Subgroups — C₃₁×#11672 #21468431

Order 7936.

334 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1332	C₂	1	4	yes	maximal, complemented
5	C₃₁×#533	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	—	2	1	no	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#494	C₄	1	1	yes	characteristic
9	C₃₁×#487	—	2	1	no	—
10	C₃₁×#467	—	2	1	no	—
11	C₃₁×#467	C₄	1	2	yes	—
12	C₄×Q₁₆×C₃₁	—	2	1	no	—
13	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₃₁	—	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	C₂×M₁₆×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	4	no	—
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₃₁×#107	—	2	1	no	—
37	C₃₁×#107	C₂×C₄	1	2	yes	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	4	1	no	—
40	C₃₁×#103	—	2	2	no	—
41	C₃₁×#103	—	2	2	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₃₁	—	2	2	no	—
47	C₂×Q₈×C₃₁	—	4	1	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₃₁	—	2	4	no	—
52	Q₁₆×C₃₁	—	4	2	no	—
53	M₁₆×C₃₁	—	8	2	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	4	1	no	—
63	C₄⋊C₄×C₃₁	—	4	1	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	1	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₂²⋊C₄×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	—	4	2	no	—
69	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₃₁	D₈	1	1	yes	characteristic
71	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₃₁	—	2	4	no	—
74	#11672	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	#102	1	1	yes	characteristic
76	Q₈×C₃₁	—	4	2	no	—
77	Q₈×C₃₁	—	4	2	no	—
78	Q₈×C₃₁	—	8	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	4	2	no	—
84	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
88	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	8	2	no	—
95	C₈×C₃₁	#104	1	2	yes	—
96	C₈×C₃₁	#131	1	2	yes	—
97	#3170	C₂×C₃₁	1	1	yes	characteristic
98	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
99	#1567	C₂×C₃₁	1	1	yes	characteristic
100	#1332	C₂×C₃₁	1	4	yes	complemented
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	#358	1	2	yes	—
104	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
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	#359	1	2	yes	—
111	C₄×C₃₁	#408	1	1	yes	characteristic
112	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
113	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
115	#533	—	2	1	no	—
116	C₂²×C₁₆	—	2	1	no	—
117	#500	C₄×C₃₁	1	1	yes	characteristic
118	#494	C₄×C₃₁	1	1	yes	characteristic
119	#487	—	2	1	no	—
120	#467	—	2	1	no	—
121	#467	C₄×C₃₁	1	2	yes	—
122	C₄×Q₁₆	—	2	1	no	—
123	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
124	#423	—	2	1	no	—
125	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₄×C₁₆	—	2	1	no	—
127	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
128	#335	—	2	1	no	—
129	#335	C₂²×C₃₁	1	2	yes	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	#1492	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
133	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
136	C₂×M₁₆	—	4	1	no	—
137	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	#107	—	2	1	no	—
151	#107	C₂×C₄×C₃₁	1	2	yes	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	#103	—	2	2	no	—
155	#103	—	2	2	no	—
156	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₃₁	#11672	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	8	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
185	C₄²	D₈×C₃₁	1	1	yes	characteristic
186	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₂³	C₃₁×#102	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
199	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
201	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
202	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	C₃₁×#104	1	2	yes	—
210	C₈	C₃₁×#131	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₃₁×#358	1	2	yes	—
214	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₃₁×#358	1	2	yes	—
220	C₄	C₃₁×#359	1	2	yes	—
221	C₄	C₃₁×#408	1	1	yes	characteristic
222	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
223	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
224	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₃₁×#1492	1	1	yes	characteristic
227	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
228	C₂	C₂×C₃₁×#359	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
74. #11672	C₃₁	direct (N × H)
100. #1332	C₂×C₃₁	semidirect
160. C₃₁	#11672	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
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1567	2
C₃₁×#1332	2
#11672	31

Subgroups — C31×#11672 #21468431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11672 #21468431