Group #34071016 — subgroups

Subgroups — #34071016

Order 19328.

126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₈×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₅₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₁₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₁₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#23751987	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₃₀₂	C₄	1	1	yes	characteristic
9	C₂×C₁₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₅₁⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₁₅₁⋊C₈	C₄	1	1	yes	characteristic
13	Dic₁₅₁⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₀₂	C₈	1	1	yes	characteristic
20	C₂×C₁₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₅₁	C₈	1	1	yes	characteristic
23	Dic₁₅₁⋊C₄	C₈	1	1	yes	characteristic
24	Dic₁₅₁⋊C₄	C₈	1	1	yes	characteristic
25	C₄×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
27	C₁₅₁⋊C₁₆	—	2	1	no	—
28	C₁₅₁⋊C₁₆	—	2	1	no	—
29	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
31	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₁₅₁	—	2	1	no	—
33	C₁₆×C₁₅₁	Q₈	1	2	yes	complemented
34	Dic₃₀₂	—	2	2	no	—
35	C₂×Dic₁₅₁	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₅₁	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₅₁	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₅₁	C₂×C₈	1	1	yes	characteristic
39	C₁₅₁⋊C₈	—	2	1	no	—
40	C₁₅₁⋊C₈	—	2	1	no	—
41	C₂×C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
44	C₈×C₁₅₁	—	2	1	no	—
45	C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₅₁	#41	1	1	yes	characteristic
47	C₂²×C₁₅₁	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₅₁	—	2	1	no	—
49	C₄×C₁₅₁	—	2	1	no	—
50	C₄×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₅₁	#131	1	1	yes	characteristic
52	Dic₁₅₁	—	2	1	no	—
53	Dic₁₅₁	—	2	1	no	—
54	Dic₁₅₁	—	2	1	no	—
55	Dic₁₅₁	M₃₂	1	2	yes	—
56	C₂×C₁₅₁	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₅₁	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₅₁	#504	1	1	yes	characteristic
59	C₁₅₁	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	151	1	no	maximal, Sylow
61	C₈×Q₈	—	151	1	no	—
62	C₄⋊C₁₆	—	151	1	no	—
63	C₄⋊C₁₆	—	151	1	no	—
64	C₄⋊C₁₆	—	151	1	no	—
65	#346	—	151	1	no	—
66	#346	D₁₅₁	1	1	yes	characteristic
67	C₄×C₁₆	—	151	1	no	—
68	C₄×Q₈	—	151	1	no	—
69	C₂×C₁₆	—	151	1	no	—
70	C₂×C₁₆	—	151	1	no	—
71	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
73	C₄⋊C₈	—	151	1	no	—
74	C₄⋊C₈	—	151	1	no	—
75	C₄⋊C₈	—	151	1	no	—
76	C₄×C₈	—	151	1	no	—
77	C₄×C₈	—	151	1	no	—
78	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
79	C₂×Q₈	—	151	1	no	—
80	C₂×C₈	—	151	1	no	—
81	C₂×C₈	—	151	1	no	—
82	C₂×C₈	C₄×D₁₅₁	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
84	C₄⋊C₄	—	151	1	no	—
85	C₄⋊C₄	—	151	1	no	—
86	C₄⋊C₄	—	151	1	no	—
87	C₄²	—	151	1	no	—
88	C₄²	—	151	1	no	—
89	C₄²	C₄×D₁₅₁	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	302	1	no	—
92	C₁₆	—	302	1	no	—
93	C₁₆	Dic₃₀₂	1	2	yes	complemented
94	Q₈	—	302	2	no	—
95	C₂×C₄	—	151	1	no	—
96	C₂×C₄	—	151	1	no	—
97	C₂×C₄	—	151	1	no	—
98	C₂×C₄	—	151	1	no	—
99	C₂×C₄	C₈×D₁₅₁	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₅₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	302	1	no	—
104	C₈	—	302	1	no	—
105	C₈	C₂×Dic₃₀₂	1	1	yes	characteristic
106	C₈	#14559004	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₁₅₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	302	1	no	—
111	C₄	—	302	1	no	—
112	C₄	—	302	1	no	—
113	C₄	—	151	2	no	—
114	C₄	C₄×Dic₃₀₂	1	1	yes	characteristic
115	C₄	#18012745	1	1	yes	characteristic
116	C₂	C₈×Dic₃₀₂	1	1	yes	characteristic
117	C₂	M₃₂×D₁₅₁	1	1	yes	characteristic
118	C₂	#23752630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₃₀₂

C₂

yes

maximal, characteristic

C₄⋊(C₁₅₁⋊C₁₆)

C₂

yes

maximal, characteristic

Dic₁₅₁⋊C₁₆

C₂

yes

maximal, characteristic

Dic₁₅₁⋊C₁₆

C₂

yes

maximal, characteristic

#23751987

C₂

yes

maximal, characteristic

C₁₆×Dic₁₅₁

C₂

yes

maximal, characteristic

#346×C₁₅₁

C₂

yes

maximal, characteristic

C₄×Dic₃₀₂

C₄

yes

characteristic

C₂×C₁₅₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₅₁⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₁₅₁⋊C₈)

C₄

yes

characteristic

Dic₁₅₁⋊C₈

C₄

yes

characteristic

Dic₁₅₁⋊C₈

C₄

yes

characteristic

C₈×Dic₁₅₁

C₂²

yes

characteristic

C₈×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₅₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

C₈

yes

characteristic

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₁₅₁

C₈

yes

characteristic

Dic₁₅₁⋊C₄

C₈

yes

characteristic

Dic₁₅₁⋊C₄

C₈

yes

characteristic

C₄×Dic₁₅₁

C₂×C₄

yes

characteristic

C₄×Dic₁₅₁

C₂×C₄

yes

characteristic

C₁₅₁⋊C₁₆

—

C₁₅₁⋊C₁₆

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

C₂×C₄

yes

characteristic

C₁₆×C₁₅₁

—

C₁₆×C₁₅₁

Q₈

yes

complemented

Dic₃₀₂

—

C₂×Dic₁₅₁

C₂×C₈

yes

characteristic

C₂×Dic₁₅₁

C₂×C₈

yes

characteristic

C₂×Dic₁₅₁

C₂×C₈

yes

characteristic

C₂×Dic₁₅₁

C₂×C₈

yes

characteristic

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂×C₄×C₁₅₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

C₂×Q₈

yes

characteristic

C₈×C₁₅₁

#41

yes

characteristic

C₂²×C₁₅₁

C₂²×C₈

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄×Q₈

yes

characteristic

C₄×C₁₅₁

#131

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

M₃₂

yes

—

C₂×C₁₅₁

C₈×Q₈

yes

characteristic

C₂×C₁₅₁

C₂×M₃₂

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#2183

—

151

maximal, Sylow

C₈×Q₈

—

151

—

C₄⋊C₁₆

—

151

—

C₄⋊C₁₆

—

151

—

C₄⋊C₁₆

—

151

—

#346

—

151

—

#346

D₁₅₁

yes

characteristic

C₄×C₁₆

—

151

—

C₄×Q₈

—

151

—

C₂×C₁₆

—

151

—

C₂×C₁₆

—

151

—

C₂×C₁₆

C₂×D₁₅₁

yes

characteristic

C₂×C₁₆

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

151

—

C₄⋊C₈

—

151

—

C₄⋊C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₄×D₁₅₁

yes

characteristic

C₂×C₈

C₂²×D₁₅₁

yes

characteristic

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄²

—

151

—

C₄²

—

151

—

C₄²

C₄×D₁₅₁

yes

characteristic

C₁₆

—

C₁₆

—

302

—

C₁₆

—

302

—

C₁₆

Dic₃₀₂

yes

complemented

Q₈

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₈×D₁₅₁

yes

characteristic

100

C₂×C₄

C₈×D₁₅₁

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

102

C₈

—

103

C₈

—

302

—

104

C₈

—

302

—

105

C₈

C₂×Dic₃₀₂

yes

characteristic

106

C₈

#14559004

yes

characteristic

107

C₂²

C₂×C₈×D₁₅₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

302

—

111

C₄

—

302

—

112

C₄

—

302

—

113

C₄

—

151

—

114

C₄

C₄×Dic₃₀₂

yes

characteristic

115

C₄

#18012745

yes

characteristic

116

C₂

C₈×Dic₃₀₂

yes

characteristic

117

C₂

M₃₂×D₁₅₁

yes

characteristic

118

C₂

#23752630

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅₁
commutator, socle, prime radical	C₂²×C₁₅₁	C₂²×C₈
Fitting	#346×C₁₅₁	C₂
cosocle	C₂×C₈×C₁₅₁	C₂³
hypercenter	#346	D₁₅₁
hypocenter	C₁₅₁	#2183
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₅₁

commutator, socle, prime radical

C₂²×C₁₅₁

C₂²×C₈

Fitting

#346×C₁₅₁

C₂

cosocle

C₂×C₈×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
33. C₁₆×C₁₅₁	Q₈	semidirect
59. C₁₅₁	#2183	semidirect
93. C₁₆	Dic₃₀₂	semidirect

H ≅ G/N

type

33. C₁₆×C₁₅₁

Q₈

semidirect

59. C₁₅₁

#2183

semidirect

93. C₁₆

Dic₃₀₂

semidirect

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

subgroup	index
C₈×Dic₃₀₂	2
C₄⋊(C₁₅₁⋊C₁₆)	2
Dic₁₅₁⋊C₁₆	2
#23751987	2
C₁₆×Dic₁₅₁	2
#346×C₁₅₁	2
#2183	151

subgroup

index

C₈×Dic₃₀₂

C₄⋊(C₁₅₁⋊C₁₆)

Dic₁₅₁⋊C₁₆

#23751987

C₁₆×Dic₁₅₁

#346×C₁₅₁

#2183

151

Subgroups — #34071016

Conjugacy classes of subgroups

Complements

Maximal subgroups