Subgroups — C₃₁×#17901 #21474660

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1583	C₂	1	4	yes	maximal, complemented
5	C₈×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#481	C₄	1	2	yes	—
14	C₃₁×#475	C₄	1	2	yes	—
15	C₃₁×#460	C₄	1	2	yes	—
16	C₃₁×#448	C₄	1	2	yes	—
17	C₈×Q₈×C₃₁	—	2	1	no	—
18	C₈×Q₈×C₃₁	—	2	1	no	—
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
22	C₃₁×#436	—	2	1	no	—
23	C₃₁×#436	C₂²	1	2	yes	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#408	—	2	1	no	—
32	C₂×C₄×C₈×C₃₁	—	2	1	no	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#334	C₂²	1	4	yes	—
35	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
36	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
37	C₃₁×#322	C₂²	1	2	yes	—
38	C₈²×C₃₁	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
42	C₃₁×#131	—	4	4	no	—
43	C₂×M₁₆×C₃₁	—	2	1	no	—
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
51	C₃₁×#124	—	2	1	no	—
52	C₃₁×#121	—	2	1	no	—
53	C₃₁×#121	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₃₁	—	2	1	no	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	—	2	1	no	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#106	C₂×C₄	1	4	yes	—
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	—	2	1	no	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	C₂×C₄	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	4	yes	—
74	C₃₁×#102	C₂×C₄	1	4	yes	—
75	C₃₁×C₂²⋊C₈	—	2	1	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₃₁	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
82	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	2	yes	—
84	C₄×C₈×C₃₁	C₂³	1	2	yes	—
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	#41×C₃₁	—	4	2	no	—
88	#41×C₃₁	—	4	2	no	—
89	C₂×Q₈×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
103	SD₁₆×C₃₁	—	2	8	no	—
104	M₁₆×C₃₁	—	4	2	no	—
105	M₁₆×C₃₁	—	4	4	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	4	2	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
113	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₃₁	—	2	2	no	—
125	C₂²⋊C₄×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	#41	1	1	yes	characteristic
131	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
132	#17901	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₃₁	—	2	4	no	—
136	Q₈×C₃₁	—	2	2	no	—
137	Q₈×C₃₁	—	4	2	no	—
138	Q₈×C₃₁	#131	1	4	yes	complemented
139	D₄×C₃₁	—	2	4	no	—
140	D₄×C₃₁	—	4	2	no	—
141	D₄×C₃₁	—	4	4	no	—
142	D₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	2	4	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	—	4	1	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
156	C₂×C₄×C₃₁	#135	1	2	yes	—
157	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
160	C₈×C₃₁	—	2	4	no	—
161	C₈×C₃₁	—	2	1	no	—
162	C₈×C₃₁	—	2	1	no	—
163	C₈×C₃₁	—	4	2	no	—
164	C₈×C₃₁	—	4	2	no	—
165	C₈×C₃₁	—	4	2	no	—
166	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
167	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
168	#3103	C₂×C₃₁	1	1	yes	characteristic
169	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#2931	C₂×C₃₁	1	1	yes	characteristic
171	#1583	C₂×C₃₁	1	4	yes	complemented
172	C₈×SD₁₆	C₂×C₃₁	1	4	yes	complemented
173	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1480	C₂×C₃₁	1	1	yes	characteristic
176	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂²×C₃₁	—	2	1	no	—
178	C₂²×C₃₁	—	2	1	no	—
179	C₂²×C₃₁	—	4	2	no	—
180	C₂²×C₃₁	—	4	2	no	—
181	C₂²×C₃₁	—	4	2	no	—
182	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
183	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₃₁	—	2	4	no	—
185	C₄×C₃₁	—	2	1	no	—
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	4	2	no	—
189	C₄×C₃₁	—	4	2	no	—
190	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
191	C₄×C₃₁	C₂×#131	1	2	yes	—
192	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
193	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
194	C₂×#131	—	2	1	no	—
195	#532	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
197	#481	C₄×C₃₁	1	2	yes	—
198	#475	C₄×C₃₁	1	2	yes	—
199	#460	C₄×C₃₁	1	2	yes	—
200	#448	C₄×C₃₁	1	2	yes	—
201	C₈×Q₈	—	2	1	no	—
202	C₈×Q₈	—	2	1	no	—
203	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
204	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
205	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
206	#436	—	2	1	no	—
207	#436	C₂²×C₃₁	1	2	yes	—
208	C₈×D₄	—	2	1	no	—
209	C₈×D₄	C₂²×C₃₁	1	2	yes	—
210	#431	C₂²×C₃₁	1	1	yes	characteristic
211	#431	C₂²×C₃₁	1	1	yes	characteristic
212	#423	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₈	—	2	1	no	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
215	#408	—	2	1	no	—
216	C₂×C₄×C₈	—	2	1	no	—
217	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
218	#334	C₂²×C₃₁	1	4	yes	—
219	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
220	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
221	#322	C₂²×C₃₁	1	2	yes	—
222	C₈²	C₂²×C₃₁	1	2	yes	—
223	C₂×C₃₁	—	2	2	no	—
224	C₂×C₃₁	—	8	2	no	—
225	C₂×C₃₁	#2929	1	1	yes	characteristic
226	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
227	C₂×C₃₁	#2955	1	1	yes	characteristic
228	C₂×#41	—	2	1	no	—
229	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
231	#131	—	4	4	no	—
232	C₂×M₁₆	—	2	1	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	2	2	no	—
236	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
238	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
239	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
240	#124	—	2	1	no	—
241	#121	—	2	1	no	—
242	#121	C₂×C₄×C₃₁	1	2	yes	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
246	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	C₂³×C₃₁	1	2	yes	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄²	—	2	1	no	—
252	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
253	#106	C₂×C₄×C₃₁	1	4	yes	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
260	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
261	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
262	#103	C₂×C₄×C₃₁	1	4	yes	—
263	#102	C₂×C₄×C₃₁	1	4	yes	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
268	C₄×C₈	—	2	2	no	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
271	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
272	C₄×C₈	C₂³×C₃₁	1	2	yes	—
273	C₄×C₈	C₂³×C₃₁	1	2	yes	—
274	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₃₁	#17901	1	1	yes	characteristic, direct product factor, Sylow
277	#41	—	4	2	no	—
278	#41	—	4	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
293	SD₁₆	—	2	8	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	4	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	4	no	—
302	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
303	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
304	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
313	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₄²	#41×C₃₁	1	1	yes	characteristic
321	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	C₃₁×#131	1	4	yes	complemented
328	D₄	—	2	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
345	C₂×C₄	C₃₁×#135	1	2	yes	—
346	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
356	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
363	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
371	C₄	C₂×C₃₁×#131	1	2	yes	—
372	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
373	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	8	2	no	—
376	C₂	C₃₁×#2929	1	1	yes	characteristic
377	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
378	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1583	C₂	semidirect
5. C₈×SD₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1570	C₂	semidirect
7. C₃₁×#1487	C₂	semidirect
9. C₃₁×#1450	C₂	semidirect
112. C₂×C₈×C₃₁	SD₁₆	semidirect
132. #17901	C₃₁	direct (N × H)
138. Q₈×C₃₁	#131	semidirect
169. C₂×C₈×Q₈	C₂×C₃₁	semidirect
171. #1583	C₂×C₃₁	semidirect
172. C₈×SD₁₆	C₂×C₃₁	semidirect
173. #1570	C₂×C₃₁	semidirect
174. #1487	C₂×C₃₁	semidirect
176. #1450	C₂×C₃₁	semidirect
276. C₃₁	#17901	direct (N × H)
302. C₂×C₈	SD₁₆×C₃₁	semidirect
327. Q₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3103	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2931	2
C₃₁×#1583	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1570	2
C₃₁×#1487	2
C₃₁×#1480	2
C₃₁×#1450	2
#17901	31

Subgroups — C31×#17901 #21474660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17901 #21474660