Group #34101059 — subgroups

Subgroups — #34101059

Order 20096.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
3	C₂×#19097694	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
6	C₂²×#11929434	C₂²	1	2	yes	—
7	#19097771	—	2	2	no	—
8	#19097771	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
11	C₄⋊#11929434	—	2	2	no	—
12	#19097694	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	—	2	1	no	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	#95×C₁₅₇	—	2	1	no	—
17	C₂×Dic₃₁₄	—	2	2	no	—
18	C₂×Dic₃₁₄	—	2	4	no	—
19	C₂×Dic₃₁₄	—	2	4	no	—
20	C₂×Dic₃₁₄	C₂×C₄	1	4	yes	—
21	C₂²×Dic₁₅₇	—	2	2	no	—
22	C₂×#11929434	—	2	4	no	—
23	C₂×#11929434	—	2	2	no	—
24	C₄⋊Dic₁₅₇	—	2	4	no	—
25	C₄⋊Dic₁₅₇	—	2	4	no	—
26	C₂²×C₄×C₁₅₇	—	2	2	no	—
27	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₅₇	—	2	2	no	—
31	C₄⋊C₄×C₁₅₇	D₄	1	4	yes	—
32	C₄²×C₁₅₇	—	4	2	no	—
33	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
34	C₄²×C₁₅₇	D₄	1	2	yes	—
35	Dic₃₁₄	—	2	8	no	—
36	Dic₃₁₄	—	4	2	no	—
37	Dic₃₁₄	—	4	2	no	—
38	C₂×Dic₁₅₇	—	2	4	no	—
39	C₂×Dic₁₅₇	—	4	4	no	—
40	#11929434	—	4	2	no	—
41	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	—	2	4	no	—
44	C₂×C₄×C₁₅₇	—	4	2	no	—
45	C₂×C₄×C₁₅₇	—	4	2	no	—
46	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
49	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	#41	1	2	yes	—
54	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₅₇	#121	1	2	yes	—
56	C₂²×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₅₇	#137	1	2	yes	—
59	C₄×C₁₅₇	—	2	2	no	—
60	C₄×C₁₅₇	—	4	4	no	—
61	C₄×C₁₅₇	#103	1	4	yes	complemented
62	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
63	C₄×C₁₅₇	#124	1	1	yes	characteristic
64	C₄×C₁₅₇	#127	1	1	yes	characteristic
65	Dic₁₅₇	—	4	4	no	—
66	C₂×C₁₅₇	#390	1	1	yes	characteristic
67	C₂×C₁₅₇	C₂×#103	1	1	yes	characteristic
68	C₂×C₁₅₇	#419	1	1	yes	characteristic
69	C₂×C₁₅₇	#461	1	2	yes	—
70	C₂×C₁₅₇	#462	1	2	yes	—
71	C₁₅₇	#1981	1	1	yes	characteristic, complemented, Sylow
72	#1981	—	157	1	no	maximal, Sylow
73	C₂×C₄⋊Q₈	—	157	1	no	—
74	C₂×C₄⋊C₈	—	157	1	no	—
75	C₂×#103	—	157	4	no	—
76	C₄×C₄⋊C₄	D₁₅₇	1	1	yes	characteristic
77	C₂²×Q₈	—	157	2	no	—
78	C₂²×C₈	—	157	2	no	—
79	C₄⋊Q₈	—	314	2	no	—
80	C₄⋊Q₈	—	157	4	no	—
81	C₂×C₄⋊C₄	—	314	1	no	—
82	C₂×C₄⋊C₄	—	314	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
86	C₄⋊C₈	—	314	2	no	—
87	#103	—	314	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	314	2	no	—
90	C₂×Q₈	—	314	4	no	—
91	C₂×Q₈	—	314	4	no	—
92	C₂×Q₈	—	157	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	314	2	no	—
95	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂×C₈	—	314	4	no	—
99	C₂×C₈	—	314	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	314	4	no	—
102	C₄⋊C₄	—	314	4	no	—
103	C₄⋊C₄	D₆₂₈	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₁₅₇	1	2	yes	—
106	C₄²	D₆₂₈	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
108	Q₈	—	314	8	no	—
109	Q₈	—	628	2	no	—
110	Q₈	—	628	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	314	4	no	—
116	C₂×C₄	—	628	4	no	—
117	C₂×C₄	#14578931	1	2	yes	—
118	C₂×C₄	#14578931	1	2	yes	—
119	C₂×C₄	#14578942	1	2	yes	—
120	C₂×C₄	#14578944	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
123	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
124	C₂×C₄	#14578954	1	2	yes	—
125	C₈	—	628	2	no	—
126	C₂²	C₂×#14578931	1	1	yes	characteristic
127	C₂²	#19097745	1	2	yes	—
128	C₂²	C₂×#14578942	1	1	yes	characteristic
129	C₂²	C₂×#14578944	1	1	yes	characteristic
130	C₂²	#19097801	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	628	4	no	—
134	C₄	#19097694	1	4	yes	complemented
135	C₄	C₄×D₆₂₈	1	2	yes	—
136	C₄	#19097759	1	1	yes	characteristic
137	C₄	#19097768	1	1	yes	characteristic
138	C₂	#23757175	1	1	yes	characteristic
139	C₂	C₂×#19097694	1	1	yes	characteristic
140	C₂	#23757281	1	1	yes	characteristic
141	C₂	#23757478	1	2	yes	—
142	C₂	#23757483	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

C₂×#19097694

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

C₂²×#11929434

C₂²

yes

—

#19097771

—

#19097771

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

—

C₄⋊#11929434

—

#19097694

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#95×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂×C₄

yes

—

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

#121

yes

—

C₂²×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#103

yes

complemented

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#462

yes

—

C₁₅₇

#1981

yes

characteristic, complemented, Sylow

#1981

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×C₄⋊C₈

—

157

—

C₂×#103

—

157

—

C₄×C₄⋊C₄

D₁₅₇

yes

characteristic

C₂²×Q₈

—

157

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

#103

—

314

—

#95

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

314

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

314

—

102

C₄⋊C₄

—

314

—

103

C₄⋊C₄

D₆₂₈

yes

—

104

C₄²

—

105

C₄²

C₄×D₁₅₇

yes

—

106

C₄²

D₆₂₈

yes

—

107

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

108

Q₈

—

314

—

109

Q₈

—

628

—

110

Q₈

—

628

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

314

—

116

C₂×C₄

—

628

—

117

C₂×C₄

#14578931

yes

—

118

C₂×C₄

#14578931

yes

—

119

C₂×C₄

#14578942

yes

—

120

C₂×C₄

#14578944

yes

—

121

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

122

C₂×C₄

C₂×D₆₂₈

yes

—

123

C₂×C₄

C₂×D₆₂₈

yes

characteristic

124

C₂×C₄

#14578954

yes

—

125

C₈

—

628

—

126

C₂²

C₂×#14578931

yes

characteristic

127

C₂²

#19097745

yes

—

128

C₂²

C₂×#14578942

yes

characteristic

129

C₂²

C₂×#14578944

yes

characteristic

130

C₂²

#19097801

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

628

—

134

C₄

#19097694

yes

complemented

135

C₄

C₄×D₆₂₈

yes

—

136

C₄

#19097759

yes

characteristic

137

C₄

#19097768

yes

characteristic

138

C₂

#23757175

yes

characteristic

139

C₂

C₂×#19097694

yes

characteristic

140

C₂

#23757281

yes

characteristic

141

C₂

#23757478

yes

—

142

C₂

#23757483

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₄×C₄⋊C₄×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₅₇
hypocenter	C₁₅₇	#1981
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₄×C₄⋊C₄×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₅₇

hypocenter

C₁₅₇

#1981

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #19097771	C₄	semidirect
61. C₄×C₁₅₇	#103	semidirect
71. C₁₅₇	#1981	semidirect
134. C₄	#19097694	semidirect

H ≅ G/N

type

8. #19097771

C₄

semidirect

61. C₄×C₁₅₇

#103

semidirect

71. C₁₅₇

#1981

semidirect

134. C₄

#19097694

semidirect

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

subgroup	index
C₂×#19097771	2
C₂×C₄⋊#11929434	2
C₂×#19097694	2
C₄×C₄⋊C₄×C₁₅₇	2
#1981	157

subgroup

index

C₄×C₄⋊C₄×C₁₅₇

157

Subgroups — #34101059

Conjugacy classes of subgroups

Complements

Maximal subgroups