Group #34101235 — subgroups

Subgroups — #34101235

Order 20096.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#14578944	C₂	1	2	yes	maximal
2	C₂×#19097771	C₂	1	1	yes	maximal, characteristic
3	C₂×#19097694	C₂	1	2	yes	maximal
4	#23757028	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
7	C₂×#14578944	—	2	4	no	—
8	C₂×#14578944	—	2	4	no	—
9	C₂×#14578944	—	2	4	no	—
10	C₂²×#11929434	C₂²	1	2	yes	—
11	#19097771	—	2	4	no	—
12	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
13	#19097694	—	2	4	no	—
14	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
17	#95×C₁₅₇	—	2	2	no	—
18	C₂×Dic₃₁₄	—	2	2	no	—
19	C₂×Dic₃₁₄	—	2	2	no	—
20	C₂×Dic₃₁₄	—	2	2	no	—
21	C₂×Dic₃₁₄	—	2	2	no	—
22	C₂×Dic₃₁₄	—	2	2	no	—
23	C₂×Dic₃₁₄	D₄	1	4	yes	—
24	C₂²×Dic₁₅₇	—	2	2	no	—
25	#14578944	—	2	16	no	—
26	C₂×#11929434	—	2	2	no	—
27	C₂×#11929434	—	2	2	no	—
28	C₂×#11929434	D₄	1	4	yes	—
29	C₄⋊Dic₁₅₇	—	2	8	no	—
30	C₂×Q₈×C₁₅₇	—	2	1	no	—
31	C₂×Q₈×C₁₅₇	—	2	2	no	—
32	C₂×Q₈×C₁₅₇	—	2	2	no	—
33	C₂×Q₈×C₁₅₇	D₄	1	2	yes	—
34	C₂²×C₄×C₁₅₇	—	2	1	no	—
35	C₂²×C₄×C₁₅₇	—	2	1	no	—
36	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₅₇	—	2	2	no	—
39	C₄²×C₁₅₇	—	2	2	no	—
40	Dic₃₁₄	—	2	4	no	—
41	Dic₃₁₄	—	2	4	no	—
42	Dic₃₁₄	—	4	2	no	—
43	Dic₃₁₄	—	4	2	no	—
44	C₂×Dic₁₅₇	—	2	4	no	—
45	C₂×Dic₁₅₇	—	4	2	no	—
46	C₂×Dic₁₅₇	—	4	2	no	—
47	#11929434	—	2	4	no	—
48	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₁₅₇	—	2	4	no	—
50	Q₈×C₁₅₇	—	4	2	no	—
51	C₂×C₄×C₁₅₇	—	2	2	no	—
52	C₂×C₄×C₁₅₇	—	2	2	no	—
53	C₂×C₄×C₁₅₇	—	2	2	no	—
54	C₂×C₄×C₁₅₇	—	4	1	no	—
55	C₂×C₄×C₁₅₇	—	4	1	no	—
56	C₂×C₄×C₁₅₇	—	4	1	no	—
57	C₂×C₄×C₁₅₇	—	4	2	no	—
58	C₂×C₄×C₁₅₇	Q₁₆	1	4	yes	complemented
59	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
63	C₂²×C₁₅₇	#120	1	1	yes	characteristic
64	C₂²×C₁₅₇	#121	1	1	yes	characteristic
65	C₂²×C₁₅₇	#127	1	1	yes	characteristic
66	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₅₇	#137	1	2	yes	—
68	C₄×C₁₅₇	—	2	4	no	—
69	C₄×C₁₅₇	—	4	2	no	—
70	C₄×C₁₅₇	—	8	1	no	—
71	C₄×C₁₅₇	#120	1	2	yes	—
72	C₄×C₁₅₇	#121	1	2	yes	—
73	Dic₁₅₇	—	4	4	no	—
74	C₂×C₁₅₇	#392	1	1	yes	characteristic
75	C₂×C₁₅₇	#451	1	2	yes	—
76	C₂×C₁₅₇	#462	1	2	yes	—
77	C₂×C₁₅₇	#494	1	1	yes	characteristic
78	C₂×C₁₅₇	#497	1	1	yes	characteristic
79	C₁₅₇	#2016	1	1	yes	characteristic, complemented, Sylow
80	#2016	—	157	1	no	maximal, Sylow
81	C₂²×Q₁₆	—	157	2	no	—
82	C₂×C₄⋊Q₈	—	157	1	no	—
83	C₂×#103	—	157	2	no	—
84	#391	D₁₅₇	1	1	yes	characteristic
85	#336	—	157	1	no	—
86	C₂²×Q₈	—	157	2	no	—
87	C₂²×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
88	C₂×Q₁₆	—	314	4	no	—
89	C₂×Q₁₆	—	314	4	no	—
90	C₂×Q₁₆	—	314	4	no	—
91	C₂²×C₈	—	157	2	no	—
92	C₄⋊Q₈	—	314	4	no	—
93	C₂×C₄⋊C₄	—	314	2	no	—
94	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
96	#103	—	314	4	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	314	2	no	—
102	C₂×Q₈	—	314	2	no	—
103	C₂×Q₈	—	314	2	no	—
104	C₂×Q₈	—	314	2	no	—
105	C₂×Q₈	—	314	2	no	—
106	C₂×Q₈	—	157	4	no	—
107	C₂×Q₈	D₆₂₈	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	314	2	no	—
111	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
113	Q₁₆	—	314	16	no	—
114	C₂×C₈	—	314	2	no	—
115	C₂×C₈	—	314	2	no	—
116	C₂×C₈	—	157	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	314	8	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	314	4	no	—
124	Q₈	—	314	4	no	—
125	Q₈	—	628	2	no	—
126	Q₈	—	628	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	314	4	no	—
135	C₂×C₄	—	628	2	no	—
136	C₂×C₄	—	628	2	no	—
137	C₂×C₄	#14578944	1	4	yes	complemented
138	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
139	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
140	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
141	C₂×C₄	#14578954	1	1	yes	characteristic
142	C₈	—	314	4	no	—
143	C₂²	#19097739	1	1	yes	characteristic
144	C₂²	#19097745	1	1	yes	characteristic
145	C₂²	#19097767	1	1	yes	characteristic
146	C₂²	C₂×#14578944	1	2	yes	—
147	C₂²	#19097801	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	628	4	no	—
152	C₄	#19097741	1	2	yes	—
153	C₄	#19097744	1	2	yes	—
154	C₂	#23757182	1	1	yes	characteristic
155	C₂	#23757428	1	2	yes	—
156	C₂	#23757483	1	2	yes	—
157	C₂	#23757652	1	1	yes	characteristic
158	C₂	#23757668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14578944

C₂

yes

maximal

C₂×#19097771

C₂

yes

maximal, characteristic

C₂×#19097694

C₂

yes

maximal

#23757028

C₂

yes

maximal, characteristic

#391×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

—

—

—

C₂²

yes

—

#19097771

—

C₂×C₄⋊Dic₁₅₇

—

#19097694

—

C₂²×Q₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#95×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

D₄

yes

—

C₂²×Dic₁₅₇

—

—

—

—

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×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

Q₁₆

yes

complemented

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

C₂×C₄×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

C₂×Q₁₆

yes

—

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

—

C₄×C₁₅₇

#121

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#392

yes

characteristic

C₂×C₁₅₇

#451

yes

—

C₂×C₁₅₇

#462

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#2016

—

157

maximal, Sylow

C₂²×Q₁₆

—

157

—

C₂×C₄⋊Q₈

—

157

—

C₂×#103

—

157

—

#391

D₁₅₇

yes

characteristic

#336

—

157

—

C₂²×Q₈

—

157

—

C₂²×Q₈

C₂×D₁₅₇

yes

characteristic

C₂×Q₁₆

—

314

—

C₂×Q₁₆

—

314

—

C₂×Q₁₆

—

314

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#103

—

314

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

314

—

102

C₂×Q₈

—

314

—

103

C₂×Q₈

—

314

—

104

C₂×Q₈

—

314

—

105

C₂×Q₈

—

314

—

106

C₂×Q₈

—

157

—

107

C₂×Q₈

D₆₂₈

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

314

—

111

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

112

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

113

Q₁₆

—

314

—

114

C₂×C₈

—

314

—

115

C₂×C₈

—

314

—

116

C₂×C₈

—

157

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

314

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

314

—

124

Q₈

—

314

—

125

Q₈

—

628

—

126

Q₈

—

628

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

314

—

135

C₂×C₄

—

628

—

136

C₂×C₄

—

628

—

137

C₂×C₄

#14578944

yes

complemented

138

C₂×C₄

D₄×D₁₅₇

yes

—

139

C₂×C₄

D₄×D₁₅₇

yes

—

140

C₂×C₄

C₂×D₆₂₈

yes

characteristic

141

C₂×C₄

#14578954

yes

characteristic

142

C₈

—

314

—

143

C₂²

#19097739

yes

characteristic

144

C₂²

#19097745

yes

characteristic

145

C₂²

#19097767

yes

characteristic

146

C₂²

C₂×#14578944

yes

—

147

C₂²

#19097801

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

628

—

152

C₄

#19097741

yes

—

153

C₄

#19097744

yes

—

154

C₂

#23757182

yes

characteristic

155

C₂

#23757428

yes

—

156

C₂

#23757483

yes

—

157

C₂

#23757652

yes

characteristic

158

C₂

#23757668

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator, cosocle	C₂²×C₄×C₁₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#391×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	#391	D₁₅₇
hypocenter	C₁₅₇	#2016
LCS	G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator, cosocle

C₂²×C₄×C₁₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#391×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
58. C₂×C₄×C₁₅₇	Q₁₆	semidirect
79. C₁₅₇	#2016	semidirect
137. C₂×C₄	#14578944	semidirect

H ≅ G/N

type

58. C₂×C₄×C₁₅₇

Q₁₆

semidirect

79. C₁₅₇

#2016

semidirect

137. C₂×C₄

#14578944

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14578944	2
C₂×#19097771	2
C₂×#19097694	2
#23757028	2
#391×C₁₅₇	2
#2016	157

subgroup

index

#391×C₁₅₇

157

Subgroups — #34101235

Conjugacy classes of subgroups

Complements

Maximal subgroups