Group #11993089 — subgroups

Subgroups — #11993089

Order 1280.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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×C₅×#322	C₂	1	1	yes	maximal, characteristic
2	#10600558	C₂	1	2	yes	maximal
3	#10600558	C₂	1	1	yes	maximal, characteristic
4	#10600557	C₂	1	1	yes	maximal, characteristic
5	#10600555	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3950	—	2	4	no	—
9	C₂×#3950	—	2	2	no	—
10	C₂×#3949	—	2	2	no	—
11	#65224	C₂²	1	1	yes	characteristic
12	#65224	C₂²	1	2	yes	—
13	#65224	C₂²	1	1	yes	characteristic
14	C₅×#322	C₄	1	4	yes	complemented
15	#64771	—	2	4	no	—
16	#8820	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	4	yes	—
18	C₂²×C₅×C₈	D₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	—	2	2	no	—
21	C₂×C₄⋊Dic₅	—	2	2	no	—
22	C₂×C₄⋊Dic₅	—	2	1	no	—
23	C₂×C₄⋊Dic₅	—	2	1	no	—
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	Q₈	1	4	yes	—
30	C₄×C₅×C₈	Q₈	1	2	yes	—
31	#3950	—	4	8	no	—
32	#3950	—	4	4	no	—
33	#3949	—	4	4	no	—
34	#1845	—	5	1	no	—
35	#1845	—	5	2	no	—
36	#1844	—	5	1	no	—
37	#1834	—	5	2	no	—
38	C₂×#322	D₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
41	C₂²×Dic₅	—	4	2	no	—
42	C₂²×Dic₅	—	4	1	no	—
43	C₂²×Dic₅	—	4	1	no	—
44	C₂×C₅×C₈	—	2	4	no	—
45	C₂×C₅×C₈	—	2	4	no	—
46	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
47	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₅×C₈	D₈	1	4	yes	—
49	C₂×C₅×C₈	Q₁₆	1	4	yes	—
50	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₅	C₂×Q₈	1	2	yes	—
53	C₄⋊Dic₅	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	4	4	no	—
56	C₄⋊Dic₅	—	4	4	no	—
57	C₄⋊Dic₅	—	4	4	no	—
58	C₄⋊Dic₅	—	4	4	no	—
59	C₄⋊Dic₅	—	4	4	no	—
60	C₂×#107	—	10	4	no	—
61	C₂×#107	—	10	2	no	—
62	C₂×#106	—	10	2	no	—
63	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
65	#384	—	5	1	no	—
66	#384	—	5	2	no	—
67	#384	—	5	1	no	—
68	#340	—	10	4	no	—
69	#322	Dic₅	1	4	yes	complemented
70	C₂³×C₅	#127	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×C₄×C₅	#137	1	4	yes	—
79	C₂×Dic₅	—	8	1	no	—
80	C₂×Dic₅	—	8	2	no	—
81	C₂×Dic₅	—	8	1	no	—
82	C₂×Dic₅	—	8	2	no	—
83	C₂×Dic₅	—	8	2	no	—
84	C₂×Dic₅	—	8	2	no	—
85	C₂×Dic₅	—	8	2	no	—
86	C₅×C₈	—	2	8	no	—
87	C₅×C₈	#107	1	8	yes	complemented
88	C₂²×C₈	D₂₀	1	4	yes	—
89	C₂²×C₈	D₂₀	1	2	yes	—
90	C₂×C₄⋊C₄	—	10	2	no	—
91	C₂×C₄⋊C₄	—	10	2	no	—
92	C₂×C₄⋊C₄	—	10	1	no	—
93	C₂×C₄⋊C₄	—	10	1	no	—
94	C₂×C₄⋊C₄	—	10	2	no	—
95	C₂×C₄⋊C₄	—	10	2	no	—
96	C₂×C₄⋊C₄	—	10	2	no	—
97	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
98	#107	—	20	8	no	—
99	#107	—	20	4	no	—
100	#106	—	20	4	no	—
101	C₄×C₈	Dic₁₀	1	4	yes	—
102	C₄×C₈	Dic₁₀	1	2	yes	—
103	C₄×C₈	C₂×Dic₅	1	4	yes	—
104	C₄×C₈	C₂×Dic₅	1	2	yes	—
105	C₂²×C₅	#384	1	1	yes	characteristic
106	C₂²×C₅	#493	1	1	yes	characteristic
107	C₂²×C₅	#494	1	1	yes	characteristic
108	C₂²×C₅	#496	1	2	yes	—
109	C₂²×C₅	#497	1	2	yes	—
110	C₄×C₅	C₂×#107	1	2	yes	—
111	C₄×C₅	#428	1	4	yes	—
112	C₄×C₅	#500	1	2	yes	—
113	C₄×C₅	#501	1	4	yes	—
114	Dic₅	—	16	1	no	—
115	Dic₅	—	16	2	no	—
116	Dic₅	—	16	1	no	—
117	C₂²×C₄	—	20	2	no	—
118	C₂²×C₄	—	20	1	no	—
119	C₂²×C₄	—	20	1	no	—
120	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
121	C₂²×C₄	C₂×D₂₀	1	2	yes	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	D₄₀	1	4	yes	—
125	C₂×C₈	Dic₂₀	1	4	yes	—
126	C₂×C₈	C₄⋊Dic₅	1	8	yes	—
127	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
128	C₄⋊C₄	—	20	2	no	—
129	C₄⋊C₄	—	20	2	no	—
130	C₄⋊C₄	—	20	4	no	—
131	C₄⋊C₄	—	20	4	no	—
132	C₄⋊C₄	—	20	4	no	—
133	C₄⋊C₄	—	20	4	no	—
134	C₄⋊C₄	—	20	4	no	—
135	C₄²	C₂×Dic₁₀	1	2	yes	—
136	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
137	C₄²	C₂²×Dic₅	1	1	yes	characteristic
138	C₂×C₅	#1721	1	1	yes	characteristic
139	C₂×C₅	#1722	1	1	yes	characteristic
140	C₂×C₅	#1723	1	2	yes	—
141	C₂×C₅	#1845	1	1	yes	characteristic
142	C₂×C₅	#1847	1	2	yes	—
143	C₂³	#4020	1	1	yes	characteristic
144	C₂×C₄	—	40	1	no	—
145	C₂×C₄	—	40	2	no	—
146	C₂×C₄	—	40	1	no	—
147	C₂×C₄	—	40	2	no	—
148	C₂×C₄	—	40	2	no	—
149	C₂×C₄	—	40	2	no	—
150	C₂×C₄	—	40	2	no	—
151	C₂×C₄	#4015	1	1	yes	characteristic
152	C₂×C₄	#4015	1	2	yes	—
153	C₂×C₄	C₂×D₄₀	1	2	yes	—
154	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
155	C₂×C₄	#4054	1	4	yes	—
156	C₂×C₄	#4055	1	4	yes	—
157	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
159	C₈	—	2	8	no	—
160	C₈	#3950	1	8	yes	complemented
161	C₅	#8820	1	1	yes	characteristic, complemented, Sylow
162	C₂²	#64984	1	1	yes	characteristic
163	C₂²	#64988	1	1	yes	characteristic
164	C₂²	#65001	1	2	yes	—
165	C₂²	#65004	1	2	yes	—
166	C₂²	#65224	1	1	yes	characteristic
167	C₄	—	80	1	no	—
168	C₄	—	80	2	no	—
169	C₄	—	80	1	no	—
170	C₄	#64971	1	2	yes	—
171	C₄	#64991	1	4	yes	—
172	C₄	C₂×#3950	1	2	yes	—
173	C₄	#65410	1	4	yes	—
174	C₂	#10598887	1	1	yes	characteristic
175	C₂	#10598891	1	1	yes	characteristic
176	C₂	#10598892	1	2	yes	—
177	C₂	#10600558	1	1	yes	characteristic
178	C₂	#10600609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#322

C₂

yes

maximal, characteristic

#10600558

C₂

yes

maximal

#10600558

C₂

yes

maximal, characteristic

#10600557

C₂

yes

maximal, characteristic

#10600555

C₂

yes

maximal

C₂×C₄×C₅×C₈

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

#65224

C₂²

yes

—

#65224

C₂²

yes

characteristic

C₅×#322

C₄

yes

—

—

C₂²×C₅×C₈

D₄

yes

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

C₄×C₅×C₈

Q₈

yes

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

Q₁₆

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄²×C₅

C₂×Q₈

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅

yes

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

Dic₅

yes

complemented

C₂³×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

#107

yes

complemented

C₂²×C₈

D₂₀

yes

—

C₂²×C₈

D₂₀

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#107

—

#107

—

100

#106

—

101

C₄×C₈

Dic₁₀

yes

—

102

C₄×C₈

Dic₁₀

yes

—

103

C₄×C₈

C₂×Dic₅

yes

—

104

C₄×C₈

C₂×Dic₅

yes

—

105

C₂²×C₅

#384

yes

characteristic

106

C₂²×C₅

#493

yes

characteristic

107

C₂²×C₅

#494

yes

characteristic

108

C₂²×C₅

#496

yes

—

109

C₂²×C₅

#497

yes

—

110

C₄×C₅

C₂×#107

yes

—

111

C₄×C₅

#428

yes

—

112

C₄×C₅

#500

yes

—

113

C₄×C₅

#501

yes

—

114

Dic₅

—

115

Dic₅

—

116

Dic₅

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×D₂₀

yes

characteristic

121

C₂²×C₄

C₂×D₂₀

yes

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

D₄₀

yes

—

125

C₂×C₈

Dic₂₀

yes

—

126

C₂×C₈

C₄⋊Dic₅

yes

—

127

C₂×C₈

C₄⋊Dic₅

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

C₂×Dic₁₀

yes

—

136

C₄²

C₂×Dic₁₀

yes

characteristic

137

C₄²

C₂²×Dic₅

yes

characteristic

138

C₂×C₅

#1721

yes

characteristic

139

C₂×C₅

#1722

yes

characteristic

140

C₂×C₅

#1723

yes

—

141

C₂×C₅

#1845

yes

characteristic

142

C₂×C₅

#1847

yes

—

143

C₂³

#4020

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#4015

yes

characteristic

152

C₂×C₄

#4015

yes

—

153

C₂×C₄

C₂×D₄₀

yes

—

154

C₂×C₄

C₂×Dic₂₀

yes

—

155

C₂×C₄

#4054

yes

—

156

C₂×C₄

#4055

yes

—

157

C₂×C₄

C₂×C₄⋊Dic₅

yes

characteristic

158

C₂×C₄

C₂×C₄⋊Dic₅

yes

—

159

C₈

—

160

C₈

#3950

yes

complemented

161

C₅

#8820

yes

characteristic, complemented, Sylow

162

C₂²

#64984

yes

characteristic

163

C₂²

#64988

yes

characteristic

164

C₂²

#65001

yes

—

165

C₂²

#65004

yes

—

166

C₂²

#65224

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#64971

yes

—

171

C₄

#64991

yes

—

172

C₄

C₂×#3950

yes

—

173

C₄

#65410

yes

—

174

C₂

#10598887

yes

characteristic

175

C₂

#10598891

yes

characteristic

176

C₂

#10598892

yes

—

177

C₂

#10600558

yes

characteristic

178

C₂

#10600609

yes

—

	subgroup	G/H
center	C₂³	#4020
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#322	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#322	D₅
hypocenter	C₅	#8820
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#4020

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₂×C₅×#322

C₂

socle, prime radical

C₂³×C₅

#127

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
14. C₅×#322	C₄	semidirect
69. #322	Dic₅	semidirect
87. C₅×C₈	#107	semidirect
160. C₈	#3950	semidirect
161. C₅	#8820	semidirect

H ≅ G/N

type

14. C₅×#322

C₄

semidirect

69. #322

Dic₅

semidirect

87. C₅×C₈

#107

semidirect

160. C₈

#3950

semidirect

161. C₅

#8820

semidirect

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

subgroup	index
C₂×C₅×#322	2
#10600558	2
#10600557	2
#10600555	2
#8820	5

subgroup

index

C₂×C₅×#322

Subgroups — #11993089

Conjugacy classes of subgroups

Complements

Maximal subgroups