Group #34101087 — subgroups

Subgroups — #34101087

Order 20096.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#23757036	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₄	—	2	1	no	—
6	#19097771	—	2	2	no	—
7	#19097771	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₅₇	—	2	1	no	—
12	#19097678	—	2	1	no	—
13	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₅₇	—	2	2	no	—
16	C₂×Dic₃₁₄	—	2	4	no	—
17	C₂×Dic₃₁₄	—	4	2	no	—
18	C₂×Dic₃₁₄	—	4	2	no	—
19	C₂²×Dic₁₅₇	—	2	2	no	—
20	C₂²×Dic₁₅₇	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	4	no	—
22	C₄⋊Dic₁₅₇	—	2	2	no	—
23	C₄⋊Dic₁₅₇	—	2	2	no	—
24	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₅₇	Q₈	1	4	yes	complemented
26	C₄×Dic₁₅₇	—	4	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	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₂×C₈×C₁₅₇	—	2	4	no	—
32	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₅₇	D₄	1	2	yes	—
34	Dic₃₁₄	—	4	8	no	—
35	C₂×Dic₁₅₇	—	2	4	no	—
36	C₂×Dic₁₅₇	—	2	4	no	—
37	C₂×Dic₁₅₇	—	4	2	no	—
38	C₂×Dic₁₅₇	—	4	2	no	—
39	C₂×Dic₁₅₇	—	4	2	no	—
40	C₂×Dic₁₅₇	—	4	2	no	—
41	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
46	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
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×Q₈	1	2	yes	—
50	C₂×C₄×C₁₅₇	#41	1	2	yes	—
51	C₈×C₁₅₇	—	4	2	no	—
52	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₅₇	#137	1	2	yes	—
57	C₄×C₁₅₇	—	2	2	no	—
58	C₄×C₁₅₇	#102	1	4	yes	—
59	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₅₇	#124	1	1	yes	characteristic
61	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₅₇	—	4	4	no	—
63	Dic₁₅₇	—	4	4	no	—
64	C₂×C₁₅₇	#391	1	1	yes	characteristic
65	C₂×C₁₅₇	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₅₇	#419	1	1	yes	characteristic
67	C₂×C₁₅₇	#474	1	2	yes	—
68	C₂×C₁₅₇	#475	1	2	yes	—
69	C₁₅₇	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	157	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	157	1	no	—
72	C₂×C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	157	1	no	—
74	#340	—	157	4	no	—
75	C₂²×Q₈	—	314	1	no	—
76	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
77	C₄⋊Q₈	—	314	2	no	—
78	C₄⋊Q₈	—	157	4	no	—
79	C₂×C₄⋊C₄	—	314	1	no	—
80	C₂×C₄⋊C₄	—	157	2	no	—
81	C₂×C₄⋊C₄	—	157	2	no	—
82	C₂×C₄²	—	314	1	no	—
83	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	314	1	no	—
86	C₂×Q₈	—	314	4	no	—
87	C₂×Q₈	—	628	2	no	—
88	C₂×Q₈	—	628	2	no	—
89	C₂²×C₄	—	314	2	no	—
90	C₂²×C₄	—	314	2	no	—
91	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
92	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	314	2	no	—
97	C₄⋊C₄	—	314	2	no	—
98	C₄⋊C₄	—	314	4	no	—
99	C₄⋊C₄	—	157	4	no	—
100	C₄⋊C₄	—	157	4	no	—
101	C₄²	—	628	2	no	—
102	C₄²	C₄×D₁₅₇	1	2	yes	—
103	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
104	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
105	Q₈	—	628	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	314	4	no	—
108	C₂×C₄	—	314	4	no	—
109	C₂×C₄	—	628	2	no	—
110	C₂×C₄	—	628	2	no	—
111	C₂×C₄	—	628	2	no	—
112	C₂×C₄	—	628	2	no	—
113	C₂×C₄	#14578931	1	2	yes	—
114	C₂×C₄	#14578931	1	2	yes	—
115	C₂×C₄	D₁₂₅₆	1	2	yes	—
116	C₂×C₄	#14578943	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
120	C₂×C₄	#14578953	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×#14578931	1	1	yes	characteristic
123	C₂²	#19097751	1	2	yes	—
124	C₂²	C₂×D₁₂₅₆	1	1	yes	characteristic
125	C₂²	C₂×#14578943	1	1	yes	characteristic
126	C₂²	#19097804	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	628	4	no	—
129	C₄	—	628	4	no	—
130	C₄	#19097693	1	4	yes	—
131	C₄	#19097736	1	2	yes	—
132	C₄	#19097758	1	1	yes	characteristic
133	C₄	#19097769	1	1	yes	characteristic
134	C₂	#23757178	1	1	yes	characteristic
135	C₂	C₂×#19097693	1	1	yes	characteristic
136	C₂	#23757283	1	1	yes	characteristic
137	C₂	#23757554	1	2	yes	—
138	C₂	#23757559	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic

#23757036

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

—

#19097771

—

#19097771

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

—

#19097678

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

C₄⋊Dic₁₅₇

Q₈

yes

complemented

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

D₈

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₂²×C₁₅₇

C₂×D₈

yes

characteristic

C₂²×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#102

yes

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#475

yes

—

C₁₅₇

#1988

yes

characteristic, complemented, Sylow

#1988

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×C₄⋊C₈

D₁₅₇

yes

characteristic

C₄×C₄⋊C₄

—

157

—

#340

—

157

—

C₂²×Q₈

—

314

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

—

314

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

#95

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

628

—

C₂×Q₈

—

628

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

characteristic

C₂²×C₄

D₆₂₈

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

100

C₄⋊C₄

—

157

—

101

C₄²

—

628

—

102

C₄²

C₄×D₁₅₇

yes

—

103

C₄²

C₂²⋊D₁₅₇

yes

—

104

C₂³

C₂×D₆₂₈

yes

characteristic

105

Q₈

—

628

—

106

C₂×C₄

—

107

C₂×C₄

—

314

—

108

C₂×C₄

—

314

—

109

C₂×C₄

—

628

—

110

C₂×C₄

—

628

—

111

C₂×C₄

—

628

—

112

C₂×C₄

—

628

—

113

C₂×C₄

#14578931

yes

—

114

C₂×C₄

#14578931

yes

—

115

C₂×C₄

D₁₂₅₆

yes

—

116

C₂×C₄

#14578943

yes

—

117

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

119

C₂×C₄

Q₈×D₁₅₇

yes

—

120

C₂×C₄

#14578953

yes

—

121

C₈

—

122

C₂²

C₂×#14578931

yes

characteristic

123

C₂²

#19097751

yes

—

124

C₂²

C₂×D₁₂₅₆

yes

characteristic

125

C₂²

C₂×#14578943

yes

characteristic

126

C₂²

#19097804

yes

—

127

C₄

—

128

C₄

—

628

—

129

C₄

—

628

—

130

C₄

#19097693

yes

—

131

C₄

#19097736

yes

—

132

C₄

#19097758

yes

characteristic

133

C₄

#19097769

yes

characteristic

134

C₂

#23757178

yes

characteristic

135

C₂

C₂×#19097693

yes

characteristic

136

C₂

#23757283

yes

characteristic

137

C₂

#23757554

yes

—

138

C₂

#23757559

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₂₈
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄⋊C₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#1988
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₂₈

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×C₄⋊C₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₇

hypocenter

C₁₅₇

#1988

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #19097771	C₄	semidirect
25. C₄⋊Dic₁₅₇	Q₈	semidirect
69. C₁₅₇	#1988	semidirect

H ≅ G/N

type

7. #19097771

C₄

semidirect

25. C₄⋊Dic₁₅₇

Q₈

semidirect

69. C₁₅₇

#1988

semidirect

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

subgroup	index
C₂×#19097771	2
C₄×C₄⋊Dic₁₅₇	2
#23757036	2
C₂×C₄⋊C₈×C₁₅₇	2
#1988	157

subgroup

index

C₂×#19097771

C₄×C₄⋊Dic₁₅₇

#23757036

C₂×C₄⋊C₈×C₁₅₇

#1988

157

Subgroups — #34101087

Conjugacy classes of subgroups

Complements

Maximal subgroups