Group #23757902 — subgroups

Subgroups — #23757902

Order 10048.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#19097770	C₂	1	2	yes	maximal, complemented
2	#19097769	C₂	1	2	yes	maximal, complemented
3	#19097761	C₂	1	2	yes	maximal, complemented
4	#19097761	C₂	1	2	yes	maximal, complemented
5	#19097750	C₂	1	1	yes	maximal, characteristic
6	#19097748	C₂	1	2	yes	maximal
7	#19097727	C₂	1	2	yes	maximal
8	C₂×#14578933	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₃₁₄	C₂²	1	2	yes	—
11	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
16	#14578933	C₂²	1	2	yes	—
17	#14578933	C₂²	1	2	yes	—
18	#14578933	C₂²	1	2	yes	—
19	#14578933	C₂²	1	2	yes	—
20	#14578933	C₂²	1	2	yes	—
21	#14578933	C₂²	1	2	yes	—
22	#14578933	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₅₇	C₂²	1	2	yes	—
24	C₄×Dic₁₅₇	C₂²	1	2	yes	—
25	C₄×Dic₁₅₇	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₁₅₇	C₂²	1	2	yes	—
31	Dic₃₁₄	—	4	2	no	—
32	Dic₃₁₄	—	4	1	no	—
33	C₂×Dic₁₅₇	—	4	1	no	—
34	C₂×Dic₁₅₇	Q₈	1	4	yes	complemented
35	C₂×Dic₁₅₇	C₂³	1	2	yes	—
36	C₂×Dic₁₅₇	C₂³	1	2	yes	—
37	C₂×Dic₁₅₇	C₂³	1	2	yes	—
38	C₂×Dic₁₅₇	C₂³	1	2	yes	—
39	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
40	Q₈×C₁₅₇	—	4	1	no	—
41	C₂×C₄×C₁₅₇	—	4	1	no	—
42	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
46	C₂²×C₁₅₇	—	2	1	no	—
47	C₂²×C₁₅₇	—	2	1	no	—
48	C₂²×C₁₅₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
50	C₄×C₁₅₇	—	2	1	no	—
51	C₄×C₁₅₇	—	2	1	no	—
52	C₄×C₁₅₇	—	2	2	no	—
53	C₄×C₁₅₇	—	2	2	no	—
54	Dic₁₅₇	—	2	2	no	—
55	Dic₁₅₇	—	2	2	no	—
56	Dic₁₅₇	—	2	2	no	—
57	Dic₁₅₇	C₂×Q₈	1	4	yes	—
58	C₂×C₁₅₇	—	2	2	no	—
59	C₂×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
62	C₁₅₇	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	157	1	no	maximal, Sylow
64	C₄⋊Q₈	—	157	2	no	—
65	C₄⋊Q₈	—	157	2	no	—
66	#125	—	157	2	no	—
67	#125	—	157	2	no	—
68	C₂²⋊Q₈	—	157	1	no	—
69	C₂²⋊Q₈	—	157	2	no	—
70	C₂²⋊Q₈	D₁₅₇	1	1	yes	characteristic
71	#117	—	157	2	no	—
72	C₂×C₄⋊C₄	—	157	1	no	—
73	C₂×Q₈	—	157	1	no	—
74	C₂×Q₈	—	157	2	no	—
75	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
76	C₂²×C₄	—	157	1	no	—
77	C₂²×C₄	—	157	1	no	—
78	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
79	C₄⋊C₄	—	157	2	no	—
80	C₄⋊C₄	—	157	2	no	—
81	C₄⋊C₄	—	157	2	no	—
82	C₄⋊C₄	—	157	2	no	—
83	C₄⋊C₄	—	157	2	no	—
84	C₄⋊C₄	—	157	1	no	—
85	C₄⋊C₄	—	157	2	no	—
86	C₄⋊C₄	—	157	2	no	—
87	C₄⋊C₄	—	157	2	no	—
88	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
90	C₂²⋊C₄	—	157	2	no	—
91	C₂²⋊C₄	C₂×D₁₅₇	1	2	yes	—
92	C₄²	—	157	2	no	—
93	C₄²	—	157	2	no	—
94	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	628	1	no	—
97	Q₈	—	628	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	628	1	no	—
100	C₂×C₄	—	157	2	no	—
101	C₂×C₄	—	157	2	no	—
102	C₂×C₄	—	157	2	no	—
103	C₂×C₄	—	157	4	no	—
104	C₂×C₄	—	157	2	no	—
105	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
106	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
107	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₁₅₇	1	2	yes	—
112	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	314	2	no	—
118	C₄	—	314	2	no	—
119	C₄	—	314	2	no	—
120	C₄	—	157	4	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
123	C₂	#19097816	1	1	yes	characteristic
124	C₂	#19097818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097770

C₂

yes

maximal, complemented

#19097769

C₂

yes

maximal, complemented

#19097761

C₂

yes

maximal, complemented

#19097761

C₂

yes

maximal, complemented

#19097750

C₂

yes

maximal, characteristic

#19097748

C₂

yes

maximal

#19097727

C₂

yes

maximal

C₂×#14578933

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₃₁₄

C₂²

yes

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

characteristic

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

#14578933

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂²⋊C₄×C₁₅₇

C₂²

yes

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

Q₈

yes

complemented

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

C₂×Q₈

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×Q₈

yes

characteristic

C₂×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

characteristic

C₁₅₇

#544

yes

characteristic, complemented, Sylow

#544

—

157

maximal, Sylow

C₄⋊Q₈

—

157

—

C₄⋊Q₈

—

157

—

#125

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

D₁₅₇

yes

characteristic

#117

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊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

characteristic

C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

C₂×D₁₅₇

yes

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

628

—

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

628

—

100

C₂×C₄

—

157

—

101

C₂×C₄

—

157

—

102

C₂×C₄

—

157

—

103

C₂×C₄

—

157

—

104

C₂×C₄

—

157

—

105

C₂×C₄

C₂²×D₁₅₇

yes

—

106

C₂×C₄

C₂²×D₁₅₇

yes

—

107

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

108

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₁₅₇

yes

—

112

C₂²

C₂³×D₁₅₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

314

—

118

C₄

—

314

—

119

C₄

—

314

—

120

C₄

—

157

—

121

C₂

—

122

C₂

C₂×Q₈×D₁₅₇

yes

characteristic

123

C₂

#19097816

yes

characteristic

124

C₂

#19097818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₂²⋊Q₈×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂³
hypercenter	C₂²⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#544
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₂²⋊Q₈×C₁₅₇

C₂

prime radical

C₂³×C₁₅₇

C₂³

hypercenter

C₂²⋊Q₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #19097770	C₂	semidirect
2. #19097769	C₂	semidirect
3. #19097761	C₂	semidirect
4. #19097761	C₂	semidirect
34. C₂×Dic₁₅₇	Q₈	semidirect
62. C₁₅₇	#544	semidirect

H ≅ G/N

type

1. #19097770

C₂

semidirect

2. #19097769

C₂

semidirect

3. #19097761

C₂

semidirect

4. #19097761

C₂

semidirect

34. C₂×Dic₁₅₇

Q₈

semidirect

62. C₁₅₇

#544

semidirect

Subgroups — #23757902

Conjugacy classes of subgroups

Complements