Group #23757442 — subgroups

Subgroups — #23757442

Order 10048.

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

normal subgroups only

This group has 168 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	#19097817	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	#19097768	C₂	1	1	yes	maximal, characteristic, complemented
5	#19097699	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097687	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14578955	—	2	1	no	—
9	#14578954	—	2	1	no	—
10	#14578954	C₂²	1	1	yes	characteristic
11	C₂×D₆₂₈	—	2	1	no	—
12	C₂×D₆₂₈	—	2	2	no	—
13	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
14	C₂×D₆₂₈	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₁₅₇	—	2	2	no	—
16	D₄×D₁₅₇	—	2	1	no	—
17	#14578943	—	2	1	no	—
18	#14578941	—	2	1	no	—
19	D₁₂₅₆	—	2	1	no	—
20	#14578939	—	2	1	no	—
21	#14578938	C₂²	1	1	yes	characteristic, complemented
22	#41×C₁₅₇	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₅₇	—	2	2	no	—
26	C₂²×D₁₅₇	—	2	1	no	—
27	C₂²×D₁₅₇	D₄	1	2	yes	—
28	Dic₃₁₄	D₄	1	1	yes	characteristic, complemented
29	D₆₂₈	—	2	2	no	—
30	D₆₂₈	—	2	1	no	—
31	D₆₂₈	—	2	1	no	—
32	D₆₂₈	—	2	1	no	—
33	D₆₂₈	—	2	1	no	—
34	D₆₂₈	—	4	4	no	—
35	D₆₂₈	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₁₅₇	—	2	2	no	—
37	C₂²⋊D₁₅₇	—	2	1	no	—
38	C₄×D₁₅₇	—	2	2	no	—
39	C₄×D₁₅₇	—	2	1	no	—
40	#11929434	—	2	1	no	—
41	Q₈×C₁₅₇	D₄	1	1	yes	characteristic, complemented
42	D₄×C₁₅₇	—	2	1	no	—
43	D₄×C₁₅₇	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₁₅₇	—	2	1	no	—
45	C₂×C₄×C₁₅₇	—	2	1	no	—
46	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
47	C₈×C₁₅₇	—	2	1	no	—
48	C₂²×C₁₅₇	—	2	1	no	—
49	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₅₇	—	2	2	no	—
51	C₂×D₁₅₇	—	2	1	no	—
52	C₂×D₁₅₇	—	4	2	no	—
53	C₂×D₁₅₇	—	4	2	no	—
54	C₂×D₁₅₇	—	4	1	no	—
55	C₂×D₁₅₇	—	8	2	no	—
56	C₄×C₁₅₇	—	2	2	no	—
57	C₄×C₁₅₇	—	2	1	no	—
58	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
59	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
60	Dic₁₅₇	—	2	1	no	—
61	C₂×C₁₅₇	—	2	1	no	—
62	C₂×C₁₅₇	—	4	1	no	—
63	C₂×C₁₅₇	#120	1	1	yes	characteristic
64	D₁₅₇	—	4	2	no	—
65	D₁₅₇	—	4	1	no	—
66	D₁₅₇	—	8	1	no	—
67	C₁₅₇	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	157	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	157	1	no	—
70	AGL(1,ℤ/8)	—	157	1	no	—
71	AGL(1,ℤ/8)	—	157	1	no	—
72	#127	—	157	1	no	—
73	#104	—	157	1	no	—
74	#104	D₁₅₇	1	1	yes	characteristic, complemented
75	#100	—	157	1	no	—
76	#41	—	314	1	no	—
77	#41	—	314	1	no	—
78	#41	—	157	1	no	—
79	#41	C₂×D₁₅₇	1	1	yes	characteristic
80	C₂×D₄	—	314	1	no	—
81	C₂×D₄	—	314	2	no	—
82	C₂×D₄	—	314	2	no	—
83	C₂×D₄	—	314	1	no	—
84	C₂×D₄	—	157	1	no	—
85	C₂×D₄	—	157	1	no	—
86	SD₁₆	—	314	1	no	—
87	SD₁₆	—	314	1	no	—
88	D₈	—	314	1	no	—
89	D₈	—	314	1	no	—
90	M₁₆	—	157	1	no	—
91	M₁₆	C₂×D₁₅₇	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₁₅₇	1	1	yes	characteristic, complemented
93	C₂³	—	314	2	no	—
94	C₂³	—	314	1	no	—
95	C₂³	—	157	2	no	—
96	Q₈	—	157	1	no	—
97	Q₈	D₆₂₈	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	314	2	no	—
100	D₄	—	314	2	no	—
101	D₄	—	314	1	no	—
102	D₄	—	314	1	no	—
103	D₄	—	314	1	no	—
104	D₄	—	314	1	no	—
105	D₄	—	314	1	no	—
106	D₄	—	628	4	no	—
107	D₄	—	157	1	no	—
108	D₄	D₆₂₈	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	314	2	no	—
112	C₂×C₄	—	314	1	no	—
113	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	314	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	314	2	no	—
118	C₂²	—	314	1	no	—
119	C₂²	—	628	2	no	—
120	C₂²	—	628	2	no	—
121	C₂²	—	628	1	no	—
122	C₂²	—	1256	2	no	—
123	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	314	1	no	—
127	C₄	D₄×D₁₅₇	1	1	yes	characteristic
128	C₄	C₂×D₆₂₈	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	628	2	no	—
132	C₂	—	628	1	no	—
133	C₂	—	1256	1	no	—
134	C₂	#19097741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097817

C₂

yes

maximal, characteristic, complemented

#19097799

C₂

yes

maximal, characteristic, complemented

#19097796

C₂

yes

maximal, characteristic, complemented

#19097768

C₂

yes

maximal, characteristic, complemented

#19097699

C₂

yes

maximal, characteristic, complemented

#19097687

C₂

yes

maximal, characteristic, complemented

#104×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578955

—

#14578954

—

#14578954

C₂²

yes

characteristic

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×D₆₂₈

C₂²

yes

characteristic, complemented

D₄×D₁₅₇

—

D₄×D₁₅₇

—

#14578943

—

#14578941

—

D₁₂₅₆

—

#14578939

—

#14578938

C₂²

yes

characteristic, complemented

#41×C₁₅₇

C₂²

yes

characteristic

M₁₆×C₁₅₇

C₂²

yes

characteristic, complemented

C₄²×C₁₅₇

C₂²

yes

characteristic, complemented

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

D₄

yes

—

Dic₃₁₄

D₄

yes

characteristic, complemented

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

D₄

yes

characteristic, complemented

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

Q₈×C₁₅₇

D₄

yes

characteristic, complemented

D₄×C₁₅₇

—

D₄×C₁₅₇

D₄

yes

characteristic, complemented

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

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

C₂×D₄

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#120

yes

characteristic

D₁₅₇

—

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#453

yes

characteristic, complemented, Sylow

#453

—

157

maximal, Sylow

H(5,𝔽₂)

—

157

—

AGL(1,ℤ/8)

—

157

—

AGL(1,ℤ/8)

—

157

—

#127

—

157

—

#104

—

157

—

#104

D₁₅₇

yes

characteristic, complemented

#100

—

157

—

#41

—

314

—

#41

—

314

—

#41

—

157

—

#41

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

SD₁₆

—

314

—

SD₁₆

—

314

—

D₈

—

314

—

D₈

—

314

—

M₁₆

—

157

—

M₁₆

C₂×D₁₅₇

yes

characteristic, complemented

C₄²

C₂×D₁₅₇

yes

characteristic, complemented

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

Q₈

—

157

—

Q₈

D₆₂₈

yes

characteristic, complemented

D₄

—

D₄

—

314

—

100

D₄

—

314

—

101

D₄

—

314

—

102

D₄

—

314

—

103

D₄

—

314

—

104

D₄

—

314

—

105

D₄

—

314

—

106

D₄

—

628

—

107

D₄

—

157

—

108

D₄

D₆₂₈

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

314

—

112

C₂×C₄

—

314

—

113

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

114

C₈

—

115

C₈

—

314

—

116

C₂²

—

117

C₂²

—

314

—

118

C₂²

—

314

—

119

C₂²

—

628

—

120

C₂²

—

628

—

121

C₂²

—

628

—

122

C₂²

—

1256

—

123

C₂²

D₄×D₁₅₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

314

—

127

C₄

D₄×D₁₅₇

yes

characteristic

128

C₄

C₂×D₆₂₈

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

628

—

132

C₂

—

628

—

133

C₂

—

1256

—

134

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	#453
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. #19097817	C₂	semidirect
2. #19097799	C₂	semidirect
3. #19097796	C₂	semidirect
4. #19097768	C₂	semidirect
5. #19097699	C₂	semidirect
6. #19097687	C₂	semidirect
7. #104×C₁₅₇	C₂	semidirect
14. C₂×D₆₂₈	C₂²	semidirect
21. #14578938	C₂²	semidirect
23. M₁₆×C₁₅₇	C₂²	semidirect
24. C₄²×C₁₅₇	C₂²	semidirect
28. Dic₃₁₄	D₄	semidirect
35. D₆₂₈	D₄	semidirect
41. Q₈×C₁₅₇	D₄	semidirect
43. D₄×C₁₅₇	D₄	semidirect
67. C₁₅₇	#453	semidirect
74. #104	D₁₅₇	semidirect
91. M₁₆	C₂×D₁₅₇	semidirect
92. C₄²	C₂×D₁₅₇	semidirect
97. Q₈	D₆₂₈	semidirect
108. D₄	D₆₂₈	semidirect

H ≅ G/N

type

1. #19097817

C₂

semidirect

2. #19097799

C₂

semidirect

3. #19097796

C₂

semidirect

4. #19097768

C₂

semidirect

5. #19097699

C₂

semidirect

6. #19097687

C₂

semidirect

7. #104×C₁₅₇

C₂

semidirect

14. C₂×D₆₂₈

C₂²

semidirect

21. #14578938

C₂²

semidirect

23. M₁₆×C₁₅₇

C₂²

semidirect

24. C₄²×C₁₅₇

C₂²

semidirect

28. Dic₃₁₄

D₄

semidirect

35. D₆₂₈

D₄

semidirect

41. Q₈×C₁₅₇

D₄

semidirect

43. D₄×C₁₅₇

D₄

semidirect

67. C₁₅₇

#453

semidirect

74. #104

D₁₅₇

semidirect

91. M₁₆

C₂×D₁₅₇

semidirect

92. C₄²

C₂×D₁₅₇

semidirect

97. Q₈

D₆₂₈

semidirect

108. D₄

D₆₂₈

semidirect

Subgroups — #23757442

Conjugacy classes of subgroups

Complements