Group #34100185 — subgroups

Subgroups — #34100185

Order 20096.

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

normal subgroups only

This group has 252 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	C₂×#19097706	C₂	1	2	yes	maximal, complemented
3	#23757255	C₂	1	2	yes	maximal
4	#23757031	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578954	C₂²	1	1	yes	characteristic
7	#19097779	—	2	4	no	—
8	C₂×#14578938	C₂²	1	2	yes	—
9	C₂×#14578937	C₄	1	2	yes	—
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₅₇	C₄	1	2	yes	—
12	#19097706	—	2	4	no	—
13	#19097689	—	2	2	no	—
14	#19097687	—	2	2	no	—
15	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
17	#14578954	—	2	2	no	—
18	#14578954	—	2	2	no	—
19	C₂×Dic₃₁₄	D₄	1	1	yes	characteristic
20	C₂×D₆₂₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₅₇	Q₈	1	2	yes	—
22	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
23	#14578938	—	2	2	no	—
24	#14578938	—	2	2	no	—
25	#14578937	—	2	4	no	—
26	C₂×#11929434	—	2	2	no	—
27	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
28	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
29	C₂×#11929434	D₄	1	4	yes	—
30	C₈×D₁₅₇	—	2	4	no	—
31	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₅₇	—	2	2	no	—
33	M₁₆×C₁₅₇	—	2	2	no	—
34	M₁₆×C₁₅₇	—	2	2	no	—
35	C₂×C₈×C₁₅₇	—	2	2	no	—
36	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
37	C₂²×D₁₅₇	C₄⋊C₄	1	2	yes	—
38	Dic₃₁₄	—	2	2	no	—
39	D₆₂₈	—	2	2	no	—
40	C₂²⋊D₁₅₇	—	2	4	no	—
41	C₂×Dic₁₅₇	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₅₇	—	2	2	no	—
43	C₄×D₁₅₇	—	2	2	no	—
44	#11929434	—	4	2	no	—
45	#11929434	C₂²⋊C₄	1	2	yes	—
46	#11929434	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	#41	1	2	yes	—
52	C₈×C₁₅₇	—	2	2	no	—
53	C₈×C₁₅₇	—	4	2	no	—
54	C₂²×C₁₅₇	—	4	1	no	—
55	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₅₇	—	2	2	no	—
58	C₂×D₁₅₇	—	4	2	no	—
59	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₅₇	#117	1	1	yes	characteristic
61	C₄×C₁₅₇	#121	1	2	yes	—
62	Dic₁₅₇	—	2	2	no	—
63	C₂×C₁₅₇	—	2	1	no	—
64	C₂×C₁₅₇	—	2	2	no	—
65	C₂×C₁₅₇	#380	1	1	yes	characteristic
66	D₁₅₇	—	4	2	no	—
67	C₁₅₇	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	157	1	no	maximal, Sylow
69	C₂×#131	—	157	1	no	—
70	C₂×#108	—	157	2	no	—
71	#413	—	157	2	no	—
72	#338	—	157	1	no	—
73	#338	D₁₅₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	157	1	no	—
75	#131	—	314	4	no	—
76	C₂×M₁₆	—	157	2	no	—
77	C₂×M₁₆	—	157	2	no	—
78	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
80	C₂²×C₈	—	157	2	no	—
81	C₂²×C₈	—	157	1	no	—
82	#108	—	314	4	no	—
83	#101	—	314	2	no	—
84	#100	—	314	2	no	—
85	#41	—	314	2	no	—
86	#41	—	314	2	no	—
87	C₂×Q₈	—	157	1	no	—
88	C₂×D₄	—	157	2	no	—
89	C₂×D₄	—	157	1	no	—
90	C₂²×C₄	—	157	2	no	—
91	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	314	4	no	—
96	M₁₆	—	314	2	no	—
97	M₁₆	—	314	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	314	4	no	—
100	C₂×C₈	—	314	2	no	—
101	C₂×C₈	—	157	1	no	—
102	C₂×C₈	—	157	1	no	—
103	C₂×C₈	—	157	4	no	—
104	C₂×C₈	C₄×D₁₅₇	1	2	yes	—
105	C₂³	—	157	2	no	—
106	C₂³	Q₈×D₁₅₇	1	1	yes	characteristic
107	Q₈	—	314	2	no	—
108	D₄	—	314	4	no	—
109	D₄	—	314	2	no	—
110	C₂×C₄	—	314	2	no	—
111	C₂×C₄	—	314	2	no	—
112	C₂×C₄	—	157	2	no	—
113	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
115	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
116	C₂×C₄	#14578954	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	628	2	no	—
120	C₈	—	157	2	no	—
121	C₈	—	157	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	314	2	no	—
124	C₂²	—	628	2	no	—
125	C₂²	C₄⋊C₄×D₁₅₇	1	1	yes	characteristic
126	C₂²	#19097749	1	2	yes	—
127	C₄	—	314	2	no	—
128	C₄	C₂²⋊C₄×D₁₅₇	1	1	yes	characteristic
129	C₄	#19097728	1	1	yes	characteristic
130	C₄	#19097744	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	628	2	no	—
134	C₂	#23757132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097779

C₂

yes

maximal, characteristic

C₂×#19097706

C₂

yes

maximal, complemented

#23757255

C₂

yes

maximal

#23757031

C₂

yes

maximal, characteristic, complemented

#338×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578954

C₂²

yes

characteristic

#19097779

—

C₂×#14578938

C₂²

yes

—

C₂×#14578937

C₄

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₈×D₁₅₇

C₄

yes

—

#19097706

—

#19097689

—

#19097687

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

#14578954

—

#14578954

—

C₂×Dic₃₁₄

D₄

yes

characteristic

C₂×D₆₂₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₅₇

Q₈

yes

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11929434

C₂×C₄

yes

characteristic

C₂×#11929434

D₄

yes

—

C₈×D₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂²×D₁₅₇

C₄⋊C₄

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

C₄⋊C₄

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

#11929434

C₂²⋊C₄

yes

—

#11929434

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂×Q₈

yes

characteristic

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

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#380

yes

characteristic

D₁₅₇

—

C₁₅₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

157

maximal, Sylow

C₂×#131

—

157

—

C₂×#108

—

157

—

#413

—

157

—

#338

—

157

—

#338

D₁₅₇

yes

characteristic, complemented

C₂×#41

—

157

—

#131

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

#108

—

314

—

#101

—

314

—

#100

—

314

—

#41

—

314

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

314

—

100

C₂×C₈

—

314

—

101

C₂×C₈

—

157

—

102

C₂×C₈

—

157

—

103

C₂×C₈

—

157

—

104

C₂×C₈

C₄×D₁₅₇

yes

—

105

C₂³

—

157

—

106

C₂³

Q₈×D₁₅₇

yes

characteristic

107

Q₈

—

314

—

108

D₄

—

314

—

109

D₄

—

314

—

110

C₂×C₄

—

314

—

111

C₂×C₄

—

314

—

112

C₂×C₄

—

157

—

113

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

114

C₂×C₄

D₄×D₁₅₇

yes

—

115

C₂×C₄

D₄×D₁₅₇

yes

characteristic

116

C₂×C₄

#14578954

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

628

—

120

C₈

—

157

—

121

C₈

—

157

—

122

C₂²

—

123

C₂²

—

314

—

124

C₂²

—

628

—

125

C₂²

C₄⋊C₄×D₁₅₇

yes

characteristic

126

C₂²

#19097749

yes

—

127

C₄

—

314

—

128

C₄

C₂²⋊C₄×D₁₅₇

yes

characteristic

129

C₄

#19097728

yes

characteristic

130

C₄

#19097744

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

628

—

134

C₂

#23757132

yes

characteristic

	subgroup	G/H
center	C₄	#19097728
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#338×C₁₅₇	C₂
socle	C₂×C₁₅₇	#380
prime radical	C₂×#14578954	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#338	D₁₅₇
hypocenter	C₁₅₇	#1813
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#19097728

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#338×C₁₅₇

C₂

C₂×C₁₅₇

C₂²

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#19097706	C₂	semidirect
4. #23757031	C₂	semidirect
5. #338×C₁₅₇	C₂	semidirect
67. C₁₅₇	#1813	semidirect
73. #338	D₁₅₇	semidirect

H ≅ G/N

type

2. C₂×#19097706

C₂

semidirect

4. #23757031

C₂

semidirect

5. #338×C₁₅₇

C₂

semidirect

67. C₁₅₇

#1813

semidirect

73. #338

D₁₅₇

semidirect

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

subgroup	index
C₂×#19097779	2
C₂×#19097706	2
#23757255	2
#23757031	2
#338×C₁₅₇	2
#1813	157

subgroup

index

#338×C₁₅₇

157

Subgroups — #34100185

Conjugacy classes of subgroups

Complements

Maximal subgroups