Group #45693115 — subgroups

Subgroups — C₁₃₁×#7231 #45693115

Order 33536.

270 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₂×C₂²⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1810×C₁₃₁	C₂	1	2	yes	maximal
3	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
4	C₂²×C₁₆×C₁₃₁	C₄	1	2	yes	—
5	C₂×#108×C₁₃₁	—	2	2	no	—
6	#407×C₁₃₁	—	2	2	no	—
7	C₂²⋊C₁₆×C₁₃₁	—	2	2	no	—
8	#341×C₁₃₁	C₄	1	4	yes	—
9	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
10	C₂×M₁₆×C₁₃₁	—	2	4	no	—
11	C₂²×C₈×C₁₃₁	—	4	2	no	—
12	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
15	C₂²×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
16	C₂×C₁₆×C₁₃₁	—	2	2	no	—
17	C₂×C₁₆×C₁₃₁	—	4	2	no	—
18	#108×C₁₃₁	—	4	4	no	—
19	C₂²⋊C₈×C₁₃₁	—	4	2	no	—
20	C₂²⋊C₈×C₁₃₁	—	4	2	no	—
21	C₂⁴×C₁₃₁	Q₁₆	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	—	4	1	no	—
23	C₂²×C₄×C₁₃₁	—	4	1	no	—
24	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₁	SD₁₆	1	2	yes	—
28	M₁₆×C₁₃₁	—	4	4	no	—
29	M₁₆×C₁₃₁	—	4	4	no	—
30	C₂×C₈×C₁₃₁	—	2	4	no	—
31	C₂×C₈×C₁₃₁	—	2	2	no	—
32	C₂×C₈×C₁₃₁	—	4	2	no	—
33	C₂×C₈×C₁₃₁	—	4	1	no	—
34	C₂×C₈×C₁₃₁	—	4	4	no	—
35	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₁₃₁	—	4	2	no	—
39	C₂³×C₁₃₁	—	2	2	no	—
40	C₂³×C₁₃₁	—	4	1	no	—
41	C₂³×C₁₃₁	—	4	1	no	—
42	C₂³×C₁₃₁	#103	1	2	yes	—
43	C₂³×C₁₃₁	#107	1	1	yes	characteristic
44	C₂×C₄×C₁₃₁	—	2	2	no	—
45	C₂×C₄×C₁₃₁	—	2	1	no	—
46	C₂×C₄×C₁₃₁	—	2	2	no	—
47	C₂×C₄×C₁₃₁	—	2	1	no	—
48	C₂×C₄×C₁₃₁	—	4	1	no	—
49	C₂×C₄×C₁₃₁	—	4	2	no	—
50	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
51	C₂×C₄×C₁₃₁	#102	1	2	yes	—
52	C₂×C₄×C₁₃₁	#106	1	1	yes	characteristic
53	C₈×C₁₃₁	—	2	1	no	—
54	C₈×C₁₃₁	—	2	2	no	—
55	C₈×C₁₃₁	—	8	4	no	—
56	C₈×C₁₃₁	#99	1	2	yes	—
57	C₂²×C₁₃₁	—	2	1	no	—
58	C₂²×C₁₃₁	—	2	1	no	—
59	C₂²×C₁₃₁	—	2	2	no	—
60	C₂²×C₁₃₁	—	2	2	no	—
61	C₂²×C₁₃₁	—	2	2	no	—
62	C₂²×C₁₃₁	—	4	4	no	—
63	C₂²×C₁₃₁	#340	1	1	yes	characteristic
64	C₂²×C₁₃₁	#368	1	2	yes	—
65	C₄×C₁₃₁	—	2	2	no	—
66	C₄×C₁₃₁	—	2	1	no	—
67	C₄×C₁₃₁	H(3,ℤ/4)	1	1	yes	characteristic
68	C₄×C₁₃₁	#342	1	1	yes	characteristic
69	C₂×C₁₃₁	—	2	2	no	—
70	C₂×C₁₃₁	—	4	2	no	—
71	C₂×C₁₃₁	#1289	1	1	yes	characteristic
72	C₂×C₁₃₁	#1381	1	1	yes	characteristic
73	C₂×C₁₃₁	#1385	1	1	yes	characteristic
74	#7231	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃₁	#7231	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₂²⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
77	#1810	C₂×C₁₃₁	1	2	yes	—
78	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
79	C₂²×C₁₆	C₄×C₁₃₁	1	2	yes	—
80	C₂×#108	—	2	2	no	—
81	#407	—	2	2	no	—
82	C₂²⋊C₁₆	—	2	2	no	—
83	#341	C₄×C₁₃₁	1	4	yes	—
84	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
85	C₂×M₁₆	—	2	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
89	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
90	C₂²×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	4	2	no	—
93	#108	—	4	4	no	—
94	C₂²⋊C₈	—	4	2	no	—
95	C₂²⋊C₈	—	4	2	no	—
96	C₂⁴	Q₁₆×C₁₃₁	1	1	yes	characteristic
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
100	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
101	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
102	C₂²×C₄	SD₁₆×C₁₃₁	1	2	yes	—
103	M₁₆	—	4	4	no	—
104	M₁₆	—	4	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₁₃₁	1	2	yes	—
113	C₁₆	—	4	2	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	#103×C₁₃₁	1	2	yes	—
118	C₂³	#107×C₁₃₁	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
126	C₂×C₄	#102×C₁₃₁	1	2	yes	—
127	C₂×C₄	#106×C₁₃₁	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	2	no	—
130	C₈	—	8	4	no	—
131	C₈	#99×C₁₃₁	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#340×C₁₃₁	1	1	yes	characteristic
139	C₂²	#368×C₁₃₁	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	H(3,ℤ/4)×C₁₃₁	1	1	yes	characteristic
143	C₄	#342×C₁₃₁	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	—	4	2	no	—
146	C₂	#1289×C₁₃₁	1	1	yes	characteristic
147	C₂	#1381×C₁₃₁	1	1	yes	characteristic
148	C₂	#1385×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊C₁₆×C₁₃₁

C₂

yes

maximal, characteristic

#1810×C₁₃₁

C₂

yes

maximal

C₂³×C₈×C₁₃₁

C₂²

yes

characteristic

C₂²×C₁₆×C₁₃₁

C₄

yes

—

C₂×#108×C₁₃₁

—

#407×C₁₃₁

—

C₂²⋊C₁₆×C₁₃₁

—

#341×C₁₃₁

C₄

yes

—

C₂³×C₄×C₁₃₁

D₄

yes

characteristic

C₂×M₁₆×C₁₃₁

—

C₂²×C₈×C₁₃₁

—

C₂²×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃₁

C₂×C₄

yes

—

C₂²×C₈×C₁₃₁

D₄

yes

—

C₂²×C₈×C₁₃₁

Q₈

yes

characteristic

C₂×C₁₆×C₁₃₁

—

C₂×C₁₆×C₁₃₁

—

#108×C₁₃₁

—

C₂²⋊C₈×C₁₃₁

—

C₂²⋊C₈×C₁₃₁

—

C₂⁴×C₁₃₁

Q₁₆

yes

characteristic

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃₁

D₈

yes

characteristic

C₂²×C₄×C₁₃₁

SD₁₆

yes

—

M₁₆×C₁₃₁

—

M₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₄²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₄⋊C₄

yes

—

C₁₆×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

#103

yes

—

C₂³×C₁₃₁

#107

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#95

yes

characteristic

C₂×C₄×C₁₃₁

#102

yes

—

C₂×C₄×C₁₃₁

#106

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

#99

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#340

yes

characteristic

C₂²×C₁₃₁

#368

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#7231

yes

characteristic, direct product factor, Sylow

C₂×C₂²⋊C₁₆

C₂×C₁₃₁

yes

characteristic

#1810

C₂×C₁₃₁

yes

—

C₂³×C₈

C₂²×C₁₃₁

yes

characteristic

C₂²×C₁₆

C₄×C₁₃₁

yes

—

C₂×#108

—

#407

—

C₂²⋊C₁₆

—

#341

C₄×C₁₃₁

yes

—

C₂³×C₄

D₄×C₁₃₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₃₁

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₃₁

yes

—

C₂²×C₈

D₄×C₁₃₁

yes

—

C₂²×C₈

Q₈×C₁₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

Q₁₆×C₁₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₁₃₁

yes

—

100

C₂²×C₄

C₄⋊C₄×C₁₃₁

yes

characteristic

101

C₂²×C₄

D₈×C₁₃₁

yes

characteristic

102

C₂²×C₄

SD₁₆×C₁₃₁

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄²×C₁₃₁

yes

characteristic

111

C₂×C₈

C₂²⋊C₄×C₁₃₁

yes

characteristic

112

C₂×C₈

C₄⋊C₄×C₁₃₁

yes

—

113

C₁₆

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

#103×C₁₃₁

yes

—

118

C₂³

#107×C₁₃₁

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#95×C₁₃₁

yes

characteristic

126

C₂×C₄

#102×C₁₃₁

yes

—

127

C₂×C₄

#106×C₁₃₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

#99×C₁₃₁

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#340×C₁₃₁

yes

characteristic

139

C₂²

#368×C₁₃₁

yes

—

140

C₄

—

141

C₄

—

142

C₄

H(3,ℤ/4)×C₁₃₁

yes

characteristic

143

C₄

#342×C₁₃₁

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#1289×C₁₃₁

yes

characteristic

147

C₂

#1381×C₁₃₁

yes

characteristic

148

C₂

#1385×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#340
commutator	C₂×C₈	C₄²×C₁₃₁
Frattini, cosocle	C₂³×C₈	C₂²×C₁₃₁
prime radical	C₂⁴×C₁₃₁	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₁

#340

commutator

C₂×C₈

C₄²×C₁₃₁

Frattini, cosocle

C₂³×C₈

C₂²×C₁₃₁

prime radical

C₂⁴×C₁₃₁

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

N	H ≅ G/N	type
74. #7231	C₁₃₁	direct (N × H)
75. C₁₃₁	#7231	direct (N × H)

H ≅ G/N

type

74. #7231

C₁₃₁

direct (N × H)

75. C₁₃₁

#7231

direct (N × H)

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

subgroup	index
C₂×C₂²⋊C₁₆×C₁₃₁	2
#1810×C₁₃₁	2
#7231	131

subgroup

index

C₂×C₂²⋊C₁₆×C₁₃₁

#1810×C₁₃₁

#7231

131

Subgroups — C₁₃₁×#7231 #45693115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#7231 #45693115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7231 #45693115