Group #34069859 — subgroups

Subgroups — #34069859

Order 19328.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#18012675	C₂	1	1	yes	maximal, characteristic
2	#23752239	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#23751971	C₂	1	2	yes	maximal
5	#427×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558988	C₂²	1	2	yes	—
7	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012675	—	2	2	no	—
10	C₄⋊(C₁₅₁⋊C₈)	—	2	4	no	—
11	C₈×Dic₁₅₁	—	2	2	no	—
12	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#117×C₁₅₁	C₂²	1	2	yes	—
14	#107×C₁₅₁	C₄	1	2	yes	complemented
15	#106×C₁₅₁	C₄	1	2	yes	complemented
16	C₂²×Dic₁₅₁	—	2	1	no	—
17	#14558988	—	2	2	no	—
18	#14558988	—	2	2	no	—
19	C₂×C₁₅₁⋊C₈	—	2	1	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₂×C₁₅₁⋊C₈	—	2	1	no	—
22	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₁₅₁	—	2	1	no	—
24	C₄×Dic₁₅₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₅₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₅₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
32	C₄²×C₁₅₁	—	2	2	no	—
33	C₂×Dic₁₅₁	—	2	4	no	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₁₅₁⋊C₈	—	2	2	no	—
36	C₁₅₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₅₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	—	2	4	no	—
39	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
45	C₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₅₁	—	2	2	no	—
48	C₂²×C₁₅₁	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₅₁	#125	1	1	yes	characteristic
51	C₄×C₁₅₁	—	4	4	no	—
52	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₅₁	#121	1	1	yes	characteristic
55	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₅₁	—	2	4	no	—
57	C₂×C₁₅₁	—	2	1	no	—
58	C₂×C₁₅₁	—	2	1	no	—
59	C₂×C₁₅₁	#389	1	1	yes	characteristic
60	C₂×C₁₅₁	#443	1	2	yes	—
61	C₁₅₁	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	151	1	no	maximal, Sylow
63	C₂×#108	—	151	1	no	—
64	#427	D₁₅₁	1	1	yes	characteristic
65	#424	—	151	2	no	—
66	C₂×C₄×C₈	—	151	1	no	—
67	#339	—	151	2	no	—
68	C₂×M₁₆	—	151	2	no	—
69	C₂²×C₈	—	151	1	no	—
70	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
71	#117	C₂×D₁₅₁	1	2	yes	—
72	C₂×C₄²	—	151	1	no	—
73	#108	—	302	2	no	—
74	#107	Dic₁₅₁	1	2	yes	complemented
75	#106	Dic₁₅₁	1	2	yes	complemented
76	C₄⋊C₈	—	302	4	no	—
77	C₄×C₈	—	302	2	no	—
78	C₂²×C₄	—	302	1	no	—
79	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
80	M₁₆	—	302	2	no	—
81	M₁₆	—	302	2	no	—
82	C₂×C₈	—	302	1	no	—
83	C₂×C₈	—	302	1	no	—
84	C₂×C₈	—	302	2	no	—
85	C₂×C₈	—	151	2	no	—
86	C₂×C₈	C₂×Dic₁₅₁	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₅₁	1	1	yes	characteristic
88	C₂×C₈	D₆₀₄	1	2	yes	—
89	C₂×C₈	Dic₃₀₂	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₅₁	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	302	1	no	—
94	C₄²	—	151	2	no	—
95	C₂³	#14559003	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	302	4	no	—
98	C₂×C₄	—	302	2	no	—
99	C₂×C₄	C₂²×Dic₁₅₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
104	C₂×C₄	#14559005	1	1	yes	characteristic
105	C₈	—	302	2	no	—
106	C₈	—	604	2	no	—
107	C₈	C₄⋊Dic₁₅₁	1	2	yes	—
108	C₈	C₄⋊Dic₁₅₁	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₅₁	1	1	yes	characteristic
111	C₂²	#18012719	1	1	yes	characteristic
112	C₂²	#18012730	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	302	4	no	—
115	C₄	C₂×C₄⋊Dic₁₅₁	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₅₁	1	1	yes	characteristic
117	C₄	#18012714	1	1	yes	characteristic
118	C₄	#18012738	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#23752111	1	1	yes	characteristic
122	C₂	#23752329	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012675

C₂

yes

maximal, characteristic

#23752239

C₂

yes

maximal

C₂×C₈×Dic₁₅₁

C₂

yes

maximal, characteristic

#23751971

C₂

yes

maximal

#427×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

—

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012675

—

C₄⋊(C₁₅₁⋊C₈)

—

C₈×Dic₁₅₁

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

#107×C₁₅₁

C₄

yes

complemented

#106×C₁₅₁

C₄

yes

complemented

C₂²×Dic₁₅₁

—

#14558988

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

Q₈

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

D₄

yes

—

C₂×C₈×C₁₅₁

Q₈

yes

—

C₄⋊C₄×C₁₅₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

—

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

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₈×C₁₅₁

C₄⋊C₄

yes

—

C₈×C₁₅₁

C₄⋊C₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₄×D₄

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₁

#125

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₁

C₄×Q₈

yes

characteristic

C₄×C₁₅₁

#121

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#389

yes

characteristic

C₂×C₁₅₁

#443

yes

—

C₁₅₁

#1945

yes

characteristic, complemented, Sylow

#1945

—

151

maximal, Sylow

C₂×#108

—

151

—

#427

D₁₅₁

yes

characteristic

#424

—

151

—

C₂×C₄×C₈

—

151

—

#339

—

151

—

C₂×M₁₆

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#117

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#108

—

302

—

#107

Dic₁₅₁

yes

complemented

#106

Dic₁₅₁

yes

complemented

C₄⋊C₈

—

302

—

C₄×C₈

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

302

—

M₁₆

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₂×C₈

C₂×Dic₁₅₁

yes

characteristic

C₂×C₈

C₂×Dic₁₅₁

yes

characteristic

C₂×C₈

D₆₀₄

yes

—

C₂×C₈

Dic₃₀₂

yes

—

C₄⋊C₄

C₂×Dic₁₅₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

302

—

C₄²

—

151

—

C₂³

#14559003

yes

characteristic

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

C₂²×Dic₁₅₁

yes

characteristic

100

C₂×C₄

D₄×D₁₅₁

yes

characteristic

101

C₂×C₄

C₂×D₆₀₄

yes

characteristic

102

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

103

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

104

C₂×C₄

#14559005

yes

characteristic

105

C₈

—

302

—

106

C₈

—

604

—

107

C₈

C₄⋊Dic₁₅₁

yes

—

108

C₈

C₄⋊Dic₁₅₁

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₅₁

yes

characteristic

111

C₂²

#18012719

yes

characteristic

112

C₂²

#18012730

yes

characteristic

113

C₄

—

114

C₄

—

302

—

115

C₄

C₂×C₄⋊Dic₁₅₁

yes

characteristic

116

C₄

Q₈×Dic₁₅₁

yes

characteristic

117

C₄

#18012714

yes

characteristic

118

C₄

#18012738

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#23752111

yes

characteristic

122

C₂

#23752329

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#427×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₄×D₄
prime radical	C₂³×C₁₅₁	#41
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#427	D₁₅₁
hypocenter	C₁₅₁	#1945
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#427×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₄×D₄

prime radical

C₂³×C₁₅₁

#41

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₅₁	C₄	semidirect
15. #106×C₁₅₁	C₄	semidirect
61. C₁₅₁	#1945	semidirect
74. #107	Dic₁₅₁	semidirect
75. #106	Dic₁₅₁	semidirect

H ≅ G/N

type

14. #107×C₁₅₁

C₄

semidirect

15. #106×C₁₅₁

C₄

semidirect

61. C₁₅₁

#1945

semidirect

74. #107

Dic₁₅₁

semidirect

75. #106

Dic₁₅₁

semidirect

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

subgroup	index
C₂×#18012675	2
#23752239	2
C₂×C₈×Dic₁₅₁	2
#23751971	2
#427×C₁₅₁	2
#1945	151

subgroup

index

C₂×C₈×Dic₁₅₁

#427×C₁₅₁

151

Subgroups — #34069859

Conjugacy classes of subgroups

Complements

Maximal subgroups