Subgroups — C₃₁×#17924 #21474683

Order 7936.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1585	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1582	C₂	1	2	yes	maximal, complemented
6	C₃₁×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1572	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#483	C₄	1	2	yes	—
15	C₃₁×#476	C₄	1	2	yes	—
16	C₃₁×#461	C₄	1	2	yes	—
17	C₃₁×#450	C₄	1	2	yes	—
18	C₃₁×#446	—	2	2	no	—
19	C₃₁×#446	C₂²	1	2	yes	complemented
20	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
21	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	—	2	2	no	—
24	C₈×D₄×C₃₁	C₂²	1	2	yes	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#433	—	2	2	no	—
27	C₃₁×#432	C₂²	1	1	yes	characteristic
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	2	yes	complemented
34	C₃₁×#334	C₂²	1	2	yes	complemented
35	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
36	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
39	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
41	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#125	—	2	1	no	—
53	C₃₁×C₂²⋊Q₈	—	2	1	no	—
54	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₃₁	C₂³	1	2	yes	—
61	C₃₁×#117	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#106	C₂×C₄	1	2	yes	—
66	C₃₁×#106	C₂×C₄	1	2	yes	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
71	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×#103	C₂×C₄	1	2	yes	—
74	C₃₁×#103	C₂×C₄	1	2	yes	—
75	C₃₁×#102	C₂×C₄	1	2	yes	—
76	C₃₁×#102	C₂×C₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	1	no	—
78	C₃₁×C₂²⋊C₈	—	2	1	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
82	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
83	C₃₁×#97	—	2	2	no	—
84	C₃₁×#97	—	2	1	no	—
85	C₃₁×#97	—	2	1	no	—
86	C₄×C₈×C₃₁	—	2	2	no	—
87	C₄×C₈×C₃₁	—	2	1	no	—
88	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	8	2	no	—
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
107	SD₁₆×C₃₁	—	2	4	no	—
108	SD₁₆×C₃₁	—	2	4	no	—
109	M₁₆×C₃₁	—	4	2	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	8	2	no	—
115	C₂×C₈×C₃₁	—	8	2	no	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₃₁	#41	1	1	yes	characteristic
135	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
136	#17924	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	8	1	no	—
140	C₂³×C₃₁	—	8	1	no	—
141	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
142	Q₈×C₃₁	—	2	2	no	—
143	Q₈×C₃₁	—	2	2	no	—
144	Q₈×C₃₁	—	8	1	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	2	2	no	—
147	D₄×C₃₁	#131	1	4	yes	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	8	2	no	—
157	C₂×C₄×C₃₁	—	8	1	no	—
158	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
164	C₈×C₃₁	—	2	2	no	—
165	C₈×C₃₁	—	2	2	no	—
166	C₈×C₃₁	—	2	2	no	—
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	2	2	no	—
169	C₈×C₃₁	—	4	1	no	—
170	C₈×C₃₁	—	4	2	no	—
171	C₈×C₃₁	—	4	2	no	—
172	#3101	C₂×C₃₁	1	1	yes	characteristic
173	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#2972	C₂×C₃₁	1	1	yes	characteristic
175	#1585	C₂×C₃₁	1	2	yes	complemented
176	#1582	C₂×C₃₁	1	2	yes	complemented
177	SD₁₆⋊C₈	C₂×C₃₁	1	2	yes	complemented
178	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
179	#1572	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1490	C₂×C₃₁	1	1	yes	characteristic
181	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	4	no	—
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	4	2	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	8	4	no	—
190	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	4	2	no	—
196	C₄×C₃₁	—	4	2	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	C₂×#131	1	2	yes	—
199	C₄×C₃₁	#577	1	2	yes	—
200	#533	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
202	#483	C₄×C₃₁	1	2	yes	—
203	#476	C₄×C₃₁	1	2	yes	—
204	#461	C₄×C₃₁	1	2	yes	—
205	#450	C₄×C₃₁	1	2	yes	—
206	#446	—	2	2	no	—
207	#446	C₂²×C₃₁	1	2	yes	complemented
208	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
209	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
210	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
211	C₈×D₄	—	2	2	no	—
212	C₈×D₄	C₂²×C₃₁	1	2	yes	—
213	C₈×D₄	C₂²×C₃₁	1	2	yes	—
214	#433	—	2	2	no	—
215	#432	C₂²×C₃₁	1	1	yes	characteristic
216	#431	C₂²×C₃₁	1	1	yes	characteristic
217	#431	C₂²×C₃₁	1	1	yes	characteristic
218	#431	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	—	2	2	no	—
220	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	#335	C₂²×C₃₁	1	2	yes	complemented
222	#334	C₂²×C₃₁	1	2	yes	complemented
223	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
224	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
225	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
226	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
227	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
228	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
229	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
230	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
231	C₂×C₃₁	—	4	1	no	—
232	C₂×C₃₁	—	4	4	no	—
233	C₂×C₃₁	#2929	1	1	yes	characteristic
234	C₂×C₃₁	#2949	1	1	yes	characteristic
235	C₂×C₃₁	#2956	1	1	yes	characteristic
236	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂³×C₄	—	2	1	no	—
238	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	2	2	no	—
243	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
245	#125	—	2	1	no	—
246	C₂²⋊Q₈	—	2	1	no	—
247	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
253	C₄×D₄	C₂³×C₃₁	1	2	yes	—
254	#117	—	2	1	no	—
255	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₂²⋊C₄	—	2	1	no	—
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	#106	C₂×C₄×C₃₁	1	2	yes	—
259	#106	C₂×C₄×C₃₁	1	2	yes	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
263	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
264	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
265	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
266	#103	C₂×C₄×C₃₁	1	2	yes	—
267	#103	C₂×C₄×C₃₁	1	2	yes	—
268	#102	C₂×C₄×C₃₁	1	2	yes	—
269	#102	C₂×C₄×C₃₁	1	2	yes	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
275	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
276	#97	—	2	2	no	—
277	#97	—	2	1	no	—
278	#97	—	2	1	no	—
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₃₁	#17924	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	—	2	1	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	8	2	no	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
301	SD₁₆	—	2	4	no	—
302	SD₁₆	—	2	4	no	—
303	M₁₆	—	4	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	8	2	no	—
309	C₂×C₈	—	8	2	no	—
310	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
311	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
313	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₄²	#41×C₃₁	1	1	yes	characteristic
329	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
330	C₂³	—	2	4	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	8	1	no	—
333	C₂³	—	8	1	no	—
334	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	1	no	—
338	D₄	—	2	4	no	—
339	D₄	—	2	2	no	—
340	D₄	C₃₁×#131	1	4	yes	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	4	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	4	no	—
372	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	C₂×C₃₁×#131	1	2	yes	—
381	C₄	C₃₁×#577	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	4	4	no	—
384	C₂	C₃₁×#2929	1	1	yes	characteristic
385	C₂	C₃₁×#2949	1	1	yes	characteristic
386	C₂	C₃₁×#2956	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	C₂×C₄×D₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2985	C₂	semidirect
4. C₃₁×#1585	C₂	semidirect
5. C₃₁×#1582	C₂	semidirect
6. C₃₁×SD₁₆⋊C₈	C₂	semidirect
7. C₈×SD₁₆×C₃₁	C₂	semidirect
8. C₃₁×#1572	C₂	semidirect
10. C₃₁×#1485	C₂	semidirect
11. C₃₁×#1453	C₂	semidirect
19. C₃₁×#446	C₂²	semidirect
20. C₈×Q₈×C₃₁	C₂²	semidirect
33. C₃₁×#335	C₂²	semidirect
34. C₃₁×#334	C₂²	semidirect
35. C₃₁×Q₈⋊C₈	C₂²	semidirect
36. C₃₁×Q₈⋊C₈	C₂²	semidirect
39. C₃₁×#322	C₂²	semidirect
40. C₃₁×#322	C₂²	semidirect
41. C₃₁×#322	C₂²	semidirect
42. C₈²×C₃₁	C₂²	semidirect
136. #17924	C₃₁	direct (N × H)
173. #2985	C₂×C₃₁	semidirect
175. #1585	C₂×C₃₁	semidirect
176. #1582	C₂×C₃₁	semidirect
177. SD₁₆⋊C₈	C₂×C₃₁	semidirect
178. C₈×SD₁₆	C₂×C₃₁	semidirect
179. #1572	C₂×C₃₁	semidirect
181. #1485	C₂×C₃₁	semidirect
182. #1453	C₂×C₃₁	semidirect
207. #446	C₂²×C₃₁	semidirect
208. C₈×Q₈	C₂²×C₃₁	semidirect
221. #335	C₂²×C₃₁	semidirect
222. #334	C₂²×C₃₁	semidirect
223. Q₈⋊C₈	C₂²×C₃₁	semidirect
224. Q₈⋊C₈	C₂²×C₃₁	semidirect
227. #322	C₂²×C₃₁	semidirect
228. #322	C₂²×C₃₁	semidirect
229. #322	C₂²×C₃₁	semidirect
230. C₈²	C₂²×C₃₁	semidirect
287. C₃₁	#17924	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×#3101	2
C₃₁×#2985	2
C₃₁×#2972	2
C₃₁×#1585	2
C₃₁×#1582	2
C₃₁×SD₁₆⋊C₈	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1572	2
C₃₁×#1490	2
C₃₁×#1485	2
C₃₁×#1453	2
#17924	31

Subgroups — C31×#17924 #21474683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17924 #21474683