Group #11751344 — subgroups

Subgroups — #11751344

Order 960.

320 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 75 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×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#91824	C₂	1	3	yes	maximal, complemented
3	#91118	C₂	1	3	yes	maximal, complemented
4	#65532	—	3	1	no	maximal, Hall
5	C₃×C₅×#41	—	2	3	no	—
6	C₂×C₃×C₅×D₄	—	2	3	no	—
7	C₂×C₃×C₅×D₄	C₂²	1	3	yes	—
8	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic, complemented
9	#6719	—	2	3	no	—
10	#6719	C₂²	1	3	yes	complemented
11	#6712	—	2	3	no	—
12	#5243	—	5	1	no	maximal, Hall
13	C₅×H(5,𝔽₂)	D₃	1	1	yes	characteristic
14	#4081	—	3	3	no	—
15	#3966	—	3	3	no	—
16	C₂³×C₃×C₅	—	2	1	no	—
17	C₂³×C₃×C₅	D₄	1	3	yes	—
18	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₃×C₅×Q₈	—	2	1	no	—
20	C₃×C₅×D₄	—	2	3	no	—
21	C₃×C₅×D₄	—	2	6	no	—
22	C₂×C₃×C₄×C₅	—	2	3	no	—
23	C₂×C₃×C₄×C₅	D₄	1	3	yes	—
24	C₂×Dic₁₅	—	2	3	no	—
25	C₂×Dic₁₅	—	2	3	no	—
26	C₃×H(5,𝔽₂)	D₅	1	1	yes	characteristic
27	#935	—	5	3	no	—
28	#834	—	5	3	no	—
29	C₅×#41	—	2	3	no	—
30	C₂×C₅×D₄	—	2	3	no	—
31	C₂×C₅×D₄	C₂×D₃	1	3	yes	—
32	C₂²×Dic₅	—	3	1	no	—
33	C₂²⋊Dic₅	—	3	3	no	—
34	C₂²⋊Dic₅	—	6	3	no	—
35	#683	—	6	3	no	—
36	#458	—	15	1	no	Sylow
37	C₂²×C₃×C₅	—	2	3	no	—
38	C₂²×C₃×C₅	—	4	1	no	—
39	C₂²×C₃×C₅	—	4	3	no	—
40	C₂²×C₃×C₅	—	8	1	no	—
41	C₂²×C₃×C₅	C₂×D₄	1	3	yes	—
42	C₃×C₄×C₅	—	2	3	no	—
43	Dic₁₅	—	4	3	no	—
44	Dic₁₅	—	4	1	no	—
45	C₃×#41	—	2	3	no	—
46	C₂×C₃×D₄	—	2	3	no	—
47	C₂×C₃×D₄	C₂×D₅	1	3	yes	—
48	C₂²×Dic₃	—	5	1	no	—
49	#217	—	5	3	no	—
50	#217	—	10	3	no	—
51	Dic₃⋊C₄	—	10	3	no	—
52	C₂³×C₅	—	2	1	no	—
53	C₂³×C₅	#67	1	3	yes	—
54	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
55	C₅×Q₈	—	2	1	no	—
56	C₅×D₄	—	2	3	no	—
57	C₅×D₄	—	2	6	no	—
58	C₂×C₄×C₅	—	2	3	no	—
59	C₂×C₄×C₅	#67	1	3	yes	—
60	C₂×Dic₅	—	6	3	no	—
61	C₂×Dic₅	—	6	3	no	—
62	H(5,𝔽₂)	D₁₅	1	1	yes	characteristic
63	#123	—	15	3	no	—
64	#99	—	15	3	no	—
65	C₂×C₃×C₅	—	2	3	no	—
66	C₂×C₃×C₅	—	4	3	no	—
67	C₂×C₃×C₅	#120	1	1	yes	characteristic
68	C₂³×C₃	—	2	1	no	—
69	C₂³×C₃	C₂²⋊D₅	1	3	yes	—
70	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
71	C₃×Q₈	—	2	1	no	—
72	C₃×D₄	—	2	3	no	—
73	C₃×D₄	—	2	6	no	—
74	C₂×C₃×C₄	—	2	3	no	—
75	C₂×C₃×C₄	C₂²⋊D₅	1	3	yes	—
76	C₂×Dic₃	—	10	3	no	—
77	C₂×Dic₃	—	10	3	no	—
78	C₂²×C₅	—	2	3	no	—
79	C₂²×C₅	—	4	1	no	—
80	C₂²×C₅	—	4	3	no	—
81	C₂²×C₅	—	8	1	no	—
82	C₂²×C₅	C₂×#67	1	3	yes	—
83	C₄×C₅	—	2	3	no	—
84	Dic₅	—	12	3	no	—
85	Dic₅	—	12	1	no	—
86	#41	—	2	3	no	—
87	C₂×D₄	—	2	3	no	—
88	C₂×D₄	C₂×D₁₅	1	3	yes	—
89	C₂²×C₄	—	15	1	no	—
90	C₄⋊C₄	—	30	3	no	—
91	C₂²⋊C₄	—	15	3	no	—
92	C₂²⋊C₄	—	30	3	no	—
93	C₃×C₅	#458	1	1	yes	characteristic, complemented, Hall
94	C₂²×C₃	—	2	3	no	—
95	C₂²×C₃	—	4	1	no	—
96	C₂²×C₃	—	4	3	no	—
97	C₂²×C₃	—	8	1	no	—
98	C₂²×C₃	C₂×C₂²⋊D₅	1	3	yes	—
99	C₃×C₄	—	2	3	no	—
100	Dic₃	—	20	3	no	—
101	Dic₃	—	20	1	no	—
102	C₂×C₅	—	2	3	no	—
103	C₂×C₅	—	4	3	no	—
104	C₂×C₅	#953	1	1	yes	characteristic
105	C₂³	—	2	1	no	—
106	C₂³	#1220	1	3	yes	—
107	C₂³	C₂²×D₁₅	1	1	yes	characteristic
108	Q₈	—	2	1	no	—
109	D₄	—	2	3	no	—
110	D₄	—	2	6	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	30	3	no	—
113	C₂×C₄	—	30	3	no	—
114	C₂×C₄	#1220	1	3	yes	—
115	C₂×C₃	—	2	3	no	—
116	C₂×C₃	—	4	3	no	—
117	C₂×C₃	#4099	1	1	yes	characteristic
118	C₅	#5243	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	2	3	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	3	no	—
122	C₂²	—	8	1	no	—
123	C₂²	C₂×#1220	1	3	yes	—
124	C₄	—	2	3	no	—
125	C₄	—	60	3	no	—
126	C₄	—	60	1	no	—
127	C₃	#65532	1	1	yes	characteristic, complemented, Sylow
128	C₂	—	2	3	no	—
129	C₂	—	4	3	no	—
130	C₂	#91842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×H(5,𝔽₂)

C₂

yes

maximal, characteristic

#91824

C₂

yes

maximal, complemented

#91118

C₂

yes

maximal, complemented

#65532

—

maximal, Hall

C₃×C₅×#41

—

C₂×C₃×C₅×D₄

—

C₂×C₃×C₅×D₄

C₂²

yes

—

C₂²×Dic₁₅

C₂²

yes

characteristic, complemented

#6719

—

#6719

C₂²

yes

—

—

C₅×H(5,𝔽₂)

D₃

yes

characteristic

#4081

—

#3966

—

C₂³×C₃×C₅

—

C₂³×C₃×C₅

D₄

yes

—

C₂³×C₃×C₅

C₂³

yes

characteristic

C₃×C₅×Q₈

—

C₃×C₅×D₄

—

C₃×C₅×D₄

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

D₄

yes

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₃×H(5,𝔽₂)

D₅

yes

characteristic

#935

—

#834

—

C₅×#41

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₃

yes

—

C₂²×Dic₅

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

#683

—

#458

—

Sylow

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×D₄

yes

—

C₃×C₄×C₅

—

Dic₁₅

—

Dic₁₅

—

C₃×#41

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×D₅

yes

—

C₂²×Dic₃

—

#217

—

#217

—

Dic₃⋊C₄

—

C₂³×C₅

—

C₂³×C₅

#67

yes

—

C₂³×C₅

C₂²×D₃

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#67

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

H(5,𝔽₂)

D₁₅

yes

characteristic

#123

—

#99

—

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#120

yes

characteristic

C₂³×C₃

—

C₂³×C₃

C₂²⋊D₅

yes

—

C₂³×C₃

C₂²×D₅

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊D₅

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×#67

yes

—

C₄×C₅

—

Dic₅

—

Dic₅

—

#41

—

C₂×D₄

—

C₂×D₄

C₂×D₁₅

yes

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₃×C₅

#458

yes

characteristic, complemented, Hall

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊D₅

yes

—

C₃×C₄

—

100

Dic₃

—

101

Dic₃

—

102

C₂×C₅

—

103

C₂×C₅

—

104

C₂×C₅

#953

yes

characteristic

105

C₂³

—

106

C₂³

#1220

yes

—

107

C₂³

C₂²×D₁₅

yes

characteristic

108

Q₈

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#1220

yes

—

115

C₂×C₃

—

116

C₂×C₃

—

117

C₂×C₃

#4099

yes

characteristic

118

C₅

#5243

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂×#1220

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₃

#65532

yes

characteristic, complemented, Sylow

128

C₂

—

129

C₂

—

130

C₂

#91842

yes

characteristic

	subgroup	G/H
center	C₂	#91842
commutator, cosocle	C₂³×C₃×C₅	C₂³
Frattini	C₂³	C₂²×D₁₅
Fitting, prime radical	C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	#120
hypercenter	H(5,𝔽₂)	D₁₅
hypocenter	C₃×C₅	#458
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

#91842

commutator, cosocle

C₂³×C₃×C₅

C₂³

Frattini

C₂³

C₂²×D₁₅

Fitting, prime radical

C₃×C₅×H(5,𝔽₂)

C₂

C₂×C₃×C₅

D₁₅

C₃×C₅

G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
2. #91824	C₂	semidirect
3. #91118	C₂	semidirect
8. C₂²×Dic₁₅	C₂²	semidirect
10. #6719	C₂²	semidirect
93. C₃×C₅	#458	semidirect
118. C₅	#5243	semidirect
127. C₃	#65532	semidirect

H ≅ G/N

type

2. #91824

C₂

semidirect

3. #91118

C₂

semidirect

8. C₂²×Dic₁₅

C₂²

semidirect

10. #6719

C₂²

semidirect

93. C₃×C₅

#458

semidirect

118. C₅

#5243

semidirect

127. C₃

#65532

semidirect

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

subgroup	index
C₃×C₅×H(5,𝔽₂)	2
#91824	2
#91118	2
#65532	3
#5243	5

subgroup

index

C₃×C₅×H(5,𝔽₂)

Subgroups — #11751344

Conjugacy classes of subgroups

Complements

Maximal subgroups