Group #34096045 — subgroups

Subgroups — #1687×C₁₅₇ #34096045

Order 20096.

198 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 68 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	#493×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#459×C₁₅₇	C₂	1	2	yes	maximal, complemented
3	#436×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#334×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂×D₈×C₁₅₇	—	2	2	no	—
7	C₂×D₈×C₁₅₇	—	2	1	no	—
8	C₂×D₈×C₁₅₇	—	2	1	no	—
9	C₂×M₁₆×C₁₅₇	—	2	1	no	—
10	#127×C₁₅₇	C₂²	1	2	yes	—
11	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
14	#102×C₁₅₇	—	2	2	no	—
15	C₂²⋊C₈×C₁₅₇	—	2	1	no	—
16	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₅₇	—	2	2	no	—
18	C₂×D₄×C₁₅₇	—	2	2	no	—
19	C₂×D₄×C₁₅₇	—	2	2	no	—
20	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	—	2	1	no	—
22	D₈×C₁₅₇	—	4	2	no	—
23	D₈×C₁₅₇	—	4	4	no	—
24	D₈×C₁₅₇	—	4	2	no	—
25	M₁₆×C₁₅₇	—	4	2	no	—
26	C₂×C₈×C₁₅₇	—	2	1	no	—
27	C₂×C₈×C₁₅₇	—	2	2	no	—
28	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
32	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂³×C₁₅₇	—	2	1	no	—
34	C₂³×C₁₅₇	—	4	2	no	—
35	D₄×C₁₅₇	—	2	2	no	—
36	D₄×C₁₅₇	—	4	2	no	—
37	D₄×C₁₅₇	—	4	2	no	—
38	D₄×C₁₅₇	—	4	2	no	—
39	D₄×C₁₅₇	—	4	2	no	—
40	D₄×C₁₅₇	—	8	2	no	—
41	C₂×C₄×C₁₅₇	—	2	1	no	—
42	C₂×C₄×C₁₅₇	—	4	1	no	—
43	C₂×C₄×C₁₅₇	—	4	1	no	—
44	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
47	C₈×C₁₅₇	—	2	1	no	—
48	C₈×C₁₅₇	—	4	1	no	—
49	C₈×C₁₅₇	—	4	2	no	—
50	C₈×C₁₅₇	D₈	1	2	yes	complemented
51	C₂²×C₁₅₇	—	4	1	no	—
52	C₂²×C₁₅₇	—	4	1	no	—
53	C₂²×C₁₅₇	—	4	1	no	—
54	C₂²×C₁₅₇	—	8	2	no	—
55	C₂²×C₁₅₇	—	8	2	no	—
56	C₂²×C₁₅₇	—	8	2	no	—
57	C₂²×C₁₅₇	#121	1	1	yes	characteristic
58	C₄×C₁₅₇	—	2	1	no	—
59	C₄×C₁₅₇	—	4	1	no	—
60	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
61	C₄×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×C₁₅₇	—	8	1	no	—
64	C₂×C₁₅₇	—	16	2	no	—
65	C₂×C₁₅₇	#459	1	1	yes	characteristic
66	C₂×C₁₅₇	#469	1	1	yes	characteristic
67	C₂×C₁₅₇	#472	1	1	yes	characteristic
68	C₁₅₇	#1687	1	1	yes	characteristic, direct product factor, Sylow
69	#1687	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	#493	C₂×C₁₅₇	1	1	yes	characteristic, complemented
71	#459	C₂×C₁₅₇	1	2	yes	complemented
72	#436	C₂×C₁₅₇	1	1	yes	characteristic, complemented
73	#334	C₂×C₁₅₇	1	1	yes	characteristic, complemented
74	#331	C₂×C₁₅₇	1	2	yes	complemented
75	C₂×D₈	—	2	2	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	2	1	no	—
78	C₂×M₁₆	—	2	1	no	—
79	#127	C₂²×C₁₅₇	1	2	yes	—
80	C₄×D₄	C₂²×C₁₅₇	1	1	yes	characteristic
81	C₄⋊C₈	C₂²×C₁₅₇	1	1	yes	characteristic
82	C₄⋊C₈	C₂²×C₁₅₇	1	2	yes	—
83	#102	—	2	2	no	—
84	C₂²⋊C₈	—	2	1	no	—
85	C₄×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	D₄×C₁₅₇	1	1	yes	characteristic
90	C₂²×C₄	—	2	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	4	no	—
93	D₈	—	4	2	no	—
94	M₁₆	—	4	2	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	D₄×C₁₅₇	1	1	yes	characteristic
98	C₂×C₈	D₄×C₁₅₇	1	1	yes	characteristic
99	C₄⋊C₄	D₄×C₁₅₇	1	1	yes	characteristic
100	C₂²⋊C₄	—	2	1	no	—
101	C₄²	C₂³×C₁₅₇	1	1	yes	characteristic
102	C₂³	—	2	1	no	—
103	C₂³	—	4	2	no	—
104	D₄	—	2	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	8	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
115	C₂×C₄	#41×C₁₅₇	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	4	1	no	—
118	C₈	—	4	2	no	—
119	C₈	D₈×C₁₅₇	1	2	yes	complemented
120	C₂²	—	4	1	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	8	2	no	—
124	C₂²	—	8	2	no	—
125	C₂²	—	8	2	no	—
126	C₂²	#121×C₁₅₇	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	1	no	—
129	C₄	C₂×D₈×C₁₅₇	1	1	yes	characteristic
130	C₄	AGL(1,ℤ/8)×C₁₅₇	1	1	yes	characteristic
131	C₄	AGL(1,ℤ/8)×C₁₅₇	1	2	yes	—
132	C₂	—	8	1	no	—
133	C₂	—	16	2	no	—
134	C₂	#459×C₁₅₇	1	1	yes	characteristic
135	C₂	#469×C₁₅₇	1	1	yes	characteristic
136	C₂	#472×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#493×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#459×C₁₅₇

C₂

yes

maximal, complemented

#436×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#334×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#331×C₁₅₇

C₂

yes

maximal, complemented

C₂×D₈×C₁₅₇

—

C₂×D₈×C₁₅₇

—

C₂×D₈×C₁₅₇

—

C₂×M₁₆×C₁₅₇

—

#127×C₁₅₇

C₂²

yes

—

C₄×D₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₂²

yes

—

#102×C₁₅₇

—

C₂²⋊C₈×C₁₅₇

—

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

—

D₈×C₁₅₇

—

D₈×C₁₅₇

—

D₈×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₄⋊C₄×C₁₅₇

D₄

yes

characteristic

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₈×C₁₅₇

D₈

yes

complemented

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₈

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, direct product factor, Sylow

#1687

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

#493

C₂×C₁₅₇

yes

characteristic, complemented

#459

C₂×C₁₅₇

yes

complemented

#436

C₂×C₁₅₇

yes

characteristic, complemented

#334

C₂×C₁₅₇

yes

characteristic, complemented

#331

C₂×C₁₅₇

yes

complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

#127

C₂²×C₁₅₇

yes

—

C₄×D₄

C₂²×C₁₅₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₅₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₅₇

yes

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₁₅₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₁₅₇

yes

characteristic

C₂²×C₄

—

D₈

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₅₇

yes

characteristic

C₂×C₈

D₄×C₁₅₇

yes

characteristic

C₄⋊C₄

D₄×C₁₅₇

yes

characteristic

100

C₂²⋊C₄

—

101

C₄²

C₂³×C₁₅₇

yes

characteristic

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

114

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

115

C₂×C₄

#41×C₁₅₇

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

D₈×C₁₅₇

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#121×C₁₅₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

C₂×D₈×C₁₅₇

yes

characteristic

130

C₄

AGL(1,ℤ/8)×C₁₅₇

yes

characteristic

131

C₄

AGL(1,ℤ/8)×C₁₅₇

yes

—

132

C₂

—

133

C₂

—

134

C₂

#459×C₁₅₇

yes

characteristic

135

C₂

#469×C₁₅₇

yes

characteristic

136

C₂

#472×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₁₅₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ C₄²×C₁₅₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₁₅₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₇ ◁ C₄²×C₁₅₇ ◁ G

N	H ≅ G/N	type
1. #493×C₁₅₇	C₂	semidirect
2. #459×C₁₅₇	C₂	semidirect
3. #436×C₁₅₇	C₂	semidirect
4. #334×C₁₅₇	C₂	semidirect
5. #331×C₁₅₇	C₂	semidirect
50. C₈×C₁₅₇	D₈	semidirect
68. C₁₅₇	#1687	direct (N × H)
69. #1687	C₁₅₇	direct (N × H)
70. #493	C₂×C₁₅₇	semidirect
71. #459	C₂×C₁₅₇	semidirect
72. #436	C₂×C₁₅₇	semidirect
73. #334	C₂×C₁₅₇	semidirect
74. #331	C₂×C₁₅₇	semidirect
119. C₈	D₈×C₁₅₇	semidirect

H ≅ G/N

type

1. #493×C₁₅₇

C₂

semidirect

2. #459×C₁₅₇

C₂

semidirect

3. #436×C₁₅₇

C₂

semidirect

4. #334×C₁₅₇

C₂

semidirect

5. #331×C₁₅₇

C₂

semidirect

50. C₈×C₁₅₇

D₈

semidirect

68. C₁₅₇

#1687

direct (N × H)

69. #1687

C₁₅₇

direct (N × H)

70. #493

C₂×C₁₅₇

semidirect

71. #459

C₂×C₁₅₇

semidirect

72. #436

C₂×C₁₅₇

semidirect

73. #334

C₂×C₁₅₇

semidirect

74. #331

C₂×C₁₅₇

semidirect

119. C₈

D₈×C₁₅₇

semidirect

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

subgroup	index
#493×C₁₅₇	2
#459×C₁₅₇	2
#436×C₁₅₇	2
#334×C₁₅₇	2
#331×C₁₅₇	2
#1687	157

subgroup

index

#493×C₁₅₇

#459×C₁₅₇

#436×C₁₅₇

#334×C₁₅₇

#331×C₁₅₇

#1687

157

Subgroups — #1687×C₁₅₇ #34096045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1687×C157 #34096045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1687×C₁₅₇ #34096045