Group #11747076 — subgroups

Subgroups — #11747076

Order 960.

248 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 72 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	#91764	C₂	1	1	yes	maximal, characteristic
2	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#6662	C₂	1	2	yes	maximal, complemented
4	C₂×#6658	C₂	1	2	yes	maximal
5	C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	#65004	—	3	1	no	maximal, Hall
7	C₂×D₆₀	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#6714	—	2	2	no	—
12	C₂×C₅×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×C₄⋊Dic₅	—	2	2	no	—
14	#6662	—	2	4	no	—
15	#6658	—	2	4	no	—
16	#5086	—	5	1	no	maximal, Hall
17	C₂×Dic₂₀	—	3	2	no	—
18	C₂×#677	—	3	2	no	—
19	#4021	—	3	1	no	—
20	#4015	D₃	1	1	yes	characteristic, complemented
21	C₅×#97	—	3	1	no	—
22	C₂²×D₁₅	—	2	1	no	—
23	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₅	—	2	1	no	—
26	D₆₀	—	2	2	no	—
27	Dic₃₀	—	2	2	no	—
28	C₂×C₃×Dic₅	—	2	2	no	—
29	C₃×Dic₁₀	—	2	2	no	—
30	C₃×Dic₁₀	—	2	2	no	—
31	C₅×C₃⋊C₈	D₄	1	4	yes	—
32	C₃×C₄⋊Q₈	—	5	1	no	—
33	C₂×#216	—	5	2	no	—
34	C₂×#215	—	5	2	no	—
35	#875	—	5	1	no	—
36	#803	D₅	1	1	yes	characteristic, complemented
37	C₂×D₂₀	—	3	1	no	—
38	C₂×Dic₁₀	—	3	1	no	—
39	C₂×Dic₁₀	C₂×D₃	1	2	yes	—
40	C₂×C₅×C₈	—	3	2	no	—
41	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	#685	—	6	2	no	—
43	C₄⋊Dic₅	—	2	2	no	—
44	Dic₂₀	—	6	4	no	—
45	#677	—	6	4	no	—
46	#497	—	15	1	no	Sylow
47	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₅	—	4	1	no	—
49	C₂×D₁₅	—	4	2	no	—
50	C₃×C₄×C₅	—	2	2	no	—
51	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
52	Dic₁₅	—	4	1	no	—
53	C₃×Dic₅	—	4	2	no	—
54	C₂×C₃×Q₈	—	5	2	no	—
55	C₂×D₁₂	—	5	1	no	—
56	C₂×Dic₆	—	5	1	no	—
57	C₃×C₄⋊C₄	—	10	2	no	—
58	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
59	#216	—	10	4	no	—
60	#215	—	10	4	no	—
61	(C₂×D₃)⋊C₄	—	10	2	no	—
62	C₂×C₃⋊C₈	C₂×D₅	1	2	yes	—
63	C₂²×D₅	—	6	1	no	—
64	C₂×C₄×C₅	#67	1	2	yes	—
65	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	—	6	1	no	—
68	D₂₀	—	6	2	no	—
69	Dic₁₀	—	2	2	no	—
70	Dic₁₀	—	2	2	no	—
71	Dic₁₀	—	6	2	no	—
72	C₅×C₈	—	3	4	no	—
73	C₂×Q₁₆	—	15	2	no	—
74	C₂×SD₁₆	—	15	2	no	—
75	C₄⋊Q₈	—	5	1	no	—
76	#124	—	15	1	no	—
77	#97	—	3	1	no	—
78	C₂×C₃×C₅	#127	1	1	yes	characteristic
79	C₂×C₃×C₅	#137	1	2	yes	—
80	D₁₅	—	8	1	no	—
81	C₂²×D₃	—	10	1	no	—
82	C₃×Q₈	—	10	2	no	—
83	C₃×Q₈	—	10	2	no	—
84	C₂×C₃×C₄	—	10	2	no	—
85	C₂×C₃×C₄	D₂₀	1	2	yes	—
86	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
87	C₂×Dic₃	—	10	1	no	—
88	D₁₂	—	10	2	no	—
89	Dic₆	—	10	2	no	—
90	C₃⋊C₈	D₂₀	1	4	yes	—
91	C₂²×C₅	C₂×#67	1	1	yes	characteristic
92	C₂×D₅	—	12	1	no	—
93	C₂×D₅	—	12	2	no	—
94	C₄×C₅	—	2	2	no	—
95	C₄×C₅	D₃×D₄	1	2	yes	—
96	Dic₅	—	4	2	no	—
97	Dic₅	—	12	1	no	—
98	C₂×Q₈	—	5	2	no	—
99	C₂×Q₈	—	15	1	no	—
100	C₂×D₄	—	15	1	no	—
101	Q₁₆	—	30	4	no	—
102	SD₁₆	—	30	4	no	—
103	C₂×C₈	—	3	2	no	—
104	C₄⋊C₄	—	10	2	no	—
105	C₂²⋊C₄	—	30	2	no	—
106	C₄²	D₃×D₅	1	1	yes	characteristic
107	C₃×C₅	#497	1	1	yes	characteristic, complemented, Hall
108	C₂²×C₃	C₂×D₂₀	1	1	yes	characteristic
109	C₂×D₃	—	20	1	no	—
110	C₂×D₃	—	20	2	no	—
111	C₃×C₄	—	2	2	no	—
112	C₃×C₄	—	20	2	no	—
113	C₃×C₄	C₂×D₂₀	1	2	yes	—
114	Dic₃	—	20	1	no	—
115	C₂×C₅	#940	1	1	yes	characteristic
116	C₂×C₅	#942	1	2	yes	—
117	D₅	—	24	1	no	—
118	C₂³	—	30	1	no	—
119	Q₈	—	10	2	no	—
120	Q₈	—	10	2	no	—
121	Q₈	—	30	2	no	—
122	D₄	—	30	2	no	—
123	C₂×C₄	—	10	2	no	—
124	C₂×C₄	—	30	1	no	—
125	C₂×C₄	#1202	1	2	yes	—
126	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
127	C₈	—	3	4	no	—
128	C₂×C₃	#4020	1	1	yes	characteristic
129	C₂×C₃	#4055	1	2	yes	—
130	D₃	—	40	1	no	—
131	C₅	#5086	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	60	1	no	—
133	C₂²	—	60	2	no	—
134	C₂²	C₂×#1202	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	20	2	no	—
137	C₄	—	60	1	no	—
138	C₄	D₃×D₂₀	1	2	yes	—
139	C₃	#65004	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	120	1	no	—
141	C₂	#91315	1	2	yes	—
142	C₂	#91458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#91764

C₂

yes

maximal, characteristic

C₃×#4015

C₂

yes

maximal, characteristic, complemented

C₂×#6662

C₂

yes

maximal, complemented

C₂×#6658

C₂

yes

maximal

C₅×#803

C₂

yes

maximal, characteristic, complemented

#65004

—

maximal, Hall

C₂×D₆₀

C₂²

yes

characteristic

C₂×Dic₃₀

C₂²

yes

characteristic

C₂×C₃×Dic₁₀

C₂²

yes

—

C₃×C₄²×C₅

C₂²

yes

characteristic

#6714

—

C₂×C₅×C₃⋊C₈

C₂²

yes

—

C₃×C₄⋊Dic₅

—

—

—

—

C₂×Dic₂₀

—

C₂×#677

—

#4021

—

#4015

D₃

yes

characteristic, complemented

C₅×#97

—

C₂²×D₁₅

—

C₂×C₃×C₄×C₅

D₄

yes

—

C₂×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₁₅

—

D₆₀

—

Dic₃₀

—

C₂×C₃×Dic₅

—

C₃×Dic₁₀

—

C₃×Dic₁₀

—

C₅×C₃⋊C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

C₂×D₃

yes

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₃

yes

characteristic

#685

—

C₄⋊Dic₅

—

Dic₂₀

—

#677

—

#497

—

Sylow

C₂²×C₃×C₅

C₂×D₄

yes

characteristic

C₂×D₁₅

—

C₂×D₁₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×D₄

yes

—

Dic₁₅

—

C₃×Dic₅

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₅

yes

characteristic

#216

—

#215

—

(C₂×D₃)⋊C₄

—

C₂×C₃⋊C₈

C₂×D₅

yes

—

C₂²×D₅

—

C₂×C₄×C₅

#67

yes

—

C₂×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#124

—

#97

—

C₂×C₃×C₅

#127

yes

characteristic

C₂×C₃×C₅

#137

yes

—

D₁₅

—

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₂₀

yes

—

C₂×C₃×C₄

C₂²×D₅

yes

characteristic

C₂×Dic₃

—

D₁₂

—

Dic₆

—

C₃⋊C₈

D₂₀

yes

—

C₂²×C₅

C₂×#67

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

D₃×D₄

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×D₄

—

101

Q₁₆

—

102

SD₁₆

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

D₃×D₅

yes

characteristic

107

C₃×C₅

#497

yes

characteristic, complemented, Hall

108

C₂²×C₃

C₂×D₂₀

yes

characteristic

109

C₂×D₃

—

110

C₂×D₃

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

C₂×D₂₀

yes

—

114

Dic₃

—

115

C₂×C₅

#940

yes

characteristic

116

C₂×C₅

#942

yes

—

117

D₅

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#1202

yes

—

126

C₂×C₄

C₂×D₃×D₅

yes

characteristic

127

C₈

—

128

C₂×C₃

#4020

yes

characteristic

129

C₂×C₃

#4055

yes

—

130

D₃

—

131

C₅

#5086

yes

characteristic, complemented, Sylow

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×#1202

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

D₃×D₂₀

yes

—

139

C₃

#65004

yes

characteristic, complemented, Sylow

140

C₂

—

120

—

141

C₂

#91315

yes

—

142

C₂

#91458

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#1202
commutator, cosocle	C₂×C₃×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂×D₃×D₅
Fitting	C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×D₆₀	C₂²
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#497
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#1202

commutator, cosocle

C₂×C₃×C₄×C₅

C₂³

Frattini

C₂×C₄

C₂×D₃×D₅

Fitting

C₃×C₄²×C₅

C₂²

socle

C₂²×C₃×C₅

C₂×D₄

prime radical

C₂×D₆₀

C₂²

hypercenter

C₄²

D₃×D₅

hypocenter

C₃×C₅

#497

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₃×#4015	C₂	semidirect
3. C₂×#6662	C₂	semidirect
5. C₅×#803	C₂	semidirect
20. #4015	D₃	semidirect
36. #803	D₅	semidirect
107. C₃×C₅	#497	semidirect
131. C₅	#5086	semidirect
139. C₃	#65004	semidirect

H ≅ G/N

type

2. C₃×#4015

C₂

semidirect

3. C₂×#6662

C₂

semidirect

5. C₅×#803

C₂

semidirect

20. #4015

D₃

semidirect

36. #803

D₅

semidirect

107. C₃×C₅

#497

semidirect

131. C₅

#5086

semidirect

139. C₃

#65004

semidirect

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

subgroup	index
#91764	2
C₃×#4015	2
C₂×#6662	2
C₂×#6658	2
C₅×#803	2
#65004	3
#5086	5

subgroup

index

Subgroups — #11747076

Conjugacy classes of subgroups

Complements

Maximal subgroups