Group #34100186 — subgroups

Subgroups — #34100186

Order 20096.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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	2	yes	maximal, complemented
2	#23757257	C₂	1	2	yes	maximal
3	#23757031	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	4	yes	—
6	#19097707	—	2	4	no	—
7	#19097690	—	2	2	no	—
8	#19097688	—	2	2	no	—
9	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
10	#131×C₁₅₇	—	2	4	no	—
11	C₂×M₁₆×C₁₅₇	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₅₇	C₄	1	2	yes	—
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	#14578938	—	2	4	no	—
16	#14578938	—	2	4	no	—
17	C₂×#11929434	—	2	4	no	—
18	#41×C₁₅₇	—	2	2	no	—
19	#41×C₁₅₇	—	2	2	no	—
20	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
21	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
22	C₂×D₄×C₁₅₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₅₇	—	2	2	no	—
26	M₁₆×C₁₅₇	—	2	4	no	—
27	C₂×C₈×C₁₅₇	—	2	4	no	—
28	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₇	D₄	1	4	yes	—
32	#11929434	—	4	4	no	—
33	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
34	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
35	Q₈×C₁₅₇	—	2	2	no	—
36	D₄×C₁₅₇	—	2	4	no	—
37	D₄×C₁₅₇	—	2	2	no	—
38	C₂×C₄×C₁₅₇	—	2	2	no	—
39	C₂×C₄×C₁₅₇	—	2	2	no	—
40	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	2	yes	—
45	C₈×C₁₅₇	—	2	2	no	—
46	C₈×C₁₅₇	C₂²⋊C₄	1	2	yes	—
47	C₈×C₁₅₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₁₅₇	—	2	2	no	—
49	C₂²×C₁₅₇	—	4	1	no	—
50	C₂²×C₁₅₇	—	4	2	no	—
51	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₁₅₇	—	2	2	no	—
54	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₁₅₇	#117	1	1	yes	characteristic
56	C₄×C₁₅₇	#121	1	2	yes	—
57	C₂×C₁₅₇	—	2	1	no	—
58	C₂×C₁₅₇	—	2	2	no	—
59	C₂×C₁₅₇	—	4	2	no	—
60	C₂×C₁₅₇	#380	1	1	yes	characteristic
61	C₁₅₇	#1813	1	1	yes	characteristic, complemented, Sylow
62	#1813	—	157	1	no	maximal, Sylow
63	C₂×#131	D₁₅₇	1	1	yes	characteristic
64	C₂×#108	—	157	2	no	—
65	#413	—	157	2	no	—
66	#338	—	157	2	no	—
67	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	—	157	4	no	—
70	C₂×M₁₆	Dic₁₅₇	1	2	yes	—
71	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
72	C₂²×C₈	Dic₁₅₇	1	2	yes	—
73	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
74	#108	—	314	4	no	—
75	#101	—	314	2	no	—
76	#100	—	314	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₆₂₈	1	1	yes	characteristic
80	C₂×D₄	D₆₂₈	1	1	yes	characteristic
81	C₂×D₄	Dic₃₁₄	1	2	yes	—
82	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	314	4	no	—
87	M₁₆	—	314	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	314	4	no	—
90	C₂×C₈	C₂×Dic₁₅₇	1	2	yes	—
91	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₅₇	1	4	yes	—
94	C₂³	C₄⋊Dic₁₅₇	1	2	yes	—
95	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
102	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
104	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
105	C₂×C₄	#14578954	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	628	4	no	—
108	C₈	C₂²⋊Dic₁₅₇	1	2	yes	—
109	C₈	C₂²⋊Dic₁₅₇	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
114	C₂²	#19097750	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
117	C₄	#19097730	1	1	yes	characteristic
118	C₄	#19097745	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#23757133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097707

C₂

yes

maximal, complemented

#23757257

C₂

yes

maximal

#23757031

C₂

yes

maximal, complemented

C₂×#131×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

#19097707

—

#19097690

—

#19097688

—

C₂×#41×C₁₅₇

C₂²

yes

characteristic

#131×C₁₅₇

—

C₂×M₁₆×C₁₅₇

C₄

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₄

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

—

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

C₂×D₄×C₁₅₇

D₄

yes

characteristic

C₂×D₄×C₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

D₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×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

—

C₈×C₁₅₇

C₂²⋊C₄

yes

—

C₈×C₁₅₇

C₂²⋊C₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

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

—

C₂×C₁₅₇

#380

yes

characteristic

C₁₅₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

157

maximal, Sylow

C₂×#131

D₁₅₇

yes

characteristic

C₂×#108

—

157

—

#413

—

157

—

#338

—

157

—

C₂×#41

C₂×D₁₅₇

yes

characteristic

#131

—

C₂×M₁₆

—

157

—

C₂×M₁₆

Dic₁₅₇

yes

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

Dic₁₅₇

yes

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#108

—

314

—

#101

—

314

—

#100

—

314

—

#41

—

#41

—

C₂×Q₈

D₆₂₈

yes

characteristic

C₂×D₄

D₆₂₈

yes

characteristic

C₂×D₄

Dic₃₁₄

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

C₂×Dic₁₅₇

yes

—

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂×Dic₁₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₂³

C₄⋊Dic₁₅₇

yes

—

C₂³

C₂×Dic₃₁₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₁₅₇

yes

—

102

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

104

C₂×C₄

C₂×D₆₂₈

yes

characteristic

105

C₂×C₄

#14578954

yes

—

106

C₈

—

107

C₈

—

628

—

108

C₈

C₂²⋊Dic₁₅₇

yes

—

109

C₈

C₂²⋊Dic₁₅₇

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊Dic₁₅₇

yes

characteristic

114

C₂²

#19097750

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

117

C₄

#19097730

yes

characteristic

118

C₄

#19097745

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#23757133

yes

characteristic

	subgroup	G/H
center	C₄	#19097730
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×#131×C₁₅₇	C₂
socle	C₂×C₁₅₇	#380
prime radical	C₂×#41×C₁₅₇	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#131	D₁₅₇
hypocenter	C₁₅₇	#1813
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#19097730

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×#131×C₁₅₇

C₂

socle

C₂×C₁₅₇

#380

prime radical

C₂×#41×C₁₅₇

C₂²

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#19097707	C₂	semidirect
3. #23757031	C₂	semidirect
61. C₁₅₇	#1813	semidirect

H ≅ G/N

type

1. C₂×#19097707

C₂

semidirect

3. #23757031

C₂

semidirect

61. C₁₅₇

#1813

semidirect

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

subgroup	index
C₂×#19097707	2
#23757257	2
#23757031	2
C₂×#131×C₁₅₇	2
#1813	157

subgroup

index

C₂×#131×C₁₅₇

157

Subgroups — #34100186

Conjugacy classes of subgroups

Complements

Maximal subgroups