Group #34101124 — subgroups

Subgroups — #34101124

Order 20096.

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

normal subgroups only

This group has 260 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	C₂×#19097771	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
3	#23757041	C₂	1	4	yes	maximal
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	C₄	1	2	yes	—
6	C₂×#14578938	C₂²	1	2	yes	—
7	#19097771	—	2	1	no	—
8	#19097771	—	2	1	no	—
9	#19097771	—	2	1	no	—
10	#19097771	—	2	1	no	—
11	C₂×C₄⋊Dic₁₅₇	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
13	#19097682	—	2	1	no	—
14	C₄×#11929434	—	2	1	no	—
15	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₅₇	—	2	1	no	—
18	C₄⋊C₈×C₁₅₇	—	2	1	no	—
19	C₂×Dic₃₁₄	—	2	2	no	—
20	C₂×Dic₃₁₄	—	2	2	no	—
21	C₂×Dic₃₁₄	—	4	4	no	—
22	C₂²×Dic₁₅₇	C₂×C₄	1	4	yes	—
23	#14578938	—	2	2	no	—
24	#14578938	Q₈	1	4	yes	complemented
25	C₂×#11929434	—	2	2	no	—
26	C₄⋊Dic₁₅₇	—	2	2	no	—
27	C₄⋊Dic₁₅₇	—	2	2	no	—
28	C₄⋊Dic₁₅₇	—	2	2	no	—
29	C₄⋊Dic₁₅₇	—	2	2	no	—
30	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₅₇	—	2	4	no	—
34	C₂×C₈×C₁₅₇	—	2	2	no	—
35	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
39	Dic₃₁₄	—	4	4	no	—
40	Dic₃₁₄	—	4	4	no	—
41	C₂×Dic₁₅₇	—	2	4	no	—
42	C₂×Dic₁₅₇	—	4	4	no	—
43	#11929434	—	2	4	no	—
44	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	—	2	2	no	—
46	C₂×C₄×C₁₅₇	—	2	2	no	—
47	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₅₇	#41	1	2	yes	—
55	C₈×C₁₅₇	—	4	2	no	—
56	C₂²×C₁₅₇	—	2	1	no	—
57	C₂²×C₁₅₇	—	2	1	no	—
58	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₅₇	—	2	1	no	—
61	C₄×C₁₅₇	—	2	1	no	—
62	C₄×C₁₅₇	#100	1	2	yes	—
63	C₄×C₁₅₇	#101	1	2	yes	—
64	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₅₇	#124	1	1	yes	characteristic
66	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₅₇	—	4	4	no	—
68	C₂×C₁₅₇	—	2	2	no	—
69	C₂×C₁₅₇	#391	1	1	yes	characteristic
70	C₂×C₁₅₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₅₇	C₂×#101	1	1	yes	characteristic
72	C₁₅₇	#1997	1	1	yes	characteristic, complemented, Sylow
73	#1997	—	157	1	no	maximal, Sylow
74	C₂×C₄⋊Q₈	—	157	1	no	—
75	#423	D₁₅₇	1	1	yes	characteristic
76	C₄×M₁₆	—	157	1	no	—
77	#343	—	157	4	no	—
78	C₂²×Q₈	—	157	2	no	—
79	C₂×M₁₆	—	157	2	no	—
80	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
81	C₄⋊Q₈	—	314	1	no	—
82	C₄⋊Q₈	—	314	1	no	—
83	C₄⋊Q₈	—	314	1	no	—
84	C₄⋊Q₈	—	314	1	no	—
85	C₂×C₄⋊C₄	—	157	2	no	—
86	C₂×C₄⋊C₄	—	157	2	no	—
87	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	314	1	no	—
91	C₄×C₈	—	314	1	no	—
92	C₂×Q₈	—	314	2	no	—
93	C₂×Q₈	—	314	2	no	—
94	C₂×Q₈	—	628	4	no	—
95	C₂²×C₄	—	157	4	no	—
96	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	314	2	no	—
101	M₁₆	—	157	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	314	2	no	—
104	C₄⋊C₄	—	314	2	no	—
105	C₄⋊C₄	—	314	2	no	—
106	C₄⋊C₄	—	314	2	no	—
107	C₄⋊C₄	—	314	2	no	—
108	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
110	C₄²	D₆₂₈	1	1	yes	characteristic
111	C₄²	D₆₂₈	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
113	Q₈	—	628	4	no	—
114	Q₈	—	628	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	314	4	no	—
118	C₂×C₄	—	628	4	no	—
119	C₂×C₄	#14578931	1	1	yes	characteristic
120	C₂×C₄	#14578931	1	1	yes	characteristic
121	C₂×C₄	#14578931	1	1	yes	characteristic
122	C₂×C₄	#14578931	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
124	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
126	C₂×C₄	#14578953	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	314	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×#14578931	1	1	yes	characteristic
132	C₂²	#19097736	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	628	4	no	—
136	C₄	#19097687	1	2	yes	—
137	C₄	#19097689	1	2	yes	—
138	C₄	#19097751	1	2	yes	—
139	C₄	#19097758	1	1	yes	characteristic
140	C₄	#19097769	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#23757178	1	1	yes	characteristic
143	C₂	C₂×#19097687	1	1	yes	characteristic
144	C₂	C₂×#19097689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₄×#14578938

C₂

yes

maximal, characteristic

#23757041

C₂

yes

maximal

#423×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₄

yes

—

C₂×#14578938

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₁₅₇

C₄

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097682

—

C₄×#11929434

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₄⋊C₈×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

C₂×C₄

yes

—

#14578938

—

#14578938

Q₈

yes

complemented

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊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

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#100

yes

—

C₄×C₁₅₇

#101

yes

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1997

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

#423

D₁₅₇

yes

characteristic

C₄×M₁₆

—

157

—

#343

—

157

—

C₂²×Q₈

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

314

—

C₄×C₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

628

—

C₂²×C₄

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

100

M₁₆

—

314

—

101

M₁₆

—

157

—

102

C₂×C₈

—

103

C₂×C₈

—

314

—

104

C₄⋊C₄

—

314

—

105

C₄⋊C₄

—

314

—

106

C₄⋊C₄

—

314

—

107

C₄⋊C₄

—

314

—

108

C₄²

C₂²⋊D₁₅₇

yes

characteristic

109

C₄²

C₂²⋊D₁₅₇

yes

characteristic

110

C₄²

D₆₂₈

yes

characteristic

111

C₄²

D₆₂₈

yes

characteristic

112

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

113

Q₈

—

628

—

114

Q₈

—

628

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

314

—

118

C₂×C₄

—

628

—

119

C₂×C₄

#14578931

yes

characteristic

120

C₂×C₄

#14578931

yes

characteristic

121

C₂×C₄

#14578931

yes

characteristic

122

C₂×C₄

#14578931

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

124

C₂×C₄

C₂×D₆₂₈

yes

characteristic

125

C₂×C₄

Q₈×D₁₅₇

yes

—

126

C₂×C₄

#14578953

yes

—

127

C₈

—

128

C₈

—

314

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×#14578931

yes

characteristic

132

C₂²

#19097736

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

628

—

136

C₄

#19097687

yes

—

137

C₄

#19097689

yes

—

138

C₄

#19097751

yes

—

139

C₄

#19097758

yes

characteristic

140

C₄

#19097769

yes

characteristic

141

C₂

—

142

C₂

#23757178

yes

characteristic

143

C₂

C₂×#19097687

yes

characteristic

144

C₂

C₂×#19097689

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14578931
commutator, prime radical	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#423×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#423	D₁₅₇
hypocenter	C₁₅₇	#1997
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#14578931

commutator, prime radical

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#423×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #14578938	Q₈	semidirect
72. C₁₅₇	#1997	semidirect

H ≅ G/N

type

24. #14578938

Q₈

semidirect

72. C₁₅₇

#1997

semidirect

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

subgroup	index
C₂×#19097771	2
C₄×#14578938	2
#23757041	2
#423×C₁₅₇	2
#1997	157

subgroup

index

#423×C₁₅₇

157

Subgroups — #34101124

Conjugacy classes of subgroups

Complements

Maximal subgroups