Subgroups — C₃₁×#21312 #21478071

Order 7936.

1250 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1989	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1988	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1987	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#533	—	2	4	no	—
10	C₃₁×#533	C₄	1	8	yes	complemented
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	2	yes	—
14	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
17	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	—	2	1	no	—
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#423	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₃₁×#95	—	2	1	no	—
30	C₃₁×#340	C₂²	1	8	yes	—
31	C₃₁×#340	C₂²	1	8	yes	—
32	C₂²×Q₈×C₃₁	—	2	1	no	—
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂³×C₄×C₃₁	—	2	2	no	—
35	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₃₁	—	2	4	no	—
39	C₂×M₁₆×C₃₁	—	2	8	no	—
40	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
44	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
45	C₃₁×#125	—	2	4	no	—
46	C₃₁×#125	C₂×C₄	1	8	yes	complemented
47	C₃₁×C₂²⋊Q₈	—	2	8	no	—
48	C₃₁×C₂²⋊Q₈	—	2	8	no	—
49	C₄×Q₈×C₃₁	—	2	8	no	—
50	C₄×Q₈×C₃₁	—	2	8	no	—
51	C₃₁×#117	—	2	4	no	—
52	C₃₁×#117	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₃₁	Q₈	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	4	4	no	—
68	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	4	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×C₄⋊C₈	—	2	4	no	—
75	C₃₁×C₄⋊C₈	—	2	4	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	4	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	8	no	—
87	C₂²×C₄×C₃₁	—	2	4	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	—	4	4	no	—
92	C₂²×C₄×C₃₁	—	4	4	no	—
93	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
94	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
103	C₂²×C₄×C₃₁	#41	1	4	yes	—
104	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₃₁	—	4	8	no	—
106	C₂×C₈×C₃₁	—	2	4	no	—
107	C₂×C₈×C₃₁	—	2	8	no	—
108	C₄⋊C₄×C₃₁	—	2	8	no	—
109	C₄⋊C₄×C₃₁	—	2	8	no	—
110	C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₄⋊C₄×C₃₁	—	2	4	no	—
112	C₄⋊C₄×C₃₁	—	2	8	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
117	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
118	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
119	C₂²⋊C₄×C₃₁	—	2	8	no	—
120	C₄²×C₃₁	—	2	8	no	—
121	C₄²×C₃₁	—	4	4	no	—
122	C₄²×C₃₁	—	4	4	no	—
123	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
124	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
125	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
126	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄²×C₃₁	C₂×D₄	1	2	yes	—
128	C₄²×C₃₁	C₂×D₄	1	2	yes	—
129	#21312	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	2	1	no	—
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
135	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₃₁	—	4	8	no	—
137	Q₈×C₃₁	—	4	8	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	8	no	—
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	2	4	no	—
145	C₂×C₄×C₃₁	—	2	8	no	—
146	C₂×C₄×C₃₁	—	4	4	no	—
147	C₂×C₄×C₃₁	—	4	16	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	4	no	—
152	C₂×C₄×C₃₁	—	4	4	no	—
153	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
159	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
160	C₂×C₄×C₃₁	#124	1	4	yes	—
161	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
162	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
163	C₂×C₄×C₃₁	#137	1	2	yes	—
164	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
167	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
168	C₈×C₃₁	—	4	4	no	—
169	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
170	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
171	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
172	#1989	C₂×C₃₁	1	4	yes	complemented
173	#1988	C₂×C₃₁	1	2	yes	complemented
174	#1987	C₂×C₃₁	1	2	yes	complemented
175	#1736	C₂×C₃₁	1	4	yes	—
176	C₂²×C₃₁	—	2	4	no	—
177	C₂²×C₃₁	—	2	4	no	—
178	C₂²×C₃₁	—	2	2	no	—
179	C₂²×C₃₁	—	2	2	no	—
180	C₂²×C₃₁	#391	1	4	yes	—
181	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
183	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
185	C₂²×C₃₁	#575	1	2	yes	—
186	C₄×C₃₁	—	2	2	no	—
187	C₄×C₃₁	—	2	2	no	—
188	C₄×C₃₁	—	2	2	no	—
189	C₄×C₃₁	—	4	8	no	—
190	C₄×C₃₁	—	4	4	no	—
191	C₄×C₃₁	—	4	4	no	—
192	C₄×C₃₁	#391	1	4	yes	—
193	C₄×C₃₁	#417	1	4	yes	—
194	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
195	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
196	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
198	#533	—	2	4	no	—
199	#533	C₄×C₃₁	1	8	yes	complemented
200	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×#125	C₂²×C₃₁	1	2	yes	—
203	C₂×C₂²⋊Q₈	—	2	1	no	—
204	C₂×C₂²⋊Q₈	—	2	1	no	—
205	C₂×C₄×Q₈	—	2	1	no	—
206	C₂×C₄×Q₈	—	2	1	no	—
207	C₂×#117	C₂²×C₃₁	1	2	yes	—
208	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
209	C₂²×C₄²	—	2	1	no	—
210	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
211	#423	—	2	4	no	—
212	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₄×C₄⋊C₄	—	2	1	no	—
215	C₄×C₄⋊C₄	—	2	1	no	—
216	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
217	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
218	C₂×#95	—	2	1	no	—
219	#340	C₂²×C₃₁	1	8	yes	—
220	#340	C₂²×C₃₁	1	8	yes	—
221	C₂×C₃₁	—	2	4	no	—
222	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
223	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
224	C₂×C₃₁	#2896	1	1	yes	characteristic
225	C₂×C₃₁	#3077	1	2	yes	—
226	C₂×C₃₁	#3078	1	2	yes	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	—	2	1	no	—
229	C₂³×C₄	—	2	2	no	—
230	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
231	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
232	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
233	C₂×M₁₆	—	2	4	no	—
234	C₂×M₁₆	—	2	8	no	—
235	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
236	C₄⋊Q₈	—	2	2	no	—
237	C₄⋊Q₈	—	2	2	no	—
238	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
239	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
240	#125	—	2	4	no	—
241	#125	C₂×C₄×C₃₁	1	8	yes	complemented
242	C₂²⋊Q₈	—	2	8	no	—
243	C₂²⋊Q₈	—	2	8	no	—
244	C₄×Q₈	—	2	8	no	—
245	C₄×Q₈	—	2	8	no	—
246	#117	—	2	4	no	—
247	#117	C₂×C₄×C₃₁	1	8	yes	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	4	no	—
254	C₂×C₄⋊C₄	Q₈×C₃₁	1	8	yes	complemented
255	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	4	4	no	—
263	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
264	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	D₄×C₃₁	1	4	yes	—
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
269	C₄⋊C₈	—	2	4	no	—
270	C₄⋊C₈	—	2	4	no	—
271	#95	—	2	2	no	—
272	#95	—	2	2	no	—
273	C₃₁	#21312	1	1	yes	characteristic, direct product factor, Sylow
274	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	4	4	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	8	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	4	no	—
288	C₂²×C₄	—	4	4	no	—
289	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
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	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
298	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
299	C₂²×C₄	#41×C₃₁	1	4	yes	—
300	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
301	M₁₆	—	4	8	no	—
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	2	8	no	—
304	C₄⋊C₄	—	2	8	no	—
305	C₄⋊C₄	—	2	8	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	8	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	4	no	—
312	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
313	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
314	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
315	C₂²⋊C₄	—	2	8	no	—
316	C₄²	—	2	8	no	—
317	C₄²	—	4	4	no	—
318	C₄²	—	4	4	no	—
319	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
320	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
321	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
322	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
323	C₄²	C₂×D₄×C₃₁	1	2	yes	—
324	C₄²	C₂×D₄×C₃₁	1	2	yes	—
325	C₂³	—	2	1	no	—
326	C₂³	—	2	1	no	—
327	C₂³	—	2	2	no	—
328	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
329	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
330	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	Q₈	—	4	8	no	—
332	Q₈	—	4	8	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	8	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	16	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
349	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
350	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
351	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
352	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
353	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
354	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
355	C₂×C₄	C₃₁×#124	1	4	yes	—
356	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
357	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
358	C₂×C₄	C₃₁×#137	1	2	yes	—
359	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
362	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
363	C₈	—	4	4	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	C₃₁×#391	1	4	yes	—
369	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
371	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
372	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
373	C₂²	C₃₁×#575	1	2	yes	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	4	8	no	—
378	C₄	—	4	4	no	—
379	C₄	—	4	4	no	—
380	C₄	C₃₁×#391	1	4	yes	—
381	C₄	C₃₁×#417	1	4	yes	—
382	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
383	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
384	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
385	C₂	—	2	4	no	—
386	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
387	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
388	C₂	C₃₁×#2896	1	1	yes	characteristic
389	C₂	C₃₁×#3077	1	2	yes	—
390	C₂	C₃₁×#3078	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	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1989	C₂	semidirect
5. C₃₁×#1988	C₂	semidirect
6. C₃₁×#1987	C₂	semidirect
10. C₃₁×#533	C₄	semidirect
43. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
44. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
46. C₃₁×#125	C₂×C₄	semidirect
59. C₂×C₄⋊C₄×C₃₁	Q₈	semidirect
129. #21312	C₃₁	direct (N × H)
172. #1989	C₂×C₃₁	semidirect
173. #1988	C₂×C₃₁	semidirect
174. #1987	C₂×C₃₁	semidirect
199. #533	C₄×C₃₁	semidirect
238. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
239. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
241. #125	C₂×C₄×C₃₁	semidirect
254. C₂×C₄⋊C₄	Q₈×C₃₁	semidirect
273. C₃₁	#21312	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×#533	2
C₂×C₃₁×#423	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1989	2
C₃₁×#1988	2
C₃₁×#1987	2
C₃₁×#1736	2
#21312	31

Subgroups — C31×#21312 #21478071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21312 #21478071