Group #34069226 — subgroups

Subgroups — #34069226

Order 19328.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊(C₁₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#23751966	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₅₁	C₄	1	2	yes	—
7	C₂×C₄×Dic₁₅₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
10	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
11	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₁	—	2	2	no	—
14	C₂²×Dic₁₅₁	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₅₁	C₂×C₄	1	2	yes	—
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	4	no	—
18	C₄×Dic₁₅₁	—	2	4	no	—
19	C₄×Dic₁₅₁	—	2	2	no	—
20	C₄×Dic₁₅₁	—	2	2	no	—
21	C₄×Dic₁₅₁	—	2	2	no	—
22	C₄×Dic₁₅₁	C₈	1	4	yes	complemented
23	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₅₁	—	2	2	no	—
27	C₂×C₈×C₁₅₁	—	2	4	no	—
28	C₄²×C₁₅₁	D₄	1	2	yes	—
29	C₄²×C₁₅₁	Q₈	1	2	yes	—
30	C₂×Dic₁₅₁	—	2	2	no	—
31	C₂×Dic₁₅₁	—	2	2	no	—
32	C₂×Dic₁₅₁	—	2	2	no	—
33	C₂×Dic₁₅₁	—	2	2	no	—
34	C₂×Dic₁₅₁	C₂×C₈	1	4	yes	—
35	C₂×Dic₁₅₁	M₁₆	1	4	yes	—
36	C₁₅₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₅₁	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₅₁	M₁₆	1	2	yes	—
43	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	#41	1	2	yes	—
46	C₂×C₄×C₁₅₁	#41	1	2	yes	—
47	C₈×C₁₅₁	—	4	2	no	—
48	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₅₁	#117	1	2	yes	—
50	C₂²×C₁₅₁	C₂²×C₈	1	1	yes	characteristic
51	C₂²×C₁₅₁	C₂×M₁₆	1	2	yes	—
52	C₂²×C₁₅₁	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₁₅₁	—	2	2	no	—
54	C₄×C₁₅₁	C₄⋊C₈	1	4	yes	—
55	C₄×C₁₅₁	#124	1	1	yes	characteristic
56	C₄×C₁₅₁	#124	1	1	yes	characteristic
57	C₄×C₁₅₁	#125	1	2	yes	—
58	Dic₁₅₁	—	2	4	no	—
59	Dic₁₅₁	—	2	4	no	—
60	C₂×C₁₅₁	#382	1	1	yes	characteristic
61	C₂×C₁₅₁	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₁₅₁	#423	1	1	yes	characteristic
63	C₂×C₁₅₁	#431	1	2	yes	—
64	C₂×C₁₅₁	#432	1	2	yes	—
65	C₁₅₁	#1831	1	1	yes	characteristic, complemented, Sylow
66	#1831	—	151	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	151	1	no	—
68	C₂×C₄⋊C₈	D₁₅₁	1	1	yes	characteristic
69	C₄³	—	151	1	no	—
70	#336	—	151	4	no	—
71	C₂²×C₈	—	151	2	no	—
72	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
73	C₂×C₄²	—	151	2	no	—
74	C₂×C₄²	—	151	1	no	—
75	C₂×C₄²	—	151	2	no	—
76	C₂×C₄²	—	151	1	no	—
77	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	302	2	no	—
80	C₂²×C₄	—	151	2	no	—
81	C₂²×C₄	—	151	2	no	—
82	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	302	2	no	—
88	C₂×C₈	—	302	4	no	—
89	C₄²	—	302	4	no	—
90	C₄²	—	302	2	no	—
91	C₄²	—	302	2	no	—
92	C₄²	—	302	2	no	—
93	C₄²	—	151	4	no	—
94	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
95	C₄²	Dic₃₀₂	1	2	yes	—
96	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	302	2	no	—
99	C₂×C₄	—	302	2	no	—
100	C₂×C₄	—	302	2	no	—
101	C₂×C₄	—	302	2	no	—
102	C₂×C₄	—	151	4	no	—
103	C₂×C₄	—	151	4	no	—
104	C₂×C₄	Dic₁₅₁⋊C₄	1	2	yes	—
105	C₂×C₄	Dic₁₅₁⋊C₄	1	2	yes	—
106	C₂×C₄	C₈×D₁₅₁	1	2	yes	—
107	C₂×C₄	#14558987	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
110	C₂×C₄	#14559003	1	2	yes	—
111	C₂×C₄	#14559005	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	604	2	no	—
114	C₂²	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
115	C₂²	#18012697	1	2	yes	—
116	C₂²	C₂×C₈×D₁₅₁	1	1	yes	characteristic
117	C₂²	M₁₆×D₁₅₁	1	2	yes	—
118	C₂²	C₂×#14558987	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	302	4	no	—
121	C₄	—	302	4	no	—
122	C₄	Dic₁₅₁⋊C₈	1	4	yes	—
123	C₄	#18012726	1	1	yes	characteristic
124	C₄	#18012728	1	1	yes	characteristic
125	C₄	#18012730	1	2	yes	—
126	C₂	#23752077	1	1	yes	characteristic
127	C₂	C₂×Dic₁₅₁⋊C₈	1	1	yes	characteristic
128	C₂	#23752235	1	1	yes	characteristic
129	C₂	#23752264	1	2	yes	—
130	C₂	#23752268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, characteristic

C₄²×Dic₁₅₁

C₂

yes

maximal, characteristic

#23751966

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₄

yes

—

C₂×C₄×Dic₁₅₁

C₄

yes

characteristic

C₂×C₄×Dic₁₅₁

C₄

yes

characteristic

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₄⋊(C₁₅₁⋊C₈)

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

C₈

yes

complemented

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

Q₈

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×C₈

yes

—

C₂×Dic₁₅₁

M₁₆

yes

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂×C₈

yes

—

C₂×C₄×C₁₅₁

M₁₆

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

#117

yes

—

C₂²×C₁₅₁

C₂²×C₈

yes

characteristic

C₂²×C₁₅₁

C₂×M₁₆

yes

—

C₂²×C₁₅₁

C₂×M₁₆

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄⋊C₈

yes

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#382

yes

characteristic

C₂×C₁₅₁

C₂×C₄⋊C₈

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#432

yes

—

C₁₅₁

#1831

yes

characteristic, complemented, Sylow

#1831

—

151

maximal, Sylow

C₂×C₄⋊C₈

—

151

—

C₂×C₄⋊C₈

D₁₅₁

yes

characteristic

C₄³

—

151

—

#336

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄²

—

302

—

C₄²

—

302

—

C₄²

—

302

—

C₄²

—

302

—

C₄²

—

151

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₄²

Dic₃₀₂

yes

—

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

100

C₂×C₄

—

302

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

151

—

103

C₂×C₄

—

151

—

104

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

105

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

106

C₂×C₄

C₈×D₁₅₁

yes

—

107

C₂×C₄

#14558987

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

109

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

110

C₂×C₄

#14559003

yes

—

111

C₂×C₄

#14559005

yes

—

112

C₈

—

113

C₈

—

604

—

114

C₂²

C₂×Dic₁₅₁⋊C₄

yes

characteristic

115

C₂²

#18012697

yes

—

116

C₂²

C₂×C₈×D₁₅₁

yes

characteristic

117

C₂²

M₁₆×D₁₅₁

yes

—

118

C₂²

C₂×#14558987

yes

characteristic

119

C₄

—

120

C₄

—

302

—

121

C₄

—

302

—

122

C₄

Dic₁₅₁⋊C₈

yes

—

123

C₄

#18012726

yes

characteristic

124

C₄

#18012728

yes

characteristic

125

C₄

#18012730

yes

—

126

C₂

#23752077

yes

characteristic

127

C₂

C₂×Dic₁₅₁⋊C₈

yes

characteristic

128

C₂

#23752235

yes

characteristic

129

C₂

#23752264

yes

—

130

C₂

#23752268

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₁
commutator	C₂²×C₁₅₁	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂²×C₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₁
hypocenter	C₁₅₁	#1831
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₁

commutator

C₂²×C₁₅₁

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂²×C₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₄×Dic₁₅₁	C₈	semidirect
65. C₁₅₁	#1831	semidirect

H ≅ G/N

type

22. C₄×Dic₁₅₁

C₈

semidirect

65. C₁₅₁

#1831

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₅₁⋊C₈)	2
C₄²×Dic₁₅₁	2
#23751966	2
C₂×C₄⋊C₈×C₁₅₁	2
#1831	151

subgroup

index

C₂×C₄⋊(C₁₅₁⋊C₈)

C₄²×Dic₁₅₁

#23751966

C₂×C₄⋊C₈×C₁₅₁

#1831

151

Subgroups — #34069226

Conjugacy classes of subgroups

Complements

Maximal subgroups