Group #12002377 — subgroups

Subgroups — #12002377

Order 1280.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2845	C₂	1	1	yes	maximal, characteristic, complemented
2	#10600346	C₂	1	3	yes	maximal
3	#10598121	C₂	1	3	yes	maximal, complemented
4	C₂×#4020	C₂²	1	1	yes	characteristic
5	C₅×#398	—	2	6	no	—
6	C₅×#382	C₂²	1	3	yes	—
7	C₂×#3941	—	2	6	no	—
8	C₂×#3935	C₂²	1	3	yes	—
9	#10360	—	5	1	no	maximal, Sylow
10	C₂²×D₂₀	—	2	3	no	—
11	C₂×C₅×C₄⋊C₄	—	2	6	no	—
12	C₂×C₅×C₄⋊C₄	D₄	1	6	yes	—
13	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
14	C₂²×#168	—	2	3	no	—
15	#4020	—	2	3	no	—
16	#4020	Q₈	1	2	yes	—
17	C₅×#95	—	2	2	no	—
18	C₅×#95	—	2	6	no	—
19	#3941	—	4	12	no	—
20	#3935	—	2	6	no	—
21	#2845	D₅	1	1	yes	characteristic, complemented
22	#1961	—	5	3	no	—
23	#1295	—	5	3	no	—
24	C₂³×D₅	—	4	1	no	—
25	C₂²×C₄×C₅	—	2	6	no	—
26	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
27	C₂×D₂₀	—	2	6	no	—
28	C₂×D₂₀	—	4	6	no	—
29	C₂×D₂₀	—	4	6	no	—
30	C₂×D₂₀	—	4	3	no	—
31	C₅×C₄⋊C₄	—	4	6	no	—
32	C₅×C₄⋊C₄	—	8	6	no	—
33	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅	#41	1	3	yes	—
35	C₂×#168	—	2	6	no	—
36	C₂×#168	—	4	3	no	—
37	C₂×#168	—	4	3	no	—
38	C₂×#127	—	5	1	no	—
39	C₂×#102	—	10	6	no	—
40	C₂×#97	—	5	3	no	—
41	#398	—	2	6	no	—
42	#382	C₂×D₅	1	3	yes	—
43	C₂³×C₅	#120	1	1	yes	characteristic
44	C₂²×D₅	—	4	2	no	—
45	C₂²×D₅	—	8	3	no	—
46	C₂²×D₅	—	8	3	no	—
47	C₂×C₄×C₅	—	2	3	no	—
48	C₂×C₄×C₅	—	2	3	no	—
49	C₂×C₄×C₅	—	4	6	no	—
50	C₂×C₄×C₅	—	4	6	no	—
51	C₂×C₄×C₅	—	4	6	no	—
52	C₂×C₄×C₅	C₂²⋊Q₈	1	3	yes	—
53	C₂×C₄×C₅	#124	1	3	yes	—
54	D₂₀	—	4	12	no	—
55	D₂₀	—	8	6	no	—
56	#168	—	4	6	no	—
57	C₂²×D₄	—	10	3	no	—
58	C₂²×C₈	—	10	3	no	—
59	#127	—	5	2	no	—
60	#127	—	10	3	no	—
61	C₂×C₄⋊C₄	—	2	6	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₅	1	6	yes	—
63	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
64	#102	—	20	12	no	—
65	#97	—	10	6	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₅	#393	1	1	yes	characteristic
69	C₂²×C₅	#453	1	3	yes	—
70	C₂²×C₅	#454	1	3	yes	—
71	C₂×D₅	—	8	6	no	—
72	C₂×D₅	—	16	3	no	—
73	C₂×D₅	—	16	1	no	—
74	C₄×C₅	—	2	3	no	—
75	C₄×C₅	—	2	3	no	—
76	C₄×C₅	—	8	6	no	—
77	C₂⁴	—	20	1	no	—
78	C₂×D₄	—	10	6	no	—
79	C₂×D₄	—	20	6	no	—
80	C₂×D₄	—	20	6	no	—
81	C₂×D₄	—	20	3	no	—
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
84	C₂×C₈	—	10	6	no	—
85	C₂×C₈	—	20	3	no	—
86	C₂×C₈	—	20	3	no	—
87	C₄⋊C₄	—	4	6	no	—
88	C₄⋊C₄	—	8	6	no	—
89	C₄²	Q₈×D₅	1	1	yes	characteristic
90	C₄²	#713	1	3	yes	—
91	C₂×C₅	#1656	1	3	yes	—
92	C₂×C₅	#1658	1	1	yes	characteristic
93	C₂×C₅	#2027	1	3	yes	—
94	D₅	—	16	2	no	—
95	C₂³	—	20	2	no	—
96	C₂³	—	40	3	no	—
97	C₂³	—	40	3	no	—
98	C₂³	#4099	1	1	yes	characteristic
99	D₄	—	20	12	no	—
100	D₄	—	40	6	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	4	6	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	#4092	1	3	yes	—
107	C₂×C₄	#4093	1	3	yes	—
108	C₈	—	20	6	no	—
109	C₅	#10360	1	1	yes	characteristic, complemented, Sylow
110	C₂²	—	40	6	no	—
111	C₂²	—	80	1	no	—
112	C₂²	—	80	3	no	—
113	C₂²	#65520	1	1	yes	characteristic
114	C₂²	#65530	1	3	yes	—
115	C₂²	#65534	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	80	2	no	—
120	C₂	#10602696	1	3	yes	—
121	C₂	#10602745	1	3	yes	—
122	C₂	#10602753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2845

C₂

yes

maximal, characteristic, complemented

#10600346

C₂

yes

maximal

#10598121

C₂

yes

maximal, complemented

C₂×#4020

C₂²

yes

characteristic

C₅×#398

—

C₅×#382

C₂²

yes

—

C₂×#3941

—

C₂×#3935

C₂²

yes

—

#10360

—

maximal, Sylow

C₂²×D₂₀

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

#4020

—

#4020

Q₈

yes

—

C₅×#95

—

C₅×#95

—

#3941

—

#3935

—

#2845

D₅

yes

characteristic, complemented

#1961

—

#1295

—

C₂³×D₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄²×C₅

#41

yes

—

—

—

—

—

—

C₂×#97

—

#398

—

#382

C₂×D₅

yes

—

C₂³×C₅

#120

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

#124

yes

—

D₂₀

—

D₂₀

—

#168

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

—

—

—

—

C₂²×C₅

#393

yes

characteristic

C₂²×C₅

#453

yes

—

C₂²×C₅

#454

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₅

yes

characteristic

C₄²

#713

yes

—

C₂×C₅

#1656

yes

—

C₂×C₅

#1658

yes

characteristic

C₂×C₅

#2027

yes

—

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#4099

yes

characteristic

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#4092

yes

—

107

C₂×C₄

#4093

yes

—

108

C₈

—

109

C₅

#10360

yes

characteristic, complemented, Sylow

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#65520

yes

characteristic

114

C₂²

#65530

yes

—

115

C₂²

#65534

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

120

C₂

#10602696

yes

—

121

C₂

#10602745

yes

—

122

C₂

#10602753

yes

characteristic

	subgroup	G/H
center	C₂³	#4099
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2845	C₂
socle	C₂³×C₅	#120
prime radical	C₂×#4020	C₂²
hypercenter	#2845	D₅
hypocenter	C₅	#10360
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#4099

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

C₂²

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
1. C₅×#2845	C₂	semidirect
3. #10598121	C₂	semidirect
21. #2845	D₅	semidirect
109. C₅	#10360	semidirect

H ≅ G/N

type

1. C₅×#2845

C₂

semidirect

3. #10598121

C₂

semidirect

21. #2845

D₅

semidirect

109. C₅

#10360

semidirect

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

subgroup	index
C₅×#2845	2
#10600346	2
#10598121	2
#10360	5

subgroup

index

Subgroups — #12002377

Conjugacy classes of subgroups

Complements

Maximal subgroups