Group #34070378 — subgroups

Subgroups — #34070378

Order 19328.

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×Dic₆₀₄	C₂	1	1	yes	maximal, characteristic
2	#23752220	C₂	1	2	yes	maximal
3	C₂×#18012657	C₂	1	2	yes	maximal
4	#23751977	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	2	yes	—
7	C₂×Dic₆₀₄	—	2	2	no	—
8	C₂×Dic₆₀₄	—	2	4	no	—
9	C₂×#14558988	C₂²	1	2	yes	—
10	#18012662	—	2	4	no	—
11	#18012657	—	2	4	no	—
12	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	#117×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₅₁	—	2	2	no	—
16	C₂×Dic₃₀₂	—	2	2	no	—
17	C₂×Dic₃₀₂	—	2	2	no	—
18	C₂×Dic₃₀₂	—	2	2	no	—
19	C₂×Dic₃₀₂	—	2	2	no	—
20	C₂×Dic₃₀₂	—	2	2	no	—
21	C₂×Dic₃₀₂	D₄	1	4	yes	—
22	C₂²×Dic₁₅₁	—	2	2	no	—
23	Dic₆₀₄	—	2	8	no	—
24	#14558988	—	2	2	no	—
25	#14558988	—	2	2	no	—
26	C₂×C₁₅₁⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₅₁	—	2	2	no	—
29	C₂×C₈×C₁₅₁	—	2	2	no	—
30	C₂×C₈×C₁₅₁	—	2	1	no	—
31	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
34	C₄²×C₁₅₁	—	2	1	no	—
35	Dic₃₀₂	—	2	4	no	—
36	Dic₃₀₂	—	2	4	no	—
37	Dic₃₀₂	—	4	2	no	—
38	Dic₃₀₂	—	4	2	no	—
39	C₂×Dic₁₅₁	—	2	4	no	—
40	C₂×Dic₁₅₁	—	4	2	no	—
41	C₂×Dic₁₅₁	—	4	2	no	—
42	C₁₅₁⋊C₈	—	4	2	no	—
43	C₂³×C₁₅₁	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	—	2	2	no	—
45	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₁	#41	1	2	yes	—
49	C₈×C₁₅₁	—	2	2	no	—
50	C₈×C₁₅₁	—	4	1	no	—
51	C₂²×C₁₅₁	—	2	2	no	—
52	C₂²×C₁₅₁	#121	1	1	yes	characteristic
53	C₂²×C₁₅₁	#123	1	2	yes	—
54	C₄×C₁₅₁	—	4	2	no	—
55	C₄×C₁₅₁	#120	1	1	yes	characteristic
56	C₄×C₁₅₁	#121	1	2	yes	—
57	C₄×C₁₅₁	#124	1	1	yes	characteristic
58	Dic₁₅₁	—	4	4	no	—
59	C₂×C₁₅₁	—	2	2	no	—
60	C₂×C₁₅₁	#394	1	1	yes	characteristic
61	C₂×C₁₅₁	#473	1	2	yes	—
62	C₁₅₁	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	151	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	151	1	no	—
65	#424	D₁₅₁	1	1	yes	characteristic
66	#419	—	151	2	no	—
67	C₂×#101	—	151	2	no	—
68	#341	—	151	1	no	—
69	C₂²×Q₈	—	151	2	no	—
70	C₂×Q₁₆	—	302	4	no	—
71	C₂×Q₁₆	—	302	2	no	—
72	C₂×M₁₆	—	151	2	no	—
73	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
75	#117	C₂×D₁₅₁	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	302	4	no	—
78	#101	—	302	4	no	—
79	C₂×Q₈	—	302	2	no	—
80	C₂×Q₈	—	302	2	no	—
81	C₂×Q₈	—	302	2	no	—
82	C₂×Q₈	—	302	2	no	—
83	C₂×Q₈	—	302	2	no	—
84	C₂×Q₈	—	151	4	no	—
85	C₂²×C₄	—	302	2	no	—
86	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
87	Q₁₆	—	302	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	302	2	no	—
90	M₁₆	—	302	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	302	2	no	—
94	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
95	C₄⋊C₄	D₆₀₄	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#14559004	1	1	yes	characteristic
99	Q₈	—	302	4	no	—
100	Q₈	—	302	4	no	—
101	Q₈	—	604	2	no	—
102	Q₈	—	604	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	302	4	no	—
105	C₂×C₄	—	604	2	no	—
106	C₂×C₄	—	604	2	no	—
107	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
110	C₂×C₄	#14559005	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	604	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#18012713	1	1	yes	characteristic
116	C₂²	#18012724	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	604	4	no	—
119	C₄	#18012707	1	1	yes	characteristic
120	C₄	#18012714	1	2	yes	—
121	C₄	#18012728	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#23752129	1	1	yes	characteristic
124	C₂	#23752487	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₀₄

C₂

yes

maximal, characteristic

#23752220

C₂

yes

maximal

C₂×#18012657

C₂

yes

maximal

#23751977

C₂

yes

maximal, characteristic

#424×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

C₂×Dic₆₀₄

—

C₂×Dic₆₀₄

—

C₂×#14558988

C₂²

yes

—

#18012662

—

#18012657

—

C₂×M₁₆×C₁₅₁

C₂²

yes

characteristic

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

D₄

yes

—

C₂²×Dic₁₅₁

—

Dic₆₀₄

—

#14558988

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#121

yes

characteristic

C₂²×C₁₅₁

#123

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

#121

yes

—

C₄×C₁₅₁

#124

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#394

yes

characteristic

C₂×C₁₅₁

#473

yes

—

C₁₅₁

#2047

yes

characteristic, complemented, Sylow

#2047

—

151

maximal, Sylow

C₂²×Q₁₆

—

151

—

#424

D₁₅₁

yes

characteristic

#419

—

151

—

C₂×#101

—

151

—

#341

—

151

—

C₂²×Q₈

—

151

—

C₂×Q₁₆

—

302

—

C₂×Q₁₆

—

302

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

characteristic

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#117

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

#103

—

302

—

#101

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

Q₁₆

—

302

—

M₁₆

—

M₁₆

—

302

—

M₁₆

—

302

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

D₆₀₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#14559004

yes

characteristic

Q₈

—

302

—

100

Q₈

—

302

—

101

Q₈

—

604

—

102

Q₈

—

604

—

103

C₂×C₄

—

104

C₂×C₄

—

302

—

105

C₂×C₄

—

604

—

106

C₂×C₄

—

604

—

107

C₂×C₄

D₄×D₁₅₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

109

C₂×C₄

C₂×D₆₀₄

yes

characteristic

110

C₂×C₄

#14559005

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

604

—

114

C₂²

—

115

C₂²

#18012713

yes

characteristic

116

C₂²

#18012724

yes

—

117

C₄

—

118

C₄

—

604

—

119

C₄

#18012707

yes

characteristic

120

C₄

#18012714

yes

—

121

C₄

#18012728

yes

characteristic

122

C₂

—

123

C₂

#23752129

yes

characteristic

124

C₂

#23752487

yes

—

	subgroup	G/H
center	C₂²	#18012713
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#424×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#121
prime radical	C₂³×C₁₅₁	#41
hypercenter	#424	D₁₅₁
hypocenter	C₁₅₁	#2047
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#18012713

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#424×C₁₅₁

C₂

C₂²×C₁₅₁

C₂³×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
62. C₁₅₁	#2047	semidirect

H ≅ G/N

type

62. C₁₅₁

#2047

semidirect

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

subgroup	index
C₂²×Dic₆₀₄	2
#23752220	2
C₂×#18012657	2
#23751977	2
#424×C₁₅₁	2
#2047	151

subgroup

index

C₂²×Dic₆₀₄

#424×C₁₅₁

151

Subgroups — #34070378

Conjugacy classes of subgroups

Complements

Maximal subgroups