Group #34100261 — subgroups

Subgroups — #34100261

Order 20096.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23757311	C₂	1	1	yes	maximal, characteristic
2	#23757228	C₂	1	1	yes	maximal, characteristic
3	#23757045	C₂	1	2	yes	maximal
4	#23757030	C₂	1	2	yes	maximal
5	C₂×#108×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	1	yes	characteristic
7	#19097730	C₂²	1	2	yes	—
8	#19097730	C₂²	1	1	yes	characteristic
9	#19097705	—	2	1	no	—
10	#19097703	—	2	1	no	—
11	#19097681	C₄	1	2	yes	complemented
12	C₈×Dic₁₅₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	#108×C₁₅₇	—	2	1	no	—
16	#108×C₁₅₇	—	2	1	no	—
17	#14578938	—	2	1	no	—
18	#14578938	—	2	1	no	—
19	C₂×#11929434	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₁₅₇	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	1	no	—
22	C₂²⋊Dic₁₅₇	—	2	1	no	—
23	C₂²⋊Dic₁₅₇	—	2	2	no	—
24	C₄×Dic₁₅₇	—	2	2	no	—
25	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₇	—	2	2	no	—
28	M₁₆×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₅₇	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₅₇	Q₈	1	1	yes	characteristic
36	C₂×Dic₁₅₇	—	2	2	no	—
37	C₂×Dic₁₅₇	—	4	2	no	—
38	#11929434	—	2	2	no	—
39	C₂³×C₁₅₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
46	C₈×C₁₅₇	—	4	2	no	—
47	C₈×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₅₇	—	4	1	no	—
52	C₂²×C₁₅₇	#117	1	1	yes	characteristic
53	C₂²×C₁₅₇	#124	1	1	yes	characteristic
54	C₂²×C₁₅₇	#125	1	1	yes	characteristic
55	C₄×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₅₇	#117	1	1	yes	characteristic
57	C₄×C₁₅₇	#124	1	1	yes	characteristic
58	C₄×C₁₅₇	#125	1	1	yes	characteristic
59	Dic₁₅₇	—	4	2	no	—
60	Dic₁₅₇	—	8	2	no	—
61	C₂×C₁₅₇	—	2	1	no	—
62	C₂×C₁₅₇	—	2	1	no	—
63	C₂×C₁₅₇	—	2	1	no	—
64	C₂×C₁₅₇	#382	1	1	yes	characteristic
65	C₁₅₇	#1833	1	1	yes	characteristic, complemented, Sylow
66	#1833	—	157	1	no	maximal, Sylow
67	C₂×#108	D₁₅₇	1	1	yes	characteristic
68	#427	—	157	1	no	—
69	#405	—	157	1	no	—
70	#344	—	157	2	no	—
71	H(3,ℤ/4)	—	157	2	no	—
72	C₂×M₁₆	—	157	1	no	—
73	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
74	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
75	#117	—	157	2	no	—
76	#117	—	157	1	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	314	1	no	—
80	#106	—	314	1	no	—
81	#97	—	157	2	no	—
82	C₄×C₈	—	157	2	no	—
83	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	314	1	no	—
87	M₁₆	—	314	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	157	2	no	—
90	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
94	C₂×C₈	Dic₃₁₄	1	1	yes	characteristic
95	C₂×C₈	Dic₃₁₄	1	1	yes	characteristic
96	C₄⋊C₄	—	314	1	no	—
97	C₄⋊C₄	—	314	2	no	—
98	C₂²⋊C₄	—	314	1	no	—
99	C₂²⋊C₄	—	314	2	no	—
100	C₄²	—	314	2	no	—
101	C₄²	—	157	2	no	—
102	C₂³	#14578955	1	1	yes	characteristic
103	C₂×C₄	—	314	2	no	—
104	C₂×C₄	—	628	2	no	—
105	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
108	C₂×C₄	#14578953	1	1	yes	characteristic
109	C₂×C₄	#14578953	1	1	yes	characteristic
110	C₂×C₄	#14578955	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	314	2	no	—
113	C₈	#14578933	1	1	yes	characteristic
114	C₈	#14578933	1	1	yes	characteristic
115	C₈	#14578933	1	1	yes	characteristic
116	C₈	#14578933	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#19097729	1	1	yes	characteristic
119	C₂²	#19097760	1	1	yes	characteristic
120	C₂²	#19097762	1	1	yes	characteristic
121	C₄	—	628	2	no	—
122	C₄	—	1256	2	no	—
123	C₄	C₂×#14578933	1	1	yes	characteristic
124	C₄	#19097729	1	1	yes	characteristic
125	C₄	#19097758	1	1	yes	characteristic
126	C₄	#19097762	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#23757138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757311

C₂

yes

maximal, characteristic

#23757228

C₂

yes

maximal, characteristic

#23757045

C₂

yes

maximal

#23757030

C₂

yes

maximal

C₂×#108×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

characteristic

#19097730

C₂²

yes

—

#19097730

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₅₇

C₄

yes

complemented

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#108×C₁₅₇

—

#108×C₁₅₇

—

#14578938

—

#14578938

—

C₂×#11929434

C₂×C₄

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

characteristic

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

#41

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

C₈×C₁₅₇

—

C₈×C₁₅₇

C₄⋊C₄

yes

characteristic

C₈×C₁₅₇

C₄⋊C₄

yes

characteristic

C₈×C₁₅₇

C₄⋊C₄

yes

characteristic

C₈×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#382

yes

characteristic

C₁₅₇

#1833

yes

characteristic, complemented, Sylow

#1833

—

157

maximal, Sylow

C₂×#108

D₁₅₇

yes

characteristic

#427

—

157

—

#405

—

157

—

#344

—

157

—

H(3,ℤ/4)

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

—

157

—

#117

—

157

—

#108

—

#108

—

#107

—

314

—

#106

—

314

—

#97

—

157

—

C₄×C₈

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

157

—

C₂×C₈

C₄×D₁₅₇

yes

characteristic

C₂×C₈

C₄×D₁₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₇

yes

characteristic

C₂×C₈

Dic₃₁₄

yes

characteristic

C₂×C₈

Dic₃₁₄

yes

characteristic

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

100

C₄²

—

314

—

101

C₄²

—

157

—

102

C₂³

#14578955

yes

characteristic

103

C₂×C₄

—

314

—

104

C₂×C₄

—

628

—

105

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

107

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

108

C₂×C₄

#14578953

yes

characteristic

109

C₂×C₄

#14578953

yes

characteristic

110

C₂×C₄

#14578955

yes

characteristic

111

C₈

—

112

C₈

—

314

—

113

C₈

#14578933

yes

characteristic

114

C₈

#14578933

yes

characteristic

115

C₈

#14578933

yes

characteristic

116

C₈

#14578933

yes

characteristic

117

C₂²

—

118

C₂²

#19097729

yes

characteristic

119

C₂²

#19097760

yes

characteristic

120

C₂²

#19097762

yes

characteristic

121

C₄

—

628

—

122

C₄

—

1256

—

123

C₄

C₂×#14578933

yes

characteristic

124

C₄

#19097729

yes

characteristic

125

C₄

#19097758

yes

characteristic

126

C₄

#19097762

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23757138

yes

characteristic

	subgroup	G/H
center	C₄	#19097729
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#108×C₁₅₇	C₂
socle	C₂×C₁₅₇	#382
prime radical	C₂³×C₁₅₇	#41
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#108	D₁₅₇
hypocenter	C₁₅₇	#1833
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#19097729

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#108×C₁₅₇

C₂

C₂×C₁₅₇

C₂³×C₁₅₇

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #19097681	C₄	semidirect
12. C₈×Dic₁₅₇	C₄	semidirect
65. C₁₅₇	#1833	semidirect

H ≅ G/N

type

11. #19097681

C₄

semidirect

12. C₈×Dic₁₅₇

C₄

semidirect

65. C₁₅₇

#1833

semidirect

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

subgroup	index
#23757311	2
#23757228	2
#23757045	2
#23757030	2
C₂×#108×C₁₅₇	2
#1833	157

subgroup

index

C₂×#108×C₁₅₇

157

Subgroups — #34100261

Conjugacy classes of subgroups

Complements

Maximal subgroups