Group #11998763 — subgroups

Subgroups — #11998763

Order 1280.

478 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 155 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×#64988	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic
3	#10601571	C₂	1	4	yes	maximal
4	#10600585	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀	—	2	2	no	—
6	C₂×#4015	C₂²	1	2	yes	—
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3951	—	2	2	no	—
10	#65213	C₂²	1	2	yes	—
11	#64988	—	2	1	no	—
12	#64988	—	2	1	no	—
13	#64988	C₄	1	4	yes	—
14	C₅×#322	—	2	1	no	—
15	C₅×C₈²	—	2	1	no	—
16	#64706	—	2	8	no	—
17	#9772	—	5	1	no	maximal, Sylow
18	C₂²×Dic₁₀	—	2	2	no	—
19	C₂²×C₅×C₈	D₄	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅	—	2	4	no	—
22	C₂×#681	—	2	2	no	—
23	C₂×Dic₂₀	—	2	4	no	—
24	C₂×Dic₂₀	—	2	4	no	—
25	C₂×Dic₂₀	—	4	4	no	—
26	C₂×Dic₂₀	—	4	4	no	—
27	#4015	—	2	4	no	—
28	#4015	—	2	2	no	—
29	#4015	C₂×C₄	1	4	yes	—
30	C₅×C₄⋊C₈	D₄	1	4	yes	—
31	C₅×C₂²⋊C₈	—	2	2	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	#3951	—	4	4	no	—
37	C₄⋊#168	—	2	4	no	—
38	C₂×#494	—	5	1	no	—
39	#1848	—	5	1	no	—
40	#1546	—	5	4	no	—
41	#1450	D₅	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	#41	1	2	yes	—
44	C₂²×Dic₅	—	4	2	no	—
45	C₂×Dic₁₀	—	2	4	no	—
46	C₂×Dic₁₀	—	4	4	no	—
47	C₂×Dic₁₀	—	4	4	no	—
48	C₂×Dic₁₀	—	4	2	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
52	C₂×C₅×C₈	D₈	1	4	yes	—
53	C₂×C₅×C₈	SD₁₆	1	4	yes	—
54	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅	C₂×D₄	1	2	yes	—
56	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
57	C₄⋊Dic₅	—	2	8	no	—
58	C₄⋊Dic₅	—	4	4	no	—
59	#681	—	4	4	no	—
60	C₂×#168	—	4	2	no	—
61	Dic₂₀	—	4	8	no	—
62	Dic₂₀	—	4	8	no	—
63	C₂²×Q₁₆	—	10	2	no	—
64	C₂×C₄⋊Q₈	—	5	2	no	—
65	#494	—	5	4	no	—
66	#494	—	10	1	no	—
67	#494	—	10	1	no	—
68	#431	C₂×D₅	1	2	yes	—
69	C₂×#108	—	10	2	no	—
70	#423	—	5	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#332	—	10	8	no	—
73	#322	—	2	1	no	—
74	C₈²	—	2	1	no	—
75	C₂³×C₅	#124	1	1	yes	characteristic
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅	#121	1	2	yes	—
80	C₂×C₄×C₅	#127	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
83	C₂×Dic₅	—	4	4	no	—
84	C₂×Dic₅	—	8	4	no	—
85	Dic₁₀	—	4	8	no	—
86	Dic₁₀	—	8	2	no	—
87	Dic₁₀	—	8	2	no	—
88	C₅×C₈	—	4	4	no	—
89	C₅×C₈	#102	1	4	yes	—
90	C₅×C₈	#102	1	4	yes	—
91	#168	—	8	2	no	—
92	C₂²×Q₈	—	10	2	no	—
93	C₂×Q₁₆	—	10	4	no	—
94	C₂×Q₁₆	—	10	4	no	—
95	C₂×Q₁₆	—	20	4	no	—
96	C₂×Q₁₆	—	20	4	no	—
97	C₂×M₁₆	—	10	2	no	—
98	C₂²×C₈	D₂₀	1	2	yes	—
99	C₄⋊Q₈	—	5	4	no	—
100	C₄⋊Q₈	—	10	4	no	—
101	C₄⋊Q₈	—	10	2	no	—
102	C₂×C₄⋊C₄	—	10	4	no	—
103	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
104	#108	—	20	4	no	—
105	C₄⋊C₈	—	10	4	no	—
106	C₄⋊C₈	D₂₀	1	4	yes	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	C₄×D₅	1	2	yes	—
110	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	#390	1	1	yes	characteristic
114	C₂²×C₅	#486	1	2	yes	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	C₂×#102	1	2	yes	—
117	C₄×C₅	#473	1	4	yes	—
118	C₄×C₅	#492	1	1	yes	characteristic
119	C₄×C₅	#493	1	1	yes	characteristic
120	Dic₅	—	8	4	no	—
121	C₂×Q₈	—	10	4	no	—
122	C₂×Q₈	—	20	2	no	—
123	C₂×Q₈	—	20	4	no	—
124	C₂×Q₈	—	20	4	no	—
125	C₂²×C₄	—	20	2	no	—
126	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
127	C₂²×C₄	#709	1	2	yes	—
128	Q₁₆	—	20	8	no	—
129	Q₁₆	—	20	8	no	—
130	M₁₆	—	20	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	20	2	no	—
133	C₂×C₈	#677	1	4	yes	—
134	C₂×C₈	D₄₀	1	4	yes	—
135	C₂×C₈	#685	1	2	yes	—
136	C₂×C₈	#685	1	2	yes	—
137	C₄⋊C₄	—	10	8	no	—
138	C₄⋊C₄	—	20	4	no	—
139	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
140	C₄²	C₂×D₂₀	1	2	yes	—
141	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₅	—	2	2	no	—
143	C₂×C₅	#1957	1	1	yes	characteristic
144	C₂×C₅	#1979	1	2	yes	—
145	C₂³	#4021	1	1	yes	characteristic
146	Q₈	—	20	8	no	—
147	Q₈	—	40	2	no	—
148	Q₈	—	40	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	20	4	no	—
151	C₂×C₄	—	40	4	no	—
152	C₂×C₄	C₄×D₂₀	1	2	yes	—
153	C₂×C₄	#4020	1	1	yes	characteristic
154	C₂×C₄	C₂×#677	1	2	yes	—
155	C₂×C₄	C₂×D₄₀	1	2	yes	—
156	C₂×C₄	C₂×#685	1	1	yes	characteristic
157	C₂×C₄	#4076	1	2	yes	—
158	C₈	—	4	4	no	—
159	C₈	—	40	2	no	—
160	C₈	#3953	1	4	yes	—
161	C₈	#3953	1	4	yes	—
162	C₅	#9772	1	1	yes	characteristic, complemented, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	#64983	1	2	yes	—
165	C₂²	#65228	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	40	4	no	—
168	C₄	#64982	1	1	yes	characteristic
169	C₄	#64984	1	1	yes	characteristic
170	C₄	C₂×#3953	1	2	yes	—
171	C₄	#65432	1	4	yes	—
172	C₂	—	2	2	no	—
173	C₂	#10600565	1	1	yes	characteristic
174	C₂	#10600697	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64988

C₂

yes

maximal, characteristic

C₅×#1450

C₂

yes

maximal, characteristic

#10601571

C₂

yes

maximal

#10600585

C₂

yes

maximal, characteristic

C₂²×Dic₂₀

—

C₂×#4015

C₂²

yes

—

C₅×#431

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

—

#65213

C₂²

yes

—

#64988

—

#64988

—

#64988

C₄

yes

—

C₅×#322

—

C₅×C₈²

—

#64706

—

#9772

—

maximal, Sylow

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×#681

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

#4015

—

#4015

C₂×C₄

yes

—

C₅×C₄⋊C₈

D₄

yes

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#681

—

C₂×#168

—

Dic₂₀

—

Dic₂₀

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₅

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#332

—

#322

—

C₈²

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#102

yes

—

C₅×C₈

#102

yes

—

#168

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₂₀

yes

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

C₂²×D₅

yes

characteristic

104

#108

—

105

C₄⋊C₈

—

106

C₄⋊C₈

D₂₀

yes

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₅

yes

—

110

C₄×C₈

C₂²⋊D₅

yes

characteristic

111

C₄×C₈

C₂²⋊D₅

yes

characteristic

112

C₂²×C₅

—

113

C₂²×C₅

#390

yes

characteristic

114

C₂²×C₅

#486

yes

—

115

C₄×C₅

—

116

C₄×C₅

C₂×#102

yes

—

117

C₄×C₅

#473

yes

—

118

C₄×C₅

#492

yes

characteristic

119

C₄×C₅

#493

yes

characteristic

120

Dic₅

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×D₂₀

yes

characteristic

127

C₂²×C₄

#709

yes

—

128

Q₁₆

—

129

Q₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#677

yes

—

134

C₂×C₈

D₄₀

yes

—

135

C₂×C₈

#685

yes

—

136

C₂×C₈

#685

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

C₂×C₄×D₅

yes

characteristic

140

C₄²

C₂×D₂₀

yes

—

141

C₄²

C₂×C₂²⋊D₅

yes

characteristic

142

C₂×C₅

—

143

C₂×C₅

#1957

yes

characteristic

144

C₂×C₅

#1979

yes

—

145

C₂³

#4021

yes

characteristic

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₄×D₂₀

yes

—

153

C₂×C₄

#4020

yes

characteristic

154

C₂×C₄

C₂×#677

yes

—

155

C₂×C₄

C₂×D₄₀

yes

—

156

C₂×C₄

C₂×#685

yes

characteristic

157

C₂×C₄

#4076

yes

—

158

C₈

—

159

C₈

—

160

C₈

#3953

yes

—

161

C₈

#3953

yes

—

162

C₅

#9772

yes

characteristic, complemented, Sylow

163

C₂²

—

164

C₂²

#64983

yes

—

165

C₂²

#65228

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

#64982

yes

characteristic

169

C₄

#64984

yes

characteristic

170

C₄

C₂×#3953

yes

—

171

C₄

#65432

yes

—

172

C₂

—

173

C₂

#10600565

yes

characteristic

174

C₂

#10600697

yes

—

	subgroup	G/H
center	C₂²	#65228
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#9772
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#65228

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
162. C₅	#9772	semidirect

H ≅ G/N

type

162. C₅

#9772

semidirect

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

subgroup	index
C₂×#64988	2
C₅×#1450	2
#10601571	2
#10600585	2
#9772	5

subgroup

index

Subgroups — #11998763

Conjugacy classes of subgroups

Complements

Maximal subgroups