Group #34100087 — subgroups

Subgroups — #34100087

Order 20096.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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	#23757257	C₂	1	3	yes	maximal, complemented
2	#23757240	C₂	1	3	yes	maximal, complemented
3	#585×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₂²	1	3	yes	complemented
5	C₂²×#11929434	C₂²	1	1	yes	characteristic, complemented
6	#19097690	—	2	3	no	—
7	#19097688	—	2	3	no	—
8	C₂²⋊#11929434	—	2	6	no	—
9	2₋¹⁺⁴×C₁₅₇	—	2	3	no	—
10	H(5,𝔽₂)×C₁₅₇	—	2	3	no	—
11	H(5,𝔽₂)×C₁₅₇	C₄	1	4	yes	—
12	C₂×#41×C₁₅₇	—	2	6	no	—
13	C₂×#41×C₁₅₇	C₂²	1	3	yes	—
14	#14578938	—	2	6	no	—
15	C₂×#11929434	—	2	3	no	—
16	C₂×#11929434	—	2	3	no	—
17	#41×C₁₅₇	—	2	4	no	—
18	#41×C₁₅₇	—	2	12	no	—
19	#41×C₁₅₇	—	2	12	no	—
20	#41×C₁₅₇	—	2	6	no	—
21	#41×C₁₅₇	—	2	6	no	—
22	C₂×Q₈×C₁₅₇	—	2	6	no	—
23	C₂×Q₈×C₁₅₇	D₄	1	3	yes	—
24	C₂×D₄×C₁₅₇	—	2	6	no	—
25	C₂×D₄×C₁₅₇	—	2	12	no	—
26	C₂×D₄×C₁₅₇	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₁₅₇	D₄	1	3	yes	—
28	C₂²×C₄×C₁₅₇	—	2	4	no	—
29	C₂²×C₄×C₁₅₇	D₄	1	6	yes	—
30	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
31	#11929434	—	4	1	no	—
32	#11929434	—	4	3	no	—
33	C₂³×C₁₅₇	—	2	4	no	—
34	C₂³×C₁₅₇	C₂²⋊C₄	1	6	yes	—
35	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₅₇	—	2	4	no	—
37	Q₈×C₁₅₇	—	2	6	no	—
38	D₄×C₁₅₇	—	2	12	no	—
39	D₄×C₁₅₇	—	2	12	no	—
40	D₄×C₁₅₇	—	2	6	no	—
41	C₂×C₄×C₁₅₇	—	2	12	no	—
42	C₂×C₄×C₁₅₇	—	2	6	no	—
43	C₂×C₄×C₁₅₇	—	2	6	no	—
44	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	6	yes	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
47	C₂²×C₁₅₇	—	2	6	no	—
48	C₂²×C₁₅₇	—	4	1	no	—
49	C₂²×C₁₅₇	—	4	6	no	—
50	C₂²×C₁₅₇	—	8	4	no	—
51	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	3	yes	—
52	C₄×C₁₅₇	—	2	6	no	—
53	C₄×C₁₅₇	#120	1	3	yes	—
54	C₄×C₁₅₇	#120	1	1	yes	characteristic
55	C₂×C₁₅₇	—	2	3	no	—
56	C₂×C₁₅₇	—	4	6	no	—
57	C₂×C₁₅₇	#379	1	1	yes	characteristic
58	C₁₅₇	#1791	1	1	yes	characteristic, complemented, Sylow
59	#1791	—	157	1	no	maximal, Sylow
60	#585	D₁₅₇	1	1	yes	characteristic
61	#413	—	157	3	no	—
62	#408	—	157	3	no	—
63	2₋¹⁺⁴	—	2	3	no	—
64	H(5,𝔽₂)	—	2	3	no	—
65	H(5,𝔽₂)	Dic₁₅₇	1	4	yes	—
66	C₂×#41	—	2	6	no	—
67	C₂×#41	C₂×D₁₅₇	1	3	yes	—
68	C₂×M₁₆	—	157	3	no	—
69	C₂²×C₈	—	157	1	no	—
70	#101	—	314	3	no	—
71	#100	—	314	3	no	—
72	C₂²⋊C₈	—	314	6	no	—
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	6	no	—
76	#41	—	2	6	no	—
77	#41	—	2	12	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂×Q₈	C₂²⋊D₁₅₇	1	3	yes	—
80	C₂×D₄	—	2	6	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	C₂×Dic₁₅₇	1	6	yes	—
83	C₂×D₄	C₂²⋊D₁₅₇	1	3	yes	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	C₂²⋊D₁₅₇	1	6	yes	—
86	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
87	M₁₆	—	314	6	no	—
88	C₂×C₈	—	314	3	no	—
89	C₂×C₈	—	314	3	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₁₅₇	1	6	yes	—
92	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	2	6	no	—
95	D₄	—	2	12	no	—
96	D₄	—	2	12	no	—
97	D₄	—	2	6	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	C₂²⋊Dic₁₅₇	1	6	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	3	yes	—
103	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	3	yes	—
104	C₈	—	628	3	no	—
105	C₈	—	628	1	no	—
106	C₂²	—	2	6	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	6	no	—
109	C₂²	—	8	4	no	—
110	C₂²	C₂×C₂²⋊Dic₁₅₇	1	3	yes	—
111	C₄	—	2	6	no	—
112	C₄	#19097740	1	3	yes	—
113	C₄	#19097740	1	1	yes	characteristic
114	C₂	—	2	3	no	—
115	C₂	—	4	6	no	—
116	C₂	#23757129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757257

C₂

yes

maximal, complemented

#23757240

C₂

yes

maximal, complemented

#585×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

complemented

C₂²×#11929434

C₂²

yes

characteristic, complemented

#19097690

—

#19097688

—

C₂²⋊#11929434

—

2₋¹⁺⁴×C₁₅₇

—

H(5,𝔽₂)×C₁₅₇

—

H(5,𝔽₂)×C₁₅₇

C₄

yes

—

C₂×#41×C₁₅₇

—

C₂×#41×C₁₅₇

C₂²

yes

—

#14578938

—

C₂×#11929434

—

C₂×#11929434

—

#41×C₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂×C₄

yes

—

C₂×D₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

—

C₄×C₁₅₇

#120

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#379

yes

characteristic

C₁₅₇

#1791

yes

characteristic, complemented, Sylow

#1791

—

157

maximal, Sylow

#585

D₁₅₇

yes

characteristic

#413

—

157

—

#408

—

157

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

Dic₁₅₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₅₇

yes

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

#101

—

314

—

#100

—

314

—

C₂²⋊C₈

—

314

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₅₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₅₇

yes

—

C₂×D₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂³

—

C₂³

C₂²⋊Dic₁₅₇

yes

—

C₂³

C₂²×Dic₁₅₇

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

103

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

104

C₈

—

628

—

105

C₈

—

628

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

—

111

C₄

—

112

C₄

#19097740

yes

—

113

C₄

#19097740

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#23757129

yes

characteristic

	subgroup	G/H
center	C₄	#19097740
commutator	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting, prime radical	#585×C₁₅₇	C₂
socle	C₂×C₁₅₇	#379
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#585	D₁₅₇
hypocenter	C₁₅₇	#1791
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₄

#19097740

commutator

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting, prime radical

#585×C₁₅₇

C₂

socle

C₂×C₁₅₇

#379

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #23757257	C₂	semidirect
2. #23757240	C₂	semidirect
4. C₂×#14578938	C₂²	semidirect
5. C₂²×#11929434	C₂²	semidirect
58. C₁₅₇	#1791	semidirect

H ≅ G/N

type

1. #23757257

C₂

semidirect

2. #23757240

C₂

semidirect

4. C₂×#14578938

C₂²

semidirect

5. C₂²×#11929434

C₂²

semidirect

58. C₁₅₇

#1791

semidirect

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

subgroup	index
#23757257	2
#23757240	2
#585×C₁₅₇	2
#1791	157

subgroup

index

#23757257

#23757240

#585×C₁₅₇

#1791

157

Subgroups — #34100087

Conjugacy classes of subgroups

Complements

Maximal subgroups