Group #45693079 — subgroups

Subgroups — C₁₃₁×#7195 #45693079

Order 33536.

222 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×#322×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆×C₁₃₁	—	2	2	no	—
7	C₂²⋊C₁₆×C₁₃₁	C₄	1	4	yes	—
8	C₄×C₁₆×C₁₃₁	—	2	2	no	—
9	#322×C₁₃₁	—	2	2	no	—
10	C₂²×C₈×C₁₃₁	C₈	1	2	yes	—
11	C₂²×C₈×C₁₃₁	C₈	1	2	yes	—
12	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₃₁	—	2	4	no	—
15	C₄×C₈×C₁₃₁	—	2	1	no	—
16	C₄×C₈×C₁₃₁	—	2	1	no	—
17	C₄×C₈×C₁₃₁	—	2	1	no	—
18	C₄×C₈×C₁₃₁	—	2	1	no	—
19	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
20	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₁₃₁	—	2	2	no	—
26	C₂×C₈×C₁₃₁	—	2	2	no	—
27	C₂×C₈×C₁₃₁	—	2	2	no	—
28	C₂×C₈×C₁₃₁	—	4	2	no	—
29	C₂×C₈×C₁₃₁	—	4	2	no	—
30	C₂×C₈×C₁₃₁	C₁₆	1	4	yes	complemented
31	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₄²×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
36	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
37	C₁₆×C₁₃₁	—	4	4	no	—
38	C₂³×C₁₃₁	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
42	C₂×C₄×C₁₃₁	#97	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₁₃₁	M₃₂	1	2	yes	—
49	C₈×C₁₃₁	—	2	2	no	—
50	C₈×C₁₃₁	—	2	2	no	—
51	C₈×C₁₃₁	—	2	4	no	—
52	C₈×C₁₃₁	—	4	2	no	—
53	C₂²×C₁₃₁	—	2	1	no	—
54	C₂²×C₁₃₁	—	2	1	no	—
55	C₂²×C₁₃₁	#336	1	1	yes	characteristic
56	C₂²×C₁₃₁	C₄×C₁₆	1	1	yes	characteristic
57	C₂²×C₁₃₁	#346	1	1	yes	characteristic
58	C₄×C₁₃₁	—	2	1	no	—
59	C₄×C₁₃₁	—	2	1	no	—
60	C₄×C₁₃₁	#344	1	2	yes	—
61	C₄×C₁₃₁	C₂²⋊C₁₆	1	2	yes	—
62	C₄×C₁₃₁	#349	1	2	yes	—
63	C₄×C₁₃₁	C₄⋊C₁₆	1	2	yes	—
64	C₂×C₁₃₁	—	2	1	no	—
65	C₂×C₁₃₁	—	2	1	no	—
66	C₂×C₁₃₁	#1281	1	1	yes	characteristic
67	C₂×C₁₃₁	#1374	1	1	yes	characteristic
68	C₂×C₁₃₁	#1377	1	1	yes	characteristic
69	#7195	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₃₁	#7195	1	1	yes	characteristic, direct product factor, Sylow
71	#2162	C₂×C₁₃₁	1	2	yes	—
72	C₂×#322	C₂×C₁₃₁	1	1	yes	characteristic
73	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
74	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
76	C₄⋊C₁₆	—	2	2	no	—
77	C₂²⋊C₁₆	C₄×C₁₃₁	1	4	yes	—
78	C₄×C₁₆	—	2	2	no	—
79	#322	—	2	2	no	—
80	C₂²×C₈	C₈×C₁₃₁	1	2	yes	—
81	C₂²×C₈	C₈×C₁₃₁	1	2	yes	—
82	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
83	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
84	C₂×C₁₆	—	2	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
90	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
91	C₄×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₁₆×C₁₃₁	1	4	yes	complemented
101	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
102	C₂×C₈	C₄⋊C₄×C₁₃₁	1	2	yes	—
103	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
104	C₄²	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
105	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
106	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
107	C₁₆	—	4	4	no	—
108	C₂³	C₄×C₈×C₁₃₁	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
112	C₂×C₄	#97×C₁₃₁	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
115	C₂×C₄	C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
116	C₂×C₄	C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₁₆×C₁₃₁	1	2	yes	—
118	C₂×C₄	M₃₂×C₁₃₁	1	2	yes	—
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	2	4	no	—
122	C₈	—	4	2	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#336×C₁₃₁	1	1	yes	characteristic
126	C₂²	C₄×C₁₆×C₁₃₁	1	1	yes	characteristic
127	C₂²	#346×C₁₃₁	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#344×C₁₃₁	1	2	yes	—
131	C₄	C₂²⋊C₁₆×C₁₃₁	1	2	yes	—
132	C₄	#349×C₁₃₁	1	2	yes	—
133	C₄	C₄⋊C₁₆×C₁₃₁	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#1281×C₁₃₁	1	1	yes	characteristic
137	C₂	#1374×C₁₃₁	1	1	yes	characteristic
138	C₂	#1377×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2162×C₁₃₁

C₂

yes

maximal

C₂×#322×C₁₃₁

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃₁

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃₁

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₃₁

—

C₂²⋊C₁₆×C₁₃₁

C₄

yes

—

C₄×C₁₆×C₁₃₁

—

#322×C₁₃₁

—

C₂²×C₈×C₁₃₁

C₈

yes

—

C₂²×C₈×C₁₃₁

C₈

yes

—

C₂²×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

D₄

yes

—

C₄×C₈×C₁₃₁

D₄

yes

characteristic

C₄×C₈×C₁₃₁

Q₈

yes

characteristic

C₂²×C₄×C₁₃₁

C₄²

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₁₆

yes

complemented

C₂×C₈×C₁₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃₁

C₄⋊C₄

yes

—

C₄²×C₁₃₁

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃₁

C₄⋊C₄

yes

characteristic

C₄²×C₁₃₁

M₁₆

yes

characteristic

C₄²×C₁₃₁

M₁₆

yes

characteristic

C₁₆×C₁₃₁

—

C₂³×C₁₃₁

C₄×C₈

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂×C₁₆

yes

—

C₂×C₄×C₁₃₁

M₃₂

yes

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#336

yes

characteristic

C₂²×C₁₃₁

C₄×C₁₆

yes

characteristic

C₂²×C₁₃₁

#346

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#344

yes

—

C₄×C₁₃₁

C₂²⋊C₁₆

yes

—

C₄×C₁₃₁

#349

yes

—

C₄×C₁₃₁

C₄⋊C₁₆

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#7195

yes

characteristic, direct product factor, Sylow

#2162

C₂×C₁₃₁

yes

—

C₂×#322

C₂×C₁₃₁

yes

characteristic

C₂×C₄×C₈

C₄×C₁₃₁

yes

characteristic

C₂×C₄×C₈

C₄×C₁₃₁

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₃₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₄×C₁₃₁

yes

—

C₄×C₁₆

—

#322

—

C₂²×C₈

C₈×C₁₃₁

yes

—

C₂²×C₈

C₈×C₁₃₁

yes

—

C₂²×C₈

C₂×C₄×C₁₃₁

yes

—

C₂×C₄²

C₂×C₄×C₁₃₁

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄×C₁₃₁

yes

—

C₄×C₈

D₄×C₁₃₁

yes

characteristic

C₄×C₈

Q₈×C₁₃₁

yes

characteristic

C₂²×C₄

C₄²×C₁₃₁

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₃₁

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₁₆×C₁₃₁

yes

complemented

101

C₂×C₈

C₂²⋊C₄×C₁₃₁

yes

—

102

C₂×C₈

C₄⋊C₄×C₁₃₁

yes

—

103

C₄²

C₂²⋊C₄×C₁₃₁

yes

characteristic

104

C₄²

C₄⋊C₄×C₁₃₁

yes

characteristic

105

C₄²

M₁₆×C₁₃₁

yes

characteristic

106

C₄²

M₁₆×C₁₃₁

yes

characteristic

107

C₁₆

—

108

C₂³

C₄×C₈×C₁₃₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#95×C₁₃₁

yes

characteristic

112

C₂×C₄

#97×C₁₃₁

yes

characteristic

113

C₂×C₄

C₂²⋊C₈×C₁₃₁

yes

characteristic

114

C₂×C₄

C₂²⋊C₈×C₁₃₁

yes

characteristic

115

C₂×C₄

C₄⋊C₈×C₁₃₁

yes

characteristic

116

C₂×C₄

C₄⋊C₈×C₁₃₁

yes

characteristic

117

C₂×C₄

C₂×C₁₆×C₁₃₁

yes

—

118

C₂×C₄

M₃₂×C₁₃₁

yes

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

#336×C₁₃₁

yes

characteristic

126

C₂²

C₄×C₁₆×C₁₃₁

yes

characteristic

127

C₂²

#346×C₁₃₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#344×C₁₃₁

yes

—

131

C₄

C₂²⋊C₁₆×C₁₃₁

yes

—

132

C₄

#349×C₁₃₁

yes

—

133

C₄

C₄⋊C₁₆×C₁₃₁

yes

—

134

C₂

—

135

C₂

—

136

C₂

#1281×C₁₃₁

yes

characteristic

137

C₂

#1374×C₁₃₁

yes

characteristic

138

C₂

#1377×C₁₃₁

yes

characteristic

	subgroup	G/H
center	C₄²×C₁₃₁	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₁₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₁
socle	C₂²×C₁₃₁	#336
prime radical	C₂³×C₁₃₁	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

subgroup

G/H

center

C₄²×C₁₃₁

C₂²⋊C₄

commutator

C₂²

C₄×C₁₆×C₁₃₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₃₁

socle

C₂²×C₁₃₁

#336

prime radical

C₂³×C₁₃₁

C₄×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

N	H ≅ G/N	type
30. C₂×C₈×C₁₃₁	C₁₆	semidirect
69. #7195	C₁₃₁	direct (N × H)
70. C₁₃₁	#7195	direct (N × H)
100. C₂×C₈	C₁₆×C₁₃₁	semidirect

H ≅ G/N

type

30. C₂×C₈×C₁₃₁

C₁₆

semidirect

69. #7195

C₁₃₁

direct (N × H)

70. C₁₃₁

#7195

direct (N × H)

100. C₂×C₈

C₁₆×C₁₃₁

semidirect

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

subgroup	index
#2162×C₁₃₁	2
C₂×#322×C₁₃₁	2
#7195	131

subgroup

index

#2162×C₁₃₁

C₂×#322×C₁₃₁

#7195

131

Subgroups — C₁₃₁×#7195 #45693079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#7195 #45693079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7195 #45693079