Group #45698387 — subgroups

Subgroups — C₁₃₁×#12503 #45698387

Order 33536.

306 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 110 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	#2252×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#2246×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2240×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1706×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₁₃₁	—	2	2	no	—
7	#500×C₁₃₁	C₂²	1	2	yes	—
8	#493×C₁₃₁	C₂²	1	2	yes	—
9	#492×C₁₃₁	—	2	2	no	—
10	#486×C₁₃₁	—	2	2	no	—
11	#366×C₁₃₁	—	2	2	no	—
12	#357×C₁₃₁	—	2	4	no	—
13	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₈²×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₁₃₁	—	4	4	no	—
17	C₂×D₈×C₁₃₁	—	2	2	no	—
18	C₂×D₈×C₁₃₁	—	2	2	no	—
19	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
20	#127×C₁₃₁	—	2	2	no	—
21	D₁₆×C₁₃₁	—	4	4	no	—
22	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
23	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
24	#107×C₁₃₁	—	2	2	no	—
25	#107×C₁₃₁	—	2	2	no	—
26	#102×C₁₃₁	—	4	4	no	—
27	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₁	—	4	2	no	—
31	C₂×D₄×C₁₃₁	—	4	2	no	—
32	C₂×D₄×C₁₃₁	—	4	2	no	—
33	C₂×D₄×C₁₃₁	—	4	2	no	—
34	SD₁₆×C₁₃₁	—	8	8	no	—
35	D₈×C₁₃₁	—	4	4	no	—
36	D₈×C₁₃₁	—	4	4	no	—
37	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
38	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
39	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₃₁	—	4	2	no	—
42	C₄⋊C₄×C₁₃₁	—	4	2	no	—
43	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₁₃₁	—	2	2	no	—
45	C₁₆×C₁₃₁	SD₁₆	1	4	yes	complemented
46	C₂³×C₁₃₁	—	8	2	no	—
47	Q₈×C₁₃₁	—	8	4	no	—
48	D₄×C₁₃₁	—	8	2	no	—
49	D₄×C₁₃₁	—	8	2	no	—
50	D₄×C₁₃₁	—	8	4	no	—
51	D₄×C₁₃₁	—	8	4	no	—
52	C₂×C₄×C₁₃₁	—	8	2	no	—
53	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
56	C₈×C₁₃₁	—	2	2	no	—
57	C₈×C₁₃₁	D₁₆	1	4	yes	complemented
58	C₈×C₁₃₁	C₂×SD₁₆	1	2	yes	—
59	C₈×C₁₃₁	#135	1	2	yes	—
60	C₂²×C₁₃₁	—	16	2	no	—
61	C₂²×C₁₃₁	—	16	2	no	—
62	C₂²×C₁₃₁	—	16	2	no	—
63	C₂²×C₁₃₁	#493	1	1	yes	characteristic
64	C₄×C₁₃₁	—	16	2	no	—
65	C₄×C₁₃₁	#492	1	1	yes	characteristic
66	C₄×C₁₃₁	#495	1	1	yes	characteristic
67	C₄×C₁₃₁	C₂×D₁₆	1	2	yes	—
68	C₄×C₁₃₁	#508	1	2	yes	—
69	C₂×C₁₃₁	—	32	2	no	—
70	C₂×C₁₃₁	#1706	1	1	yes	characteristic
71	C₂×C₁₃₁	#2246	1	1	yes	characteristic
72	C₂×C₁₃₁	#2249	1	1	yes	characteristic
73	#12503	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃₁	#12503	1	1	yes	characteristic, direct product factor, Sylow
75	#2252	C₂×C₁₃₁	1	1	yes	characteristic, complemented
76	#2246	C₂×C₁₃₁	1	1	yes	characteristic
77	#2240	C₂×C₁₃₁	1	2	yes	complemented
78	#1706	C₂×C₁₃₁	1	2	yes	complemented
79	C₈×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
80	C₂×D₁₆	—	2	2	no	—
81	#500	C₂²×C₁₃₁	1	2	yes	—
82	#493	C₂²×C₁₃₁	1	2	yes	—
83	#492	—	2	2	no	—
84	#486	—	2	2	no	—
85	#366	—	2	2	no	—
86	#357	—	2	4	no	—
87	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
88	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
89	C₈²	C₂²×C₁₃₁	1	1	yes	characteristic
90	C₂×SD₁₆	—	4	4	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	2	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	#127	—	2	2	no	—
95	D₁₆	—	4	4	no	—
96	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
97	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#102	—	4	4	no	—
101	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
102	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
103	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	—	4	2	no	—
108	SD₁₆	—	8	8	no	—
109	D₈	—	4	4	no	—
110	D₈	—	4	4	no	—
111	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
112	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
113	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	SD₁₆×C₁₃₁	1	4	yes	complemented
120	C₂³	—	8	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	8	2	no	—
123	D₄	—	8	2	no	—
124	D₄	—	8	4	no	—
125	D₄	—	8	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
128	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
129	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₁₆×C₁₃₁	1	4	yes	complemented
132	C₈	C₂×SD₁₆×C₁₃₁	1	2	yes	—
133	C₈	#135×C₁₃₁	1	2	yes	—
134	C₂²	—	16	2	no	—
135	C₂²	—	16	2	no	—
136	C₂²	—	16	2	no	—
137	C₂²	#493×C₁₃₁	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	#492×C₁₃₁	1	1	yes	characteristic
140	C₄	#495×C₁₃₁	1	1	yes	characteristic
141	C₄	C₂×D₁₆×C₁₃₁	1	2	yes	—
142	C₄	#508×C₁₃₁	1	2	yes	—
143	C₂	—	32	2	no	—
144	C₂	#1706×C₁₃₁	1	1	yes	characteristic
145	C₂	#2246×C₁₃₁	1	1	yes	characteristic
146	C₂	#2249×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2252×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#2246×C₁₃₁

C₂

yes

maximal, characteristic

#2240×C₁₃₁

C₂

yes

maximal, complemented

#1706×C₁₃₁

C₂

yes

maximal, complemented

C₈×C₁₆×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₁₃₁

—

#500×C₁₃₁

C₂²

yes

—

#493×C₁₃₁

C₂²

yes

—

#492×C₁₃₁

—

#486×C₁₃₁

—

#366×C₁₃₁

—

#357×C₁₃₁

—

C₄×C₁₆×C₁₃₁

C₂²

yes

characteristic

C₄×C₁₆×C₁₃₁

C₂²

yes

characteristic

C₈²×C₁₃₁

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

—

#127×C₁₃₁

—

D₁₆×C₁₃₁

—

C₂×C₁₆×C₁₃₁

D₄

yes

—

C₂×C₁₆×C₁₃₁

D₄

yes

—

#107×C₁₃₁

—

#107×C₁₃₁

—

#102×C₁₃₁

—

C₄×C₈×C₁₃₁

D₄

yes

characteristic

C₄×C₈×C₁₃₁

D₄

yes

characteristic

C₄×C₈×C₁₃₁

C₂³

yes

characteristic

C₂×Q₈×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

SD₁₆×C₁₃₁

—

D₈×C₁₃₁

—

D₈×C₁₃₁

—

C₂×C₈×C₁₃₁

D₈

yes

—

C₂×C₈×C₁₃₁

D₈

yes

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

C₂×D₄

yes

characteristic

C₁₆×C₁₃₁

—

C₁₆×C₁₃₁

SD₁₆

yes

complemented

C₂³×C₁₃₁

—

Q₈×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#127

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₈

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

D₁₆

yes

complemented

C₈×C₁₃₁

C₂×SD₁₆

yes

—

C₈×C₁₃₁

#135

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#493

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

C₂×D₁₆

yes

—

C₄×C₁₃₁

#508

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#12503

yes

characteristic, direct product factor, Sylow

#2252

C₂×C₁₃₁

yes

characteristic, complemented

#2246

C₂×C₁₃₁

yes

characteristic

#2240

C₂×C₁₃₁

yes

complemented

#1706

C₂×C₁₃₁

yes

complemented

C₈×C₁₆

C₂×C₁₃₁

yes

characteristic, complemented

C₂×D₁₆

—

#500

C₂²×C₁₃₁

yes

—

#493

C₂²×C₁₃₁

yes

—

—

—

—

—

C₄×C₁₆

C₂²×C₁₃₁

yes

characteristic

C₄×C₁₆

C₂²×C₁₃₁

yes

characteristic

C₈²

C₂²×C₁₃₁

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

D₄×C₁₃₁

yes

—

C₂×C₁₆

D₄×C₁₃₁

yes

—

#107

—

#107

—

100

#102

—

101

C₄×C₈

D₄×C₁₃₁

yes

characteristic

102

C₄×C₈

D₄×C₁₃₁

yes

characteristic

103

C₄×C₈

C₂³×C₁₃₁

yes

characteristic

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

109

D₈

—

110

D₈

—

111

C₂×C₈

D₈×C₁₃₁

yes

—

112

C₂×C₈

D₈×C₁₃₁

yes

—

113

C₂×C₈

C₂×D₄×C₁₃₁

yes

characteristic

114

C₂×C₈

C₂×D₄×C₁₃₁

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

C₂×D₄×C₁₃₁

yes

characteristic

118

C₁₆

—

119

C₁₆

SD₁₆×C₁₃₁

yes

complemented

120

C₂³

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

#127×C₁₃₁

yes

characteristic

128

C₂×C₄

C₂×D₈×C₁₃₁

yes

characteristic

129

C₂×C₄

C₂×D₈×C₁₃₁

yes

characteristic

130

C₈

—

131

C₈

D₁₆×C₁₃₁

yes

complemented

132

C₈

C₂×SD₁₆×C₁₃₁

yes

—

133

C₈

#135×C₁₃₁

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#493×C₁₃₁

yes

characteristic

138

C₄

—

139

C₄

#492×C₁₃₁

yes

characteristic

140

C₄

#495×C₁₃₁

yes

characteristic

141

C₄

C₂×D₁₆×C₁₃₁

yes

—

142

C₄

#508×C₁₃₁

yes

—

143

C₂

—

144

C₂

#1706×C₁₃₁

yes

characteristic

145

C₂

#2246×C₁₃₁

yes

characteristic

146

C₂

#2249×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃₁
prime radical	#2246×C₁₃₁	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₄²×C₁₃₁ ◁ C₈²×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₃₁

prime radical

#2246×C₁₃₁

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₁ ◁ C₄²×C₁₃₁ ◁ C₈²×C₁₃₁ ◁ G

N	H ≅ G/N	type
1. #2252×C₁₃₁	C₂	semidirect
3. #2240×C₁₃₁	C₂	semidirect
4. #1706×C₁₃₁	C₂	semidirect
5. C₈×C₁₆×C₁₃₁	C₂	semidirect
45. C₁₆×C₁₃₁	SD₁₆	semidirect
57. C₈×C₁₃₁	D₁₆	semidirect
73. #12503	C₁₃₁	direct (N × H)
74. C₁₃₁	#12503	direct (N × H)
75. #2252	C₂×C₁₃₁	semidirect
77. #2240	C₂×C₁₃₁	semidirect
78. #1706	C₂×C₁₃₁	semidirect
79. C₈×C₁₆	C₂×C₁₃₁	semidirect
119. C₁₆	SD₁₆×C₁₃₁	semidirect
131. C₈	D₁₆×C₁₃₁	semidirect

H ≅ G/N

type

1. #2252×C₁₃₁

C₂

semidirect

3. #2240×C₁₃₁

C₂

semidirect

4. #1706×C₁₃₁

C₂

semidirect

5. C₈×C₁₆×C₁₃₁

C₂

semidirect

45. C₁₆×C₁₃₁

SD₁₆

semidirect

57. C₈×C₁₃₁

D₁₆

semidirect

73. #12503

C₁₃₁

direct (N × H)

74. C₁₃₁

#12503

direct (N × H)

75. #2252

C₂×C₁₃₁

semidirect

77. #2240

C₂×C₁₃₁

semidirect

78. #1706

C₂×C₁₃₁

semidirect

79. C₈×C₁₆

C₂×C₁₃₁

semidirect

119. C₁₆

SD₁₆×C₁₃₁

semidirect

131. C₈

D₁₆×C₁₃₁

semidirect

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

subgroup	index
#2252×C₁₃₁	2
#2246×C₁₃₁	2
#2240×C₁₃₁	2
#1706×C₁₃₁	2
C₈×C₁₆×C₁₃₁	2
#12503	131

subgroup

index

#2252×C₁₃₁

#2246×C₁₃₁

#2240×C₁₃₁

#1706×C₁₃₁

C₈×C₁₆×C₁₃₁

#12503

131

Subgroups — C₁₃₁×#12503 #45698387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#12503 #45698387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#12503 #45698387