Group #14293572 — subgroups

Subgroups — #14293572

Order 1920.

424 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 116 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	#11750800	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64984	C₂	1	1	yes	maximal, characteristic
3	#11746890	C₂	1	4	yes	maximal, complemented
4	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598853	—	3	1	no	maximal, Hall
6	C₂×Dic₆₀	—	2	2	no	—
7	#91758	C₂²	1	2	yes	—
8	C₂×C₃×D₄₀	—	2	2	no	—
9	C₃×#4020	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	#90965	—	2	8	no	—
13	#71275	—	5	1	no	maximal, Hall
14	#64988	—	3	1	no	—
15	#64984	D₃	1	1	yes	characteristic
16	#64983	—	3	4	no	—
17	C₅×C₈²	—	3	1	no	—
18	C₂×Dic₃₀	—	2	2	no	—
19	C₂×C₃×D₂₀	—	2	4	no	—
20	C₂×C₃×D₂₀	—	2	2	no	—
21	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
22	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₁₅	—	2	4	no	—
24	Dic₆₀	—	4	4	no	—
25	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
26	C₃×D₄₀	—	4	4	no	—
27	C₃×#493	—	5	1	no	—
28	#5070	—	5	4	no	—
29	#4699	—	5	1	no	—
30	C₈×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
31	C₂×Dic₂₀	—	6	2	no	—
32	C₂×D₄₀	—	2	2	no	—
33	#4020	C₂×D₃	1	2	yes	—
34	#4015	—	3	2	no	—
35	C₄×C₅×C₈	—	3	2	no	—
36	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	#3953	—	6	8	no	—
38	#1709	—	15	1	no	Sylow
39	C₂²×C₃×D₅	—	4	2	no	—
40	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
42	C₂×Dic₁₅	—	4	2	no	—
43	Dic₃₀	—	4	4	no	—
44	C₃×D₂₀	—	4	4	no	—
45	C₃×D₂₀	—	4	8	no	—
46	C₃×C₅×C₈	D₈	1	4	yes	—
47	C₅×C₃⋊C₈	—	2	4	no	—
48	C₂×C₃×D₈	—	10	2	no	—
49	C₃×#127	—	5	2	no	—
50	C₂×Dic₁₂	—	10	2	no	—
51	#869	—	5	2	no	—
52	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
53	#832	—	10	8	no	—
54	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
55	C₂×D₂₀	—	2	4	no	—
56	C₂×D₂₀	—	2	2	no	—
57	C₂×Dic₁₀	—	6	2	no	—
58	C₂×C₅×C₈	—	3	4	no	—
59	C₂×C₅×C₈	#67	1	2	yes	—
60	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
61	C₄⋊Dic₅	—	6	4	no	—
62	Dic₂₀	—	12	4	no	—
63	D₄₀	—	4	4	no	—
64	#494	—	15	1	no	—
65	#493	—	5	1	no	—
66	#486	—	15	4	no	—
67	C₈²	—	3	1	no	—
68	C₂²×C₃×C₅	#127	1	1	yes	characteristic
69	C₂×C₃×D₅	—	8	2	no	—
70	C₂×C₃×D₅	—	8	4	no	—
71	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
72	C₃×C₄×C₅	#135	1	4	yes	—
73	Dic₁₅	—	8	2	no	—
74	C₂×C₃×D₄	—	10	4	no	—
75	C₂×C₃×D₄	—	10	2	no	—
76	C₂×Dic₆	—	10	2	no	—
77	C₃×D₈	—	20	4	no	—
78	C₂×C₃×C₈	D₂₀	1	2	yes	—
79	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
80	C₄⋊Dic₃	—	10	4	no	—
81	C₂×C₃⋊C₈	D₂₀	1	4	yes	—
82	Dic₁₂	—	20	4	no	—
83	C₂²×D₅	—	4	2	no	—
84	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
85	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
86	C₂×Dic₅	—	12	2	no	—
87	D₂₀	—	4	8	no	—
88	D₂₀	—	4	4	no	—
89	Dic₁₀	—	12	4	no	—
90	C₅×C₈	—	6	4	no	—
91	C₅×C₈	#213	1	4	yes	—
92	C₂×Q₁₆	—	30	2	no	—
93	C₂×D₈	—	10	2	no	—
94	C₄⋊Q₈	—	15	2	no	—
95	#127	—	5	2	no	—
96	#102	—	30	8	no	—
97	C₄×C₈	—	3	2	no	—
98	C₄×C₈	D₃×D₅	1	1	yes	characteristic
99	C₂×C₃×C₅	#493	1	1	yes	characteristic
100	C₂×C₃×C₅	#495	1	2	yes	—
101	C₃×D₅	—	16	2	no	—
102	C₂³×C₃	—	20	2	no	—
103	C₃×D₄	—	20	4	no	—
104	C₃×D₄	—	20	8	no	—
105	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
106	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
107	C₂×Dic₃	—	20	2	no	—
108	Dic₆	—	20	4	no	—
109	C₃×C₈	D₄₀	1	4	yes	—
110	C₃⋊C₈	—	2	4	no	—
111	C₂²×C₅	#940	1	1	yes	characteristic
112	C₂×D₅	—	8	4	no	—
113	C₂×D₅	—	8	2	no	—
114	C₄×C₅	#912	1	4	yes	—
115	C₄×C₅	C₂×#213	1	2	yes	—
116	Dic₅	—	24	2	no	—
117	C₂×Q₈	—	30	2	no	—
118	C₂×D₄	—	10	4	no	—
119	C₂×D₄	—	10	2	no	—
120	Q₁₆	—	60	4	no	—
121	D₈	—	20	4	no	—
122	C₂×C₈	—	3	4	no	—
123	C₂×C₈	#1202	1	2	yes	—
124	C₄⋊C₄	—	30	4	no	—
125	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
126	C₃×C₅	#1709	1	1	yes	characteristic, complemented, Hall
127	C₂²×C₃	—	40	2	no	—
128	C₂²×C₃	—	40	4	no	—
129	C₂²×C₃	#4020	1	1	yes	characteristic
130	C₃×C₄	C₂×D₄₀	1	2	yes	—
131	C₃×C₄	#4050	1	4	yes	—
132	Dic₃	—	40	2	no	—
133	C₂×C₅	#5073	1	1	yes	characteristic
134	C₂×C₅	#5155	1	2	yes	—
135	D₅	—	16	2	no	—
136	C₂³	—	20	2	no	—
137	Q₈	—	60	4	no	—
138	D₄	—	20	8	no	—
139	D₄	—	20	4	no	—
140	C₂×C₄	—	60	2	no	—
141	C₂×C₄	D₃×D₂₀	1	2	yes	—
142	C₂×C₄	C₂×#1202	1	1	yes	characteristic
143	C₈	—	6	4	no	—
144	C₈	#6653	1	4	yes	—
145	C₂×C₃	—	80	2	no	—
146	C₂×C₃	#64984	1	1	yes	characteristic
147	C₂×C₃	#64985	1	2	yes	—
148	C₅	#71275	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	40	4	no	—
150	C₂²	—	40	2	no	—
151	C₂²	#91458	1	1	yes	characteristic
152	C₄	—	120	2	no	—
153	C₄	#91273	1	4	yes	—
154	C₄	C₂×#6653	1	2	yes	—
155	C₃	#10598853	1	1	yes	characteristic, complemented, Sylow
156	C₂	—	80	2	no	—
157	C₂	#11747035	1	1	yes	characteristic
158	C₂	#11747971	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750800

C₂

yes

maximal, characteristic, complemented

C₃×#64984

C₂

yes

maximal, characteristic

#11746890

C₂

yes

maximal, complemented

C₅×C₈×C₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#10598853

—

maximal, Hall

C₂×Dic₆₀

—

#91758

C₂²

yes

—

C₂×C₃×D₄₀

—

C₃×#4020

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#64983

—

C₅×C₈²

—

C₂×Dic₃₀

—

C₂×C₃×D₂₀

—

C₂×C₃×D₂₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

Dic₆₀

—

C₂×C₅×C₃⋊C₈

D₄

yes

—

C₃×D₄₀

—

C₃×#493

—

#5070

—

#4699

—

C₈×C₃⋊C₈

D₅

yes

characteristic, complemented

C₂×Dic₂₀

—

C₂×D₄₀

—

#4020

C₂×D₃

yes

—

#4015

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

C₂²×C₃×D₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

Dic₃₀

—

C₃×D₂₀

—

C₃×D₂₀

—

C₃×C₅×C₈

D₈

yes

—

C₅×C₃⋊C₈

—

C₂×C₃×D₈

—

C₃×#127

—

C₂×Dic₁₂

—

#869

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#832

—

C₄×C₃⋊C₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

D₄₀

—

#494

—

#493

—

#486

—

C₈²

—

C₂²×C₃×C₅

#127

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₃×C₄×C₅

C₂×D₈

yes

—

C₃×C₄×C₅

#135

yes

—

Dic₁₅

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×D₈

—

C₂×C₃×C₈

D₂₀

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₂₀

yes

—

Dic₁₂

—

C₂²×D₅

—

C₂×C₄×C₅

D₃×D₄

yes

—

C₂×C₄×C₅

C₂×#67

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#213

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#102

—

C₄×C₈

—

C₄×C₈

D₃×D₅

yes

characteristic

C₂×C₃×C₅

#493

yes

characteristic

100

C₂×C₃×C₅

#495

yes

—

101

C₃×D₅

—

102

C₂³×C₃

—

103

C₃×D₄

—

104

C₃×D₄

—

105

C₂×C₃×C₄

C₂×D₂₀

yes

—

106

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

107

C₂×Dic₃

—

108

Dic₆

—

109

C₃×C₈

D₄₀

yes

—

110

C₃⋊C₈

—

111

C₂²×C₅

#940

yes

characteristic

112

C₂×D₅

—

113

C₂×D₅

—

114

C₄×C₅

#912

yes

—

115

C₄×C₅

C₂×#213

yes

—

116

Dic₅

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

Q₁₆

—

121

D₈

—

122

C₂×C₈

—

123

C₂×C₈

#1202

yes

—

124

C₄⋊C₄

—

125

C₄²

C₂×D₃×D₅

yes

characteristic

126

C₃×C₅

#1709

yes

characteristic, complemented, Hall

127

C₂²×C₃

—

128

C₂²×C₃

—

129

C₂²×C₃

#4020

yes

characteristic

130

C₃×C₄

C₂×D₄₀

yes

—

131

C₃×C₄

#4050

yes

—

132

Dic₃

—

133

C₂×C₅

#5073

yes

characteristic

134

C₂×C₅

#5155

yes

—

135

D₅

—

136

C₂³

—

137

Q₈

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

D₃×D₂₀

yes

—

142

C₂×C₄

C₂×#1202

yes

characteristic

143

C₈

—

144

C₈

#6653

yes

—

145

C₂×C₃

—

146

C₂×C₃

#64984

yes

characteristic

147

C₂×C₃

#64985

yes

—

148

C₅

#71275

yes

characteristic, complemented, Sylow

149

C₂²

—

150

C₂²

—

151

C₂²

#91458

yes

characteristic

152

C₄

—

120

—

153

C₄

#91273

yes

—

154

C₄

C₂×#6653

yes

—

155

C₃

#10598853

yes

characteristic, complemented, Sylow

156

C₂

—

157

C₂

#11747035

yes

characteristic

158

C₂

#11747971

yes

—

	subgroup	G/H
center	C₂²	#91458
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	#127
prime radical	C₃×#64984	C₂
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1709
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91458

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

C₂²×C₃×C₅

C₂

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1709

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
1. #11750800	C₂	semidirect
3. #11746890	C₂	semidirect
4. C₅×C₈×C₃⋊C₈	C₂	semidirect
30. C₈×C₃⋊C₈	D₅	semidirect
126. C₃×C₅	#1709	semidirect
148. C₅	#71275	semidirect
155. C₃	#10598853	semidirect

H ≅ G/N

type

1. #11750800

C₂

semidirect

3. #11746890

C₂

semidirect

4. C₅×C₈×C₃⋊C₈

C₂

semidirect

30. C₈×C₃⋊C₈

D₅

semidirect

126. C₃×C₅

#1709

semidirect

148. C₅

#71275

semidirect

155. C₃

#10598853

semidirect

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

subgroup	index
#11750800	2
C₃×#64984	2
#11746890	2
C₅×C₈×C₃⋊C₈	2
#10598853	3
#71275	5

subgroup

index

C₅×C₈×C₃⋊C₈

Subgroups — #14293572

Conjugacy classes of subgroups

Complements

Maximal subgroups