Group #11993090 — subgroups

Subgroups — #11993090

Order 1280.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₅×#1845	C₂	1	1	yes	maximal, characteristic
2	#10600557	C₂	1	1	yes	maximal, characteristic
3	C₂×#64675	C₂	1	1	yes	maximal, characteristic
4	#10600370	C₂	1	2	yes	maximal
5	#10600367	C₂	1	2	yes	maximal
6	C₂×C₅×#107	—	2	2	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	C₂²	1	2	yes	—
9	C₂×#3949	—	2	2	no	—
10	C₂×#3939	—	2	4	no	—
11	#65224	C₂²	1	1	yes	characteristic
12	#65224	C₂²	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	2	yes	—
14	#64749	—	2	4	no	—
15	#64675	C₄	1	4	yes	complemented
16	#8820	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
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	C₂²×#168	D₄	1	4	yes	—
27	C₅×#107	—	4	4	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	Q₈	1	2	yes	—
30	#3949	—	4	4	no	—
31	#3939	—	4	8	no	—
32	C₄×#168	C₂×C₄	1	4	yes	—
33	C₄×#168	Q₈	1	4	yes	—
34	#1845	—	5	2	no	—
35	#1845	D₅	1	1	yes	characteristic
36	#1844	—	5	1	no	—
37	#1834	—	5	2	no	—
38	C₂×#322	—	5	1	no	—
39	C₂²×C₄×C₅	—	4	2	no	—
40	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅	—	4	1	no	—
43	C₂²×Dic₅	—	4	1	no	—
44	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₅×C₈	D₈	1	4	yes	—
46	C₂×C₅×C₈	Q₁₆	1	4	yes	—
47	C₅×C₄⋊C₄	—	4	4	no	—
48	C₅×C₄⋊C₄	—	4	4	no	—
49	C₅×C₄⋊C₄	—	4	4	no	—
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	C₂×#168	—	2	4	no	—
58	C₂×#168	—	2	4	no	—
59	C₂×#168	C₄⋊C₄	1	8	yes	—
60	C₂×#107	—	2	2	no	—
61	C₂×#107	—	10	4	no	—
62	C₂×#106	—	10	2	no	—
63	C₂×C₄×C₈	—	5	2	no	—
64	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
65	#384	—	5	1	no	—
66	#384	—	5	1	no	—
67	#384	C₂×D₅	1	2	yes	—
68	#340	—	10	4	no	—
69	#322	—	5	4	no	—
70	C₂³×C₅	#127	1	1	yes	characteristic
71	C₂×C₄×C₅	—	8	2	no	—
72	C₂×C₄×C₅	—	8	2	no	—
73	C₂×C₄×C₅	—	8	2	no	—
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
77	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₅	#137	1	4	yes	—
82	C₂×Dic₅	—	8	1	no	—
83	C₂×Dic₅	—	8	1	no	—
84	C₂×Dic₅	—	8	2	no	—
85	C₂×Dic₅	—	8	2	no	—
86	C₅×C₈	#107	1	8	yes	complemented
87	#168	—	2	8	no	—
88	C₂²×C₈	—	5	4	no	—
89	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	—	10	2	no	—
94	C₂×C₄⋊C₄	—	10	2	no	—
95	C₂×C₄⋊C₄	—	10	1	no	—
96	C₂×C₄⋊C₄	—	10	1	no	—
97	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
98	#107	—	4	4	no	—
99	#107	—	20	8	no	—
100	#106	—	20	4	no	—
101	C₄×C₈	—	5	4	no	—
102	C₄×C₈	—	5	4	no	—
103	C₄×C₈	Dic₁₀	1	2	yes	—
104	C₄×C₈	C₄×D₅	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	—	16	2	no	—
111	C₄×C₅	C₂×#107	1	2	yes	—
112	C₄×C₅	#428	1	4	yes	—
113	C₄×C₅	#500	1	2	yes	—
114	C₄×C₅	#501	1	4	yes	—
115	Dic₅	—	16	1	no	—
116	Dic₅	—	16	1	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	20	1	no	—
119	C₂²×C₄	—	20	1	no	—
120	C₂²×C₄	D₄×D₅	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
122	C₂×C₈	—	5	8	no	—
123	C₂×C₈	—	10	4	no	—
124	C₂×C₈	—	10	4	no	—
125	C₂×C₈	#683	1	4	yes	—
126	C₂×C₈	#686	1	4	yes	—
127	C₂×C₈	#689	1	4	yes	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	—	20	4	no	—
132	C₄⋊C₄	—	20	4	no	—
133	C₄⋊C₄	—	20	2	no	—
134	C₄⋊C₄	—	20	2	no	—
135	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
136	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
137	C₄²	Q₈×D₅	1	2	yes	—
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	#4086	1	1	yes	characteristic
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	40	1	no	—
148	C₂×C₄	—	40	1	no	—
149	C₂×C₄	—	40	2	no	—
150	C₂×C₄	—	40	2	no	—
151	C₂×C₄	#4034	1	2	yes	—
152	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
153	C₂×C₄	#4057	1	4	yes	—
154	C₂×C₄	#4064	1	4	yes	—
155	C₂×C₄	C₂×#683	1	1	yes	characteristic
156	C₂×C₄	C₂×#686	1	2	yes	—
157	C₂×C₄	C₂×#689	1	2	yes	—
158	C₂×C₄	#4090	1	1	yes	characteristic
159	C₈	—	10	8	no	—
160	C₈	#3939	1	8	yes	complemented
161	C₅	#8820	1	1	yes	characteristic, complemented, Sylow
162	C₂²	#65265	1	1	yes	characteristic
163	C₂²	#65362	1	1	yes	characteristic
164	C₂²	#65381	1	1	yes	characteristic
165	C₂²	#65443	1	2	yes	—
166	C₂²	#65479	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	80	1	no	—
169	C₄	—	80	1	no	—
170	C₄	#65165	1	4	yes	—
171	C₄	#65170	1	4	yes	—
172	C₄	C₂×#3939	1	2	yes	—
173	C₄	#65367	1	2	yes	—
174	C₂	#10600370	1	1	yes	characteristic
175	C₂	#10601401	1	2	yes	—
176	C₂	#10602073	1	1	yes	characteristic
177	C₂	#10602094	1	1	yes	characteristic
178	C₂	#10602160	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1845

C₂

yes

maximal, characteristic

#10600557

C₂

yes

maximal, characteristic

C₂×#64675

C₂

yes

maximal, characteristic

#10600370

C₂

yes

maximal

#10600367

C₂

yes

maximal

C₂×C₅×#107

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

C₂×#3949

—

C₂×#3939

—

#65224

C₂²

yes

characteristic

#65224

C₂²

yes

characteristic

C₂×C₄×#168

C₂²

yes

—

#64749

—

#64675

C₄

yes

complemented

#8820

—

maximal, Sylow

C₂²×C₅×C₈

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

D₄

yes

—

C₅×#107

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

#3949

—

#3939

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

Q₈

yes

—

#1845

—

#1845

D₅

yes

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

Q₁₆

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

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

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

C₂×D₅

yes

—

#340

—

#322

—

C₂³×C₅

#127

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

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

—

C₅×C₈

#107

yes

complemented

#168

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅

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

—

102

C₄×C₈

—

103

C₄×C₈

Dic₁₀

yes

—

104

C₄×C₈

C₄×D₅

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

—

111

C₄×C₅

C₂×#107

yes

—

112

C₄×C₅

#428

yes

—

113

C₄×C₅

#500

yes

—

114

C₄×C₅

#501

yes

—

115

Dic₅

—

116

Dic₅

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

D₄×D₅

yes

—

121

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#683

yes

—

126

C₂×C₈

#686

yes

—

127

C₂×C₈

#689

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

characteristic

136

C₄²

C₂×C₄×D₅

yes

characteristic

137

C₄²

Q₈×D₅

yes

—

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

#4086

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

#4034

yes

—

152

C₂×C₄

D₅×C₄⋊C₄

yes

—

153

C₂×C₄

#4057

yes

—

154

C₂×C₄

#4064

yes

—

155

C₂×C₄

C₂×#683

yes

characteristic

156

C₂×C₄

C₂×#686

yes

—

157

C₂×C₄

C₂×#689

yes

—

158

C₂×C₄

#4090

yes

characteristic

159

C₈

—

160

C₈

#3939

yes

complemented

161

C₅

#8820

yes

characteristic, complemented, Sylow

162

C₂²

#65265

yes

characteristic

163

C₂²

#65362

yes

characteristic

164

C₂²

#65381

yes

characteristic

165

C₂²

#65443

yes

—

166

C₂²

#65479

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#65165

yes

—

171

C₄

#65170

yes

—

172

C₄

C₂×#3939

yes

—

173

C₄

#65367

yes

—

174

C₂

#10600370

yes

characteristic

175

C₂

#10601401

yes

—

176

C₂

#10602073

yes

characteristic

177

C₂

#10602094

yes

characteristic

178

C₂

#10602160

yes

—

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1845	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1845	D₅
hypocenter	C₅	#8820
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#4086

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
15. #64675	C₄	semidirect
86. C₅×C₈	#107	semidirect
160. C₈	#3939	semidirect
161. C₅	#8820	semidirect

H ≅ G/N

type

15. #64675

C₄

semidirect

86. C₅×C₈

#107

semidirect

160. C₈

#3939

semidirect

161. C₅

#8820

semidirect

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

subgroup	index
C₅×#1845	2
#10600557	2
C₂×#64675	2
#10600370	2
#10600367	2
#8820	5

subgroup

index

Subgroups — #11993090

Conjugacy classes of subgroups

Complements

Maximal subgroups