Subgroups — C₃₁×#20662 #21477421

Order 7936.

1438 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2311	C₂	1	2	yes	maximal
5	C₃₁×#1809	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1807	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#131	—	2	4	no	—
10	C₂×C₃₁×#131	—	2	4	no	—
11	C₂×C₃₁×#131	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#427	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	—	2	4	no	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×#106	C₂²	1	2	yes	—
23	C₂×C₃₁×#106	C₂²	1	2	yes	—
24	C₃₁×#391	—	2	2	no	—
25	C₃₁×#390	—	2	2	no	—
26	C₃₁×#386	—	2	4	no	—
27	C₃₁×#380	C₂²	1	8	yes	—
28	C₃₁×#340	C₂²	1	8	yes	—
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×#41×C₃₁	Q₈	1	4	yes	—
35	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₃₁	D₄	1	2	yes	—
37	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#131	—	2	32	no	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	2	4	no	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₃₁	—	2	4	no	—
49	C₂²×C₈×C₃₁	—	2	4	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
52	C₂²×C₈×C₃₁	D₄	1	4	yes	—
53	C₂²×C₈×C₃₁	D₄	1	4	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#117	—	2	4	no	—
58	C₃₁×#117	—	2	4	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₃₁×#107	—	2	4	no	—
68	C₃₁×#106	—	2	4	no	—
69	C₃₁×#106	—	2	4	no	—
70	C₃₁×#106	—	2	4	no	—
71	C₃₁×#95	—	2	8	no	—
72	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
73	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
74	#41×C₃₁	—	2	8	no	—
75	#41×C₃₁	—	2	8	no	—
76	#41×C₃₁	—	2	8	no	—
77	#41×C₃₁	—	2	8	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
81	C₂×D₄×C₃₁	—	2	4	no	—
82	C₂×D₄×C₃₁	—	2	4	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	C₄⋊C₄	1	8	yes	—
86	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	4	no	—
92	C₂²×C₄×C₃₁	—	2	4	no	—
93	C₂²×C₄×C₃₁	—	4	4	no	—
94	C₂²×C₄×C₃₁	—	8	4	no	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
102	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₃₁	#41	1	2	yes	—
104	C₂²×C₄×C₃₁	#41	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₃₁	—	2	8	no	—
107	M₁₆×C₃₁	—	2	16	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	16	no	—
111	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	4	no	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	4	4	no	—
124	C₂²⋊C₄×C₃₁	—	4	4	no	—
125	C₂²⋊C₄×C₃₁	—	8	8	no	—
126	C₄²×C₃₁	—	4	4	no	—
127	#20662	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	2	4	no	—
129	C₂³×C₃₁	—	2	2	no	—
130	C₂³×C₃₁	—	2	2	no	—
131	C₂³×C₃₁	—	2	8	no	—
132	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
133	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
134	C₂³×C₃₁	#121	1	2	yes	—
135	C₂³×C₃₁	#121	1	2	yes	—
136	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₃₁	—	2	8	no	—
138	D₄×C₃₁	—	2	16	no	—
139	D₄×C₃₁	—	2	8	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	2	4	no	—
145	C₂×C₄×C₃₁	—	2	4	no	—
146	C₂×C₄×C₃₁	—	4	4	no	—
147	C₂×C₄×C₃₁	—	4	4	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	4	8	no	—
150	C₂×C₄×C₃₁	—	4	4	no	—
151	C₂×C₄×C₃₁	—	8	8	no	—
152	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₃₁	#117	1	4	yes	—
156	C₂×C₄×C₃₁	#121	1	2	yes	—
157	C₂×C₄×C₃₁	#121	1	2	yes	—
158	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
159	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
160	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₈×C₃₁	—	2	4	no	—
167	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
168	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
169	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
170	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
171	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#2311	C₂×C₃₁	1	2	yes	—
173	#1809	C₂×C₃₁	1	4	yes	complemented
174	#1807	C₂×C₃₁	1	4	yes	complemented
175	#1736	C₂×C₃₁	1	2	yes	complemented
176	C₂²×C₃₁	—	2	4	no	—
177	C₂²×C₃₁	—	2	4	no	—
178	C₂²×C₃₁	—	2	4	no	—
179	C₂²×C₃₁	—	2	8	no	—
180	C₂²×C₃₁	—	2	4	no	—
181	C₂²×C₃₁	—	4	4	no	—
182	C₂²×C₃₁	—	4	4	no	—
183	C₂²×C₃₁	#380	1	4	yes	—
184	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
186	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
187	C₂²×C₃₁	#577	1	4	yes	—
188	C₄×C₃₁	—	2	4	no	—
189	C₄×C₃₁	—	8	4	no	—
190	C₄×C₃₁	—	8	4	no	—
191	C₄×C₃₁	#380	1	4	yes	—
192	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
194	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
195	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#131	—	2	4	no	—
198	C₂×#131	—	2	4	no	—
199	C₂×#131	C₄×C₃₁	1	8	yes	complemented
200	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
201	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
202	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
203	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×#117	C₂²×C₃₁	1	2	yes	—
205	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
206	#427	—	2	4	no	—
207	C₂×#107	C₂²×C₃₁	1	2	yes	—
208	C₂×#106	—	2	4	no	—
209	C₂×#106	C₂²×C₃₁	1	2	yes	—
210	C₂×#106	C₂²×C₃₁	1	2	yes	—
211	C₂×#106	C₂²×C₃₁	1	2	yes	—
212	#391	—	2	2	no	—
213	#390	—	2	2	no	—
214	#386	—	2	4	no	—
215	#380	C₂²×C₃₁	1	8	yes	—
216	#340	C₂²×C₃₁	1	8	yes	—
217	C₂×C₃₁	—	2	4	no	—
218	C₂×C₃₁	—	4	4	no	—
219	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
220	C₂×C₃₁	#2912	1	2	yes	—
221	C₂×C₃₁	#3057	1	2	yes	—
222	C₂×C₃₁	#3060	1	2	yes	—
223	C₂×#41	—	2	2	no	—
224	C₂×#41	—	2	2	no	—
225	C₂×#41	—	2	2	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
228	C₂×#41	Q₈×C₃₁	1	4	yes	—
229	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
230	C₂²×D₄	D₄×C₃₁	1	2	yes	—
231	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
232	C₂³×C₄	—	2	2	no	—
233	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
234	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
235	#131	—	2	32	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	4	no	—
239	C₂×M₁₆	—	2	4	no	—
240	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
241	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
242	C₂²×C₈	—	2	4	no	—
243	C₂²×C₈	—	2	4	no	—
244	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
245	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
246	C₂²×C₈	D₄×C₃₁	1	4	yes	—
247	C₂²×C₈	D₄×C₃₁	1	4	yes	—
248	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
249	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	#117	—	2	4	no	—
252	#117	—	2	4	no	—
253	C₂×C₄⋊C₄	—	2	4	no	—
254	C₂×C₄⋊C₄	—	2	4	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
257	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	8	no	—
260	C₂×C₄²	—	2	2	no	—
261	#107	—	2	4	no	—
262	#106	—	2	4	no	—
263	#106	—	2	4	no	—
264	#106	—	2	4	no	—
265	#95	—	2	8	no	—
266	C₃₁	#20662	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
268	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
269	#41	—	2	8	no	—
270	#41	—	2	8	no	—
271	#41	—	2	8	no	—
272	#41	—	2	8	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
276	C₂×D₄	—	2	4	no	—
277	C₂×D₄	—	2	4	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₄⋊C₄×C₃₁	1	8	yes	—
281	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	4	4	no	—
289	C₂²×C₄	—	8	4	no	—
290	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
294	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
297	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
298	C₂²×C₄	#41×C₃₁	1	2	yes	—
299	C₂²×C₄	#41×C₃₁	1	2	yes	—
300	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
301	M₁₆	—	2	8	no	—
302	M₁₆	—	2	16	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	16	no	—
306	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	complemented
307	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
308	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
309	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
311	C₂×C₈	C₂×D₄×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	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	4	no	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₂²⋊C₄	—	8	8	no	—
321	C₄²	—	4	4	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	2	2	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	2	8	no	—
326	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
327	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
328	C₂³	C₃₁×#121	1	2	yes	—
329	C₂³	C₃₁×#121	1	2	yes	—
330	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	Q₈	—	2	8	no	—
332	D₄	—	2	16	no	—
333	D₄	—	2	8	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	8	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	8	8	no	—
346	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
347	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
348	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
349	C₂×C₄	C₃₁×#117	1	4	yes	—
350	C₂×C₄	C₃₁×#121	1	2	yes	—
351	C₂×C₄	C₃₁×#121	1	2	yes	—
352	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
353	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
354	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
359	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
360	C₈	—	2	4	no	—
361	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
362	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
363	C₂²	—	2	4	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	2	8	no	—
367	C₂²	—	2	4	no	—
368	C₂²	—	4	4	no	—
369	C₂²	—	4	4	no	—
370	C₂²	C₃₁×#380	1	4	yes	—
371	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
372	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
373	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
374	C₂²	C₃₁×#577	1	4	yes	—
375	C₄	—	2	4	no	—
376	C₄	—	8	4	no	—
377	C₄	—	8	4	no	—
378	C₄	C₃₁×#380	1	4	yes	—
379	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
380	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
381	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
382	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
383	C₂	—	2	4	no	—
384	C₂	—	4	4	no	—
385	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
386	C₂	C₃₁×#2912	1	2	yes	—
387	C₂	C₃₁×#3057	1	2	yes	—
388	C₂	C₃₁×#3060	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#427	C₂	semidirect
3. C₂²×C₃₁×#106	C₂	semidirect
5. C₃₁×#1809	C₂	semidirect
6. C₃₁×#1807	C₂	semidirect
7. C₃₁×#1736	C₂	semidirect
11. C₂×C₃₁×#131	C₄	semidirect
47. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
51. C₂²×C₈×C₃₁	C₂×C₄	semidirect
111. C₂×C₈×C₃₁	C₂²⋊C₄	semidirect
127. #20662	C₃₁	direct (N × H)
170. C₂×#427	C₂×C₃₁	semidirect
171. C₂²×#106	C₂×C₃₁	semidirect
173. #1809	C₂×C₃₁	semidirect
174. #1807	C₂×C₃₁	semidirect
175. #1736	C₂×C₃₁	semidirect
199. C₂×#131	C₄×C₃₁	semidirect
241. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
245. C₂²×C₈	C₂×C₄×C₃₁	semidirect
266. C₃₁	#20662	direct (N × H)
306. C₂×C₈	C₂²⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#131	2
C₂×C₃₁×#427	2
C₂²×C₃₁×#106	2
C₃₁×#2311	2
C₃₁×#1809	2
C₃₁×#1807	2
C₃₁×#1736	2
#20662	31

Subgroups — C31×#20662 #21477421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20662 #21477421