Group #11756445 — subgroups

Subgroups — #11756445

Order 960.

344 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#91779	C₂	1	4	yes	maximal
3	#91778	C₂	1	4	yes	maximal
4	#91778	C₂	1	4	yes	maximal
5	#91759	C₂	1	2	yes	maximal
6	C₅⋊#564	—	3	1	no	maximal, Hall
7	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₅	C₂²	1	2	yes	—
11	C₄⋊Dic₁₅	C₂²	1	2	yes	—
12	C₄⋊Dic₁₅	C₂²	1	4	yes	—
13	#6712	C₂²	1	4	yes	—
14	#6712	C₂²	1	4	yes	—
15	#6712	C₂²	1	4	yes	—
16	#6712	C₂²	1	4	yes	—
17	C₄×Dic₁₅	C₂²	1	4	yes	—
18	C₃⋊#564	—	5	1	no	maximal, Hall
19	C₅×#125	D₃	1	1	yes	characteristic
20	#4036	—	3	4	no	—
21	#4035	—	3	4	no	—
22	#4035	—	3	4	no	—
23	#4016	—	3	2	no	—
24	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
25	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
27	C₂×Dic₁₅	C₂³	1	4	yes	—
28	C₂×Dic₁₅	C₂³	1	4	yes	—
29	C₃×#125	D₅	1	1	yes	characteristic
30	#890	—	5	4	no	—
31	#889	—	5	4	no	—
32	#889	—	5	4	no	—
33	#870	—	5	2	no	—
34	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
35	C₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
36	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
37	C₄⋊Dic₅	—	3	2	no	—
38	C₄⋊Dic₅	—	3	2	no	—
39	C₄⋊Dic₅	—	3	4	no	—
40	#683	—	3	4	no	—
41	#683	—	3	4	no	—
42	#683	—	3	4	no	—
43	#683	—	3	4	no	—
44	C₄×Dic₅	—	3	4	no	—
45	#564	—	15	1	no	Sylow
46	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
47	C₃×C₄×C₅	—	2	1	no	—
48	C₃×C₄×C₅	—	2	4	no	—
49	C₃×C₄×C₅	—	2	2	no	—
50	Dic₁₅	—	2	4	no	—
51	Dic₁₅	—	2	4	no	—
52	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
53	C₃×C₄⋊C₄	C₂×D₅	1	4	yes	—
54	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
55	C₄⋊Dic₃	—	5	2	no	—
56	C₄⋊Dic₃	—	5	2	no	—
57	C₄⋊Dic₃	—	5	4	no	—
58	Dic₃⋊C₄	—	5	4	no	—
59	Dic₃⋊C₄	—	5	4	no	—
60	Dic₃⋊C₄	—	5	4	no	—
61	Dic₃⋊C₄	—	5	4	no	—
62	C₄×Dic₃	—	5	4	no	—
63	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
64	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
66	C₂×Dic₅	—	3	4	no	—
67	C₂×Dic₅	—	3	4	no	—
68	#125	—	15	4	no	—
69	#125	—	15	4	no	—
70	#125	—	15	4	no	—
71	#125	—	15	2	no	—
72	#125	D₁₅	1	1	yes	characteristic
73	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
74	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
75	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
76	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
78	C₂×Dic₃	—	5	4	no	—
79	C₂×Dic₃	—	5	4	no	—
80	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
81	C₄×C₅	—	2	1	no	—
82	C₄×C₅	—	2	4	no	—
83	C₄×C₅	—	2	2	no	—
84	Dic₅	—	6	4	no	—
85	Dic₅	—	6	4	no	—
86	C₄⋊C₄	—	15	4	no	—
87	C₄⋊C₄	—	15	4	no	—
88	C₄⋊C₄	—	15	4	no	—
89	C₄⋊C₄	—	15	4	no	—
90	C₄⋊C₄	—	15	4	no	—
91	C₄⋊C₄	—	15	2	no	—
92	C₄⋊C₄	—	15	2	no	—
93	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
94	C₄⋊C₄	C₂×D₁₅	1	4	yes	—
95	C₄²	—	15	4	no	—
96	C₄²	C₂×D₁₅	1	1	yes	characteristic
97	C₃×C₅	#564	1	1	yes	characteristic, complemented, Hall
98	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
99	C₃×C₄	—	2	1	no	—
100	C₃×C₄	—	2	4	no	—
101	C₃×C₄	—	2	2	no	—
102	Dic₃	—	10	4	no	—
103	Dic₃	—	10	4	no	—
104	C₂×C₅	#1007	1	1	yes	characteristic
105	C₂×C₅	#1010	1	2	yes	—
106	C₂×C₄	—	15	4	no	—
107	C₂×C₄	—	15	4	no	—
108	C₂×C₄	C₂²×D₁₅	1	4	yes	—
109	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₁₅	1	2	yes	—
111	C₂×C₃	#4147	1	1	yes	characteristic
112	C₂×C₃	#4150	1	2	yes	—
113	C₅	C₃⋊#564	1	1	yes	characteristic, complemented, Sylow
114	C₂²	C₂³×D₁₅	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	4	no	—
117	C₄	—	2	2	no	—
118	C₄	—	30	4	no	—
119	C₄	—	30	4	no	—
120	C₃	C₅⋊#564	1	1	yes	characteristic, complemented, Sylow
121	C₂	#92098	1	1	yes	characteristic
122	C₂	#92101	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#125

C₂

yes

maximal, characteristic

#91779

C₂

yes

maximal

#91778

C₂

yes

maximal

#91778

C₂

yes

maximal

#91759

C₂

yes

maximal

C₅⋊#564

—

maximal, Hall

C₃×C₅×C₄⋊C₄

C₂²

yes

—

C₃×C₅×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₅

C₂²

yes

characteristic

C₄⋊Dic₁₅

C₂²

yes

—

C₄⋊Dic₁₅

C₂²

yes

—

C₄⋊Dic₁₅

C₂²

yes

—

#6712

C₂²

yes

—

#6712

C₂²

yes

—

#6712

C₂²

yes

—

#6712

C₂²

yes

—

C₄×Dic₁₅

C₂²

yes

—

C₃⋊#564

—

maximal, Hall

C₅×#125

D₃

yes

—

—

—

—

C₂×C₃×C₄×C₅

C₂³

yes

—

C₂×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×C₃×C₄×C₅

C₂³

yes

—

C₂×Dic₁₅

C₂³

yes

—

C₂×Dic₁₅

C₂³

yes

—

C₃×#125

D₅

yes

—

—

—

—

C₅×C₄⋊C₄

C₂×D₃

yes

—

C₅×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₅

C₂×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

C₄×Dic₅

—

#564

—

Sylow

C₂²×C₃×C₅

C₂⁴

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

Dic₁₅

—

Dic₁₅

—

C₃×C₄⋊C₄

C₂×D₅

yes

—

C₃×C₄⋊C₄

C₂×D₅

yes

—

C₃×C₄²

C₂×D₅

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₄×C₅

C₂²×D₃

yes

—

C₂×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×C₄×C₅

C₂²×D₃

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

—

—

—

—

D₁₅

yes

characteristic

C₂×C₃×C₅

2₋¹⁺⁴

yes

characteristic

C₂×C₃×C₅

2₋¹⁺⁴

yes

—

C₂×C₃×C₄

C₂²×D₅

yes

—

C₂×C₃×C₄

C₂²×D₅

yes

characteristic

C₂×C₃×C₄

C₂²×D₅

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂²×C₅

C₂³×D₃

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₅

yes

—

C₄⋊C₄

C₂×D₁₅

yes

—

C₄²

—

C₄²

C₂×D₁₅

yes

characteristic

C₃×C₅

#564

yes

characteristic, complemented, Hall

C₂²×C₃

C₂³×D₅

yes

characteristic

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

Dic₃

—

103

Dic₃

—

104

C₂×C₅

#1007

yes

characteristic

105

C₂×C₅

#1010

yes

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×D₁₅

yes

—

109

C₂×C₄

C₂²×D₁₅

yes

characteristic

110

C₂×C₄

C₂²×D₁₅

yes

—

111

C₂×C₃

#4147

yes

characteristic

112

C₂×C₃

#4150

yes

—

113

C₅

C₃⋊#564

yes

characteristic, complemented, Sylow

114

C₂²

C₂³×D₁₅

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₃

C₅⋊#564

yes

characteristic, complemented, Sylow

121

C₂

#92098

yes

characteristic

122

C₂

#92101

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, prime radical, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×#125	C₂
hypercenter	#125	D₁₅
hypocenter	C₃×C₅	#564
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅

commutator, socle, prime radical, cosocle

C₂²×C₃×C₅

C₂⁴

Fitting

C₃×C₅×#125

C₂

D₁₅

C₃×C₅

G ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
97. C₃×C₅	#564	semidirect
113. C₅	C₃⋊#564	semidirect
120. C₃	C₅⋊#564	semidirect

H ≅ G/N

type

97. C₃×C₅

#564

semidirect

113. C₅

C₃⋊#564

semidirect

120. C₃

C₅⋊#564

semidirect

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

subgroup	index
C₃×C₅×#125	2
#91779	2
#91778	2
#91759	2
C₅⋊#564	3
C₃⋊#564	5

subgroup

index

C₃×C₅×#125

Subgroups — #11756445

Conjugacy classes of subgroups

Complements

Maximal subgroups