Group #34099853 — subgroups

Subgroups — #34099853

Order 20096.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#19097730	C₂	1	2	yes	maximal
2	#23757036	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097730	—	2	4	no	—
5	#19097730	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₅₇	—	2	2	no	—
8	C₂×C₄×Dic₁₅₇	—	2	2	no	—
9	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₅₇	—	2	4	no	—
11	C₂²×C₈×C₁₅₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₅₇	—	2	4	no	—
14	C₄⋊Dic₁₅₇	—	2	4	no	—
15	C₄⋊Dic₁₅₇	C₂×C₄	1	8	yes	—
16	C₂²⋊Dic₁₅₇	—	2	8	no	—
17	C₄×Dic₁₅₇	—	2	8	no	—
18	C₂⁴×C₁₅₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₅₇	D₄	1	4	yes	—
22	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₅₇	—	2	8	no	—
26	C₂×C₈×C₁₅₇	—	2	4	no	—
27	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₅₇	—	2	8	no	—
29	C₂×Dic₁₅₇	—	4	8	no	—
30	C₂³×C₁₅₇	—	2	4	no	—
31	C₂³×C₁₅₇	C₂²⋊C₄	1	4	yes	—
32	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
33	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₄²	1	4	yes	—
35	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₅₇	C₄⋊C₄	1	8	yes	—
38	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₅₇	—	2	4	no	—
44	C₂²×C₁₅₇	—	2	8	no	—
45	C₂²×C₁₅₇	—	2	4	no	—
46	C₂²×C₁₅₇	#95	1	4	yes	—
47	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₅₇	#135	1	4	yes	—
50	C₄×C₁₅₇	#95	1	4	yes	—
51	C₄×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₁₅₇	—	4	8	no	—
55	C₂×C₁₅₇	—	2	4	no	—
56	C₂×C₁₅₇	C₂×#95	1	1	yes	characteristic
57	C₂×C₁₅₇	#416	1	4	yes	—
58	C₂×C₁₅₇	#427	1	2	yes	—
59	C₁₅₇	#1735	1	1	yes	characteristic, complemented, Sylow
60	#1735	—	157	1	no	maximal, Sylow
61	C₂³×C₈	D₁₅₇	1	1	yes	characteristic
62	C₂×#117	—	157	2	no	—
63	#340	—	157	4	no	—
64	C₂³×C₄	C₂×D₁₅₇	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₁₅₇	1	4	yes	complemented
67	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
68	#117	—	314	4	no	—
69	#117	—	157	8	no	—
70	C₂×C₄⋊C₄	—	157	4	no	—
71	C₂×C₂²⋊C₄	—	314	2	no	—
72	C₂×C₄²	—	314	2	no	—
73	C₂⁴	D₆₂₈	1	1	yes	characteristic
74	C₂²×C₄	—	314	4	no	—
75	C₂²×C₄	C₄×D₁₅₇	1	4	yes	—
76	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
77	C₂²×C₄	C₂²⋊D₁₅₇	1	4	yes	—
78	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
79	C₂²×C₄	Dic₃₁₄	1	2	yes	—
80	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
84	C₄⋊C₄	—	314	4	no	—
85	C₄⋊C₄	—	157	8	no	—
86	C₂²⋊C₄	—	314	8	no	—
87	C₄²	—	314	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	#14578931	1	4	yes	—
90	C₂³	C₄⋊Dic₁₅₇	1	2	yes	—
91	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
92	C₂×C₄	—	314	8	no	—
93	C₂×C₄	—	628	8	no	—
94	C₂×C₄	C₄×Dic₁₅₇	1	4	yes	—
95	C₂×C₄	#14578931	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₁₅₇	1	4	yes	—
97	C₂×C₄	#14578933	1	8	yes	—
98	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
100	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
102	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#19097678	1	4	yes	—
107	C₂²	C₂×#14578931	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
109	C₂²	#19097793	1	4	yes	—
110	C₄	—	628	8	no	—
111	C₄	#19097678	1	4	yes	—
112	C₄	C₂×C₄×Dic₁₅₇	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
114	C₄	C₂×#14578933	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#19097678	1	1	yes	characteristic
117	C₂	#23757270	1	4	yes	—
118	C₂	#23757311	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097730

C₂

yes

maximal

#23757036

C₂

yes

maximal, complemented

C₂³×C₈×C₁₅₇

C₂

yes

maximal, characteristic

#19097730

—

#19097730

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₅₇

—

C₂×C₄×Dic₁₅₇

—

C₂³×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

—

C₂²×C₈×C₁₅₇

C₄

yes

complemented

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂⁴×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

Q₈

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄²

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#95

yes

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#135

yes

—

C₄×C₁₅₇

#95

yes

—

C₄×C₁₅₇

C₂×C₄²

yes

characteristic

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×#95

yes

characteristic

C₂×C₁₅₇

#416

yes

—

C₂×C₁₅₇

#427

yes

—

C₁₅₇

#1735

yes

characteristic, complemented, Sylow

#1735

—

157

maximal, Sylow

C₂³×C₈

D₁₅₇

yes

characteristic

C₂×#117

—

157

—

#340

—

157

—

C₂³×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₁₅₇

yes

complemented

C₂²×C₈

C₂×D₁₅₇

yes

—

#117

—

314

—

#117

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

314

—

C₂×C₄²

—

314

—

C₂⁴

D₆₂₈

yes

characteristic

C₂²×C₄

—

314

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

D₆₂₈

yes

characteristic

C₂²×C₄

Dic₃₁₄

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

314

—

C₄²

—

314

—

C₂³

—

C₂³

#14578931

yes

—

C₂³

C₄⋊Dic₁₅₇

yes

—

C₂³

C₂×D₆₂₈

yes

characteristic

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

C₄×Dic₁₅₇

yes

—

C₂×C₄

#14578931

yes

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

#14578933

yes

—

C₂×C₄

C₄⋊Dic₁₅₇

yes

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

100

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

102

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#19097678

yes

—

107

C₂²

C₂×#14578931

yes

—

108

C₂²

C₂×C₄⋊Dic₁₅₇

yes

characteristic

109

C₂²

#19097793

yes

—

110

C₄

—

628

—

111

C₄

#19097678

yes

—

112

C₄

C₂×C₄×Dic₁₅₇

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

114

C₄

C₂×#14578933

yes

—

115

C₂

—

116

C₂

C₂×#19097678

yes

characteristic

117

C₂

#23757270

yes

—

118

C₂

#23757311

yes

—

	subgroup	G/H
center	C₂²×C₄	D₆₂₈
commutator	C₄×C₁₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂³×C₈×C₁₅₇	C₂
socle	C₂³×C₁₅₇	C₂×D₄
prime radical	C₂⁴×C₁₅₇	D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂³×C₈	D₁₅₇
hypocenter	C₁₅₇	#1735
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₆₂₈

commutator

C₄×C₁₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂³×C₈×C₁₅₇

C₂

socle

C₂³×C₁₅₇

C₂×D₄

prime radical

C₂⁴×C₁₅₇

D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂³×C₈

D₁₅₇

hypocenter

C₁₅₇

#1735

LCS

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #23757036	C₂	semidirect
5. #19097730	C₄	semidirect
11. C₂²×C₈×C₁₅₇	C₄	semidirect
59. C₁₅₇	#1735	semidirect
66. C₂²×C₈	Dic₁₅₇	semidirect

H ≅ G/N

type

2. #23757036

C₂

semidirect

5. #19097730

C₄

semidirect

11. C₂²×C₈×C₁₅₇

C₄

semidirect

59. C₁₅₇

#1735

semidirect

66. C₂²×C₈

Dic₁₅₇

semidirect

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

subgroup	index
C₂×#19097730	2
#23757036	2
C₂³×C₈×C₁₅₇	2
#1735	157

subgroup

index

C₂×#19097730

#23757036

C₂³×C₈×C₁₅₇

#1735

157

Subgroups — #34099853

Conjugacy classes of subgroups

Complements

Maximal subgroups