Group #34070059 — subgroups

Subgroups — #34070059

Order 19328.

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

normal subgroups only

This group has 292 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	C₄×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#23751974	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₁	—	2	1	no	—
8	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
9	C₁₅₁⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₁₅₁	—	2	1	no	—
11	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₅₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₅₁	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	—	2	4	no	—
19	C₄⋊Dic₁₅₁	—	2	2	no	—
20	C₄⋊Dic₁₅₁	Q₈	1	4	yes	complemented
21	C₄×Dic₁₅₁	—	4	2	no	—
22	C₂×Q₈×C₁₅₁	—	2	4	no	—
23	C₂×Q₈×C₁₅₁	—	4	4	no	—
24	C₂²×C₄×C₁₅₁	—	2	2	no	—
25	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₁	—	2	4	no	—
29	C₄⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄⋊C₄×C₁₅₁	C₂×C₄	1	4	yes	—
31	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
32	C₄²×C₁₅₁	D₄	1	2	yes	—
33	C₂×Dic₁₅₁	—	2	4	no	—
34	C₂×Dic₁₅₁	—	4	2	no	—
35	C₂×Dic₁₅₁	—	4	2	no	—
36	C₁₅₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₅₁	—	4	8	no	—
39	C₂×C₄×C₁₅₁	—	2	2	no	—
40	C₂×C₄×C₁₅₁	—	2	4	no	—
41	C₂×C₄×C₁₅₁	—	4	4	no	—
42	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₅₁	SD₁₆	1	2	yes	—
45	C₂×C₄×C₁₅₁	Q₁₆	1	2	yes	—
46	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₅₁	#41	1	2	yes	—
50	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₅₁	—	2	2	no	—
56	C₄×C₁₅₁	—	4	4	no	—
57	C₄×C₁₅₁	#103	1	4	yes	—
58	C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
59	C₄×C₁₅₁	#124	1	1	yes	characteristic
60	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₁₅₁	—	4	4	no	—
62	C₂×C₁₅₁	#391	1	1	yes	characteristic
63	C₂×C₁₅₁	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₅₁	#418	1	1	yes	characteristic
65	C₂×C₁₅₁	#476	1	2	yes	—
66	C₂×C₁₅₁	#477	1	2	yes	—
67	C₁₅₁	#1987	1	1	yes	characteristic, complemented, Sylow
68	#1987	—	151	1	no	maximal, Sylow
69	C₂×C₄⋊Q₈	D₁₅₁	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	151	1	no	—
71	C₄×C₄⋊C₄	—	151	1	no	—
72	#340	—	151	4	no	—
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	151	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₁₅₁	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	151	2	no	—
79	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
80	C₂×C₄²	—	302	1	no	—
81	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
82	C₄⋊C₈	—	302	2	no	—
83	#95	—	302	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	302	2	no	—
88	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
91	C₂×C₈	—	302	2	no	—
92	C₂×C₈	—	302	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	302	2	no	—
96	C₄⋊C₄	—	151	4	no	—
97	C₄⋊C₄	C₂×Dic₁₅₁	1	4	yes	—
98	C₄²	—	604	2	no	—
99	C₄²	C₂×Dic₁₅₁	1	2	yes	—
100	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	302	4	no	—
107	C₂×C₄	—	604	2	no	—
108	C₂×C₄	—	604	2	no	—
109	C₂×C₄	C₂²⋊Dic₁₅₁	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₁₅₁	1	2	yes	—
111	C₂×C₄	#14558991	1	2	yes	—
112	C₂×C₄	#14558994	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₅₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
116	C₂×C₄	#14559005	1	2	yes	—
117	C₈	—	604	2	no	—
118	C₂²	C₂×C₂²⋊Dic₁₅₁	1	1	yes	characteristic
119	C₂²	#18012715	1	2	yes	—
120	C₂²	C₂×#14558991	1	1	yes	characteristic
121	C₂²	C₂×#14558994	1	1	yes	characteristic
122	C₂²	#18012765	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	604	4	no	—
126	C₄	#18012663	1	4	yes	—
127	C₄	Q₈×Dic₁₅₁	1	2	yes	—
128	C₄	#18012728	1	1	yes	characteristic
129	C₄	#18012737	1	1	yes	characteristic
130	C₂	#23752119	1	1	yes	characteristic
131	C₂	C₂×#18012663	1	1	yes	characteristic
132	C₂	#23752217	1	1	yes	characteristic
133	C₂	#23752499	1	2	yes	—
134	C₂	#23752504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

—

C₄⋊(C₁₅₁⋊C₈)

—

C₁₅₁⋊#95

—

C₂²×Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

Q₈

yes

complemented

C₄×Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

Q₁₆

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

—

C₂²×C₁₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₁

C₂×Q₁₆

yes

characteristic

C₂²×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#103

yes

—

C₄×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

#124

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#477

yes

—

C₁₅₁

#1987

yes

characteristic, complemented, Sylow

#1987

—

151

maximal, Sylow

C₂×C₄⋊Q₈

D₁₅₁

yes

characteristic

C₂×C₄⋊C₈

—

151

—

C₄×C₄⋊C₄

—

151

—

#340

—

151

—

C₂²×Q₈

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₅₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

302

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

302

—

#95

—

302

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂×Dic₁₅₁

yes

—

C₄²

—

604

—

C₄²

C₂×Dic₁₅₁

yes

—

100

C₄²

C₂²⋊D₁₅₁

yes

—

101

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

302

—

107

C₂×C₄

—

604

—

108

C₂×C₄

—

604

—

109

C₂×C₄

C₂²⋊Dic₁₅₁

yes

—

110

C₂×C₄

C₂²⋊Dic₁₅₁

yes

—

111

C₂×C₄

#14558991

yes

—

112

C₂×C₄

#14558994

yes

—

113

C₂×C₄

C₂²×Dic₁₅₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

115

C₂×C₄

Q₈×D₁₅₁

yes

—

116

C₂×C₄

#14559005

yes

—

117

C₈

—

604

—

118

C₂²

C₂×C₂²⋊Dic₁₅₁

yes

characteristic

119

C₂²

#18012715

yes

—

120

C₂²

C₂×#14558991

yes

characteristic

121

C₂²

C₂×#14558994

yes

characteristic

122

C₂²

#18012765

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

604

—

126

C₄

#18012663

yes

—

127

C₄

Q₈×Dic₁₅₁

yes

—

128

C₄

#18012728

yes

characteristic

129

C₄

#18012737

yes

characteristic

130

C₂

#23752119

yes

characteristic

131

C₂

C₂×#18012663

yes

characteristic

132

C₂

#23752217

yes

characteristic

133

C₂

#23752499

yes

—

134

C₂

#23752504

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×C₄⋊Q₈×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅₁
hypocenter	C₁₅₁	#1987
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×C₄⋊Q₈×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₅₁

hypocenter

C₁₅₁

#1987

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₅₁	C₄	semidirect
20. C₄⋊Dic₁₅₁	Q₈	semidirect
67. C₁₅₁	#1987	semidirect
76. C₄⋊Q₈	Dic₁₅₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₅₁

C₄

semidirect

20. C₄⋊Dic₁₅₁

Q₈

semidirect

67. C₁₅₁

#1987

semidirect

76. C₄⋊Q₈

Dic₁₅₁

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
C₄×C₄⋊Dic₁₅₁	2
#23751974	2
C₂×C₄⋊Q₈×C₁₅₁	2
#1987	151

subgroup

index

C₂×C₄⋊(C₁₅₁⋊C₈)

C₄×C₄⋊Dic₁₅₁

#23751974

C₂×C₄⋊Q₈×C₁₅₁

#1987

151

Subgroups — #34070059

Conjugacy classes of subgroups

Complements

Maximal subgroups