Subgroups — C₃₁×#7159 #21463918

Order 7936.

222 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×M₃₂	C₄	1	2	yes	—
5	C₃₁×#431	—	2	1	no	—
6	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#346	—	2	1	no	—
9	C₄×C₁₆×C₃₁	—	2	1	no	—
10	C₃₁×#335	—	2	1	no	—
11	C₃₁×#335	—	2	1	no	—
12	C₃₁×#335	C₄	1	2	yes	—
13	C₃₁×#334	C₄	1	2	yes	—
14	C₂²×C₈×C₃₁	—	4	1	no	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
18	C₃₁×M₃₂	—	2	2	no	—
19	C₃₁×M₃₂	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₃₁	—	2	2	no	—
21	C₃₁×C₄⋊C₈	—	2	2	no	—
22	C₃₁×C₄⋊C₈	—	2	2	no	—
23	C₃₁×C₄⋊C₈	—	4	1	no	—
24	C₃₁×C₂²⋊C₈	—	4	1	no	—
25	C₄×C₈×C₃₁	—	4	1	no	—
26	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	4	2	no	—
35	C₂×C₈×C₃₁	—	4	2	no	—
36	C₂×C₈×C₃₁	—	8	2	no	—
37	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
43	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₃₁	D₈	1	1	yes	characteristic
46	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
47	C₁₆×C₃₁	—	2	4	no	—
48	#7159	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₃₁	#106	1	1	yes	characteristic
50	C₂×C₄×C₃₁	—	2	1	no	—
51	C₂×C₄×C₃₁	—	2	1	no	—
52	C₂×C₄×C₃₁	—	4	1	no	—
53	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
60	C₂×C₄×C₃₁	#108	1	2	yes	—
61	C₈×C₃₁	—	2	1	no	—
62	C₈×C₃₁	—	2	1	no	—
63	C₈×C₃₁	—	8	2	no	—
64	C₈×C₃₁	—	8	2	no	—
65	C₈×C₃₁	C₄×C₈	1	1	yes	characteristic
66	C₈×C₃₁	#97	1	1	yes	characteristic
67	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
68	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
69	#1564	C₂×C₃₁	1	1	yes	characteristic
70	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
71	C₂²×C₃₁	—	2	1	no	—
72	C₂²×C₃₁	—	2	1	no	—
73	C₂²×C₃₁	#334	1	2	yes	—
74	C₂²×C₃₁	#340	1	1	yes	characteristic
75	C₄×C₃₁	—	2	1	no	—
76	C₄×C₃₁	—	2	1	no	—
77	C₄×C₃₁	—	2	1	no	—
78	C₄×C₃₁	#335	1	2	yes	—
79	C₄×C₃₁	#336	1	1	yes	characteristic
80	C₄×C₃₁	#341	1	1	yes	characteristic
81	C₄×C₃₁	#361	1	1	yes	characteristic
82	C₄×C₃₁	#362	1	1	yes	characteristic
83	C₂×M₃₂	C₄×C₃₁	1	2	yes	—
84	#431	—	2	1	no	—
85	C₂×C₄⋊C₈	—	2	1	no	—
86	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
87	#346	—	2	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	#335	—	2	1	no	—
90	#335	—	2	1	no	—
91	#335	C₄×C₃₁	1	2	yes	—
92	#334	C₄×C₃₁	1	2	yes	—
93	C₂×C₃₁	—	2	2	no	—
94	C₂×C₃₁	#1277	1	1	yes	characteristic
95	C₂×C₃₁	#1387	1	1	yes	characteristic
96	C₂×C₃₁	#1389	1	1	yes	characteristic
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	C₈×C₃₁	1	4	yes	complemented
103	C₂×C₁₆	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₂²⋊C₈	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
111	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
112	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
113	C₃₁	#7159	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
116	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
126	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
127	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
128	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
129	C₄²	D₈×C₃₁	1	1	yes	characteristic
130	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
131	C₁₆	—	2	4	no	—
132	C₂³	C₃₁×#106	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
137	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
138	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
139	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
140	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
141	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
142	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
143	C₂×C₄	C₃₁×#108	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₈	C₄×C₈×C₃₁	1	1	yes	characteristic
149	C₈	C₃₁×#97	1	1	yes	characteristic
150	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	C₃₁×#334	1	2	yes	—
154	C₂²	C₃₁×#340	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	C₃₁×#335	1	2	yes	—
159	C₄	C₃₁×#336	1	1	yes	characteristic
160	C₄	C₃₁×#341	1	1	yes	characteristic
161	C₄	C₃₁×#361	1	1	yes	characteristic
162	C₄	C₃₁×#362	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	C₃₁×#1277	1	1	yes	characteristic
165	C₂	C₃₁×#1387	1	1	yes	characteristic
166	C₂	C₃₁×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#102
commutator	C₈	C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
socle	C₂²×C₃₁	#340
prime radical	C₂³×C₃₁	#106
LCS	G ▷ 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
19. C₃₁×M₃₂	C₈	semidirect
48. #7159	C₃₁	direct (N × H)
102. M₃₂	C₈×C₃₁	semidirect
113. C₃₁	#7159	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×M₃₂	2
C₃₁×#1564	2
C₂×C₃₁×#335	2
#7159	31

Subgroups — C31×#7159 #21463918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7159 #21463918