Group #23757202 — subgroups

Subgroups — #23757202

Order 10048.

170 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14578933	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14578931	C₂	1	1	yes	maximal, characteristic
3	C₂×#14578931	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578931	C₂	1	1	yes	maximal, characteristic
5	#19097678	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097678	C₂	1	1	yes	maximal, characteristic, complemented
7	#95×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₅₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
12	#14578933	—	2	2	no	—
13	#14578931	—	2	2	no	—
14	#14578931	—	2	2	no	—
15	#14578931	—	2	2	no	—
16	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂²×D₁₅₇	—	2	1	no	—
20	C₂²×D₁₅₇	—	2	1	no	—
21	C₂²×D₁₅₇	—	2	1	no	—
22	C₂²×D₁₅₇	—	2	1	no	—
23	C₂²×D₁₅₇	—	2	1	no	—
24	C₂²×D₁₅₇	—	2	1	no	—
25	C₂²×D₁₅₇	D₄	1	2	yes	—
26	C₂×Dic₁₅₇	—	2	1	no	—
27	C₂×Dic₁₅₇	—	2	1	no	—
28	C₂×Dic₁₅₇	—	2	1	no	—
29	C₂×Dic₁₅₇	—	2	1	no	—
30	C₂×Dic₁₅₇	—	2	1	no	—
31	C₂×Dic₁₅₇	—	2	1	no	—
32	C₂×Dic₁₅₇	—	2	1	no	—
33	C₂×Dic₁₅₇	—	2	1	no	—
34	C₂×Dic₁₅₇	D₄	1	2	yes	—
35	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	—	2	1	no	—
37	C₂×C₄×C₁₅₇	—	2	1	no	—
38	C₂×C₄×C₁₅₇	—	2	1	no	—
39	C₂×C₄×C₁₅₇	—	2	1	no	—
40	C₂×C₄×C₁₅₇	—	2	1	no	—
41	C₂×C₄×C₁₅₇	—	2	1	no	—
42	C₂×C₄×C₁₅₇	—	2	1	no	—
43	C₂×C₄×C₁₅₇	—	2	1	no	—
44	C₂×C₄×C₁₅₇	—	2	1	no	—
45	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	#41	1	1	yes	characteristic
48	C₂²×C₁₅₇	#41	1	1	yes	characteristic
49	C₂²×C₁₅₇	#41	1	1	yes	characteristic
50	C₂²×C₁₅₇	#41	1	1	yes	characteristic
51	C₂²×C₁₅₇	#41	1	1	yes	characteristic
52	C₂×D₁₅₇	—	2	2	no	—
53	C₂×D₁₅₇	—	2	2	no	—
54	C₂×D₁₅₇	—	2	2	no	—
55	C₂×D₁₅₇	—	4	1	no	—
56	C₂×D₁₅₇	—	4	1	no	—
57	C₂×D₁₅₇	—	4	1	no	—
58	C₂×D₁₅₇	—	4	1	no	—
59	C₄×C₁₅₇	—	4	1	no	—
60	C₄×C₁₅₇	—	4	1	no	—
61	C₄×C₁₅₇	—	4	1	no	—
62	Dic₁₅₇	—	2	2	no	—
63	Dic₁₅₇	—	4	1	no	—
64	Dic₁₅₇	—	4	1	no	—
65	C₂×C₁₅₇	#121	1	1	yes	characteristic
66	C₂×C₁₅₇	#123	1	1	yes	characteristic
67	C₂×C₁₅₇	#123	1	1	yes	characteristic
68	C₂×C₁₅₇	#123	1	1	yes	characteristic
69	C₂×C₁₅₇	#124	1	1	yes	characteristic
70	C₂×C₁₅₇	#126	1	1	yes	characteristic
71	C₂×C₁₅₇	#126	1	1	yes	characteristic
72	D₁₅₇	—	4	2	no	—
73	C₁₅₇	#397	1	1	yes	characteristic, complemented, Sylow
74	#397	—	157	1	no	maximal, Sylow
75	C₂×C₄⋊C₄	—	157	1	no	—
76	C₂×C₂²⋊C₄	—	157	1	no	—
77	C₂×C₂²⋊C₄	—	157	1	no	—
78	C₂×C₂²⋊C₄	—	157	1	no	—
79	#95	—	157	1	no	—
80	#95	—	157	1	no	—
81	#95	D₁₅₇	1	1	yes	characteristic, complemented
82	C₂⁴	—	157	1	no	—
83	C₂²×C₄	—	157	1	no	—
84	C₂²×C₄	—	157	1	no	—
85	C₂²×C₄	—	157	1	no	—
86	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
89	C₄⋊C₄	—	314	2	no	—
90	C₂²⋊C₄	—	314	2	no	—
91	C₂²⋊C₄	—	314	2	no	—
92	C₂²⋊C₄	—	314	2	no	—
93	C₂³	—	314	1	no	—
94	C₂³	—	314	1	no	—
95	C₂³	—	314	1	no	—
96	C₂³	—	314	1	no	—
97	C₂³	—	314	1	no	—
98	C₂³	—	314	1	no	—
99	C₂³	—	157	2	no	—
100	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	314	1	no	—
111	C₂×C₄	—	314	1	no	—
112	C₂×C₄	—	314	1	no	—
113	C₂×C₄	—	314	1	no	—
114	C₂×C₄	—	314	1	no	—
115	C₂×C₄	—	314	1	no	—
116	C₂×C₄	—	314	1	no	—
117	C₂×C₄	—	314	1	no	—
118	C₂×C₄	—	157	2	no	—
119	C₂²	—	314	2	no	—
120	C₂²	—	314	2	no	—
121	C₂²	—	314	2	no	—
122	C₂²	—	628	1	no	—
123	C₂²	—	628	1	no	—
124	C₂²	—	628	1	no	—
125	C₂²	—	628	1	no	—
126	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
127	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
128	C₂²	#14578953	1	1	yes	characteristic
129	C₂²	#14578954	1	1	yes	characteristic
130	C₂²	#14578954	1	1	yes	characteristic
131	C₂²	#14578954	1	1	yes	characteristic
132	C₂²	#14578955	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	314	2	no	—
137	C₄	—	628	1	no	—
138	C₄	—	628	1	no	—
139	C₂	—	628	2	no	—
140	C₂	#19097744	1	1	yes	characteristic
141	C₂	#19097753	1	1	yes	characteristic
142	C₂	#19097756	1	1	yes	characteristic
143	C₂	#19097756	1	1	yes	characteristic
144	C₂	#19097757	1	1	yes	characteristic
145	C₂	#19097765	1	1	yes	characteristic
146	C₂	#19097766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578933

C₂

yes

maximal, characteristic, complemented

C₂×#14578931

C₂

yes

maximal, characteristic

C₂×#14578931

C₂

yes

maximal, characteristic

C₂×#14578931

C₂

yes

maximal, characteristic

#19097678

C₂

yes

maximal, characteristic, complemented

#19097678

C₂

yes

maximal, characteristic, complemented

#95×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

—

—

—

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

D₄

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

D₁₅₇

—

C₁₅₇

#397

yes

characteristic, complemented, Sylow

#397

—

157

maximal, Sylow

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

#95

—

157

—

#95

—

157

—

#95

D₁₅₇

yes

characteristic, complemented

C₂⁴

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

100

C₂³

C₂²×D₁₅₇

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

314

—

111

C₂×C₄

—

314

—

112

C₂×C₄

—

314

—

113

C₂×C₄

—

314

—

114

C₂×C₄

—

314

—

115

C₂×C₄

—

314

—

116

C₂×C₄

—

314

—

117

C₂×C₄

—

314

—

118

C₂×C₄

—

157

—

119

C₂²

—

314

—

120

C₂²

—

314

—

121

C₂²

—

314

—

122

C₂²

—

628

—

123

C₂²

—

628

—

124

C₂²

—

628

—

125

C₂²

—

628

—

126

C₂²

D₄×D₁₅₇

yes

characteristic

127

C₂²

D₄×D₁₅₇

yes

characteristic

128

C₂²

#14578953

yes

characteristic

129

C₂²

#14578954

yes

characteristic

130

C₂²

#14578954

yes

characteristic

131

C₂²

#14578954

yes

characteristic

132

C₂²

#14578955

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

314

—

137

C₄

—

628

—

138

C₄

—

628

—

139

C₂

—

628

—

140

C₂

#19097744

yes

characteristic

141

C₂

#19097753

yes

characteristic

142

C₂

#19097756

yes

characteristic

143

C₂

#19097756

yes

characteristic

144

C₂

#19097757

yes

characteristic

145

C₂

#19097765

yes

characteristic

146

C₂

#19097766

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₇
commutator, socle, cosocle	C₂³×C₁₅₇	C₂³
Fitting	#95×C₁₅₇	C₂
prime radical	C₂³×D₁₅₇	C₂²
hypercenter	#95	D₁₅₇
hypocenter	C₁₅₇	#397
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₇

commutator, socle, cosocle

C₂³×C₁₅₇

C₂³

Fitting

#95×C₁₅₇

C₂

prime radical

C₂³×D₁₅₇

C₂²

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
1. C₂×#14578933	C₂	semidirect
5. #19097678	C₂	semidirect
6. #19097678	C₂	semidirect
7. #95×C₁₅₇	C₂	semidirect
73. C₁₅₇	#397	semidirect
81. #95	D₁₅₇	semidirect

H ≅ G/N

type

1. C₂×#14578933

C₂

semidirect

5. #19097678

C₂

semidirect

6. #19097678

C₂

semidirect

7. #95×C₁₅₇

C₂

semidirect

73. C₁₅₇

#397

semidirect

81. #95

D₁₅₇

semidirect

Subgroups — #23757202

Conjugacy classes of subgroups

Complements