Subgroups — C₃₁×#17388 #21474147

Order 7936.

976 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	C₂×C₃₁×#542	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1472	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1468	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1461	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1458	C₂	1	2	yes	maximal, complemented
8	C₃₁×#542	—	2	4	no	—
9	C₃₁×#542	—	2	4	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₃₁×#126	C₄	1	2	yes	—
12	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
13	C₂×C₃₁×#124	C₄	1	2	yes	—
14	C₂×C₃₁×#123	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₃₁×#436	—	2	4	no	—
22	C₈×D₄×C₃₁	—	2	4	no	—
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₃₁×#407	C₂²	1	4	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
27	C₃₁×#324	C₂²	1	4	yes	—
28	C₃₁×#324	C₂²	1	8	yes	—
29	C₃₁×#323	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	—	2	4	no	—
38	C₃₁×C₄⋊Q₈	—	2	4	no	—
39	C₃₁×#126	—	2	8	no	—
40	C₃₁×#124	—	2	4	no	—
41	C₃₁×#124	—	2	8	no	—
42	C₃₁×#123	—	2	4	no	—
43	C₃₁×#123	—	2	4	no	—
44	C₃₁×C₂²⋊Q₈	—	2	4	no	—
45	C₃₁×C₂²⋊Q₈	—	2	4	no	—
46	C₃₁×C₂²⋊Q₈	—	2	4	no	—
47	C₃₁×#121	—	2	4	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	D₄	1	4	yes	—
51	C₄×D₄×C₃₁	D₄	1	4	yes	—
52	C₃₁×#117	—	2	2	no	—
53	C₃₁×#117	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	—	2	2	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×C₂²⋊C₈	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	2	4	no	—
75	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
76	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
77	C₄×C₈×C₃₁	—	2	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	4	no	—
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	8	2	no	—
85	C₂×D₄×C₃₁	—	2	4	no	—
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	—	8	2	no	—
88	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	4	no	—
91	C₂²×C₄×C₃₁	—	8	2	no	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₃₁	—	8	4	no	—
104	C₂×C₈×C₃₁	—	2	4	no	—
105	C₂×C₈×C₃₁	—	2	4	no	—
106	C₂×C₈×C₃₁	—	8	4	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₄⋊C₄×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
116	C₂²⋊C₄×C₃₁	—	2	4	no	—
117	C₂²⋊C₄×C₃₁	—	2	4	no	—
118	C₂²⋊C₄×C₃₁	—	2	4	no	—
119	C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂²⋊C₄×C₃₁	—	2	4	no	—
121	C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	4	no	—
126	C₄²×C₃₁	C₂×D₄	1	2	yes	—
127	C₄²×C₃₁	C₂×D₄	1	2	yes	—
128	#17388	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	4	no	—
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	1	no	—
135	C₂³×C₃₁	—	4	1	no	—
136	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
138	Q₈×C₃₁	—	4	4	no	—
139	Q₈×C₃₁	—	8	4	no	—
140	D₄×C₃₁	—	2	8	no	—
141	D₄×C₃₁	—	8	4	no	—
142	C₂×C₄×C₃₁	—	2	1	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	2	no	—
148	C₂×C₄×C₃₁	—	2	4	no	—
149	C₂×C₄×C₃₁	—	4	4	no	—
150	C₂×C₄×C₃₁	—	4	4	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	4	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂×C₄×C₃₁	#131	1	4	yes	—
162	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	C₈×C₃₁	—	4	4	no	—
165	C₈×C₃₁	—	4	4	no	—
166	C₂×#542	C₂×C₃₁	1	1	yes	characteristic
167	#2975	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#2972	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1472	C₂×C₃₁	1	4	yes	complemented
170	#1468	C₂×C₃₁	1	4	yes	complemented
171	#1461	C₂×C₃₁	1	2	yes	complemented
172	#1458	C₂×C₃₁	1	2	yes	complemented
173	C₂²×C₃₁	—	2	1	no	—
174	C₂²×C₃₁	—	2	1	no	—
175	C₂²×C₃₁	—	2	4	no	—
176	C₂²×C₃₁	—	2	4	no	—
177	C₂²×C₃₁	—	2	4	no	—
178	C₂²×C₃₁	—	4	1	no	—
179	C₂²×C₃₁	—	4	1	no	—
180	C₂²×C₃₁	—	4	1	no	—
181	C₂²×C₃₁	—	8	4	no	—
182	C₂²×C₃₁	—	8	2	no	—
183	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₃₁	C₂×#131	1	2	yes	—
185	C₄×C₃₁	—	2	1	no	—
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	2	4	no	—
188	C₄×C₃₁	—	2	2	no	—
189	C₄×C₃₁	—	4	1	no	—
190	C₄×C₃₁	—	4	2	no	—
191	C₄×C₃₁	—	4	2	no	—
192	C₄×C₃₁	—	4	1	no	—
193	C₄×C₃₁	—	4	1	no	—
194	C₄×C₃₁	#408	1	4	yes	—
195	#542	—	2	4	no	—
196	#542	—	2	4	no	—
197	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
198	C₂×#126	C₄×C₃₁	1	2	yes	—
199	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
200	C₂×#124	C₄×C₃₁	1	2	yes	—
201	C₂×#123	C₂²×C₃₁	1	2	yes	—
202	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
204	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
207	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
208	#436	—	2	4	no	—
209	C₈×D₄	—	2	4	no	—
210	#431	C₂²×C₃₁	1	2	yes	—
211	#431	C₂²×C₃₁	1	2	yes	—
212	#407	C₂²×C₃₁	1	4	yes	—
213	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
214	#324	C₂²×C₃₁	1	4	yes	—
215	#324	C₂²×C₃₁	1	8	yes	—
216	#323	C₂²×C₃₁	1	4	yes	—
217	C₂×C₃₁	—	2	2	no	—
218	C₂×C₃₁	—	4	4	no	—
219	C₂×C₃₁	—	8	1	no	—
220	C₂×C₃₁	C₂×#408	1	1	yes	characteristic
221	C₂×C₃₁	#2883	1	1	yes	characteristic
222	C₂×C₃₁	#2889	1	1	yes	characteristic
223	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
224	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
225	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
226	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
227	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
229	C₂×M₁₆	—	2	4	no	—
230	C₂²×C₈	—	2	4	no	—
231	C₄⋊Q₈	—	2	4	no	—
232	#126	—	2	8	no	—
233	#124	—	2	4	no	—
234	#124	—	2	8	no	—
235	#123	—	2	4	no	—
236	#123	—	2	4	no	—
237	C₂²⋊Q₈	—	2	4	no	—
238	C₂²⋊Q₈	—	2	4	no	—
239	C₂²⋊Q₈	—	2	4	no	—
240	#121	—	2	4	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×D₄	D₄×C₃₁	1	4	yes	—
244	C₄×D₄	D₄×C₃₁	1	4	yes	—
245	#117	—	2	2	no	—
246	#117	—	2	2	no	—
247	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
250	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
251	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
253	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
254	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
256	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
257	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
258	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
259	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
260	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
263	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	2	4	no	—
267	C₂²⋊C₈	—	2	4	no	—
268	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
269	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₃₁	#17388	1	1	yes	characteristic, direct product factor, Sylow
273	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
274	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	2	4	no	—
278	C₂×Q₈	—	8	2	no	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	4	no	—
281	C₂×D₄	—	8	2	no	—
282	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	8	2	no	—
286	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
296	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
297	M₁₆	—	8	4	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	8	4	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	4	no	—
315	C₂²⋊C₄	—	2	1	no	—
316	C₂²⋊C₄	—	2	1	no	—
317	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	4	no	—
320	C₄²	C₂×D₄×C₃₁	1	2	yes	—
321	C₄²	C₂×D₄×C₃₁	1	2	yes	—
322	C₂³	—	2	4	no	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	4	1	no	—
328	C₂³	—	4	1	no	—
329	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
330	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
331	Q₈	—	4	4	no	—
332	Q₈	—	8	4	no	—
333	D₄	—	2	8	no	—
334	D₄	—	8	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
352	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
353	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
354	C₂×C₄	C₃₁×#131	1	4	yes	—
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₈	—	4	4	no	—
358	C₈	—	4	4	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	4	no	—
362	C₂²	—	2	4	no	—
363	C₂²	—	2	4	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	8	4	no	—
368	C₂²	—	8	2	no	—
369	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂²	C₂×C₃₁×#131	1	2	yes	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	4	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	C₃₁×#408	1	4	yes	—
381	C₂	—	2	2	no	—
382	C₂	—	4	4	no	—
383	C₂	—	8	1	no	—
384	C₂	C₂×C₃₁×#408	1	1	yes	characteristic
385	C₂	C₃₁×#2883	1	1	yes	characteristic
386	C₂	C₃₁×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2975	C₂	semidirect
3. C₃₁×#2972	C₂	semidirect
4. C₃₁×#1472	C₂	semidirect
5. C₃₁×#1468	C₂	semidirect
6. C₃₁×#1461	C₂	semidirect
7. C₃₁×#1458	C₂	semidirect
128. #17388	C₃₁	direct (N × H)
167. #2975	C₂×C₃₁	semidirect
168. #2972	C₂×C₃₁	semidirect
169. #1472	C₂×C₃₁	semidirect
170. #1468	C₂×C₃₁	semidirect
171. #1461	C₂×C₃₁	semidirect
172. #1458	C₂×C₃₁	semidirect
272. C₃₁	#17388	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#542	2
C₃₁×#2975	2
C₃₁×#2972	2
C₃₁×#1472	2
C₃₁×#1468	2
C₃₁×#1461	2
C₃₁×#1458	2
#17388	31

Subgroups — C31×#17388 #21474147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17388 #21474147