Group #34116374 — subgroups

Subgroups — #34116374

Order 20096.

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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×#19097845	C₂	1	2	yes	maximal, complemented
2	#23758170	C₂	1	4	yes	maximal, complemented
3	C₂×#19097779	C₂	1	1	yes	maximal, characteristic
4	#19097845	—	2	2	no	—
5	#19097845	C₂²	1	4	yes	—
6	C₂×#14578957	—	2	2	no	—
7	C₂×#14578954	C₄	1	1	yes	characteristic
8	#19097779	—	2	2	no	—
9	#19097779	—	2	2	no	—
10	C₂×#14578937	C₂²	1	2	yes	—
11	C₂²×#11929434	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₅₇	C₄	1	2	yes	—
13	C₂×M₁₆×C₁₅₇	C₄	1	1	yes	characteristic
14	#14578957	—	2	4	no	—
15	#14578954	—	2	2	no	—
16	#14578954	—	2	2	no	—
17	C₂×Dic₃₁₄	C₈	1	1	yes	characteristic
18	C₂×D₆₂₈	C₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₅₇	C₈	1	2	yes	—
20	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
21	#14578937	—	2	4	no	—
22	C₂×#11929434	C₂×C₄	1	1	yes	characteristic
23	C₂×#11929434	D₄	1	2	yes	—
24	C₂×#11929434	D₄	1	2	yes	—
25	C₂×#11929434	C₂³	1	1	yes	characteristic
26	C₈×D₁₅₇	—	2	2	no	—
27	C₈×D₁₅₇	—	2	2	no	—
28	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	M₁₆×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
31	C₂²×D₁₅₇	C₂×C₈	1	2	yes	—
32	Dic₃₁₄	—	2	2	no	—
33	D₆₂₈	—	2	2	no	—
34	C₂²⋊D₁₅₇	—	2	4	no	—
35	C₂×Dic₁₅₇	C₂×C₈	1	2	yes	—
36	C₄×D₁₅₇	—	2	2	no	—
37	C₄×D₁₅₇	—	2	2	no	—
38	#11929434	C₂²⋊C₄	1	1	yes	characteristic
39	#11929434	C₂²⋊C₄	1	1	yes	characteristic
40	#11929434	C₂×D₄	1	2	yes	—
41	C₂³×C₁₅₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	M₁₆	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₅₇	—	2	2	no	—
47	C₂²×C₁₅₇	—	4	1	no	—
48	C₂²×C₁₅₇	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₁₅₇	—	2	2	no	—
51	C₂×D₁₅₇	—	4	2	no	—
52	C₄×C₁₅₇	C₂²⋊C₈	1	2	yes	—
53	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₁₅₇	C₂×M₁₆	1	1	yes	characteristic
55	Dic₁₅₇	—	2	2	no	—
56	C₂×C₁₅₇	—	2	1	no	—
57	C₂×C₁₅₇	—	2	2	no	—
58	C₂×C₁₅₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	D₁₅₇	—	4	2	no	—
60	C₁₅₇	#2115	1	1	yes	characteristic, complemented, Sylow
61	#2115	—	157	1	no	maximal, Sylow
62	C₂×#131	—	157	1	no	—
63	C₂×M₃₂	—	157	2	no	—
64	#349	—	157	4	no	—
65	C₂×#41	—	157	1	no	—
66	#131	—	314	2	no	—
67	#131	—	314	2	no	—
68	C₂×M₁₆	—	157	2	no	—
69	C₂×M₁₆	#10597845	1	1	yes	characteristic
70	C₂²×C₈	—	157	2	no	—
71	C₂²×C₈	—	157	1	no	—
72	M₃₂	—	314	2	no	—
73	M₃₂	—	157	4	no	—
74	C₂×C₁₆	—	314	2	no	—
75	#41	—	314	2	no	—
76	#41	—	314	2	no	—
77	C₂×Q₈	—	157	1	no	—
78	C₂×D₄	—	157	2	no	—
79	C₂×D₄	—	157	1	no	—
80	C₂²×C₄	—	157	2	no	—
81	C₂²×C₄	C₂×#10597845	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	314	4	no	—
84	C₂×C₈	—	314	2	no	—
85	C₂×C₈	—	314	2	no	—
86	C₂×C₈	—	157	2	no	—
87	C₂×C₈	—	157	2	no	—
88	C₂×C₈	—	157	1	no	—
89	C₂×C₈	—	157	1	no	—
90	C₂×C₈	C₂×#10597845	1	2	yes	—
91	C₁₆	—	314	4	no	—
92	C₂³	—	157	2	no	—
93	C₂³	D₁₅₇⋊C₈	1	1	yes	characteristic
94	Q₈	—	314	2	no	—
95	D₄	—	314	4	no	—
96	D₄	—	314	2	no	—
97	C₂×C₄	—	314	2	no	—
98	C₂×C₄	—	314	2	no	—
99	C₂×C₄	—	157	2	no	—
100	C₂×C₄	C₂²⋊#10597845	1	2	yes	—
101	C₂×C₄	D₁₅₇⋊C₈	1	1	yes	characteristic
102	C₂×C₄	#14578964	1	2	yes	—
103	C₂×C₄	C₂²×#10597845	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	157	2	no	—
106	C₈	—	157	1	no	—
107	C₈	—	157	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	314	2	no	—
110	C₂²	—	628	2	no	—
111	C₂²	#19097828	1	2	yes	—
112	C₂²	C₂×D₁₅₇⋊C₈	1	1	yes	characteristic
113	C₄	—	314	2	no	—
114	C₄	#19097828	1	2	yes	—
115	C₄	C₂×C₂²⋊#10597845	1	1	yes	characteristic
116	C₄	C₂×#14578964	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	628	2	no	—
120	C₂	C₂×#19097828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097845

C₂

yes

maximal, complemented

#23758170

C₂

yes

maximal, complemented

C₂×#19097779

C₂

yes

maximal, characteristic

#19097845

—

#19097845

C₂²

yes

—

C₂×#14578957

—

C₂×#14578954

C₄

yes

—

—

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂×C₈×D₁₅₇

C₄

yes

—

C₂×M₁₆×C₁₅₇

C₄

yes

—

—

—

C₂×Dic₃₁₄

C₈

yes

characteristic

C₂×D₆₂₈

C₈

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₈

yes

—

C₂×C₄×D₁₅₇

C₂×C₄

yes

—

#14578937

—

C₂×#11929434

C₂×C₄

yes

characteristic

C₂×#11929434

D₄

yes

—

C₂×#11929434

D₄

yes

—

C₂×#11929434

C₂³

yes

characteristic

C₈×D₁₅₇

—

C₈×D₁₅₇

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂²×D₁₅₇

C₂×C₈

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

C₂×C₈

yes

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

C₂²⋊C₄

yes

characteristic

#11929434

C₂²⋊C₄

yes

characteristic

#11929434

C₂×D₄

yes

—

C₂³×C₁₅₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂²⋊C₈

yes

—

C₂²×C₁₅₇

C₂²×C₈

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂²⋊C₈

yes

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

C₂×M₁₆

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₂²⋊C₈

yes

characteristic

D₁₅₇

—

C₁₅₇

#2115

yes

characteristic, complemented, Sylow

#2115

—

157

maximal, Sylow

C₂×#131

—

157

—

C₂×M₃₂

—

157

—

#349

—

157

—

C₂×#41

—

157

—

#131

—

314

—

#131

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

#10597845

yes

characteristic

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

M₃₂

—

314

—

M₃₂

—

157

—

C₂×C₁₆

—

314

—

#41

—

314

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×#10597845

yes

characteristic

M₁₆

—

M₁₆

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₂×#10597845

yes

—

C₁₆

—

314

—

C₂³

—

157

—

C₂³

D₁₅₇⋊C₈

yes

characteristic

Q₈

—

314

—

D₄

—

314

—

D₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

100

C₂×C₄

C₂²⋊#10597845

yes

—

101

C₂×C₄

D₁₅₇⋊C₈

yes

characteristic

102

C₂×C₄

#14578964

yes

—

103

C₂×C₄

C₂²×#10597845

yes

characteristic

104

C₈

—

105

C₈

—

157

—

106

C₈

—

157

—

107

C₈

—

157

—

108

C₂²

—

109

C₂²

—

314

—

110

C₂²

—

628

—

111

C₂²

#19097828

yes

—

112

C₂²

C₂×D₁₅₇⋊C₈

yes

characteristic

113

C₄

—

314

—

114

C₄

#19097828

yes

—

115

C₄

C₂×C₂²⋊#10597845

yes

characteristic

116

C₄

C₂×#14578964

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

628

—

120

C₂

C₂×#19097828

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#10597845
commutator	C₂²×C₁₅₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10597845
Fitting	C₂×M₁₆×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₂×C₂²⋊C₈
prime radical	C₂×#14578954	C₄
cosocle	C₂×#11929434	C₂³
hypercenter	C₂×M₁₆	#10597845
hypocenter	C₁₅₇	#2115
LCS	G ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₂²⋊#10597845

commutator

C₂²×C₁₅₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10597845

Fitting

C₂×M₁₆×C₁₅₇

C₄

socle

C₂×C₁₅₇

C₂×C₂²⋊C₈

C₄

C₂³

C₂×M₁₆

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#19097845	C₂	semidirect
2. #23758170	C₂	semidirect
60. C₁₅₇	#2115	semidirect

H ≅ G/N

type

1. C₂×#19097845

C₂

semidirect

2. #23758170

C₂

semidirect

60. C₁₅₇

#2115

semidirect

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

subgroup	index
C₂×#19097845	2
#23758170	2
C₂×#19097779	2
#2115	157

subgroup

index

157

Subgroups — #34116374

Conjugacy classes of subgroups

Complements

Maximal subgroups