Group #34069901 — subgroups

Subgroups — #34069901

Order 19328.

184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23752261	C₂	1	2	yes	maximal
2	C₂×#18012675	C₂	1	1	yes	maximal, characteristic
3	#23752167	C₂	1	1	yes	maximal, characteristic
4	#23751984	C₂	1	2	yes	maximal
5	C₂×#108×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558988	C₂²	1	2	yes	—
7	C₂×#14558988	C₂²	1	1	yes	characteristic
8	#18012698	C₂²	1	1	yes	characteristic
9	#18012675	—	2	1	no	—
10	#18012675	—	2	1	no	—
11	#18012675	—	2	4	no	—
12	#18012649	—	2	1	no	—
13	C₈×Dic₁₅₁	—	2	1	no	—
14	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
16	#108×C₁₅₁	C₄	1	4	yes	complemented
17	#14558988	—	2	2	no	—
18	#14558988	—	2	2	no	—
19	#14558988	—	2	1	no	—
20	#14558988	D₄	1	2	yes	—
21	C₂×C₁₅₁⋊C₈	—	2	1	no	—
22	C₂×C₁₅₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₅₁	—	2	1	no	—
24	C₂²⋊Dic₁₅₁	Q₈	1	2	yes	—
25	C₄×Dic₁₅₁	—	2	1	no	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₁	—	2	2	no	—
28	M₁₆×C₁₅₁	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₁₅₁	—	2	2	no	—
30	C₂×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₁	Q₈	1	2	yes	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₁₅₁⋊C₈	—	2	2	no	—
36	C₁₅₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
44	C₈×C₁₅₁	—	2	4	no	—
45	C₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₅₁	—	4	1	no	—
48	C₂²×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₅₁	#121	1	1	yes	characteristic
54	C₄×C₁₅₁	#125	1	1	yes	characteristic
55	Dic₁₅₁	—	4	2	no	—
56	C₂×C₁₅₁	—	2	1	no	—
57	C₂×C₁₅₁	—	2	1	no	—
58	C₂×C₁₅₁	—	2	1	no	—
59	C₂×C₁₅₁	#389	1	1	yes	characteristic
60	C₁₅₁	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	151	1	no	maximal, Sylow
62	#430	—	151	2	no	—
63	C₂×#108	—	151	1	no	—
64	C₂×#108	D₁₅₁	1	1	yes	characteristic
65	#405	—	151	1	no	—
66	#344	—	151	2	no	—
67	C₂×M₁₆	—	151	2	no	—
68	C₂×M₁₆	—	151	1	no	—
69	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
70	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
71	#117	—	151	1	no	—
72	#108	—	302	4	no	—
73	#108	—	302	1	no	—
74	#108	—	302	1	no	—
75	#108	Dic₁₅₁	1	4	yes	complemented
76	#97	—	302	1	no	—
77	C₄×C₈	—	302	1	no	—
78	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	302	1	no	—
81	M₁₆	—	302	2	no	—
82	M₁₆	—	302	2	no	—
83	M₁₆	—	151	2	no	—
84	M₁₆	C₂×Dic₁₅₁	1	4	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	302	1	no	—
87	C₂×C₈	—	302	2	no	—
88	C₂×C₈	C₂×Dic₁₅₁	1	2	yes	—
89	C₂×C₈	D₆₀₄	1	1	yes	characteristic
90	C₂×C₈	D₆₀₄	1	1	yes	characteristic
91	C₂×C₈	Dic₃₀₂	1	2	yes	—
92	C₄⋊C₄	—	302	1	no	—
93	C₂²⋊C₄	—	151	2	no	—
94	C₄²	—	302	1	no	—
95	C₂³	Q₈×D₁₅₁	1	1	yes	characteristic
96	C₂×C₄	—	302	2	no	—
97	C₂×C₄	C₂²×Dic₁₅₁	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
99	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
101	C₂×C₄	#14559003	1	1	yes	characteristic
102	C₂×C₄	#14559005	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	302	2	no	—
105	C₈	—	604	2	no	—
106	C₈	C₄⋊Dic₁₅₁	1	2	yes	—
107	C₈	C₄⋊Dic₁₅₁	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	Q₈×Dic₁₅₁	1	1	yes	characteristic
110	C₂²	#18012719	1	1	yes	characteristic
111	C₂²	#18012738	1	1	yes	characteristic
112	C₄	—	604	2	no	—
113	C₄	C₂×C₄⋊Dic₁₅₁	1	1	yes	characteristic
114	C₄	D₄×Dic₁₅₁	1	1	yes	characteristic
115	C₄	#18012714	1	1	yes	characteristic
116	C₄	#18012730	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#23752111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752261

C₂

yes

maximal

C₂×#18012675

C₂

yes

maximal, characteristic

#23752167

C₂

yes

maximal, characteristic

#23751984

C₂

yes

maximal

C₂×#108×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

—

C₂×#14558988

C₂²

yes

characteristic

#18012698

C₂²

yes

—

—

—

—

C₈×Dic₁₅₁

—

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#108×C₁₅₁

C₄

yes

—

—

—

D₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

Q₈

yes

—

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

C₂×C₄

yes

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

—

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

Q₈

yes

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

C₂×C₄×C₁₅₁

yes

C₂×C₄×C₁₅₁

yes

C₈×C₁₅₁

—

C₈×C₁₅₁

C₄⋊C₄

yes

—

C₈×C₁₅₁

C₄⋊C₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₄×Q₈

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₁

C₄×D₄

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#389

yes

characteristic

C₁₅₁

#1953

yes

characteristic, complemented, Sylow

#1953

—

151

maximal, Sylow

#430

—

151

—

C₂×#108

—

151

—

C₂×#108

D₁₅₁

yes

characteristic

#405

—

151

—

#344

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#117

—

151

—

#108

—

302

—

#108

—

302

—

#108

—

302

—

#108

Dic₁₅₁

yes

complemented

#97

—

302

—

C₄×C₈

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

—

302

—

M₁₆

—

302

—

M₁₆

—

302

—

M₁₆

—

151

—

M₁₆

C₂×Dic₁₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

C₂×Dic₁₅₁

yes

—

C₂×C₈

D₆₀₄

yes

characteristic

C₂×C₈

D₆₀₄

yes

characteristic

C₂×C₈

Dic₃₀₂

yes

—

C₄⋊C₄

—

302

—

C₂²⋊C₄

—

151

—

C₄²

—

302

—

C₂³

Q₈×D₁₅₁

yes

characteristic

C₂×C₄

—

302

—

C₂×C₄

C₂²×Dic₁₅₁

yes

characteristic

C₂×C₄

D₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂×D₆₀₄

yes

characteristic

100

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

101

C₂×C₄

#14559003

yes

characteristic

102

C₂×C₄

#14559005

yes

characteristic

103

C₈

—

104

C₈

—

302

—

105

C₈

—

604

—

106

C₈

C₄⋊Dic₁₅₁

yes

—

107

C₈

C₄⋊Dic₁₅₁

yes

—

108

C₂²

—

109

C₂²

Q₈×Dic₁₅₁

yes

characteristic

110

C₂²

#18012719

yes

characteristic

111

C₂²

#18012738

yes

characteristic

112

C₄

—

604

—

113

C₄

C₂×C₄⋊Dic₁₅₁

yes

characteristic

114

C₄

D₄×Dic₁₅₁

yes

characteristic

115

C₄

#18012714

yes

characteristic

116

C₄

#18012730

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#23752111

yes

characteristic

	subgroup	G/H
center	C₄	#18012714
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×#108×C₁₅₁	C₂
socle	C₂×C₁₅₁	#389
prime radical	C₂³×C₁₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×#108	D₁₅₁
hypocenter	C₁₅₁	#1953
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#18012714

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×#108×C₁₅₁

C₂

socle

C₂×C₁₅₁

#389

prime radical

C₂³×C₁₅₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
16. #108×C₁₅₁	C₄	semidirect
60. C₁₅₁	#1953	semidirect
75. #108	Dic₁₅₁	semidirect

H ≅ G/N

type

16. #108×C₁₅₁

C₄

semidirect

60. C₁₅₁

#1953

semidirect

75. #108

Dic₁₅₁

semidirect

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

subgroup	index
#23752261	2
C₂×#18012675	2
#23752167	2
#23751984	2
C₂×#108×C₁₅₁	2
#1953	151

subgroup

index

C₂×#108×C₁₅₁

151

Subgroups — #34069901

Conjugacy classes of subgroups

Complements

Maximal subgroups