Group #34100172 — subgroups

Subgroups — #34100172

Order 20096.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×#11929434	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097706	C₂	1	2	yes	maximal, complemented
3	#23757237	C₂	1	1	yes	maximal, characteristic
4	#23757037	C₂	1	2	yes	maximal, complemented
5	#407×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂²×#11929434	—	2	4	no	—
8	C₂²×#11929434	C₄	1	4	yes	—
9	C₂²×#11929434	C₂²	1	1	yes	characteristic
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	#19097706	—	2	4	no	—
12	C₂²⋊#11929434	—	2	2	no	—
13	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₅₇	—	2	2	no	—
16	#14578938	—	2	2	no	—
17	#14578938	—	2	2	no	—
18	C₂×#11929434	—	2	8	no	—
19	C₂×#11929434	—	2	2	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	C₂×C₄	1	2	yes	—
22	C₂×#11929434	C₂×C₄	1	2	yes	—
23	C₂×#11929434	D₄	1	4	yes	—
24	C₂⁴×C₁₅₇	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	—	2	2	no	—
26	C₂²×C₄×C₁₅₇	—	2	2	no	—
27	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₅₇	—	2	2	no	—
32	M₁₆×C₁₅₇	—	2	2	no	—
33	C₂×C₈×C₁₅₇	—	2	2	no	—
34	#11929434	—	2	2	no	—
35	#11929434	—	4	2	no	—
36	#11929434	C₂²⋊C₄	1	4	yes	—
37	C₂³×C₁₅₇	—	2	4	no	—
38	C₂³×C₁₅₇	—	2	2	no	—
39	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	—	2	2	no	—
44	C₂×C₄×C₁₅₇	—	2	4	no	—
45	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	#41	1	2	yes	—
50	C₈×C₁₅₇	—	4	2	no	—
51	C₂²×C₁₅₇	—	2	8	no	—
52	C₂²×C₁₅₇	—	2	2	no	—
53	C₂²×C₁₅₇	—	2	2	no	—
54	C₂²×C₁₅₇	—	2	2	no	—
55	C₂²×C₁₅₇	#108	1	4	yes	complemented
56	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₅₇	—	2	2	no	—
59	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₅₇	#117	1	1	yes	characteristic
61	C₄×C₁₅₇	#121	1	2	yes	—
62	C₂×C₁₅₇	—	2	4	no	—
63	C₂×C₁₅₇	—	2	2	no	—
64	C₂×C₁₅₇	#380	1	1	yes	characteristic
65	C₂×C₁₅₇	C₂×#108	1	2	yes	—
66	C₁₅₇	#1810	1	1	yes	characteristic, complemented, Sylow
67	#1810	—	157	1	no	maximal, Sylow
68	C₂³×C₈	—	157	1	no	—
69	C₂×#108	—	157	2	no	—
70	#407	—	157	1	no	—
71	#407	D₁₅₇	1	1	yes	characteristic
72	#341	—	157	2	no	—
73	C₂³×C₄	C₂×D₁₅₇	1	1	yes	characteristic
74	C₂×M₁₆	—	157	2	no	—
75	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
76	C₂²×C₈	—	314	4	no	—
77	C₂²×C₈	—	157	4	no	—
78	C₂²×C₈	—	157	1	no	—
79	C₂²×C₈	—	157	1	no	—
80	#108	—	314	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	314	2	no	—
83	C₂⁴	Dic₃₁₄	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
88	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	314	2	no	—
93	M₁₆	—	314	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	314	2	no	—
96	C₂×C₈	—	314	8	no	—
97	C₂×C₈	—	314	2	no	—
98	C₂×C₈	—	157	2	no	—
99	C₂×C₈	—	157	2	no	—
100	C₂×C₈	—	157	4	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	#14578933	1	2	yes	—
104	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#14578933	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
113	C₂×C₄	#14578953	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	314	2	no	—
116	C₈	—	628	2	no	—
117	C₈	—	157	4	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#19097706	1	4	yes	complemented
123	C₂²	C₂×#14578933	1	1	yes	characteristic
124	C₂²	#19097748	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
127	C₄	#19097727	1	1	yes	characteristic
128	C₄	#19097743	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#23757131	1	1	yes	characteristic
132	C₂	C₂×#19097706	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#11929434

C₂

yes

maximal, characteristic

C₂×#19097706

C₂

yes

maximal, complemented

#23757237

C₂

yes

maximal, characteristic

#23757037

C₂

yes

maximal, complemented

#407×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

—

C₂²×#11929434

C₄

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂²×#11929434

C₂²

yes

characteristic

#19097706

—

C₂²⋊#11929434

—

C₂³×C₄×C₁₅₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²⋊C₈×C₁₅₇

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

D₄

yes

—

C₂⁴×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

#11929434

—

#11929434

—

#11929434

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#108

yes

complemented

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#380

yes

characteristic

C₂×C₁₅₇

C₂×#108

yes

—

C₁₅₇

#1810

yes

characteristic, complemented, Sylow

#1810

—

157

maximal, Sylow

C₂³×C₈

—

157

—

C₂×#108

—

157

—

#407

—

157

—

#407

D₁₅₇

yes

characteristic

#341

—

157

—

C₂³×C₄

C₂×D₁₅₇

yes

characteristic

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

—

314

—

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

#108

—

314

—

C₂²⋊C₈

—

C₂²⋊C₈

—

314

—

C₂⁴

Dic₃₁₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

Dic₃₁₄

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

100

C₂×C₈

—

157

—

101

C₂³

—

102

C₂³

—

103

C₂³

#14578933

yes

—

104

C₂³

C₂×Dic₃₁₄

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14578933

yes

—

110

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

111

C₂×C₄

D₄×D₁₅₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

113

C₂×C₄

#14578953

yes

—

114

C₈

—

115

C₈

—

314

—

116

C₈

—

628

—

117

C₈

—

157

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#19097706

yes

complemented

123

C₂²

C₂×#14578933

yes

characteristic

124

C₂²

#19097748

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₁₅₇

yes

characteristic

127

C₄

#19097727

yes

characteristic

128

C₄

#19097743

yes

—

129

C₂

—

130

C₂

—

131

C₂

#23757131

yes

characteristic

132

C₂

C₂×#19097706

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#407×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₅₇	Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#407	D₁₅₇
hypocenter	C₁₅₇	#1810
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#407×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₅₇

Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#19097706	C₂	semidirect
4. #23757037	C₂	semidirect
55. C₂²×C₁₅₇	#108	semidirect
66. C₁₅₇	#1810	semidirect
122. C₂²	#19097706	semidirect

H ≅ G/N

type

2. C₂×#19097706

C₂

semidirect

4. #23757037

C₂

semidirect

55. C₂²×C₁₅₇

#108

semidirect

66. C₁₅₇

#1810

semidirect

122. C₂²

#19097706

semidirect

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

subgroup	index
C₂³×#11929434	2
C₂×#19097706	2
#23757237	2
#23757037	2
#407×C₁₅₇	2
#1810	157

subgroup

index

#407×C₁₅₇

157

Subgroups — #34100172

Conjugacy classes of subgroups

Complements

Maximal subgroups