Subgroups — C₃₁×#8688 #21465447

Order 7936.

414 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1547	C₂	1	2	yes	maximal
5	C₃₁×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	—	2	1	no	—
7	C₃₁×#532	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	1	no	—
9	C₃₁×#436	C₄	1	2	yes	—
10	C₈×D₄×C₃₁	—	2	1	no	—
11	C₈×D₄×C₃₁	C₄	1	2	yes	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₃₁×#423	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₃₁×#408	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#341	—	2	1	no	—
18	C₃₁×#341	—	2	1	no	—
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#335	—	2	1	no	—
21	C₃₁×#332	—	2	2	no	—
22	C₃₁×#331	—	2	2	no	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₃₁×#131	—	4	4	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
33	C₃₁×#127	D₄	1	1	yes	characteristic
34	C₃₁×#124	—	2	1	no	—
35	C₃₁×#121	—	2	1	no	—
36	C₃₁×#121	Q₈	1	2	yes	—
37	C₄×D₄×C₃₁	—	2	1	no	—
38	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₃₁×C₂²⋊C₈	—	2	2	no	—
47	C₃₁×C₂²⋊C₈	—	2	2	no	—
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
52	#41×C₃₁	—	4	2	no	—
53	#41×C₃₁	—	4	2	no	—
54	C₂×Q₈×C₃₁	—	2	1	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	1	no	—
57	C₂×D₄×C₃₁	—	2	1	no	—
58	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
63	M₁₆×C₃₁	—	4	4	no	—
64	M₁₆×C₃₁	—	4	2	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	4	4	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	8	2	no	—
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₁	#41	1	1	yes	characteristic
86	#8688	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
89	Q₈×C₃₁	—	4	2	no	—
90	D₄×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	4	2	no	—
93	D₄×C₃₁	—	4	1	no	—
94	D₄×C₃₁	—	4	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	4	1	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#135	1	2	yes	—
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	2	1	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	C₈×C₃₁	—	8	2	no	—
116	#2931	C₂×C₃₁	1	1	yes	characteristic
117	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1547	C₂×C₃₁	1	2	yes	—
120	#1292	C₂×C₃₁	1	2	yes	complemented
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	#380	1	1	yes	characteristic
127	C₂²×C₃₁	#478	1	2	yes	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	#416	1	1	yes	characteristic
133	C₄×C₃₁	#418	1	1	yes	characteristic
134	C₄×C₃₁	#463	1	2	yes	—
135	C₄×C₃₁	#472	1	1	yes	characteristic
136	C₄×C₃₁	#473	1	1	yes	characteristic
137	C₂×#131	—	2	1	no	—
138	#532	C₂²×C₃₁	1	1	yes	characteristic
139	#436	—	2	1	no	—
140	#436	C₄×C₃₁	1	2	yes	—
141	C₈×D₄	—	2	1	no	—
142	C₈×D₄	C₄×C₃₁	1	2	yes	—
143	#423	C₂²×C₃₁	1	2	yes	—
144	#423	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	#408	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	#341	—	2	1	no	—
149	#341	—	2	1	no	—
150	#335	—	2	1	no	—
151	#335	—	2	1	no	—
152	#332	—	2	2	no	—
153	#331	—	2	2	no	—
154	C₂×C₃₁	—	2	2	no	—
155	C₂×C₃₁	—	8	2	no	—
156	C₂×C₃₁	#1804	1	1	yes	characteristic
157	C₂×C₃₁	#1813	1	1	yes	characteristic
158	C₂×C₃₁	#1816	1	1	yes	characteristic
159	C₂×#41	—	2	1	no	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
169	#127	D₄×C₃₁	1	1	yes	characteristic
170	#124	—	2	1	no	—
171	#121	—	2	1	no	—
172	#121	Q₈×C₃₁	1	2	yes	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
188	C₃₁	#8688	1	1	yes	characteristic, direct product factor, Sylow
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	8	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₄²	#41×C₃₁	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
240	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
241	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
242	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₄	C₃₁×#135	1	2	yes	—
244	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	8	2	no	—
251	C₈	—	8	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₃₁×#380	1	1	yes	characteristic
258	C₂²	C₃₁×#478	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	C₃₁×#416	1	1	yes	characteristic
264	C₄	C₃₁×#418	1	1	yes	characteristic
265	C₄	C₃₁×#463	1	2	yes	—
266	C₄	C₃₁×#472	1	1	yes	characteristic
267	C₄	C₃₁×#473	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	2	no	—
270	C₂	C₃₁×#1804	1	1	yes	characteristic
271	C₂	C₃₁×#1813	1	1	yes	characteristic
272	C₂	C₃₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1837	C₂	semidirect
3. C₂×C₃₁×#335	C₂	semidirect
5. C₃₁×#1292	C₂	semidirect
86. #8688	C₃₁	direct (N × H)
117. #1837	C₂×C₃₁	semidirect
118. C₂×#335	C₂×C₃₁	semidirect
120. #1292	C₂×C₃₁	semidirect
188. C₃₁	#8688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2931	2
C₃₁×#1837	2
C₂×C₃₁×#335	2
C₃₁×#1547	2
C₃₁×#1292	2
#8688	31

Subgroups — C31×#8688 #21465447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8688 #21465447