Group #34080914 — subgroups

Subgroups — #34080914

Order 19328.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#23753025	C₂	1	4	yes	maximal, complemented
2	#23752484	C₂	1	4	yes	maximal, complemented
3	#23752394	C₂	1	4	yes	maximal, complemented
4	#23752330	C₂	1	2	yes	maximal, complemented
5	#494×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#18012787	C₂²	1	4	yes	—
7	#18012771	—	2	8	no	—
8	#18012770	C₂²	1	8	yes	—
9	#18012762	—	2	4	no	—
10	#18012761	C₂²	1	2	yes	—
11	Q₁₆×D₁₅₁	—	2	4	no	—
12	#18012745	C₂²	1	4	yes	—
13	#18012675	C₂²	1	2	yes	—
14	#18012667	C₂²	1	4	yes	—
15	#18012658	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₅₁	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
18	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	#14559005	—	2	8	no	—
20	#14559004	—	4	4	no	—
21	#14559004	C₂³	1	4	yes	—
22	Q₈×D₁₅₁	—	2	8	no	—
23	Dic₆₀₄	D₄	1	2	yes	—
24	#14558994	—	2	8	no	—
25	#14558993	D₄	1	4	yes	—
26	#14558991	—	2	8	no	—
27	D₁₂₀₈	D₄	1	2	yes	—
28	#14558988	C₂³	1	4	yes	—
29	#14558987	—	2	4	no	—
30	C₈×D₁₅₁	—	2	4	no	—
31	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	—
32	Q₁₆×C₁₅₁	—	2	4	no	—
33	Q₁₆×C₁₅₁	—	2	4	no	—
34	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₅₁	—	4	2	no	—
36	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
37	Dic₃₀₂	—	4	4	no	—
38	Dic₃₀₂	C₂×D₄	1	4	yes	—
39	D₆₀₄	—	4	4	no	—
40	D₆₀₄	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₅₁	—	2	4	no	—
42	C₄×D₁₅₁	—	2	4	no	—
43	C₄×D₁₅₁	—	4	8	no	—
44	C₁₅₁⋊C₈	—	2	4	no	—
45	Q₈×C₁₅₁	—	2	8	no	—
46	C₂×C₄×C₁₅₁	—	2	1	no	—
47	C₂×C₄×C₁₅₁	—	2	4	no	—
48	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₅₁	C₂×D₄	1	2	yes	—
50	C₈×C₁₅₁	C₂×D₄	1	2	yes	—
51	C₂²×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₅₁	—	2	4	no	—
53	C₄×C₁₅₁	—	2	2	no	—
54	C₄×C₁₅₁	—	4	4	no	—
55	C₄×C₁₅₁	C₂²×D₄	1	2	yes	—
56	Dic₁₅₁	—	2	4	no	—
57	C₂×C₁₅₁	—	2	1	no	—
58	C₂×C₁₅₁	D₄²	1	1	yes	characteristic
59	D₁₅₁	—	4	4	no	—
60	C₁₅₁	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	151	1	no	maximal, Sylow
62	#578	—	151	4	no	—
63	#494	D₁₅₁	1	1	yes	characteristic, complemented
64	#473	—	151	4	no	—
65	#456	—	151	4	no	—
66	#443	—	151	2	no	—
67	2₋¹⁺⁴	—	151	4	no	—
68	#137	—	302	8	no	—
69	#137	—	151	8	no	—
70	#135	—	302	4	no	—
71	#135	—	151	2	no	—
72	C₂×Q₁₆	—	302	4	no	—
73	C₂×Q₁₆	C₂×D₁₅₁	1	4	yes	—
74	#131	—	151	4	no	—
75	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	—
76	#108	—	151	2	no	—
77	#104	—	151	4	no	—
78	#101	—	151	4	no	—
79	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
80	#41	—	302	8	no	—
81	#41	—	604	4	no	—
82	#41	—	151	4	no	—
83	C₂×Q₈	—	302	8	no	—
84	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	302	8	no	—
88	Q₁₆	—	151	2	no	—
89	SD₁₆	—	302	8	no	—
90	SD₁₆	—	151	4	no	—
91	D₈	—	151	2	no	—
92	M₁₆	—	302	4	no	—
93	M₁₆	—	151	4	no	—
94	C₂×C₈	—	302	4	no	—
95	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	604	4	no	—
100	Q₈	—	151	4	no	—
101	D₄	—	302	4	no	—
102	D₄	—	604	4	no	—
103	D₄	—	151	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	302	4	no	—
107	C₂×C₄	—	604	8	no	—
108	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
109	C₈	—	302	4	no	—
110	C₈	D₄×D₁₅₁	1	2	yes	—
111	C₈	D₄×D₁₅₁	1	2	yes	—
112	C₂²	—	302	4	no	—
113	C₂²	#18012784	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	302	4	no	—
117	C₄	C₂×D₄×D₁₅₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	604	4	no	—
120	C₂	#23752847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23753025

C₂

yes

maximal, complemented

#23752484

C₂

yes

maximal, complemented

#23752394

C₂

yes

maximal, complemented

#23752330

C₂

yes

maximal, complemented

#494×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#18012787

C₂²

yes

—

#18012771

—

#18012770

C₂²

yes

—

#18012762

—

#18012761

C₂²

yes

—

Q₁₆×D₁₅₁

—

#18012745

C₂²

yes

—

#18012675

C₂²

yes

—

#18012667

C₂²

yes

—

#18012658

C₂²

yes

—

C₂×Q₁₆×C₁₅₁

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₅₁

—

Dic₆₀₄

D₄

yes

—

#14558994

—

#14558993

D₄

yes

—

#14558991

—

D₁₂₀₈

D₄

yes

—

#14558988

C₂³

yes

—

#14558987

—

C₈×D₁₅₁

—

C₂×Q₈×C₁₅₁

C₂³

yes

—

Q₁₆×C₁₅₁

—

Q₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

C₂×D₄

yes

—

D₆₀₄

—

D₆₀₄

C₂×D₄

yes

—

C₂²⋊D₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂×D₄

yes

—

C₈×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

H(5,𝔽₂)

yes

characteristic

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂²×D₄

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

D₄²

yes

characteristic

D₁₅₁

—

C₁₅₁

#3293

yes

characteristic, complemented, Sylow

#3293

—

151

maximal, Sylow

#578

—

151

—

#494

D₁₅₁

yes

characteristic, complemented

#473

—

151

—

#456

—

151

—

#443

—

151

—

2₋¹⁺⁴

—

151

—

#137

—

302

—

#137

—

151

—

#135

—

302

—

#135

—

151

—

C₂×Q₁₆

—

302

—

C₂×Q₁₆

C₂×D₁₅₁

yes

—

#131

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

—

#108

—

151

—

#104

—

151

—

#101

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

#41

—

302

—

#41

—

604

—

#41

—

151

—

C₂×Q₈

—

302

—

C₂×Q₈

C₂²×D₁₅₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

302

—

Q₁₆

—

151

—

SD₁₆

—

302

—

SD₁₆

—

151

—

D₈

—

151

—

M₁₆

—

302

—

M₁₆

—

151

—

C₂×C₈

—

302

—

C₂×C₈

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅₁

yes

characteristic

Q₈

—

Q₈

—

604

—

100

Q₈

—

151

—

101

D₄

—

302

—

102

D₄

—

604

—

103

D₄

—

151

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

302

—

107

C₂×C₄

—

604

—

108

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

109

C₈

—

302

—

110

C₈

D₄×D₁₅₁

yes

—

111

C₈

D₄×D₁₅₁

yes

—

112

C₂²

—

302

—

113

C₂²

#18012784

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

302

—

117

C₄

C₂×D₄×D₁₅₁

yes

—

118

C₂

—

119

C₂

—

604

—

120

C₂

#23752847

yes

characteristic

	subgroup	G/H
center	C₂	#23752847
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#494×C₁₅₁	C₂
socle	C₂×C₁₅₁	D₄²
hypercenter	#494	D₁₅₁
hypocenter	C₁₅₁	#3293
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#23752847

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#494×C₁₅₁

C₂

socle

C₂×C₁₅₁

D₄²

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #23753025	C₂	semidirect
2. #23752484	C₂	semidirect
3. #23752394	C₂	semidirect
4. #23752330	C₂	semidirect
5. #494×C₁₅₁	C₂	semidirect
60. C₁₅₁	#3293	semidirect
63. #494	D₁₅₁	semidirect

H ≅ G/N

type

1. #23753025

C₂

semidirect

2. #23752484

C₂

semidirect

3. #23752394

C₂

semidirect

4. #23752330

C₂

semidirect

5. #494×C₁₅₁

C₂

semidirect

60. C₁₅₁

#3293

semidirect

63. #494

D₁₅₁

semidirect

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

subgroup	index
#23753025	2
#23752484	2
#23752394	2
#23752330	2
#494×C₁₅₁	2
#3293	151

subgroup

index

#494×C₁₅₁

151

Subgroups — #34080914

Conjugacy classes of subgroups

Complements

Maximal subgroups