Group #23757447 — subgroups

Subgroups — #23757447

Order 10048.

146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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	#19097818	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097799	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097796	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097759	C₂	1	1	yes	maximal, characteristic, complemented
5	#19097699	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097689	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14578954	—	2	1	no	—
9	#14578954	C₂²	1	1	yes	characteristic
10	#14578953	—	2	2	no	—
11	#14578953	—	2	1	no	—
12	C₂×Dic₃₁₄	—	2	1	no	—
13	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
14	Q₈×D₁₅₇	—	2	1	no	—
15	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
16	#14578943	—	2	1	no	—
17	#14578941	—	2	1	no	—
18	D₁₂₅₆	—	2	1	no	—
19	#14578939	—	2	1	no	—
20	#14578938	C₂²	1	1	yes	characteristic
21	#14578931	—	2	2	no	—
22	#41×C₁₅₇	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
24	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₅₇	—	2	1	no	—
26	Dic₃₁₄	—	2	2	no	—
27	Dic₃₁₄	—	2	1	no	—
28	Dic₃₁₄	—	2	1	no	—
29	Dic₃₁₄	D₄	1	1	yes	characteristic
30	D₆₂₈	—	2	1	no	—
31	D₆₂₈	—	2	1	no	—
32	D₆₂₈	D₄	1	1	yes	characteristic
33	C₂²⋊D₁₅₇	—	2	2	no	—
34	C₂²⋊D₁₅₇	—	2	1	no	—
35	C₂×Dic₁₅₇	—	2	2	no	—
36	C₂×Dic₁₅₇	D₄	1	2	yes	—
37	C₄×D₁₅₇	—	2	2	no	—
38	C₄×D₁₅₇	—	2	1	no	—
39	#11929434	—	2	1	no	—
40	Q₈×C₁₅₇	D₄	1	1	yes	characteristic
41	D₄×C₁₅₇	—	2	1	no	—
42	D₄×C₁₅₇	D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	—	2	1	no	—
44	C₂×C₄×C₁₅₇	—	2	1	no	—
45	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
46	C₈×C₁₅₇	—	2	1	no	—
47	C₂²×C₁₅₇	—	2	1	no	—
48	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₅₇	—	2	1	no	—
50	C₂×D₁₅₇	—	4	2	no	—
51	C₂×D₁₅₇	—	4	1	no	—
52	C₄×C₁₅₇	—	2	1	no	—
53	C₄×C₁₅₇	—	4	1	no	—
54	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
55	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
56	Dic₁₅₇	—	2	1	no	—
57	Dic₁₅₇	—	2	2	no	—
58	C₂×C₁₅₇	—	2	1	no	—
59	C₂×C₁₅₇	—	4	1	no	—
60	C₂×C₁₅₇	#120	1	1	yes	characteristic
61	D₁₅₇	—	4	1	no	—
62	D₁₅₇	—	8	1	no	—
63	C₁₅₇	#454	1	1	yes	characteristic, complemented, Sylow
64	#454	—	157	1	no	maximal, Sylow
65	2₋¹⁺⁴	—	157	1	no	—
66	AGL(1,ℤ/8)	—	157	1	no	—
67	AGL(1,ℤ/8)	—	157	1	no	—
68	#124	—	157	1	no	—
69	#104	—	157	1	no	—
70	#104	D₁₅₇	1	1	yes	characteristic, complemented
71	#101	—	157	1	no	—
72	#41	—	314	1	no	—
73	#41	—	314	1	no	—
74	#41	—	314	2	no	—
75	#41	—	157	1	no	—
76	#41	C₂×D₁₅₇	1	1	yes	characteristic
77	C₂×Q₈	—	314	1	no	—
78	C₂×Q₈	—	314	1	no	—
79	C₂×Q₈	—	157	1	no	—
80	C₂×D₄	—	157	1	no	—
81	SD₁₆	—	314	1	no	—
82	SD₁₆	—	314	1	no	—
83	D₈	—	314	1	no	—
84	D₈	—	314	1	no	—
85	M₁₆	—	157	1	no	—
86	M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
87	C₂²⋊C₄	—	314	2	no	—
88	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
89	C₂³	—	314	1	no	—
90	Q₈	—	314	2	no	—
91	Q₈	—	314	1	no	—
92	Q₈	—	314	1	no	—
93	Q₈	—	157	1	no	—
94	Q₈	D₆₂₈	1	1	yes	characteristic
95	D₄	—	2	1	no	—
96	D₄	—	314	2	no	—
97	D₄	—	314	1	no	—
98	D₄	—	314	1	no	—
99	D₄	—	314	1	no	—
100	D₄	—	157	1	no	—
101	D₄	D₆₂₈	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	314	2	no	—
105	C₂×C₄	—	314	2	no	—
106	C₂×C₄	—	314	1	no	—
107	C₂×C₄	—	157	2	no	—
108	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	314	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	314	1	no	—
113	C₂²	—	628	2	no	—
114	C₂²	—	628	1	no	—
115	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	314	1	no	—
119	C₄	—	314	2	no	—
120	C₄	D₄×D₁₅₇	1	1	yes	characteristic
121	C₄	C₂×D₆₂₈	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	4	1	no	—
124	C₂	—	628	1	no	—
125	C₂	—	1256	1	no	—
126	C₂	#19097741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097818

C₂

yes

maximal, characteristic, complemented

#19097799

C₂

yes

maximal, characteristic, complemented

#19097796

C₂

yes

maximal, characteristic, complemented

#19097759

C₂

yes

maximal, characteristic, complemented

#19097699

C₂

yes

maximal, characteristic, complemented

#19097689

C₂

yes

maximal, characteristic, complemented

#104×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

—

#14578954

C₂²

yes

characteristic

#14578953

—

#14578953

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

Q₈×D₁₅₇

—

C₂×D₆₂₈

C₂²

yes

characteristic

#14578943

—

#14578941

—

D₁₂₅₆

—

#14578939

—

#14578938

C₂²

yes

characteristic

#14578931

—

#41×C₁₅₇

C₂²

yes

characteristic

M₁₆×C₁₅₇

C₂²

yes

characteristic

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

D₄

yes

characteristic

D₆₂₈

—

D₆₂₈

—

D₆₂₈

D₄

yes

characteristic

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

D₄

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

Q₈×C₁₅₇

D₄

yes

characteristic

D₄×C₁₅₇

—

D₄×C₁₅₇

D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

C₂×D₄

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#120

yes

characteristic

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#454

yes

characteristic, complemented, Sylow

#454

—

157

maximal, Sylow

2₋¹⁺⁴

—

157

—

AGL(1,ℤ/8)

—

157

—

AGL(1,ℤ/8)

—

157

—

#124

—

157

—

#104

—

157

—

#104

D₁₅₇

yes

characteristic, complemented

#101

—

157

—

#41

—

314

—

#41

—

314

—

#41

—

314

—

#41

—

157

—

#41

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

SD₁₆

—

314

—

SD₁₆

—

314

—

D₈

—

314

—

D₈

—

314

—

M₁₆

—

157

—

M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

314

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

314

—

Q₈

—

314

—

Q₈

—

314

—

Q₈

—

314

—

Q₈

—

157

—

Q₈

D₆₂₈

yes

characteristic

D₄

—

D₄

—

314

—

D₄

—

314

—

D₄

—

314

—

D₄

—

314

—

100

D₄

—

157

—

101

D₄

D₆₂₈

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

314

—

105

C₂×C₄

—

314

—

106

C₂×C₄

—

314

—

107

C₂×C₄

—

157

—

108

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

109

C₈

—

110

C₈

—

314

—

111

C₂²

—

112

C₂²

—

314

—

113

C₂²

—

628

—

114

C₂²

—

628

—

115

C₂²

D₄×D₁₅₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

314

—

119

C₄

—

314

—

120

C₄

D₄×D₁₅₇

yes

characteristic

121

C₄

C₂×D₆₂₈

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

—

628

—

125

C₂

—

1256

—

126

C₂

#19097741

yes

characteristic

	subgroup	G/H
center	C₂	#19097741
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting	#104×C₁₅₇	C₂
socle	C₂×C₁₅₇	#120
hypercenter	#104	D₁₅₇
hypocenter	C₁₅₇	#454
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#19097741

commutator, cosocle

C₂×C₄×C₁₅₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₇

Fitting

#104×C₁₅₇

C₂

C₂×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #19097818	C₂	semidirect
2. #19097799	C₂	semidirect
3. #19097796	C₂	semidirect
4. #19097759	C₂	semidirect
5. #19097699	C₂	semidirect
6. #19097689	C₂	semidirect
7. #104×C₁₅₇	C₂	semidirect
63. C₁₅₇	#454	semidirect
70. #104	D₁₅₇	semidirect

H ≅ G/N

type

1. #19097818

C₂

semidirect

2. #19097799

C₂

semidirect

3. #19097796

C₂

semidirect

4. #19097759

C₂

semidirect

5. #19097699

C₂

semidirect

6. #19097689

C₂

semidirect

7. #104×C₁₅₇

C₂

semidirect

63. C₁₅₇

#454

semidirect

70. #104

D₁₅₇

semidirect

Subgroups — #23757447

Conjugacy classes of subgroups

Complements