Group #34100177 — subgroups

Subgroups — #34100177

Order 20096.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#14578937	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097706	C₂	1	2	yes	maximal, complemented
3	#23757236	C₂	1	2	yes	maximal
4	#23757038	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
7	C₂×#14578938	C₂²	1	2	yes	—
8	C₂×#14578937	—	2	4	no	—
9	C₂×#14578937	C₄	1	4	yes	—
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	#19097706	—	2	4	no	—
12	#19097683	—	2	4	no	—
13	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂³×D₁₅₇	Q₈	1	1	yes	characteristic
16	C₂²×Dic₁₅₇	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₁₅₇	—	2	2	no	—
18	C₂×C₄×D₁₅₇	—	2	2	no	—
19	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₅₇	D₄	1	2	yes	—
21	#14578938	—	2	2	no	—
22	#14578938	—	2	2	no	—
23	#14578937	—	2	8	no	—
24	C₂×#11929434	—	2	2	no	—
25	C₂×#11929434	C₂×C₄	1	2	yes	—
26	C₂×#11929434	D₄	1	4	yes	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₅₇	—	2	2	no	—
29	M₁₆×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₂×C₈×C₁₅₇	—	2	2	no	—
32	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
33	C₂²×D₁₅₇	—	2	2	no	—
34	C₂²×D₁₅₇	—	4	2	no	—
35	C₂²×D₁₅₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₅₇	—	2	2	no	—
37	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
38	C₄×D₁₅₇	—	2	2	no	—
39	C₄×D₁₅₇	—	2	2	no	—
40	C₄×D₁₅₇	—	2	4	no	—
41	#11929434	—	4	2	no	—
42	#11929434	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	#41	1	2	yes	—
48	C₈×C₁₅₇	—	2	2	no	—
49	C₈×C₁₅₇	—	4	2	no	—
50	C₂²×C₁₅₇	—	2	2	no	—
51	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₁₅₇	—	2	2	no	—
54	C₂×D₁₅₇	—	2	4	no	—
55	C₂×D₁₅₇	—	4	4	no	—
56	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₅₇	#117	1	1	yes	characteristic
58	C₄×C₁₅₇	#121	1	2	yes	—
59	Dic₁₅₇	—	2	2	no	—
60	C₂×C₁₅₇	—	2	2	no	—
61	C₂×C₁₅₇	#380	1	1	yes	characteristic
62	C₂×C₁₅₇	#430	1	2	yes	—
63	D₁₅₇	—	4	2	no	—
64	C₁₅₇	#1811	1	1	yes	characteristic, complemented, Sylow
65	#1811	—	157	1	no	maximal, Sylow
66	C₂²×M₁₆	—	157	1	no	—
67	C₂×#108	—	157	2	no	—
68	#407	—	157	2	no	—
69	#341	—	157	1	no	—
70	#341	D₁₅₇	1	1	yes	characteristic, complemented
71	C₂³×C₄	—	157	1	no	—
72	C₂×M₁₆	—	314	4	no	—
73	C₂×M₁₆	—	157	2	no	—
74	C₂×M₁₆	—	157	4	no	—
75	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
76	C₂²×C₈	—	157	1	no	—
77	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
78	#108	—	314	4	no	—
79	C₂²⋊C₈	—	314	4	no	—
80	C₂⁴	—	157	1	no	—
81	C₂²×C₄	—	314	2	no	—
82	C₂²×C₄	—	314	2	no	—
83	C₂²×C₄	—	157	1	no	—
84	C₂²×C₄	—	157	2	no	—
85	C₂²×C₄	—	157	2	no	—
86	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	314	8	no	—
90	M₁₆	—	314	2	no	—
91	M₁₆	—	314	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	314	2	no	—
95	C₂×C₈	—	157	2	no	—
96	C₂×C₈	—	157	4	no	—
97	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
98	C₂³	—	314	2	no	—
99	C₂³	—	628	2	no	—
100	C₂³	—	157	2	no	—
101	C₂³	Q₈×D₁₅₇	1	1	yes	characteristic
102	C₂×C₄	—	314	2	no	—
103	C₂×C₄	—	314	2	no	—
104	C₂×C₄	—	314	2	no	—
105	C₂×C₄	—	314	4	no	—
106	C₂×C₄	—	157	2	no	—
107	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
109	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
110	C₂×C₄	#14578954	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	628	2	no	—
114	C₈	—	157	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	314	4	no	—
117	C₂²	—	314	2	no	—
118	C₂²	—	628	4	no	—
119	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
120	C₂²	#19097749	1	2	yes	—
121	C₄	—	314	2	no	—
122	C₄	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
123	C₄	#19097728	1	1	yes	characteristic
124	C₄	#19097744	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	628	2	no	—
127	C₂	#23757132	1	1	yes	characteristic
128	C₂	#23757320	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14578937

C₂

yes

maximal, characteristic

C₂×#19097706

C₂

yes

maximal, complemented

#23757236

C₂

yes

maximal

#23757038

C₂

yes

maximal, characteristic, complemented

#341×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×#14578938

C₂²

yes

—

C₂×#14578937

—

C₂×#14578937

C₄

yes

—

C₂²×#11929434

C₂²

yes

characteristic

#19097706

—

#19097683

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂³×D₁₅₇

Q₈

yes

characteristic

C₂²×Dic₁₅₇

Q₈

yes

characteristic

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

—

C₂×C₄×D₁₅₇

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11929434

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

C₄⋊C₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

#11929434

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

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

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#380

yes

characteristic

C₂×C₁₅₇

#430

yes

—

D₁₅₇

—

C₁₅₇

#1811

yes

characteristic, complemented, Sylow

#1811

—

157

maximal, Sylow

C₂²×M₁₆

—

157

—

C₂×#108

—

157

—

#407

—

157

—

#341

—

157

—

#341

D₁₅₇

yes

characteristic, complemented

C₂³×C₄

—

157

—

C₂×M₁₆

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#108

—

314

—

C₂²⋊C₈

—

314

—

C₂⁴

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

—

C₂³

—

314

—

C₂³

—

628

—

100

C₂³

—

157

—

101

C₂³

Q₈×D₁₅₇

yes

characteristic

102

C₂×C₄

—

314

—

103

C₂×C₄

—

314

—

104

C₂×C₄

—

314

—

105

C₂×C₄

—

314

—

106

C₂×C₄

—

157

—

107

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

108

C₂×C₄

D₄×D₁₅₇

yes

—

109

C₂×C₄

D₄×D₁₅₇

yes

characteristic

110

C₂×C₄

#14578954

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

628

—

114

C₈

—

157

—

115

C₂²

—

116

C₂²

—

314

—

117

C₂²

—

314

—

118

C₂²

—

628

—

119

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

120

C₂²

#19097749

yes

—

121

C₄

—

314

—

122

C₄

C₂²⋊C₄×D₁₅₇

yes

characteristic

123

C₄

#19097728

yes

characteristic

124

C₄

#19097744

yes

—

125

C₂

—

126

C₂

—

628

—

127

C₂

#23757132

yes

characteristic

128

C₂

#23757320

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#341×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₅₇	Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#341	D₁₅₇
hypocenter	C₁₅₇	#1811
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#341×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×D₁₅₇

Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#19097706	C₂	semidirect
4. #23757038	C₂	semidirect
5. #341×C₁₅₇	C₂	semidirect
64. C₁₅₇	#1811	semidirect
70. #341	D₁₅₇	semidirect

H ≅ G/N

type

2. C₂×#19097706

C₂

semidirect

4. #23757038

C₂

semidirect

5. #341×C₁₅₇

C₂

semidirect

64. C₁₅₇

#1811

semidirect

70. #341

D₁₅₇

semidirect

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

subgroup	index
C₂²×#14578937	2
C₂×#19097706	2
#23757236	2
#23757038	2
#341×C₁₅₇	2
#1811	157

subgroup

index

#341×C₁₅₇

157

Subgroups — #34100177

Conjugacy classes of subgroups

Complements

Maximal subgroups