Group #34068169 — subgroups

Subgroups — #34068169

Order 19328.

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

normal subgroups only

This group has 240 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	#23752588	C₂	1	1	yes	maximal, characteristic
2	#23752412	C₂	1	2	yes	maximal, complemented
3	#23751956	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₁₅₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558991	—	2	2	no	—
7	C₂×#14558989	—	2	2	no	—
8	#18012736	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
10	#18012659	—	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×D₆₀₄	—	2	1	no	—
18	C₂×D₆₀₄	—	2	2	no	—
19	#14558991	—	4	4	no	—
20	#14558989	—	4	4	no	—
21	C₂×C₁₅₁⋊C₈	—	2	1	no	—
22	C₂×C₁₅₁⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
24	C₂×D₄×C₁₅₁	—	2	2	no	—
25	C₂×D₄×C₁₅₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	—	2	2	no	—
28	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
30	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	—	2	2	no	—
32	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
33	C₂²×D₁₅₁	—	4	1	no	—
34	D₆₀₄	—	4	2	no	—
35	D₆₀₄	—	4	2	no	—
36	D₆₀₄	—	4	2	no	—
37	C₁₅₁⋊C₈	—	2	2	no	—
38	C₁₅₁⋊C₈	—	4	2	no	—
39	C₂³×C₁₅₁	—	2	2	no	—
40	Q₈×C₁₅₁	—	2	2	no	—
41	Q₈×C₁₅₁	—	2	2	no	—
42	D₄×C₁₅₁	—	4	2	no	—
43	D₄×C₁₅₁	—	4	2	no	—
44	D₄×C₁₅₁	—	4	2	no	—
45	C₂×C₄×C₁₅₁	—	2	2	no	—
46	C₂×C₄×C₁₅₁	—	2	2	no	—
47	C₂×C₄×C₁₅₁	—	8	2	no	—
48	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
50	C₂²×C₁₅₁	—	4	4	no	—
51	C₂²×C₁₅₁	—	4	2	no	—
52	C₂²×C₁₅₁	#120	1	1	yes	characteristic
53	C₂×D₁₅₁	—	8	2	no	—
54	C₂×D₁₅₁	—	8	1	no	—
55	C₄×C₁₅₁	—	2	1	no	—
56	C₄×C₁₅₁	—	4	2	no	—
57	C₄×C₁₅₁	—	4	2	no	—
58	C₄×C₁₅₁	#135	1	2	yes	—
59	C₄×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₅₁	—	8	2	no	—
61	C₂×C₁₅₁	#449	1	2	yes	—
62	C₂×C₁₅₁	#453	1	1	yes	characteristic
63	D₁₅₁	—	16	1	no	—
64	C₁₅₁	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	151	1	no	maximal, Sylow
66	#559	D₁₅₁	1	1	yes	characteristic, complemented
67	#493	—	151	1	no	—
68	#460	—	151	2	no	—
69	#331	—	151	1	no	—
70	Q₈⋊C₈	—	151	2	no	—
71	C₂×SD₁₆	—	302	2	no	—
72	C₂×D₈	—	302	2	no	—
73	#127	—	151	1	no	—
74	#127	C₂×D₁₅₁	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₅₁	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	151	2	no	—
80	#102	—	302	2	no	—
81	C₄×C₈	—	151	1	no	—
82	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	302	2	no	—
85	C₂×D₄	—	302	1	no	—
86	C₂×D₄	C₂²⋊D₁₅₁	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	604	4	no	—
89	D₈	—	604	4	no	—
90	C₂×C₈	—	302	1	no	—
91	C₂×C₈	—	302	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	604	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	604	2	no	—
106	D₄	—	604	2	no	—
107	D₄	—	604	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
113	C₈	—	302	2	no	—
114	C₈	—	604	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	1208	2	no	—
118	C₂²	—	1208	1	no	—
119	C₂²	#18012708	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#18012759	1	2	yes	—
124	C₄	#18012764	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	2416	1	no	—
127	C₂	#23752357	1	2	yes	—
128	C₂	#23752380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752588

C₂

yes

maximal, characteristic

#23752412

C₂

yes

maximal, complemented

#23751956

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14558991

—

C₂×#14558989

—

#18012736

C₂²

yes

characteristic

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012659

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

#124×C₁₅₁

—

#123×C₁₅₁

—

C₄×Q₈×C₁₅₁

C₂²

yes

—

C₄×D₄×C₁₅₁

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

#14558991

—

#14558989

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

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

C₂²×D₁₅₁

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

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

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#135

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#449

yes

—

C₂×C₁₅₁

#453

yes

characteristic

D₁₅₁

—

C₁₅₁

#1620

yes

characteristic, complemented, Sylow

#1620

—

151

maximal, Sylow

#559

D₁₅₁

yes

characteristic, complemented

#493

—

151

—

#460

—

151

—

#331

—

151

—

Q₈⋊C₈

—

151

—

C₂×SD₁₆

—

302

—

C₂×D₈

—

302

—

#127

—

151

—

#127

C₂×D₁₅₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₅₁

yes

—

C₄×D₄

—

C₄⋊C₈

—

151

—

#102

—

302

—

C₄×C₈

—

151

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

C₂×D₄

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

—

SD₁₆

—

604

—

D₈

—

604

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

C₂³

—

604

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

604

—

106

D₄

—

604

—

107

D₄

—

604

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

113

C₈

—

302

—

114

C₈

—

604

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

1208

—

118

C₂²

—

1208

—

119

C₂²

#18012708

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#18012759

yes

—

124

C₄

#18012764

yes

—

125

C₂

—

126

C₂

—

2416

—

127

C₂

#23752357

yes

—

128

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	#23752588	C₂
hypercenter	#559	D₁₅₁
hypocenter	C₁₅₁	#1620
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

C₂²×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #23752412	C₂	semidirect
3. #23751956	C₂	semidirect
4. Q₈⋊(C₁₅₁⋊C₈)	C₂	semidirect
5. #559×C₁₅₁	C₂	semidirect
64. C₁₅₁	#1620	semidirect
66. #559	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752412

C₂

semidirect

3. #23751956

C₂

semidirect

4. Q₈⋊(C₁₅₁⋊C₈)

C₂

semidirect

5. #559×C₁₅₁

C₂

semidirect

64. C₁₅₁

#1620

semidirect

66. #559

D₁₅₁

semidirect

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

subgroup	index
#23752588	2
#23752412	2
#23751956	2
Q₈⋊(C₁₅₁⋊C₈)	2
#559×C₁₅₁	2
#1620	151

subgroup

index

#23752588

#23752412

#23751956

Q₈⋊(C₁₅₁⋊C₈)

#559×C₁₅₁

#1620

151

Subgroups — #34068169

Conjugacy classes of subgroups

Complements

Maximal subgroups