Subgroups — C₃₁×#11646 #21468405

Order 7936.

314 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1332	C₂	1	4	yes	maximal, complemented
5	C₃₁×#533	—	2	1	no	—
6	C₃₁×#500	C₄	1	1	yes	characteristic
7	C₃₁×#494	C₄	1	1	yes	characteristic
8	C₃₁×#487	—	2	1	no	—
9	C₃₁×#467	—	2	1	no	—
10	C₃₁×#467	C₄	1	2	yes	—
11	C₄×Q₁₆×C₃₁	—	2	1	no	—
12	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊C₁₆	—	2	1	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#335	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₃₁	—	2	1	no	—
21	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₃₁	—	4	1	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	—	2	1	no	—
26	C₃₁×C₄⋊Q₈	—	2	1	no	—
27	C₃₁×#125	—	4	1	no	—
28	C₃₁×C₂²⋊Q₈	—	2	2	no	—
29	C₃₁×C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈×C₃₁	—	2	2	no	—
31	C₄×Q₈×C₃₁	—	4	1	no	—
32	C₃₁×#117	—	4	1	no	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₃₁×#107	—	2	1	no	—
36	C₃₁×#107	C₂×C₄	1	2	yes	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	4	1	no	—
39	C₃₁×#103	—	2	2	no	—
40	C₃₁×#103	—	2	2	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₃₁	—	2	2	no	—
46	C₂×Q₈×C₃₁	—	4	1	no	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
50	Q₁₆×C₃₁	—	2	4	no	—
51	Q₁₆×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	4	1	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	—	8	2	no	—
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	4	1	no	—
62	C₄⋊C₄×C₃₁	—	4	1	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	1	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₂²⋊C₄×C₃₁	—	4	2	no	—
67	C₄²×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₃₁	D₈	1	1	yes	characteristic
70	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₃₁	—	4	2	no	—
73	#11646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	#102	1	1	yes	characteristic
75	Q₈×C₃₁	—	4	2	no	—
76	Q₈×C₃₁	—	4	2	no	—
77	Q₈×C₃₁	—	8	2	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	4	1	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
87	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	8	2	no	—
95	C₈×C₃₁	#131	1	2	yes	—
96	#3170	C₂×C₃₁	1	1	yes	characteristic
97	#2162	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
99	#1332	C₂×C₃₁	1	4	yes	complemented
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	#358	1	2	yes	—
103	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	8	2	no	—
108	C₄×C₃₁	—	8	2	no	—
109	C₄×C₃₁	#358	1	2	yes	—
110	C₄×C₃₁	#408	1	1	yes	characteristic
111	C₄×C₃₁	#421	1	1	yes	characteristic
112	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
113	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
114	#533	—	2	1	no	—
115	#500	C₄×C₃₁	1	1	yes	characteristic
116	#494	C₄×C₃₁	1	1	yes	characteristic
117	#487	—	2	1	no	—
118	#467	—	2	1	no	—
119	#467	C₄×C₃₁	1	2	yes	—
120	C₄×Q₁₆	—	2	1	no	—
121	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
122	C₂×C₄⋊C₈	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	C₄⋊C₁₆	—	2	1	no	—
125	C₂²⋊C₁₆	—	2	1	no	—
126	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
127	#335	—	2	1	no	—
128	#335	C₂²×C₃₁	1	2	yes	—
129	C₂×C₃₁	—	2	2	no	—
130	C₂×C₃₁	#1480	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
132	C₂×C₃₁	#2143	1	1	yes	characteristic
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
135	C₂²×C₈	—	4	1	no	—
136	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	#125	—	4	1	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	4	1	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	4	1	no	—
145	#117	—	4	1	no	—
146	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	#107	—	2	1	no	—
149	#107	C₂×C₄×C₃₁	1	2	yes	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	4	1	no	—
152	#103	—	2	2	no	—
153	#103	—	2	2	no	—
154	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
156	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	C₃₁	#11646	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
183	C₄²	D₈×C₃₁	1	1	yes	characteristic
184	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₁₆	—	4	2	no	—
187	C₂³	C₃₁×#102	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
197	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
198	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
199	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
200	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	C₃₁×#131	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₃₁×#358	1	2	yes	—
212	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	C₃₁×#358	1	2	yes	—
219	C₄	C₃₁×#408	1	1	yes	characteristic
220	C₄	C₃₁×#421	1	1	yes	characteristic
221	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
222	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₃₁×#1480	1	1	yes	characteristic
225	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
226	C₂	C₃₁×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂³×C₃₁	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1332	C₂	semidirect
73. #11646	C₃₁	direct (N × H)
99. #1332	C₂×C₃₁	semidirect
158. C₃₁	#11646	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×#3170	2
C₃₁×#2162	2
C₂×C₃₁×#335	2
C₃₁×#1332	2
#11646	31

Subgroups — C31×#11646 #21468405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11646 #21468405