Group #12008337 — subgroups

Subgroups — #12008337

Order 1280.

260 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₅×#1450	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	2	yes	maximal
3	#10597929	C₂	1	2	yes	maximal, complemented
4	#10597928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#431	C₄	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#322	C₄	1	2	yes	—
9	C₅×C₈²	C₂²	1	1	yes	characteristic
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	C₂²⋊#672	C₄	1	4	yes	—
12	C₄⋊#672	C₄	1	4	yes	—
13	C₄×#672	C₂²	1	2	yes	—
14	C₄×#672	C₂²	1	2	yes	—
15	#11412	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₅×C₄⋊C₈	C₈	1	4	yes	—
19	C₅×C₂²⋊C₈	C₈	1	4	yes	—
20	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×#672	C₂×C₄	1	4	yes	—
26	C₂×#672	C₂×C₄	1	4	yes	—
27	#2162	—	5	2	no	—
28	#1450	D₅	1	1	yes	characteristic
29	#1312	—	5	2	no	—
30	#1311	—	5	1	no	—
31	C₈×C₁₆	—	5	1	no	—
32	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
34	C₂×C₅×C₈	—	2	4	no	—
35	C₂×C₅×C₈	C₄²	1	2	yes	—
36	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
37	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
38	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
39	C₄²×C₅	C₄²	1	1	yes	characteristic
40	C₄²×C₅	C₂×C₈	1	2	yes	—
41	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
42	#672	—	2	4	no	—
43	#672	—	2	4	no	—
44	#431	Dic₅	1	2	yes	—
45	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
46	C₄⋊C₁₆	—	5	4	no	—
47	C₂²⋊C₁₆	—	5	4	no	—
48	C₄×C₁₆	—	5	2	no	—
49	C₄×C₁₆	—	5	2	no	—
50	#322	Dic₅	1	2	yes	—
51	C₈²	C₂×D₅	1	1	yes	characteristic
52	C₈²	C₂×D₅	1	1	yes	characteristic
53	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
60	C₅×C₈	—	2	2	no	—
61	C₅×C₈	—	2	4	no	—
62	C₅×C₈	—	2	4	no	—
63	C₅×C₈	#131	1	4	yes	—
64	C₂²×C₈	C₄×D₅	1	2	yes	—
65	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
66	C₂×C₁₆	—	5	4	no	—
67	C₂×C₁₆	—	5	4	no	—
68	C₄⋊C₈	#168	1	4	yes	—
69	C₂²⋊C₈	#168	1	4	yes	—
70	C₄×C₈	C₄×D₅	1	2	yes	—
71	C₄×C₈	C₂×Dic₅	1	2	yes	—
72	C₄×C₈	C₂×Dic₅	1	2	yes	—
73	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
74	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
75	C₂²×C₅	—	2	2	no	—
76	C₂²×C₅	—	2	1	no	—
77	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	—	2	1	no	—
80	C₄×C₅	#405	1	2	yes	—
81	C₄×C₅	#504	1	4	yes	—
82	C₂²×C₄	C₂×#168	1	2	yes	—
83	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	C₂×#168	1	4	yes	—
86	C₂×C₈	C₂×#168	1	4	yes	—
87	C₂×C₈	C₄×Dic₅	1	2	yes	—
88	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
89	C₄²	C₂×#168	1	2	yes	—
90	C₄²	C₄×Dic₅	1	1	yes	characteristic
91	C₄²	C₂²×Dic₅	1	1	yes	characteristic
92	C₁₆	—	10	4	no	—
93	C₁₆	—	10	4	no	—
94	C₂×C₅	—	4	1	no	—
95	C₂×C₅	#1450	1	1	yes	characteristic
96	C₂×C₅	#2108	1	2	yes	—
97	C₂³	C₄×#168	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	C₄×#168	1	1	yes	characteristic
101	C₂×C₄	C₄×#168	1	2	yes	—
102	C₂×C₄	C₂²×#168	1	2	yes	—
103	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	#4047	1	4	yes	—
108	C₅	#11412	1	1	yes	characteristic, complemented, Sylow
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂×C₄×#168	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	#65390	1	2	yes	—
115	C₄	#65429	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	#10600517	1	1	yes	characteristic
118	C₂	#10600674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1450

C₂

yes

maximal, characteristic

#10600537

C₂

yes

maximal

#10597929

C₂

yes

maximal, complemented

#10597928

C₂

yes

maximal, characteristic, complemented

C₈×#672

C₂

yes

maximal, characteristic, complemented

C₅×#431

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#322

C₄

yes

—

C₅×C₈²

C₂²

yes

characteristic

C₅×C₈²

C₂²

yes

characteristic

C₂²⋊#672

C₄

yes

—

C₄⋊#672

C₄

yes

—

C₄×#672

C₂²

yes

—

C₄×#672

C₂²

yes

—

#11412

—

maximal, Sylow

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

C₈

yes

—

C₅×C₂²⋊C₈

C₈

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

—

C₂×#672

C₂×C₄

yes

—

#2162

—

#1450

D₅

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₂²×C₄×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

—

C₄²×C₅

C₄²

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

C₄²×C₅

C₂²×C₄

yes

—

—

Dic₅

yes

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

Dic₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₈²

C₂×D₅

yes

characteristic

C₂³×C₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×C₈

yes

—

C₂×C₄×C₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#131

yes

—

C₂²×C₈

C₄×D₅

yes

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

#168

yes

—

C₂²⋊C₈

#168

yes

—

C₄×C₈

C₄×D₅

yes

—

C₄×C₈

C₂×Dic₅

yes

—

C₄×C₈

C₂×Dic₅

yes

—

C₄×C₈

C₂×Dic₅

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄×C₈

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#405

yes

—

C₄×C₅

#504

yes

—

C₂²×C₄

C₂×#168

yes

—

C₂²×C₄

C₄×Dic₅

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×#168

yes

—

C₂×C₈

C₂×#168

yes

—

C₂×C₈

C₄×Dic₅

yes

—

C₂×C₈

C₂×C₄×D₅

yes

—

C₄²

C₂×#168

yes

—

C₄²

C₄×Dic₅

yes

characteristic

C₄²

C₂²×Dic₅

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

#1450

yes

characteristic

C₂×C₅

#2108

yes

—

C₂³

C₄×#168

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₄×#168

yes

characteristic

101

C₂×C₄

C₄×#168

yes

—

102

C₂×C₄

C₂²×#168

yes

—

103

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

#4047

yes

—

108

C₅

#11412

yes

characteristic, complemented, Sylow

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×#168

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#65390

yes

—

115

C₄

#65429

yes

—

116

C₂

—

117

C₂

#10600517

yes

characteristic

118

C₂

#10600674

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅
commutator, socle	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×#1450	C₂
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#11412
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ #1450

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₅

commutator, socle

C₂²×C₅

C₂×C₄×C₈

Fitting

C₅×#1450

C₂

prime radical

C₂³×C₅

C₄×C₈

cosocle

C₄×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₄×C₈ ◁ #1450

N	H ≅ G/N	type
3. #10597929	C₂	semidirect
4. #10597928	C₂	semidirect
5. C₈×#672	C₂	semidirect
108. C₅	#11412	semidirect

H ≅ G/N

type

3. #10597929

C₂

semidirect

4. #10597928

C₂

semidirect

5. C₈×#672

C₂

semidirect

108. C₅

#11412

semidirect

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

subgroup	index
C₅×#1450	2
#10600537	2
#10597929	2
#10597928	2
C₈×#672	2
#11412	5

subgroup

index

Subgroups — #12008337

Conjugacy classes of subgroups

Complements

Maximal subgroups