Group #34101404 — subgroups

Subgroups — #34101404

Order 20096.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₆₂₈	C₂	1	1	yes	maximal, characteristic
2	#23757281	C₂	1	2	yes	maximal
3	C₂×#19097689	C₂	1	2	yes	maximal
4	#23757038	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
7	C₂×Dic₆₂₈	—	2	2	no	—
8	C₂×Dic₆₂₈	—	2	4	no	—
9	C₂×#14578938	C₂²	1	2	yes	—
10	#19097694	—	2	4	no	—
11	#19097689	—	2	4	no	—
12	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	#117×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₅₇	—	2	2	no	—
16	C₂×Dic₃₁₄	—	2	2	no	—
17	C₂×Dic₃₁₄	—	2	2	no	—
18	C₂×Dic₃₁₄	—	2	2	no	—
19	C₂×Dic₃₁₄	—	2	2	no	—
20	C₂×Dic₃₁₄	—	2	2	no	—
21	C₂×Dic₃₁₄	D₄	1	4	yes	—
22	C₂²×Dic₁₅₇	—	2	2	no	—
23	Dic₆₂₈	—	2	8	no	—
24	#14578938	—	2	2	no	—
25	#14578938	—	2	2	no	—
26	C₂×#11929434	—	2	2	no	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₅₇	—	2	2	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	—	2	1	no	—
31	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₅₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
34	C₄²×C₁₅₇	—	2	1	no	—
35	Dic₃₁₄	—	2	4	no	—
36	Dic₃₁₄	—	2	4	no	—
37	Dic₃₁₄	—	4	2	no	—
38	Dic₃₁₄	—	4	2	no	—
39	C₂×Dic₁₅₇	—	2	4	no	—
40	C₂×Dic₁₅₇	—	4	2	no	—
41	C₂×Dic₁₅₇	—	4	2	no	—
42	#11929434	—	4	2	no	—
43	C₂³×C₁₅₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	—	2	2	no	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	#41	1	2	yes	—
49	C₈×C₁₅₇	—	2	2	no	—
50	C₈×C₁₅₇	—	4	1	no	—
51	C₂²×C₁₅₇	—	2	2	no	—
52	C₂²×C₁₅₇	#121	1	1	yes	characteristic
53	C₂²×C₁₅₇	#123	1	2	yes	—
54	C₄×C₁₅₇	—	4	2	no	—
55	C₄×C₁₅₇	#120	1	1	yes	characteristic
56	C₄×C₁₅₇	#121	1	2	yes	—
57	C₄×C₁₅₇	#124	1	1	yes	characteristic
58	Dic₁₅₇	—	4	4	no	—
59	C₂×C₁₅₇	—	2	2	no	—
60	C₂×C₁₅₇	#394	1	1	yes	characteristic
61	C₂×C₁₅₇	#473	1	2	yes	—
62	C₁₅₇	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	157	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	157	1	no	—
65	#424	D₁₅₇	1	1	yes	characteristic
66	#419	—	157	2	no	—
67	C₂×#101	—	157	2	no	—
68	#341	—	157	1	no	—
69	C₂²×Q₈	—	157	2	no	—
70	C₂×Q₁₆	—	314	4	no	—
71	C₂×Q₁₆	—	314	2	no	—
72	C₂×M₁₆	—	157	2	no	—
73	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
75	#117	C₂×D₁₅₇	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	314	4	no	—
78	#101	—	314	4	no	—
79	C₂×Q₈	—	314	2	no	—
80	C₂×Q₈	—	314	2	no	—
81	C₂×Q₈	—	314	2	no	—
82	C₂×Q₈	—	314	2	no	—
83	C₂×Q₈	—	314	2	no	—
84	C₂×Q₈	—	157	4	no	—
85	C₂²×C₄	—	314	2	no	—
86	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
87	Q₁₆	—	314	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	314	2	no	—
90	M₁₆	—	314	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	314	2	no	—
94	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
95	C₄⋊C₄	D₆₂₈	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#14578954	1	1	yes	characteristic
99	Q₈	—	314	4	no	—
100	Q₈	—	314	4	no	—
101	Q₈	—	628	2	no	—
102	Q₈	—	628	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	314	4	no	—
105	C₂×C₄	—	628	2	no	—
106	C₂×C₄	—	628	2	no	—
107	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
110	C₂×C₄	#14578955	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	628	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#19097745	1	1	yes	characteristic
116	C₂²	#19097756	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	628	4	no	—
119	C₄	#19097739	1	1	yes	characteristic
120	C₄	#19097746	1	2	yes	—
121	C₄	#19097760	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#23757190	1	1	yes	characteristic
124	C₂	#23757548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₂₈

C₂

yes

maximal, characteristic

#23757281

C₂

yes

maximal

C₂×#19097689

C₂

yes

maximal

#23757038

C₂

yes

maximal, characteristic

#424×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

C₂×Dic₆₂₈

—

C₂×Dic₆₂₈

—

C₂×#14578938

C₂²

yes

—

#19097694

—

#19097689

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

D₄

yes

—

C₂²×Dic₁₅₇

—

Dic₆₂₈

—

#14578938

—

#14578938

—

C₂×#11929434

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

#41

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#121

yes

characteristic

C₂²×C₁₅₇

#123

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

#121

yes

—

C₄×C₁₅₇

#124

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#394

yes

characteristic

C₂×C₁₅₇

#473

yes

—

C₁₅₇

#2047

yes

characteristic, complemented, Sylow

#2047

—

157

maximal, Sylow

C₂²×Q₁₆

—

157

—

#424

D₁₅₇

yes

characteristic

#419

—

157

—

C₂×#101

—

157

—

#341

—

157

—

C₂²×Q₈

—

157

—

C₂×Q₁₆

—

314

—

C₂×Q₁₆

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

#103

—

314

—

#101

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

Q₁₆

—

314

—

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

C₂²⋊D₁₅₇

yes

—

C₄⋊C₄

D₆₂₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#14578954

yes

characteristic

Q₈

—

314

—

100

Q₈

—

314

—

101

Q₈

—

628

—

102

Q₈

—

628

—

103

C₂×C₄

—

104

C₂×C₄

—

314

—

105

C₂×C₄

—

628

—

106

C₂×C₄

—

628

—

107

C₂×C₄

D₄×D₁₅₇

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

109

C₂×C₄

C₂×D₆₂₈

yes

characteristic

110

C₂×C₄

#14578955

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

628

—

114

C₂²

—

115

C₂²

#19097745

yes

characteristic

116

C₂²

#19097756

yes

—

117

C₄

—

118

C₄

—

628

—

119

C₄

#19097739

yes

characteristic

120

C₄

#19097746

yes

—

121

C₄

#19097760

yes

characteristic

122

C₂

—

123

C₂

#23757190

yes

characteristic

124

C₂

#23757548

yes

—

	subgroup	G/H
center	C₂²	#19097745
commutator, cosocle	C₂²×C₄×C₁₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#424×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#121
prime radical	C₂³×C₁₅₇	#41
hypercenter	#424	D₁₅₇
hypocenter	C₁₅₇	#2047
LCS	G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#19097745

commutator, cosocle

C₂²×C₄×C₁₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#424×C₁₅₇

C₂

C₂²×C₁₅₇

C₂³×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
62. C₁₅₇	#2047	semidirect

H ≅ G/N

type

62. C₁₅₇

#2047

semidirect

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

subgroup	index
C₂²×Dic₆₂₈	2
#23757281	2
C₂×#19097689	2
#23757038	2
#424×C₁₅₇	2
#2047	157

subgroup

index

C₂²×Dic₆₂₈

#424×C₁₅₇

157

Subgroups — #34101404

Conjugacy classes of subgroups

Complements

Maximal subgroups