Subgroups — C₃₁×#17855 #21474614

Order 7936.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×SD₁₆⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₈×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1478	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₃₁	—	2	2	no	—
14	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
15	C₄×SD₁₆×C₃₁	C₄	1	2	yes	—
16	C₄×SD₁₆×C₃₁	C₄	1	2	yes	—
17	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
18	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
19	C₈×D₄×C₃₁	—	2	2	no	—
20	C₈×D₄×C₃₁	C₂²	1	4	yes	—
21	C₃₁×#431	—	2	2	no	—
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₃₁×#431	C₂²	1	1	yes	characteristic
25	C₃₁×#431	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#106	C₄	1	1	yes	characteristic
27	C₂×C₃₁×#103	C₄	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₄	1	1	yes	characteristic
29	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#335	C₂²	1	2	yes	complemented
32	C₃₁×#334	C₂²	1	2	yes	complemented
33	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
34	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
35	C₃₁×#322	C₂²	1	2	yes	complemented
36	C₈²×C₃₁	C₂²	1	2	yes	complemented
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	—	2	1	no	—
40	C₂×SD₁₆×C₃₁	C₈	1	8	yes	complemented
41	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₃₁	—	2	4	no	—
44	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₃₁	—	2	1	no	—
47	C₄×Q₈×C₃₁	—	2	1	no	—
48	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₄×D₄×C₃₁	—	2	1	no	—
51	C₄×D₄×C₃₁	—	2	1	no	—
52	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	—	2	1	no	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#106	C₂×C₄	1	2	yes	—
61	C₃₁×#106	C₂×C₄	1	2	yes	—
62	C₃₁×C₄⋊C₈	—	2	4	no	—
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₃₁×#103	C₂×C₄	1	2	yes	—
68	C₃₁×#103	C₂×C₄	1	2	yes	—
69	C₃₁×#102	C₂×C₄	1	2	yes	—
70	C₃₁×#102	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊C₈	—	2	1	no	—
72	C₃₁×C₂²⋊C₈	—	2	1	no	—
73	C₃₁×C₂²⋊C₈	—	2	1	no	—
74	C₃₁×C₂²⋊C₈	—	2	4	no	—
75	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
76	C₄×C₈×C₃₁	—	2	4	no	—
77	C₄×C₈×C₃₁	—	2	1	no	—
78	C₄×C₈×C₃₁	—	2	1	no	—
79	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	2	yes	—
82	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	2	yes	—
85	C₂⁴×C₃₁	—	2	1	no	—
86	C₂×Q₈×C₃₁	—	2	1	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	complemented
89	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
93	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	8	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	C₂×C₈	1	8	yes	complemented
104	SD₁₆×C₃₁	C₂×C₈	1	8	yes	complemented
105	C₂×C₈×C₃₁	—	2	2	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	—	8	4	no	—
110	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
111	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
121	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₁	#41	1	1	yes	characteristic
129	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
130	#17855	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	8	1	no	—
134	C₂³×C₃₁	—	8	1	no	—
135	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₃₁	—	2	2	no	—
137	Q₈×C₃₁	—	2	4	no	—
138	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
139	D₄×C₃₁	—	2	2	no	—
140	D₄×C₃₁	—	2	4	no	—
141	D₄×C₃₁	C₂²×C₈	1	4	yes	—
142	C₂×C₄×C₃₁	—	2	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	—	4	1	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	8	2	no	—
150	C₂×C₄×C₃₁	—	8	1	no	—
151	C₂×C₄×C₃₁	—	8	1	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	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₃₁	#131	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	1	no	—
161	C₈×C₃₁	—	2	1	no	—
162	C₈×C₃₁	—	2	4	no	—
163	C₈×C₃₁	—	4	2	no	—
164	C₈×C₃₁	—	4	2	no	—
165	C₈×C₃₁	—	4	2	no	—
166	C₈×C₃₁	C₂²×C₈	1	2	yes	—
167	C₈×C₃₁	C₂²×C₈	1	2	yes	—
168	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#2972	C₂×C₃₁	1	1	yes	characteristic
170	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
171	SD₁₆⋊C₈	C₂×C₃₁	1	4	yes	complemented
172	C₈×SD₁₆	C₂×C₃₁	1	4	yes	complemented
173	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#1479	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1478	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	—	2	4	no	—
180	C₂²×C₃₁	—	4	2	no	—
181	C₂²×C₃₁	—	4	2	no	—
182	C₂²×C₃₁	—	8	4	no	—
183	C₂²×C₃₁	C₈×D₄	1	2	yes	—
184	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₃₁	—	2	4	no	—
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	2	1	no	—
189	C₄×C₃₁	—	4	2	no	—
190	C₄×C₃₁	—	4	2	no	—
191	C₄×C₃₁	C₈×D₄	1	2	yes	—
192	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
193	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
194	C₄×C₃₁	#577	1	2	yes	—
195	C₂²×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
198	C₈×Q₈	—	2	2	no	—
199	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
200	C₄×SD₁₆	C₄×C₃₁	1	2	yes	—
201	C₄×SD₁₆	C₄×C₃₁	1	2	yes	—
202	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
203	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
204	C₈×D₄	—	2	2	no	—
205	C₈×D₄	C₂²×C₃₁	1	4	yes	—
206	#431	—	2	2	no	—
207	#431	C₂²×C₃₁	1	1	yes	characteristic
208	#431	C₂²×C₃₁	1	1	yes	characteristic
209	#431	C₂²×C₃₁	1	1	yes	characteristic
210	#431	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#106	C₄×C₃₁	1	1	yes	characteristic
212	C₂×#103	C₄×C₃₁	1	1	yes	characteristic
213	C₂×#102	C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₂²⋊C₈	—	2	2	no	—
215	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
216	#335	C₂²×C₃₁	1	2	yes	complemented
217	#334	C₂²×C₃₁	1	2	yes	complemented
218	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	complemented
219	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
220	#322	C₂²×C₃₁	1	2	yes	complemented
221	C₈²	C₂²×C₃₁	1	2	yes	complemented
222	C₂×C₃₁	—	2	2	no	—
223	C₂×C₃₁	—	4	4	no	—
224	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
225	C₂×C₃₁	#2949	1	1	yes	characteristic
226	C₂×C₃₁	#2957	1	1	yes	characteristic
227	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₂³×C₄	—	2	1	no	—
230	C₂×SD₁₆	C₈×C₃₁	1	8	yes	complemented
231	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
232	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
233	C₂²×C₈	—	2	4	no	—
234	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
239	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
240	C₄×D₄	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
243	C₄×D₄	C₂³×C₃₁	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₂²⋊C₄	—	2	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
250	#106	C₂×C₄×C₃₁	1	2	yes	—
251	#106	C₂×C₄×C₃₁	1	2	yes	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
255	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
256	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
257	#103	C₂×C₄×C₃₁	1	2	yes	—
258	#103	C₂×C₄×C₃₁	1	2	yes	—
259	#102	C₂×C₄×C₃₁	1	2	yes	—
260	#102	C₂×C₄×C₃₁	1	2	yes	—
261	C₂²⋊C₈	—	2	1	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	2	4	no	—
265	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
266	C₄×C₈	—	2	4	no	—
267	C₄×C₈	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
270	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₄×C₈	C₂³×C₃₁	1	2	yes	—
272	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₃₁	1	2	yes	—
275	C₃₁	#17855	1	1	yes	characteristic, direct product factor, Sylow
276	C₂⁴	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	complemented
280	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
284	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	8	2	no	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
294	SD₁₆	C₂×C₈×C₃₁	1	8	yes	complemented
295	SD₁₆	C₂×C₈×C₃₁	1	8	yes	complemented
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	—	8	4	no	—
301	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
302	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
312	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₄²	#41×C₃₁	1	1	yes	characteristic
320	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	8	1	no	—
324	C₂³	—	8	1	no	—
325	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
329	D₄	—	2	4	no	—
330	D₄	—	2	2	no	—
331	D₄	C₂²×C₈×C₃₁	1	4	yes	—
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	4	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
346	C₂×C₄	C₃₁×#131	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₈	C₂²×C₈×C₃₁	1	2	yes	—
357	C₈	C₂²×C₈×C₃₁	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	4	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	4	no	—
364	C₂²	C₈×D₄×C₃₁	1	2	yes	—
365	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
366	C₄	—	2	4	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	C₈×D₄×C₃₁	1	2	yes	—
373	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
374	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
375	C₄	C₃₁×#577	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	4	4	no	—
378	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
379	C₂	C₃₁×#2949	1	1	yes	characteristic
380	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ 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
1. C₃₁×#2982	C₂	semidirect
4. C₃₁×SD₁₆⋊C₈	C₂	semidirect
5. C₈×SD₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1564	C₂	semidirect
7. C₃₁×#1479	C₂	semidirect
9. C₃₁×#1450	C₂	semidirect
14. C₈×Q₈×C₃₁	C₂²	semidirect
31. C₃₁×#335	C₂²	semidirect
32. C₃₁×#334	C₂²	semidirect
33. C₃₁×Q₈⋊C₈	C₂²	semidirect
35. C₃₁×#322	C₂²	semidirect
36. C₈²×C₃₁	C₂²	semidirect
40. C₂×SD₁₆×C₃₁	C₈	semidirect
88. C₂×Q₈×C₃₁	C₂×C₈	semidirect
103. SD₁₆×C₃₁	C₂×C₈	semidirect
104. SD₁₆×C₃₁	C₂×C₈	semidirect
110. C₂×C₈×C₃₁	C₂×C₈	semidirect
130. #17855	C₃₁	direct (N × H)
168. #2982	C₂×C₃₁	semidirect
171. SD₁₆⋊C₈	C₂×C₃₁	semidirect
172. C₈×SD₁₆	C₂×C₃₁	semidirect
173. #1564	C₂×C₃₁	semidirect
174. #1479	C₂×C₃₁	semidirect
176. #1450	C₂×C₃₁	semidirect
199. C₈×Q₈	C₂²×C₃₁	semidirect
216. #335	C₂²×C₃₁	semidirect
217. #334	C₂²×C₃₁	semidirect
218. Q₈⋊C₈	C₂²×C₃₁	semidirect
220. #322	C₂²×C₃₁	semidirect
221. C₈²	C₂²×C₃₁	semidirect
230. C₂×SD₁₆	C₈×C₃₁	semidirect
275. C₃₁	#17855	direct (N × H)
279. C₂×Q₈	C₂×C₈×C₃₁	semidirect
294. SD₁₆	C₂×C₈×C₃₁	semidirect
295. SD₁₆	C₂×C₈×C₃₁	semidirect
301. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2982	2
C₃₁×#2972	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×SD₁₆⋊C₈	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1564	2
C₃₁×#1479	2
C₃₁×#1478	2
C₃₁×#1450	2
#17855	31

Subgroups — C31×#17855 #21474614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17855 #21474614