	4	yes	—
327	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	C₃₁×#424	1	2	yes	—
345	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	C₃₁×#424	1	2	yes	—
353	C₄	C₂×C₃₁×#131	1	2	yes	—
354	C₄	C₃₁×#576	1	1	yes	characteristic
355	C₄	C₃₁×#578	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
359	C₂	C₃₁×#2913	1	1	yes	characteristic
360	C₂	C₃₁×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1574	C₂	semidirect
5. C₃₁×#1573	C₂	semidirect
6. C₃₁×#1572	C₂	semidirect
7. C₃₁×#1571	C₂	semidirect
8. C₃₁×#1565	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1564	C₂	semidirect
11. C₂×C₃₁×#335	C₂	semidirect
127. #17812	C₃₁	direct (N × H)
162. #1574	C₂×C₃₁	semidirect
163. #1573	C₂×C₃₁	semidirect
164. #1572	C₂×C₃₁	semidirect
165. #1571	C₂×C₃₁	semidirect
166. #1565	C₂×C₃₁	semidirect
167. #1565	C₂×C₃₁	semidirect
168. #1564	C₂×C₃₁	semidirect
169. C₂×#335	C₂×C₃₁	semidirect
268. C₃₁	#17812	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×#2989	2
C₃₁×#2988	2
C₃₁×#2922	2
C₃₁×#1574	2
C₃₁×#1573	2
C₃₁×#1572	2
C₃₁×#1571	2
C₃₁×#1565	2
C₃₁×#1564	2
C₂×C₃₁×#335	2
#17812	31

Subgroups — C31×#17812 #21474571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17812 #21474571