Group #34100017 — subgroups

Subgroups — #34100017

Order 20096.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19097707	C₂	1	1	yes	maximal, characteristic
2	#23757229	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#23757032	C₂	1	2	yes	maximal
5	#427×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097707	C₄	1	4	yes	complemented
10	#19097682	—	2	2	no	—
11	C₄×#11929434	—	2	2	no	—
12	C₈×Dic₁₅₇	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	#117×C₁₅₇	C₂²	1	2	yes	—
15	#107×C₁₅₇	C₄	1	2	yes	complemented
16	#106×C₁₅₇	C₄	1	2	yes	complemented
17	C₂²×Dic₁₅₇	—	2	1	no	—
18	#14578938	—	2	2	no	—
19	#14578938	C₂×C₄	1	4	yes	—
20	C₂×#11929434	—	2	2	no	—
21	C₂×#11929434	C₂×C₄	1	2	yes	—
22	C₂×#11929434	D₄	1	1	yes	characteristic
23	C₂×#11929434	D₄	1	1	yes	characteristic
24	C₂×#11929434	Q₈	1	2	yes	—
25	C₄×Dic₁₅₇	—	2	1	no	—
26	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
34	C₄²×C₁₅₇	—	2	2	no	—
35	C₂×Dic₁₅₇	—	2	4	no	—
36	C₂×Dic₁₅₇	—	2	2	no	—
37	#11929434	—	2	4	no	—
38	#11929434	C₄⋊C₄	1	2	yes	—
39	#11929434	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₅₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	—	2	4	no	—
42	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
48	C₈×C₁₅₇	C₄²	1	2	yes	—
49	C₈×C₁₅₇	C₄²	1	2	yes	—
50	C₂²×C₁₅₇	—	2	2	no	—
51	C₂²×C₁₅₇	#117	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₅₇	—	4	4	no	—
55	C₄×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
58	C₄×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
59	Dic₁₅₇	—	2	4	no	—
60	C₂×C₁₅₇	—	2	1	no	—
61	C₂×C₁₅₇	—	2	1	no	—
62	C₂×C₁₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₅₇	#443	1	2	yes	—
64	C₁₅₇	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	157	1	no	maximal, Sylow
66	C₂×#108	—	157	1	no	—
67	#427	D₁₅₇	1	1	yes	characteristic
68	#405	—	157	2	no	—
69	C₂×C₄×C₈	—	157	1	no	—
70	#339	—	157	2	no	—
71	C₂×M₁₆	—	157	2	no	—
72	C₂²×C₈	—	157	1	no	—
73	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
74	#117	C₂×D₁₅₇	1	2	yes	—
75	C₂×C₄²	—	157	1	no	—
76	#108	—	157	4	no	—
77	#107	Dic₁₅₇	1	2	yes	complemented
78	#106	Dic₁₅₇	1	2	yes	complemented
79	#97	—	314	2	no	—
80	C₄×C₈	—	314	2	no	—
81	C₄×C₈	—	157	4	no	—
82	C₂²×C₄	—	314	1	no	—
83	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
84	M₁₆	—	314	2	no	—
85	M₁₆	—	157	4	no	—
86	C₂×C₈	—	314	2	no	—
87	C₂×C₈	—	157	2	no	—
88	C₂×C₈	—	157	1	no	—
89	C₂×C₈	—	157	1	no	—
90	C₂×C₈	—	157	2	no	—
91	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
92	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
93	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	314	1	no	—
99	C₄²	—	157	2	no	—
100	C₂³	#14578953	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	314	4	no	—
103	C₂×C₄	—	314	2	no	—
104	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
109	C₂×C₄	#14578955	1	1	yes	characteristic
110	C₈	—	314	4	no	—
111	C₈	—	157	2	no	—
112	C₈	—	157	2	no	—
113	C₈	C₄×Dic₁₅₇	1	2	yes	—
114	C₈	C₄×Dic₁₅₇	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#19097729	1	1	yes	characteristic
117	C₂²	D₄×Dic₁₅₇	1	1	yes	characteristic
118	C₂²	#19097736	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	314	4	no	—
121	C₄	C₂×C₄×Dic₁₅₇	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
123	C₄	#19097733	1	1	yes	characteristic
124	C₄	Q₈×Dic₁₅₇	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₁₅₇	1	1	yes	characteristic
128	C₂	#23757390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097707

C₂

yes

maximal, characteristic

#23757229

C₂

yes

maximal

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757032

C₂

yes

maximal

#427×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097707

C₄

yes

complemented

#19097682

—

C₄×#11929434

—

C₈×Dic₁₅₇

C₄

yes

complemented

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

—

#107×C₁₅₇

C₄

yes

complemented

#106×C₁₅₇

C₄

yes

complemented

C₂²×Dic₁₅₇

—

#14578938

—

#14578938

C₂×C₄

yes

—

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

D₄

yes

characteristic

C₂×#11929434

D₄

yes

characteristic

C₂×#11929434

Q₈

yes

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

#11929434

C₄⋊C₄

yes

—

#11929434

C₄⋊C₄

yes

—

C₂³×C₁₅₇

#41

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

C₄²

yes

—

C₈×C₁₅₇

C₄²

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#117

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₄²

yes

characteristic

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₇

C₄×D₄

yes

characteristic

C₄×C₁₅₇

C₄×Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

#443

yes

—

C₁₅₇

#1772

yes

characteristic, complemented, Sylow

#1772

—

157

maximal, Sylow

C₂×#108

—

157

—

#427

D₁₅₇

yes

characteristic

#405

—

157

—

C₂×C₄×C₈

—

157

—

#339

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

#108

—

157

—

#107

Dic₁₅₇

yes

complemented

#106

Dic₁₅₇

yes

complemented

#97

—

314

—

C₄×C₈

—

314

—

C₄×C₈

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

314

—

C₄²

—

157

—

100

C₂³

#14578953

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

314

—

104

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

105

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

106

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

107

C₂×C₄

D₄×D₁₅₇

yes

characteristic

108

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

109

C₂×C₄

#14578955

yes

characteristic

110

C₈

—

314

—

111

C₈

—

157

—

112

C₈

—

157

—

113

C₈

C₄×Dic₁₅₇

yes

—

114

C₈

C₄×Dic₁₅₇

yes

—

115

C₂²

—

116

C₂²

#19097729

yes

characteristic

117

C₂²

D₄×Dic₁₅₇

yes

characteristic

118

C₂²

#19097736

yes

characteristic

119

C₄

—

120

C₄

—

314

—

121

C₄

C₂×C₄×Dic₁₅₇

yes

characteristic

122

C₄

C₄⋊C₄×D₁₅₇

yes

characteristic

123

C₄

#19097733

yes

characteristic

124

C₄

Q₈×Dic₁₅₇

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₁₅₇

yes

characteristic

128

C₂

#23757390

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#427×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₄×D₄
prime radical	C₂³×C₁₅₇	#41
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#427	D₁₅₇
hypocenter	C₁₅₇	#1772
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₄×C₁₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#427×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₄×D₄

prime radical

C₂³×C₁₅₇

#41

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #19097707	C₄	semidirect
12. C₈×Dic₁₅₇	C₄	semidirect
15. #107×C₁₅₇	C₄	semidirect
16. #106×C₁₅₇	C₄	semidirect
64. C₁₅₇	#1772	semidirect
77. #107	Dic₁₅₇	semidirect
78. #106	Dic₁₅₇	semidirect

H ≅ G/N

type

9. #19097707

C₄

semidirect

12. C₈×Dic₁₅₇

C₄

semidirect

15. #107×C₁₅₇

C₄

semidirect

16. #106×C₁₅₇

C₄

semidirect

64. C₁₅₇

#1772

semidirect

77. #107

Dic₁₅₇

semidirect

78. #106

Dic₁₅₇

semidirect

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

subgroup	index
C₂×#19097707	2
#23757229	2
C₂×C₈×Dic₁₅₇	2
#23757032	2
#427×C₁₅₇	2
#1772	157

subgroup

index

C₂×C₈×Dic₁₅₇

#427×C₁₅₇

157

Subgroups — #34100017

Conjugacy classes of subgroups

Complements

Maximal subgroups