Group #34068231 — subgroups

Subgroups — #34068231

Order 19328.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23752584	C₂	1	1	yes	maximal, characteristic
2	#23752507	C₂	1	2	yes	maximal, complemented
3	#23751956	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₁₅₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558992	—	2	2	no	—
7	#18012739	C₂²	1	1	yes	characteristic
8	#18012672	—	2	2	no	—
9	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
10	#18012663	—	2	2	no	—
11	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₁₅₁	C₂²	1	1	yes	characteristic
13	#124×C₁₅₁	—	2	2	no	—
14	#123×C₁₅₁	—	2	2	no	—
15	C₄×Q₈×C₁₅₁	C₂²	1	2	yes	—
16	C₄×D₄×C₁₅₁	—	2	2	no	—
17	C₂×Dic₃₀₂	—	2	1	no	—
18	#14558992	—	4	4	no	—
19	C₂×C₁₅₁⋊C₈	—	2	1	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
23	C₂×D₄×C₁₅₁	—	2	2	no	—
24	C₂×D₄×C₁₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
29	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄²×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
32	Dic₃₀₂	—	4	2	no	—
33	C₂×Dic₁₅₁	—	4	1	no	—
34	C₁₅₁⋊C₈	—	2	2	no	—
35	C₁₅₁⋊C₈	—	4	2	no	—
36	C₂³×C₁₅₁	—	2	2	no	—
37	Q₈×C₁₅₁	—	2	2	no	—
38	Q₈×C₁₅₁	—	2	2	no	—
39	D₄×C₁₅₁	—	4	2	no	—
40	D₄×C₁₅₁	—	4	2	no	—
41	D₄×C₁₅₁	—	4	2	no	—
42	C₂×C₄×C₁₅₁	—	2	2	no	—
43	C₂×C₄×C₁₅₁	—	2	2	no	—
44	C₂×C₄×C₁₅₁	—	8	2	no	—
45	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
47	C₂²×C₁₅₁	—	4	4	no	—
48	C₂²×C₁₅₁	—	4	2	no	—
49	C₂²×C₁₅₁	#120	1	1	yes	characteristic
50	C₄×C₁₅₁	—	2	1	no	—
51	C₄×C₁₅₁	—	4	2	no	—
52	C₄×C₁₅₁	—	4	2	no	—
53	C₄×C₁₅₁	#135	1	2	yes	—
54	C₄×C₁₅₁	#137	1	2	yes	—
55	Dic₁₅₁	—	8	1	no	—
56	C₂×C₁₅₁	—	8	2	no	—
57	C₂×C₁₅₁	#452	1	2	yes	—
58	C₂×C₁₅₁	#453	1	1	yes	characteristic
59	C₁₅₁	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	151	1	no	maximal, Sylow
61	#559	D₁₅₁	1	1	yes	characteristic
62	#492	—	151	1	no	—
63	#477	—	151	2	no	—
64	#331	—	151	1	no	—
65	Q₈⋊C₈	—	151	2	no	—
66	C₂×SD₁₆	—	302	2	no	—
67	C₄⋊Q₈	—	151	1	no	—
68	#127	C₂×D₁₅₁	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂×D₁₅₁	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	302	2	no	—
74	C₄⋊C₈	—	151	2	no	—
75	#103	—	302	2	no	—
76	C₄×C₈	—	151	1	no	—
77	C₂×Q₈	—	302	1	no	—
78	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₁₅₁	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	604	4	no	—
83	C₂×C₈	—	302	1	no	—
84	C₂×C₈	—	302	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	302	2	no	—
87	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	604	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	604	1	no	—
103	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
105	C₈	—	302	2	no	—
106	C₈	—	604	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#18012708	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	1208	1	no	—
114	C₄	#18012759	1	2	yes	—
115	C₄	#18012769	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#23752374	1	2	yes	—
118	C₂	#23752380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752584

C₂

yes

maximal, characteristic

#23752507

C₂

yes

maximal, complemented

#23751956

C₂

yes

maximal, characteristic

Q₈⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558992

—

#18012739

C₂²

yes

characteristic

#18012672

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012663

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

#124×C₁₅₁

—

#123×C₁₅₁

—

C₄×Q₈×C₁₅₁

C₂²

yes

—

C₄×D₄×C₁₅₁

—

C₂×Dic₃₀₂

—

#14558992

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₂²⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#135

yes

—

C₄×C₁₅₁

#137

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#452

yes

—

C₂×C₁₅₁

#453

yes

characteristic

C₁₅₁

#1630

yes

characteristic, complemented, Sylow

#1630

—

151

maximal, Sylow

#559

D₁₅₁

yes

characteristic

#492

—

151

—

#477

—

151

—

#331

—

151

—

Q₈⋊C₈

—

151

—

C₂×SD₁₆

—

302

—

C₄⋊Q₈

—

151

—

#127

C₂×D₁₅₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₅₁

yes

—

C₄×D₄

—

#107

—

302

—

C₄⋊C₈

—

151

—

#103

—

302

—

C₄×C₈

—

151

—

C₂×Q₈

—

302

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

—

SD₁₆

—

604

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

604

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

604

—

103

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

105

C₈

—

302

—

106

C₈

—

604

—

107

C₂²

—

108

C₂²

—

109

C₂²

#18012708

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

1208

—

114

C₄

#18012759

yes

—

115

C₄

#18012769

yes

—

116

C₂

—

117

C₂

#23752374

yes

—

118

C₂

#23752380

yes

characteristic

	subgroup	G/H
center	C₂²	#18012708
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#559×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#120
prime radical	#127×C₁₅₁	C₂²
hypercenter	#559	D₁₅₁
hypocenter	C₁₅₁	#1630
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#18012708

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#559×C₁₅₁

C₂

socle

C₂²×C₁₅₁

#120

prime radical

#127×C₁₅₁

C₂²

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #23752507	C₂	semidirect
4. Q₈⋊(C₁₅₁⋊C₈)	C₂	semidirect
59. C₁₅₁	#1630	semidirect

H ≅ G/N

type

2. #23752507

C₂

semidirect

4. Q₈⋊(C₁₅₁⋊C₈)

C₂

semidirect

59. C₁₅₁

#1630

semidirect

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

subgroup	index
#23752584	2
#23752507	2
#23751956	2
Q₈⋊(C₁₅₁⋊C₈)	2
#559×C₁₅₁	2
#1630	151

subgroup

index

#23752584

#23752507

#23751956

Q₈⋊(C₁₅₁⋊C₈)

#559×C₁₅₁

#1630

151

Subgroups — #34068231

Conjugacy classes of subgroups

Complements

Maximal subgroups