Group #11991756 — subgroups

Subgroups — #11991756

Order 1280.

656 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂×C₅×#564	C₂	1	1	yes	maximal, characteristic
2	#10602534	C₂	1	3	yes	maximal
3	C₂×#64763	C₂	1	3	yes	maximal
4	C₅×#564	—	2	6	no	—
5	C₅×#564	C₄	1	4	yes	complemented
6	C₂×C₅×#125	—	2	6	no	—
7	C₂×C₅×#125	C₂²	1	3	yes	—
8	#65224	C₂²	1	1	yes	characteristic
9	C₂×#3935	C₂²	1	3	yes	—
10	#64763	—	2	6	no	—
11	#64749	—	2	6	no	—
12	#8509	—	5	1	no	maximal, Sylow
13	C₅×#125	—	2	24	no	—
14	C₅×#125	—	2	24	no	—
15	C₅×#125	—	2	6	no	—
16	C₅×#125	C₂×C₄	1	6	yes	—
17	C₅×#125	D₄	1	6	yes	—
18	C₂×C₅×C₄⋊C₄	—	2	6	no	—
19	C₂×C₅×C₄⋊C₄	—	2	6	no	—
20	C₂×C₅×C₄⋊C₄	D₄	1	6	yes	—
21	C₂×C₄²×C₅	—	2	2	no	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	—	2	3	no	—
24	C₂²×#168	—	2	3	no	—
25	#3935	—	2	6	no	—
26	C₂×#564	D₅	1	1	yes	characteristic
27	#1992	—	5	3	no	—
28	C₂×#330	—	5	3	no	—
29	C₂²×C₄×C₅	—	2	6	no	—
30	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
31	C₂²×Dic₅	—	4	1	no	—
32	C₅×C₄⋊C₄	—	2	24	no	—
33	C₅×C₄⋊C₄	—	2	12	no	—
34	C₅×C₄⋊C₄	—	2	12	no	—
35	C₅×C₄⋊C₄	—	2	6	no	—
36	C₅×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
37	C₄²×C₅	—	2	8	no	—
38	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅	C₂×D₄	1	3	yes	—
40	C₄⋊Dic₅	—	4	6	no	—
41	C₂×#168	—	2	6	no	—
42	C₂×#168	—	4	3	no	—
43	C₂×#168	—	4	3	no	—
44	#564	—	2	6	no	—
45	#564	Dic₅	1	4	yes	complemented
46	C₂×#125	—	2	6	no	—
47	C₂×#125	C₂×D₅	1	3	yes	—
48	C₂×#97	—	5	3	no	—
49	#384	—	5	1	no	—
50	#340	—	10	6	no	—
51	#330	—	10	6	no	—
52	C₂³×C₅	#120	1	1	yes	characteristic
53	C₂×C₄×C₅	—	2	3	no	—
54	C₂×C₄×C₅	—	2	3	no	—
55	C₂×C₄×C₅	—	2	12	no	—
56	C₂×C₄×C₅	—	4	6	no	—
57	C₂×C₄×C₅	—	4	6	no	—
58	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
59	C₂×C₄×C₅	#120	1	3	yes	—
60	C₂×Dic₅	—	8	3	no	—
61	#168	—	4	6	no	—
62	C₂²×C₈	—	10	3	no	—
63	#125	—	2	24	no	—
64	#125	—	2	24	no	—
65	#125	—	2	6	no	—
66	#125	C₂×Dic₅	1	6	yes	—
67	#125	C₂²⋊D₅	1	6	yes	—
68	C₂×C₄⋊C₄	—	2	6	no	—
69	C₂×C₄⋊C₄	—	2	6	no	—
70	C₂×C₄⋊C₄	—	10	3	no	—
71	C₂×C₄⋊C₄	C₂²⋊D₅	1	6	yes	—
72	C₂×C₄²	—	2	2	no	—
73	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
74	#97	—	10	6	no	—
75	C₂²×C₅	#379	1	1	yes	characteristic
76	C₂²×C₅	#454	1	3	yes	—
77	C₂²×C₅	#456	1	3	yes	—
78	C₄×C₅	—	2	3	no	—
79	C₄×C₅	—	2	3	no	—
80	C₄×C₅	—	4	12	no	—
81	Dic₅	—	16	1	no	—
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	—	20	1	no	—
84	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
85	C₂×C₈	—	10	6	no	—
86	C₂×C₈	—	20	3	no	—
87	C₂×C₈	—	20	3	no	—
88	C₄⋊C₄	—	2	24	no	—
89	C₄⋊C₄	—	2	12	no	—
90	C₄⋊C₄	—	2	12	no	—
91	C₄⋊C₄	—	2	6	no	—
92	C₄⋊C₄	—	20	6	no	—
93	C₄⋊C₄	C₂²⋊Dic₅	1	12	yes	—
94	C₄²	—	2	8	no	—
95	C₄²	C₂²×Dic₅	1	1	yes	characteristic
96	C₄²	C₂×C₂²⋊D₅	1	3	yes	—
97	C₂×C₅	#1658	1	1	yes	characteristic
98	C₂×C₅	#1663	1	3	yes	—
99	C₂×C₅	#1793	1	3	yes	—
100	C₂³	#4099	1	1	yes	characteristic
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	40	3	no	—
107	C₂×C₄	C₂×C₂²⋊Dic₅	1	3	yes	—
108	C₂×C₄	#4099	1	3	yes	—
109	C₈	—	20	6	no	—
110	C₅	#8509	1	1	yes	characteristic, complemented, Sylow
111	C₂²	#65534	1	3	yes	—
112	C₂²	#65535	1	3	yes	—
113	C₂²	#65536	1	1	yes	characteristic
114	C₄	—	2	3	no	—
115	C₄	—	2	3	no	—
116	C₄	—	4	12	no	—
117	C₄	—	80	1	no	—
118	C₂	#10602753	1	1	yes	characteristic
119	C₂	#10602754	1	3	yes	—
120	C₂	#10602936	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#564

C₂

yes

maximal, characteristic

#10602534

C₂

yes

maximal

C₂×#64763

C₂

yes

maximal

C₅×#564

—

C₅×#564

C₄

yes

complemented

C₂×C₅×#125

—

C₂×C₅×#125

C₂²

yes

—

#65224

C₂²

yes

characteristic

C₂×#3935

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₅×#125

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂²×#168

—

#3935

—

C₂×#564

D₅

yes

characteristic

#1992

—

C₂×#330

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂²⋊C₄

yes

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄⋊Dic₅

—

—

—

—

—

Dic₅

yes

complemented

C₂×#125

—

C₂×#125

C₂×D₅

yes

—

C₂×#97

—

#384

—

#340

—

#330

—

C₂³×C₅

#120

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅

#120

yes

—

C₂×Dic₅

—

#168

—

C₂²×C₈

—

—

—

—

C₂×Dic₅

yes

—

#125

C₂²⋊D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#97

—

C₂²×C₅

#379

yes

characteristic

C₂²×C₅

#454

yes

—

C₂²×C₅

#456

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊Dic₅

yes

—

C₄²

—

C₄²

C₂²×Dic₅

yes

characteristic

C₄²

C₂×C₂²⋊D₅

yes

—

C₂×C₅

#1658

yes

characteristic

C₂×C₅

#1663

yes

—

C₂×C₅

#1793

yes

—

100

C₂³

#4099

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₂²⋊Dic₅

yes

—

108

C₂×C₄

#4099

yes

—

109

C₈

—

110

C₅

#8509

yes

characteristic, complemented, Sylow

111

C₂²

#65534

yes

—

112

C₂²

#65535

yes

—

113

C₂²

#65536

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₂

#10602753

yes

characteristic

119

C₂

#10602754

yes

—

120

C₂

#10602936

yes

—

	subgroup	G/H
center	C₂³	#4099
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#564	C₂
socle, prime radical	C₂³×C₅	#120
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#564	D₅
hypocenter	C₅	#8509
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

subgroup

G/H

center

C₂³

#4099

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₂×C₅×#564

C₂

socle, prime radical

C₂³×C₅

#120

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

N	H ≅ G/N	type
5. C₅×#564	C₄	semidirect
45. #564	Dic₅	semidirect
110. C₅	#8509	semidirect

H ≅ G/N

type

5. C₅×#564

C₄

semidirect

45. #564

Dic₅

semidirect

110. C₅

#8509

semidirect

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

subgroup	index
C₂×C₅×#564	2
#10602534	2
C₂×#64763	2
#8509	5

subgroup

index

C₂×C₅×#564

#10602534

C₂×#64763

#8509

Subgroups — #11991756

Conjugacy classes of subgroups

Complements

Maximal subgroups