Group #34100181 — subgroups

Subgroups — #34100181

Order 20096.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#19097779	C₂	1	1	yes	maximal, characteristic
2	#23757309	C₂	1	2	yes	maximal, complemented
3	#23757246	C₂	1	2	yes	maximal
4	#23757030	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578954	C₂²	1	1	yes	characteristic
7	#19097779	—	2	2	no	—
8	#19097779	—	2	2	no	—
9	C₂×#14578937	C₄	1	2	yes	complemented
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₅₇	C₄	1	2	yes	complemented
12	#19097730	C₂²	1	2	yes	—
13	#19097704	—	2	2	no	—
14	#19097702	—	2	2	no	—
15	#19097685	—	2	4	no	—
16	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
17	#117×C₁₅₇	C₂²	1	2	yes	—
18	#14578954	—	2	2	no	—
19	#14578954	—	2	2	no	—
20	C₂×Dic₃₁₄	Q₈	1	1	yes	characteristic
21	C₂×D₆₂₈	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₅₇	D₄	1	2	yes	—
23	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
24	#14578937	—	2	4	no	—
25	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
26	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
27	C₂×#11929434	D₄	1	2	yes	—
28	C₂×#11929434	D₄	1	2	yes	—
29	C₈×D₁₅₇	—	2	2	no	—
30	C₈×D₁₅₇	—	2	2	no	—
31	C₄⋊Dic₁₅₇	—	2	2	no	—
32	C₂²⋊Dic₁₅₇	—	2	2	no	—
33	C₄×Dic₁₅₇	—	2	2	no	—
34	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₅₇	—	2	2	no	—
36	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₁₅₇	—	2	2	no	—
38	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
39	C₄²×C₁₅₇	—	2	2	no	—
40	C₂²×D₁₅₇	C₄⋊C₄	1	2	yes	—
41	Dic₃₁₄	—	2	2	no	—
42	D₆₂₈	—	2	2	no	—
43	C₂²⋊D₁₅₇	—	2	4	no	—
44	C₂×Dic₁₅₇	—	4	2	no	—
45	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
46	C₄×D₁₅₇	—	2	2	no	—
47	C₄×D₁₅₇	—	2	2	no	—
48	#11929434	C₂²⋊C₄	1	1	yes	characteristic
49	#11929434	C₂²⋊C₄	1	1	yes	characteristic
50	#11929434	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₅₇	—	4	2	no	—
53	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₅₇	#41	1	2	yes	—
57	C₈×C₁₅₇	—	2	2	no	—
58	C₂²×C₁₅₇	—	4	1	no	—
59	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₅₇	#121	1	2	yes	—
61	C₂×D₁₅₇	—	2	2	no	—
62	C₂×D₁₅₇	—	4	2	no	—
63	C₄×C₁₅₇	—	8	2	no	—
64	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₁₅₇	#117	1	1	yes	characteristic
66	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₅₇	—	2	2	no	—
68	Dic₁₅₇	—	8	2	no	—
69	C₂×C₁₅₇	—	2	2	no	—
70	C₂×C₁₅₇	—	2	1	no	—
71	C₂×C₁₅₇	#380	1	1	yes	characteristic
72	D₁₅₇	—	4	2	no	—
73	C₁₅₇	#1812	1	1	yes	characteristic, complemented, Sylow
74	#1812	—	157	1	no	maximal, Sylow
75	C₂×#131	—	157	1	no	—
76	#427	—	157	2	no	—
77	#410	—	157	2	no	—
78	H(3,ℤ/4)	—	157	1	no	—
79	H(3,ℤ/4)	D₁₅₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	157	1	no	—
81	#131	—	314	2	no	—
82	#131	—	314	2	no	—
83	C₂×M₁₆	—	157	2	no	—
84	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
85	C₂²×C₈	—	157	2	no	—
86	C₂²×C₈	—	157	1	no	—
87	#117	—	157	2	no	—
88	#117	C₂×D₁₅₇	1	2	yes	—
89	#107	—	314	2	no	—
90	#106	—	314	2	no	—
91	#99	—	314	4	no	—
92	#41	—	314	2	no	—
93	#41	—	314	2	no	—
94	C₂×Q₈	—	157	1	no	—
95	C₂×D₄	—	157	2	no	—
96	C₂×D₄	—	157	1	no	—
97	C₂²×C₄	—	157	2	no	—
98	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	314	4	no	—
101	C₂×C₈	—	314	2	no	—
102	C₂×C₈	—	314	2	no	—
103	C₂×C₈	—	157	1	no	—
104	C₂×C₈	—	157	1	no	—
105	C₂×C₈	—	157	2	no	—
106	C₂×C₈	—	157	2	no	—
107	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	314	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	314	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	314	2	no	—
114	C₂³	—	157	2	no	—
115	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
116	Q₈	—	314	2	no	—
117	D₄	—	314	4	no	—
118	D₄	—	314	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	314	2	no	—
121	C₂×C₄	—	314	2	no	—
122	C₂×C₄	—	628	2	no	—
123	C₂×C₄	—	157	2	no	—
124	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
126	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
127	C₂×C₄	#14578954	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	157	1	no	—
130	C₈	—	157	1	no	—
131	C₈	—	157	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	314	2	no	—
134	C₂²	—	628	2	no	—
135	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
136	C₂²	#19097744	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	314	2	no	—
139	C₄	—	1256	2	no	—
140	C₄	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
141	C₄	#19097728	1	1	yes	characteristic
142	C₄	#19097749	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	628	2	no	—
146	C₂	#23757132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097779

C₂

yes

maximal, characteristic

#23757309

C₂

yes

maximal, complemented

#23757246

C₂

yes

maximal

#23757030

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578954

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₈×D₁₅₇

C₄

yes

complemented

#19097730

C₂²

yes

—

#19097704

—

#19097702

—

#19097685

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

—

#14578954

—

#14578954

—

C₂×Dic₃₁₄

Q₈

yes

characteristic

C₂×D₆₂₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₅₇

D₄

yes

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

—

#14578937

—

C₂×#11929434

C₂×C₄

yes

characteristic

C₂×#11929434

C₂×C₄

yes

characteristic

C₂×#11929434

D₄

yes

—

C₂×#11929434

D₄

yes

—

C₈×D₁₅₇

—

C₈×D₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₂²×D₁₅₇

C₄⋊C₄

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

C₂²⋊C₄

yes

characteristic

#11929434

C₂²⋊C₄

yes

characteristic

#11929434

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#121

yes

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

#117

yes

characteristic

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#380

yes

characteristic

D₁₅₇

—

C₁₅₇

#1812

yes

characteristic, complemented, Sylow

#1812

—

157

maximal, Sylow

C₂×#131

—

157

—

#427

—

157

—

#410

—

157

—

H(3,ℤ/4)

—

157

—

H(3,ℤ/4)

D₁₅₇

yes

characteristic, complemented

C₂×#41

—

157

—

#131

—

314

—

#131

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

#117

—

157

—

#117

C₂×D₁₅₇

yes

—

#107

—

314

—

#106

—

314

—

#99

—

314

—

#41

—

314

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

100

M₁₆

—

314

—

101

C₂×C₈

—

314

—

102

C₂×C₈

—

314

—

103

C₂×C₈

—

157

—

104

C₂×C₈

—

157

—

105

C₂×C₈

—

157

—

106

C₂×C₈

—

157

—

107

C₂×C₈

C₄×D₁₅₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

314

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

314

—

112

C₄²

—

113

C₄²

—

314

—

114

C₂³

—

157

—

115

C₂³

D₄×D₁₅₇

yes

characteristic

116

Q₈

—

314

—

117

D₄

—

314

—

118

D₄

—

314

—

119

C₂×C₄

—

120

C₂×C₄

—

314

—

121

C₂×C₄

—

314

—

122

C₂×C₄

—

628

—

123

C₂×C₄

—

157

—

124

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

125

C₂×C₄

D₄×D₁₅₇

yes

—

126

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

127

C₂×C₄

#14578954

yes

—

128

C₈

—

129

C₈

—

157

—

130

C₈

—

157

—

131

C₈

—

157

—

132

C₂²

—

133

C₂²

—

314

—

134

C₂²

—

628

—

135

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

136

C₂²

#19097744

yes

—

137

C₄

—

138

C₄

—

314

—

139

C₄

—

1256

—

140

C₄

C₂²⋊C₄×D₁₅₇

yes

characteristic

141

C₄

#19097728

yes

characteristic

142

C₄

#19097749

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

628

—

146

C₂

#23757132

yes

characteristic

	subgroup	G/H
center	C₄	#19097728
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	H(3,ℤ/4)×C₁₅₇	C₂
socle	C₂×C₁₅₇	#380
prime radical	C₂×#14578954	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₅₇
hypocenter	C₁₅₇	#1812
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19097728

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

H(3,ℤ/4)×C₁₅₇

C₂

C₂×C₁₅₇

C₂²

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #23757309	C₂	semidirect
4. #23757030	C₂	semidirect
5. H(3,ℤ/4)×C₁₅₇	C₂	semidirect
9. C₂×#14578937	C₄	semidirect
11. C₂×C₈×D₁₅₇	C₄	semidirect
73. C₁₅₇	#1812	semidirect
79. H(3,ℤ/4)	D₁₅₇	semidirect

H ≅ G/N

type

2. #23757309

C₂

semidirect

4. #23757030

C₂

semidirect

5. H(3,ℤ/4)×C₁₅₇

C₂

semidirect

9. C₂×#14578937

C₄

semidirect

11. C₂×C₈×D₁₅₇

C₄

semidirect

73. C₁₅₇

#1812

semidirect

79. H(3,ℤ/4)

D₁₅₇

semidirect

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

subgroup	index
C₂×#19097779	2
#23757309	2
#23757246	2
#23757030	2
H(3,ℤ/4)×C₁₅₇	2
#1812	157

subgroup

index

H(3,ℤ/4)×C₁₅₇

157

Subgroups — #34100181

Conjugacy classes of subgroups

Complements

Maximal subgroups