Group #23758022 — subgroups

Subgroups — #23758022

Order 10048.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19097769	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097766	C₂	1	2	yes	maximal, complemented
3	#19097765	C₂	1	2	yes	maximal, complemented
4	#19097761	C₂	1	2	yes	maximal, complemented
5	#19097760	C₂	1	1	yes	maximal, characteristic
6	#19097756	C₂	1	2	yes	maximal
7	#19097749	C₂	1	2	yes	maximal, complemented
8	#19097747	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
11	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
14	#14578933	C₂²	1	2	yes	—
15	#14578933	C₂²	1	2	yes	—
16	#14578933	C₂²	1	2	yes	—
17	#14578933	C₂²	1	2	yes	—
18	#14578933	C₂²	1	2	yes	—
19	#14578931	C₂²	1	2	yes	—
20	#14578931	C₂²	1	2	yes	—
21	#14578931	C₂²	1	2	yes	—
22	#14578931	C₂²	1	2	yes	—
23	#14578931	C₂²	1	2	yes	—
24	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
30	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
31	C₂²×D₁₅₇	C₂³	1	2	yes	—
32	Dic₃₁₄	—	4	1	no	—
33	D₆₂₈	—	4	1	no	—
34	C₂×Dic₁₅₇	C₂³	1	2	yes	—
35	C₂×Dic₁₅₇	C₂³	1	2	yes	—
36	C₂×Dic₁₅₇	C₂³	1	2	yes	—
37	C₄×D₁₅₇	—	4	2	no	—
38	Q₈×C₁₅₇	—	4	1	no	—
39	Q₈×C₁₅₇	—	4	1	no	—
40	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
41	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
45	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₅₇	—	2	2	no	—
47	C₂×D₁₅₇	—	2	4	no	—
48	C₄×C₁₅₇	—	2	1	no	—
49	C₄×C₁₅₇	—	2	2	no	—
50	C₄×C₁₅₇	—	2	1	no	—
51	C₄×C₁₅₇	—	2	1	no	—
52	C₄×C₁₅₇	—	2	2	no	—
53	Dic₁₅₇	—	2	2	no	—
54	Dic₁₅₇	—	2	2	no	—
55	Dic₁₅₇	—	2	2	no	—
56	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₅₇	2₋¹⁺⁴	1	2	yes	—
58	D₁₅₇	—	4	2	no	—
59	C₁₅₇	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	157	1	no	maximal, Sylow
61	C₄⋊Q₈	—	157	1	no	—
62	C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic, complemented
63	#126	—	157	2	no	—
64	#126	—	157	2	no	—
65	#125	—	157	2	no	—
66	#124	—	157	1	no	—
67	#123	—	157	2	no	—
68	C₂²⋊Q₈	—	157	2	no	—
69	C₂²⋊Q₈	—	157	2	no	—
70	C₂×Q₈	—	157	1	no	—
71	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
73	C₂×D₄	—	157	1	no	—
74	C₂²×C₄	—	157	2	no	—
75	C₄⋊C₄	—	157	2	no	—
76	C₄⋊C₄	—	157	2	no	—
77	C₄⋊C₄	—	157	2	no	—
78	C₄⋊C₄	—	157	2	no	—
79	C₄⋊C₄	—	157	2	no	—
80	C₄⋊C₄	—	157	2	no	—
81	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
82	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
83	C₂²⋊C₄	—	157	2	no	—
84	C₂²⋊C₄	—	157	2	no	—
85	C₂²⋊C₄	—	157	2	no	—
86	C₂²⋊C₄	—	157	2	no	—
87	C₂²⋊C₄	—	157	2	no	—
88	C₄²	—	157	1	no	—
89	C₄²	—	157	1	no	—
90	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
91	C₂³	—	157	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	628	1	no	—
95	D₄	—	628	1	no	—
96	C₂×C₄	—	628	2	no	—
97	C₂×C₄	—	157	2	no	—
98	C₂×C₄	—	157	2	no	—
99	C₂×C₄	—	157	2	no	—
100	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
103	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
105	C₂²	—	314	2	no	—
106	C₂²	—	314	4	no	—
107	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	314	2	no	—
114	C₄	—	314	2	no	—
115	C₄	—	314	2	no	—
116	C₂	—	628	2	no	—
117	C₂	#19097816	1	1	yes	characteristic
118	C₂	#19097819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19097769

C₂

yes

maximal, characteristic, complemented

#19097766

C₂

yes

maximal, complemented

#19097765

C₂

yes

maximal, complemented

#19097761

C₂

yes

maximal, complemented

#19097760

C₂

yes

maximal, characteristic

#19097756

C₂

yes

maximal

#19097749

C₂

yes

maximal, complemented

#19097747

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

—

D₁₅₇

—

C₁₅₇

#563

yes

characteristic, complemented, Sylow

#563

—

157

maximal, Sylow

C₄⋊Q₈

—

157

—

C₄⋊Q₈

D₁₅₇

yes

characteristic, complemented

#126

—

157

—

#126

—

157

—

#125

—

157

—

#124

—

157

—

#123

—

157

—

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₄²

—

157

—

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

157

—

Q₈

—

Q₈

—

Q₈

—

628

—

D₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

100

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

101

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

102

C₂×C₄

C₂²×D₁₅₇

yes

—

103

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

104

C₂×C₄

C₂²×D₁₅₇

yes

—

105

C₂²

—

314

—

106

C₂²

—

314

—

107

C₂²

C₂³×D₁₅₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

314

—

114

C₄

—

314

—

115

C₄

—

314

—

116

C₂

—

628

—

117

C₂

#19097816

yes

characteristic

118

C₂

#19097819

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₅₇	C₂
prime radical	C₂×D₆₂₈	C₂²
hypercenter	C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#563
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₅₇

C₂

prime radical

C₂×D₆₂₈

C₂²

hypercenter

C₄⋊Q₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #19097769	C₂	semidirect
2. #19097766	C₂	semidirect
3. #19097765	C₂	semidirect
4. #19097761	C₂	semidirect
7. #19097749	C₂	semidirect
8. #19097747	C₂	semidirect
9. C₄⋊Q₈×C₁₅₇	C₂	semidirect
59. C₁₅₇	#563	semidirect
62. C₄⋊Q₈	D₁₅₇	semidirect

H ≅ G/N

type

1. #19097769

C₂

semidirect

2. #19097766

C₂

semidirect

3. #19097765

C₂

semidirect

4. #19097761

C₂

semidirect

7. #19097749

C₂

semidirect

8. #19097747

C₂

semidirect

9. C₄⋊Q₈×C₁₅₇

C₂

semidirect

59. C₁₅₇

#563

semidirect

62. C₄⋊Q₈

D₁₅₇

semidirect

Subgroups — #23758022

Conjugacy classes of subgroups

Complements