Group #34100912 — subgroups

Subgroups — #34100912

Order 20096.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19097707	C₂	1	2	yes	maximal
2	#23757297	C₂	1	1	yes	maximal, characteristic
3	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
4	#23757038	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578938	C₂²	1	2	yes	—
7	C₂×#14578938	C₂²	1	1	yes	characteristic
8	C₂×#14578938	C₂²	1	1	yes	characteristic
9	#19097707	—	2	4	no	—
10	C₄⋊#11929434	—	2	1	no	—
11	C₄⋊#11929434	—	2	1	no	—
12	#19097682	—	2	1	no	—
13	C₄×#11929434	—	2	1	no	—
14	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₅₇	C₄	1	2	yes	—
17	C₄⋊C₈×C₁₅₇	C₄	1	2	yes	—
18	#14578938	—	2	2	no	—
19	#14578938	—	2	1	no	—
20	#14578938	—	2	1	no	—
21	#14578938	—	2	2	no	—
22	#14578938	D₄	1	2	yes	—
23	#14578938	Q₈	1	2	yes	—
24	C₂×#11929434	—	2	1	no	—
25	C₂×#11929434	—	2	1	no	—
26	C₂×#11929434	—	2	2	no	—
27	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₂×C₈×C₁₅₇	—	2	2	no	—
32	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
33	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
37	#11929434	—	2	2	no	—
38	#11929434	—	2	2	no	—
39	#11929434	—	4	2	no	—
40	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	4	1	no	—
43	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
53	C₈×C₁₅₇	—	2	4	no	—
54	C₂²×C₁₅₇	—	2	1	no	—
55	C₂²×C₁₅₇	—	2	1	no	—
56	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₅₇	#125	1	1	yes	characteristic
58	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₁₅₇	—	2	1	no	—
60	C₄×C₁₅₇	—	2	1	no	—
61	C₄×C₁₅₇	—	2	1	no	—
62	C₄×C₁₅₇	#108	1	2	yes	complemented
63	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₅₇	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₁₅₇	#121	1	1	yes	characteristic
66	C₄×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₅₇	—	2	1	no	—
68	C₂×C₁₅₇	—	2	1	no	—
69	C₂×C₁₅₇	#389	1	1	yes	characteristic
70	C₂×C₁₅₇	C₂×#108	1	1	yes	characteristic
71	C₂×C₁₅₇	#430	1	1	yes	characteristic
72	C₁₅₇	#1950	1	1	yes	characteristic, complemented, Sylow
73	#1950	—	157	1	no	maximal, Sylow
74	C₂×#108	—	157	2	no	—
75	#423	—	157	1	no	—
76	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
77	C₄×M₁₆	—	157	1	no	—
78	#341	—	157	2	no	—
79	C₂×M₁₆	—	157	2	no	—
80	C₂×M₁₆	—	157	1	no	—
81	C₂×M₁₆	—	157	1	no	—
82	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
83	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
84	#108	—	314	4	no	—
85	C₄⋊C₈	—	314	1	no	—
86	C₄⋊C₈	—	314	1	no	—
87	C₄⋊C₈	Dic₁₅₇	1	2	yes	—
88	C₄⋊C₈	Dic₁₅₇	1	2	yes	—
89	#97	—	314	1	no	—
90	C₄×C₈	—	314	1	no	—
91	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
92	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
94	M₁₆	—	314	1	no	—
95	M₁₆	—	314	1	no	—
96	M₁₆	—	314	2	no	—
97	M₁₆	—	314	2	no	—
98	M₁₆	—	157	2	no	—
99	M₁₆	—	157	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	314	1	no	—
103	C₂×C₈	—	314	1	no	—
104	C₂×C₈	—	314	2	no	—
105	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	—
106	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
107	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
108	C₄²	D₆₂₈	1	1	yes	characteristic
109	C₄²	D₆₂₈	1	1	yes	characteristic
110	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
114	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
115	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
121	C₂×C₄	#14578953	1	1	yes	characteristic
122	C₂×C₄	#14578955	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	314	2	no	—
125	C₈	—	314	2	no	—
126	C₈	—	628	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
130	C₂²	#19097762	1	1	yes	characteristic
131	C₂²	#19097770	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#19097707	1	2	yes	complemented
136	C₄	D₄×Dic₁₅₇	1	1	yes	characteristic
137	C₄	Q₈×Dic₁₅₇	1	1	yes	characteristic
138	C₄	#19097746	1	1	yes	characteristic
139	C₄	#19097751	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#23757172	1	1	yes	characteristic
143	C₂	C₂×#19097707	1	1	yes	characteristic
144	C₂	#23757322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097707

C₂

yes

maximal

#23757297

C₂

yes

maximal, characteristic

C₄×#14578938

C₂

yes

maximal, characteristic

#23757038

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂×#14578938

C₂²

yes

characteristic

C₂×#14578938

C₂²

yes

—

—

—

—

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

C₄

yes

—

C₄⋊C₈×C₁₅₇

C₄

yes

—

—

—

—

—

D₄

yes

—

#14578938

Q₈

yes

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

—

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

—

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#125

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#108

yes

complemented

C₄×C₁₅₇

C₄×D₄

yes

characteristic

C₄×C₁₅₇

C₄×Q₈

yes

characteristic

C₄×C₁₅₇

#121

yes

characteristic

C₄×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1950

—

157

maximal, Sylow

C₂×#108

—

157

—

#423

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

C₄×M₁₆

—

157

—

#341

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#108

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₄⋊C₈

Dic₁₅₇

yes

—

C₄⋊C₈

Dic₁₅₇

yes

—

#97

—

314

—

C₄×C₈

—

314

—

C₂²×C₄

Dic₃₁₄

yes

characteristic

C₂²×C₄

Dic₃₁₄

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

157

—

M₁₆

—

157

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

314

—

103

C₂×C₈

—

314

—

104

C₂×C₈

—

314

—

105

C₂×C₈

C₂×Dic₁₅₇

yes

—

106

C₄²

C₂×Dic₁₅₇

yes

characteristic

107

C₄²

C₂×Dic₁₅₇

yes

characteristic

108

C₄²

D₆₂₈

yes

characteristic

109

C₄²

D₆₂₈

yes

characteristic

110

C₂³

C₂×Dic₃₁₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

114

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

115

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

117

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

118

C₂×C₄

D₄×D₁₅₇

yes

characteristic

119

C₂×C₄

C₂×D₆₂₈

yes

characteristic

120

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

121

C₂×C₄

#14578953

yes

characteristic

122

C₂×C₄

#14578955

yes

characteristic

123

C₈

—

124

C₈

—

314

—

125

C₈

—

314

—

126

C₈

—

628

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₄⋊Dic₁₅₇

yes

characteristic

130

C₂²

#19097762

yes

characteristic

131

C₂²

#19097770

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#19097707

yes

complemented

136

C₄

D₄×Dic₁₅₇

yes

characteristic

137

C₄

Q₈×Dic₁₅₇

yes

characteristic

138

C₄

#19097746

yes

characteristic

139

C₄

#19097751

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#23757172

yes

characteristic

143

C₂

C₂×#19097707

yes

characteristic

144

C₂

#23757322

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1950
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₆₂₈

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×C₄⋊C₈×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₇

hypocenter

C₁₅₇

#1950

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
62. C₄×C₁₅₇	#108	semidirect
72. C₁₅₇	#1950	semidirect
135. C₄	#19097707	semidirect

H ≅ G/N

type

62. C₄×C₁₅₇

#108

semidirect

72. C₁₅₇

#1950

semidirect

135. C₄

#19097707

semidirect

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

subgroup	index
C₂×#19097707	2
#23757297	2
C₄×#14578938	2
#23757038	2
C₂×C₄⋊C₈×C₁₅₇	2
#1950	157

subgroup

index

C₂×C₄⋊C₈×C₁₅₇

157

Subgroups — #34100912

Conjugacy classes of subgroups

Complements

Maximal subgroups