Group #23757851 — subgroups

Subgroups — #23757851

Order 10048.

294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#14578954	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097750	C₂	1	2	yes	maximal, complemented
3	#19097747	C₂	1	2	yes	maximal, complemented
4	#19097745	C₂	1	2	yes	maximal, complemented
5	#19097743	C₂	1	4	yes	maximal, complemented
6	#19097742	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14578954	—	2	4	no	—
10	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
11	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₅₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₅₇	C₂²	1	4	yes	—
14	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
18	#14578933	C₂²	1	4	yes	—
19	C₂²⋊Dic₁₅₇	C₂²	1	4	yes	—
20	#14578931	C₂²	1	4	yes	—
21	#41×C₁₅₇	—	2	4	no	—
22	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₁₅₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₅₇	C₂³	1	2	yes	—
28	Dic₃₁₄	—	2	2	no	—
29	D₆₂₈	—	2	2	no	—
30	C₂²⋊D₁₅₇	—	2	4	no	—
31	C₂²⋊D₁₅₇	—	4	4	no	—
32	C₂×Dic₁₅₇	—	4	2	no	—
33	C₂×Dic₁₅₇	C₂³	1	2	yes	—
34	C₂×Dic₁₅₇	C₂³	1	4	yes	—
35	C₄×D₁₅₇	—	2	4	no	—
36	C₂³×C₁₅₇	C₂³	1	2	yes	—
37	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
38	Q₈×C₁₅₇	—	2	2	no	—
39	D₄×C₁₅₇	—	2	4	no	—
40	D₄×C₁₅₇	—	2	2	no	—
41	C₂×C₄×C₁₅₇	—	2	4	no	—
42	C₂×C₄×C₁₅₇	D₄	1	4	yes	complemented
43	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
46	C₂²×C₁₅₇	—	2	1	no	—
47	C₂²×C₁₅₇	—	2	1	no	—
48	C₂²×C₁₅₇	—	2	2	no	—
49	C₂²×C₁₅₇	—	2	2	no	—
50	C₂²×C₁₅₇	—	2	2	no	—
51	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₅₇	—	2	2	no	—
54	C₂×D₁₅₇	—	2	2	no	—
55	C₂×D₁₅₇	—	2	2	no	—
56	C₄×C₁₅₇	—	2	2	no	—
57	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
58	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
59	Dic₁₅₇	—	2	2	no	—
60	Dic₁₅₇	—	2	4	no	—
61	C₂×C₁₅₇	—	2	2	no	—
62	C₂×C₁₅₇	—	4	2	no	—
63	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₅₇	—	4	2	no	—
67	C₁₅₇	#537	1	1	yes	characteristic, complemented, Sylow
68	#537	—	157	1	no	maximal, Sylow
69	C₂×#41	—	157	1	no	—
70	C₂×#41	D₁₅₇	1	1	yes	characteristic, complemented
71	C₂²⋊Q₈	—	157	2	no	—
72	C₂²⋊Q₈	—	157	2	no	—
73	#121	—	157	2	no	—
74	#121	—	157	4	no	—
75	#121	—	157	2	no	—
76	C₂×C₄⋊C₄	—	157	1	no	—
77	#41	—	2	4	no	—
78	#41	—	314	4	no	—
79	C₂×Q₈	—	157	1	no	—
80	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
81	C₂×D₄	—	157	2	no	—
82	C₂×D₄	—	157	4	no	—
83	C₂×D₄	—	157	1	no	—
84	C₂×D₄	C₂×D₁₅₇	1	2	yes	—
85	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic
86	C₂²×C₄	—	157	2	no	—
87	C₂²×C₄	—	157	2	no	—
88	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
89	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
90	C₄⋊C₄	—	157	2	no	—
91	C₄⋊C₄	—	157	2	no	—
92	C₄⋊C₄	—	157	4	no	—
93	C₂²⋊C₄	—	157	4	no	—
94	C₂²⋊C₄	—	157	4	no	—
95	C₂³	—	157	2	no	—
96	C₂³	C₂²×D₁₅₇	1	2	yes	—
97	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	Q₈	—	314	2	no	—
100	D₄	—	2	4	no	—
101	D₄	—	2	2	no	—
102	D₄	—	314	4	no	—
103	D₄	—	314	2	no	—
104	D₄	—	628	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	314	4	no	—
107	C₂×C₄	—	628	2	no	—
108	C₂×C₄	—	157	4	no	—
109	C₂×C₄	—	157	2	no	—
110	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	complemented
111	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
112	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	314	2	no	—
120	C₂²	—	314	2	no	—
121	C₂²	—	314	2	no	—
122	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
123	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	314	2	no	—
126	C₄	—	314	4	no	—
127	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
128	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	628	2	no	—
132	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
133	C₂	#19097817	1	1	yes	characteristic
134	C₂	#19097818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578954

C₂

yes

maximal, characteristic, complemented

#19097750

C₂

yes

maximal, complemented

#19097747

C₂

yes

maximal, complemented

#19097745

C₂

yes

maximal, complemented

#19097743

C₂

yes

maximal, complemented

#19097742

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₂²

yes

—

C₂×C₂²⋊D₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

—

#14578931

C₂²

yes

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

complemented

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

characteristic

D₁₅₇

—

C₁₅₇

#537

yes

characteristic, complemented, Sylow

#537

—

157

maximal, Sylow

C₂×#41

—

157

—

C₂×#41

D₁₅₇

yes

characteristic, complemented

C₂²⋊Q₈

—

157

—

C₂²⋊Q₈

—

157

—

#121

—

157

—

#121

—

157

—

#121

—

157

—

C₂×C₄⋊C₄

—

157

—

#41

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

314

—

100

D₄

—

101

D₄

—

102

D₄

—

314

—

103

D₄

—

314

—

104

D₄

—

628

—

105

C₂×C₄

—

106

C₂×C₄

—

314

—

107

C₂×C₄

—

628

—

108

C₂×C₄

—

157

—

109

C₂×C₄

—

157

—

110

C₂×C₄

C₂²⋊D₁₅₇

yes

complemented

111

C₂×C₄

C₂²×D₁₅₇

yes

—

112

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

113

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

314

—

120

C₂²

—

314

—

121

C₂²

—

314

—

122

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

123

C₂²

C₂³×D₁₅₇

yes

characteristic

124

C₄

—

125

C₄

—

314

—

126

C₄

—

314

—

127

C₄

C₂×C₂²⋊D₁₅₇

yes

—

128

C₄

C₂×C₂²⋊D₁₅₇

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

628

—

132

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

133

C₂

#19097817

yes

characteristic

134

C₂

#19097818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₂×#41×C₁₅₇	C₂
hypercenter	C₂×#41	D₁₅₇
hypocenter	C₁₅₇	#537
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₂×#41×C₁₅₇

C₂

hypercenter

C₂×#41

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14578954	C₂	semidirect
2. #19097750	C₂	semidirect
3. #19097747	C₂	semidirect
4. #19097745	C₂	semidirect
5. #19097743	C₂	semidirect
6. #19097742	C₂	semidirect
7. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
8. C₂×#41×C₁₅₇	C₂	semidirect
42. C₂×C₄×C₁₅₇	D₄	semidirect
67. C₁₅₇	#537	semidirect
70. C₂×#41	D₁₅₇	semidirect
110. C₂×C₄	C₂²⋊D₁₅₇	semidirect

H ≅ G/N

type

1. C₂×#14578954

C₂

semidirect

2. #19097750

C₂

semidirect

3. #19097747

C₂

semidirect

4. #19097745

C₂

semidirect

5. #19097743

C₂

semidirect

6. #19097742

C₂

semidirect

7. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

8. C₂×#41×C₁₅₇

C₂

semidirect

42. C₂×C₄×C₁₅₇

D₄

semidirect

67. C₁₅₇

#537

semidirect

70. C₂×#41

D₁₅₇

semidirect

110. C₂×C₄

C₂²⋊D₁₅₇

semidirect

Subgroups — #23757851

Conjugacy classes of subgroups

Complements