Group #34070664 — subgroups

Subgroups — #34070664

Order 19328.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	1	yes	maximal, characteristic, complemented
2	#23752239	C₂	1	2	yes	maximal, complemented
3	#23752230	C₂	1	2	yes	maximal
4	#23752180	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14559004	C₂²	1	1	yes	characteristic
7	C₂×#14558988	C₂²	1	2	yes	—
8	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
9	#18012675	—	2	2	no	—
10	C₄⋊(C₁₅₁⋊C₈)	—	2	4	no	—
11	#18012667	—	2	4	no	—
12	(C₂×D₁₅₁)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
14	#117×C₁₅₁	C₂²	1	2	yes	—
15	#14559004	—	2	2	no	—
16	#14559004	—	2	2	no	—
17	C₂×Dic₃₀₂	Q₈	1	1	yes	characteristic
18	C₂×D₆₀₄	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
20	C₂×C₄×D₁₅₁	—	2	1	no	—
21	#14558988	—	2	2	no	—
22	#14558988	—	2	2	no	—
23	C₂×C₁₅₁⋊C₈	—	2	1	no	—
24	C₂×C₁₅₁⋊C₈	—	2	2	no	—
25	C₂×C₁₅₁⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₁	—	2	2	no	—
28	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₅₁	—	2	2	no	—
30	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	D₄	1	4	yes	—
32	C₂²×D₁₅₁	—	2	1	no	—
33	Dic₃₀₂	—	2	2	no	—
34	D₆₀₄	—	2	2	no	—
35	C₂²⋊D₁₅₁	—	4	2	no	—
36	C₂×Dic₁₅₁	—	2	1	no	—
37	C₄×D₁₅₁	—	4	1	no	—
38	C₄×D₁₅₁	—	4	1	no	—
39	C₁₅₁⋊C₈	—	4	1	no	—
40	C₁₅₁⋊C₈	—	4	2	no	—
41	C₂³×C₁₅₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	—	2	4	no	—
43	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₅₁	#41	1	2	yes	—
47	C₈×C₁₅₁	—	2	2	no	—
48	C₂²×C₁₅₁	—	4	1	no	—
49	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₅₁	#123	1	2	yes	—
51	C₂×D₁₅₁	—	4	2	no	—
52	C₂×D₁₅₁	—	4	1	no	—
53	C₄×C₁₅₁	—	4	4	no	—
54	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₅₁	#123	1	1	yes	characteristic
56	C₄×C₁₅₁	#127	1	1	yes	characteristic
57	Dic₁₅₁	—	4	1	no	—
58	C₂×C₁₅₁	—	2	2	no	—
59	C₂×C₁₅₁	—	2	1	no	—
60	C₂×C₁₅₁	#399	1	1	yes	characteristic
61	D₁₅₁	—	8	1	no	—
62	C₁₅₁	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	151	1	no	maximal, Sylow
64	#430	—	151	1	no	—
65	#424	—	151	2	no	—
66	#421	—	151	2	no	—
67	#408	—	151	1	no	—
68	H(3,ℤ/4)	D₁₅₁	1	1	yes	characteristic, complemented
69	C₂×#41	—	151	1	no	—
70	C₂×M₁₆	—	151	2	no	—
71	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
72	C₂²×C₈	—	151	1	no	—
73	#117	C₂×D₁₅₁	1	2	yes	—
74	#108	—	302	2	no	—
75	C₄⋊C₈	—	302	4	no	—
76	#104	—	302	4	no	—
77	C₂²⋊C₈	—	302	2	no	—
78	#41	—	302	2	no	—
79	#41	—	302	2	no	—
80	C₂×Q₈	—	151	1	no	—
81	C₂×D₄	—	302	1	no	—
82	C₂×D₄	—	151	1	no	—
83	C₂²×C₄	—	302	1	no	—
84	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	302	2	no	—
87	M₁₆	—	302	2	no	—
88	C₂×C₈	—	302	1	no	—
89	C₂×C₈	—	302	2	no	—
90	C₂×C₈	—	302	2	no	—
91	C₂×C₈	D₆₀₄	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₆₀₄	1	4	yes	—
95	C₂³	—	302	1	no	—
96	C₂³	#14559003	1	1	yes	characteristic
97	Q₈	—	302	2	no	—
98	D₄	—	302	2	no	—
99	D₄	—	604	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	302	1	no	—
102	C₂×C₄	—	604	1	no	—
103	C₂×C₄	—	604	1	no	—
104	C₂×C₄	C₂×D₆₀₄	1	2	yes	—
105	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
107	C₂×C₄	#14559003	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	604	1	no	—
110	C₈	—	604	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	604	2	no	—
113	C₂²	—	604	1	no	—
114	C₂²	#18012719	1	1	yes	characteristic
115	C₂²	#18012723	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	604	1	no	—
118	C₄	#18012719	1	2	yes	—
119	C₄	#18012723	1	1	yes	characteristic
120	C₄	#18012736	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1208	1	no	—
124	C₂	#23752149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752261

C₂

yes

maximal, characteristic, complemented

#23752239

C₂

yes

maximal, complemented

#23752230

C₂

yes

maximal

#23752180

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559004

C₂²

yes

characteristic

C₂×#14558988

C₂²

yes

—

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012675

—

C₄⋊(C₁₅₁⋊C₈)

—

#18012667

—

(C₂×D₁₅₁)⋊C₈

—

C₂×M₁₆×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

#14559004

—

#14559004

—

C₂×Dic₃₀₂

Q₈

yes

characteristic

C₂×D₆₀₄

Q₈

yes

characteristic

C₂×C₂²⋊D₁₅₁

—

C₂×C₄×D₁₅₁

—

#14558988

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

—

C₂²×D₁₅₁

—

Dic₃₀₂

—

D₆₀₄

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₁

#123

yes

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#399

yes

characteristic

D₁₅₁

—

C₁₅₁

#2098

yes

characteristic, complemented, Sylow

#2098

—

151

maximal, Sylow

#430

—

151

—

#424

—

151

—

#421

—

151

—

#408

—

151

—

H(3,ℤ/4)

D₁₅₁

yes

characteristic, complemented

C₂×#41

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

#117

C₂×D₁₅₁

yes

—

#108

—

302

—

C₄⋊C₈

—

302

—

#104

—

302

—

C₂²⋊C₈

—

302

—

#41

—

302

—

#41

—

302

—

C₂×Q₈

—

151

—

C₂×D₄

—

302

—

C₂×D₄

—

151

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

—

302

—

M₁₆

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

D₆₀₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₆₀₄

yes

—

C₂³

—

302

—

C₂³

#14559003

yes

characteristic

Q₈

—

302

—

D₄

—

302

—

D₄

—

604

—

100

C₂×C₄

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

604

—

103

C₂×C₄

—

604

—

104

C₂×C₄

C₂×D₆₀₄

yes

—

105

C₂×C₄

C₂×D₆₀₄

yes

characteristic

106

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

107

C₂×C₄

#14559003

yes

—

108

C₈

—

109

C₈

—

604

—

110

C₈

—

604

—

111

C₂²

—

112

C₂²

—

604

—

113

C₂²

—

604

—

114

C₂²

#18012719

yes

characteristic

115

C₂²

#18012723

yes

—

116

C₄

—

117

C₄

—

604

—

118

C₄

#18012719

yes

—

119

C₄

#18012723

yes

characteristic

120

C₄

#18012736

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1208

—

124

C₂

#23752149

yes

characteristic

	subgroup	G/H
center	C₄	#18012736
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	H(3,ℤ/4)×C₁₅₁	C₂
socle	C₂×C₁₅₁	#399
prime radical	C₂×#14559004	C₂²
hypercenter	H(3,ℤ/4)	D₁₅₁
hypocenter	C₁₅₁	#2098
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#18012736

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

H(3,ℤ/4)×C₁₅₁

C₂

C₂×C₁₅₁

C₂²

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #23752261	C₂	semidirect
2. #23752239	C₂	semidirect
5. H(3,ℤ/4)×C₁₅₁	C₂	semidirect
62. C₁₅₁	#2098	semidirect
68. H(3,ℤ/4)	D₁₅₁	semidirect

H ≅ G/N

type

1. #23752261

C₂

semidirect

2. #23752239

C₂

semidirect

5. H(3,ℤ/4)×C₁₅₁

C₂

semidirect

62. C₁₅₁

#2098

semidirect

68. H(3,ℤ/4)

D₁₅₁

semidirect

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

subgroup	index
#23752261	2
#23752239	2
#23752230	2
#23752180	2
H(3,ℤ/4)×C₁₅₁	2
#2098	151

subgroup

index

H(3,ℤ/4)×C₁₅₁

151

Subgroups — #34070664

Conjugacy classes of subgroups

Complements

Maximal subgroups