Group #34069235 — subgroups

Subgroups — #34069235

Order 19328.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23752250	C₂	1	1	yes	maximal, characteristic
2	#23752167	C₂	1	1	yes	maximal, characteristic
3	#23751984	C₂	1	2	yes	maximal
4	#23751969	C₂	1	2	yes	maximal
5	C₂×#108×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558988	C₂²	1	1	yes	characteristic
7	#18012698	C₂²	1	2	yes	—
8	#18012698	C₂²	1	1	yes	characteristic
9	#18012673	—	2	1	no	—
10	#18012671	—	2	1	no	—
11	#18012649	C₄	1	2	yes	complemented
12	C₈×Dic₁₅₁	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	#108×C₁₅₁	—	2	1	no	—
16	#108×C₁₅₁	—	2	1	no	—
17	#14558988	—	2	1	no	—
18	#14558988	—	2	1	no	—
19	C₂×C₁₅₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₁₅₁	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	1	no	—
22	C₂²⋊Dic₁₅₁	—	2	1	no	—
23	C₂²⋊Dic₁₅₁	—	2	2	no	—
24	C₄×Dic₁₅₁	—	2	2	no	—
25	C₄×Dic₁₅₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₁	—	2	2	no	—
28	M₁₆×C₁₅₁	—	2	2	no	—
29	C₂×C₈×C₁₅₁	—	2	2	no	—
30	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₅₁	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₅₁	Q₈	1	1	yes	characteristic
36	C₂×Dic₁₅₁	—	2	2	no	—
37	C₂×Dic₁₅₁	—	4	2	no	—
38	C₁₅₁⋊C₈	—	2	2	no	—
39	C₂³×C₁₅₁	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂²×C₄	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	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
46	C₈×C₁₅₁	—	4	2	no	—
47	C₈×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₅₁	—	4	1	no	—
52	C₂²×C₁₅₁	#117	1	1	yes	characteristic
53	C₂²×C₁₅₁	#124	1	1	yes	characteristic
54	C₂²×C₁₅₁	#125	1	1	yes	characteristic
55	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₅₁	#117	1	1	yes	characteristic
57	C₄×C₁₅₁	#124	1	1	yes	characteristic
58	C₄×C₁₅₁	#125	1	1	yes	characteristic
59	Dic₁₅₁	—	4	2	no	—
60	Dic₁₅₁	—	8	2	no	—
61	C₂×C₁₅₁	—	2	1	no	—
62	C₂×C₁₅₁	—	2	1	no	—
63	C₂×C₁₅₁	—	2	1	no	—
64	C₂×C₁₅₁	#382	1	1	yes	characteristic
65	C₁₅₁	#1833	1	1	yes	characteristic, complemented, Sylow
66	#1833	—	151	1	no	maximal, Sylow
67	C₂×#108	D₁₅₁	1	1	yes	characteristic
68	#427	—	151	1	no	—
69	#405	—	151	1	no	—
70	#344	—	151	2	no	—
71	H(3,ℤ/4)	—	151	2	no	—
72	C₂×M₁₆	—	151	1	no	—
73	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
74	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
75	#117	—	151	2	no	—
76	#117	—	151	1	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	302	1	no	—
80	#106	—	302	1	no	—
81	#97	—	151	2	no	—
82	C₄×C₈	—	151	2	no	—
83	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	302	1	no	—
87	M₁₆	—	302	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	151	2	no	—
90	C₂×C₈	C₄×D₁₅₁	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₅₁	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
94	C₂×C₈	Dic₃₀₂	1	1	yes	characteristic
95	C₂×C₈	Dic₃₀₂	1	1	yes	characteristic
96	C₄⋊C₄	—	302	1	no	—
97	C₄⋊C₄	—	302	2	no	—
98	C₂²⋊C₄	—	302	1	no	—
99	C₂²⋊C₄	—	302	2	no	—
100	C₄²	—	302	2	no	—
101	C₄²	—	151	2	no	—
102	C₂³	#14559005	1	1	yes	characteristic
103	C₂×C₄	—	302	2	no	—
104	C₂×C₄	—	604	2	no	—
105	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
108	C₂×C₄	#14559003	1	1	yes	characteristic
109	C₂×C₄	#14559003	1	1	yes	characteristic
110	C₂×C₄	#14559005	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	302	2	no	—
113	C₈	Dic₁₅₁⋊C₄	1	1	yes	characteristic
114	C₈	Dic₁₅₁⋊C₄	1	1	yes	characteristic
115	C₈	Dic₁₅₁⋊C₄	1	1	yes	characteristic
116	C₈	Dic₁₅₁⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#18012697	1	1	yes	characteristic
119	C₂²	#18012728	1	1	yes	characteristic
120	C₂²	#18012730	1	1	yes	characteristic
121	C₄	—	604	2	no	—
122	C₄	—	1208	2	no	—
123	C₄	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
124	C₄	#18012697	1	1	yes	characteristic
125	C₄	#18012726	1	1	yes	characteristic
126	C₄	#18012730	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#23752077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752250

C₂

yes

maximal, characteristic

#23752167

C₂

yes

maximal, characteristic

#23751984

C₂

yes

maximal

#23751969

C₂

yes

maximal

C₂×#108×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

characteristic

#18012698

C₂²

yes

—

#18012698

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₅₁

C₄

yes

complemented

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#108×C₁₅₁

—

#108×C₁₅₁

—

#14558988

—

#14558988

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

D₄

yes

characteristic

C₂×C₈×C₁₅₁

Q₈

yes

characteristic

C₂×C₈×C₁₅₁

Q₈

yes

characteristic

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

C₂×C₄×C₁₅₁

yes

C₂×C₄×C₁₅₁

yes

C₂×C₄×C₁₅₁

yes

C₈×C₁₅₁

—

C₈×C₁₅₁

C₄⋊C₄

yes

characteristic

C₈×C₁₅₁

C₄⋊C₄

yes

characteristic

C₈×C₁₅₁

C₄⋊C₄

yes

characteristic

C₈×C₁₅₁

C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#382

yes

characteristic

C₁₅₁

#1833

yes

characteristic, complemented, Sylow

#1833

—

151

maximal, Sylow

C₂×#108

D₁₅₁

yes

characteristic

#427

—

151

—

#405

—

151

—

#344

—

151

—

H(3,ℤ/4)

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#117

—

151

—

#117

—

151

—

#108

—

#108

—

#107

—

302

—

#106

—

302

—

#97

—

151

—

C₄×C₈

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

302

—

M₁₆

—

302

—

C₂×C₈

—

C₂×C₈

—

151

—

C₂×C₈

C₄×D₁₅₁

yes

characteristic

C₂×C₈

C₄×D₁₅₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₅₁

yes

characteristic

C₂×C₈

Dic₃₀₂

yes

characteristic

C₂×C₈

Dic₃₀₂

yes

characteristic

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

100

C₄²

—

302

—

101

C₄²

—

151

—

102

C₂³

#14559005

yes

characteristic

103

C₂×C₄

—

302

—

104

C₂×C₄

—

604

—

105

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

107

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

108

C₂×C₄

#14559003

yes

characteristic

109

C₂×C₄

#14559003

yes

characteristic

110

C₂×C₄

#14559005

yes

characteristic

111

C₈

—

112

C₈

—

302

—

113

C₈

Dic₁₅₁⋊C₄

yes

characteristic

114

C₈

Dic₁₅₁⋊C₄

yes

characteristic

115

C₈

Dic₁₅₁⋊C₄

yes

characteristic

116

C₈

Dic₁₅₁⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#18012697

yes

characteristic

119

C₂²

#18012728

yes

characteristic

120

C₂²

#18012730

yes

characteristic

121

C₄

—

604

—

122

C₄

—

1208

—

123

C₄

C₂×Dic₁₅₁⋊C₄

yes

characteristic

124

C₄

#18012697

yes

characteristic

125

C₄

#18012726

yes

characteristic

126

C₄

#18012730

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23752077

yes

characteristic

	subgroup	G/H
center	C₄	#18012697
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×#108×C₁₅₁	C₂
socle	C₂×C₁₅₁	#382
prime radical	C₂³×C₁₅₁	#41
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×#108	D₁₅₁
hypocenter	C₁₅₁	#1833
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#18012697

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×#108×C₁₅₁

C₂

C₂×C₁₅₁

C₂³×C₁₅₁

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
11. #18012649	C₄	semidirect
12. C₈×Dic₁₅₁	C₄	semidirect
65. C₁₅₁	#1833	semidirect

H ≅ G/N

type

11. #18012649

C₄

semidirect

12. C₈×Dic₁₅₁

C₄

semidirect

65. C₁₅₁

#1833

semidirect

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

subgroup	index
#23752250	2
#23752167	2
#23751984	2
#23751969	2
C₂×#108×C₁₅₁	2
#1833	151

subgroup

index

C₂×#108×C₁₅₁

151

Subgroups — #34069235

Conjugacy classes of subgroups

Complements

Maximal subgroups