Group #11993065 — subgroups

Subgroups — #11993065

Order 1280.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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×#1831	C₂	1	1	yes	maximal, characteristic
2	C₂×#64699	C₂	1	2	yes	maximal
3	C₂×#64700	C₂	1	2	yes	maximal
4	#10600187	C₂	1	1	yes	maximal, characteristic
5	#10600181	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#168	—	2	2	no	—
10	C₂×C₄⋊#168	C₂²	1	2	yes	—
11	C₂×C₄×#168	C₂²	1	2	yes	—
12	C₅×#336	—	2	2	no	—
13	#64744	C₄	1	2	yes	—
14	#64744	C₄	1	2	yes	—
15	#64700	—	2	4	no	—
16	#64699	—	2	4	no	—
17	#8812	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	—	2	1	no	—
19	C₂²×C₅×C₈	—	2	1	no	—
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	D₄	1	2	yes	—
23	C₂×C₄²×C₅	Q₈	1	2	yes	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂²×#168	—	2	2	no	—
26	C₂²×#168	C₂×C₄	1	4	yes	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₄⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	4	4	no	—
32	C₄×#168	D₄	1	4	yes	—
33	C₄×#168	Q₈	1	4	yes	—
34	#1842	—	5	1	no	—
35	#1831	D₅	1	1	yes	characteristic
36	#1744	—	5	1	no	—
37	C₂×#335	—	5	2	no	—
38	C₂×#334	—	5	2	no	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₅×C₈	—	4	1	no	—
45	C₂×C₅×C₈	—	4	1	no	—
46	C₂×C₅×C₈	—	4	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₄²×C₅	—	2	4	no	—
49	C₄²×C₅	—	2	2	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	C₂×D₄	1	2	yes	—
54	C₄²×C₅	C₂×Q₈	1	2	yes	—
55	C₂×#168	—	2	4	no	—
56	C₂×#168	—	2	4	no	—
57	C₂×#168	—	4	2	no	—
58	C₂×#168	—	4	4	no	—
59	C₂×#168	C₄⋊C₄	1	8	yes	—
60	C₂×C₄⋊C₈	—	5	2	no	—
61	C₂×C₄⋊C₈	—	10	2	no	—
62	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
63	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
64	C₂×C₄×C₈	—	5	2	no	—
65	C₄³	C₂×D₅	1	1	yes	characteristic
66	#336	—	2	2	no	—
67	#336	—	5	2	no	—
68	#336	—	5	2	no	—
69	#335	—	10	4	no	—
70	#334	—	10	4	no	—
71	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	#108	1	4	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅	#127	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	#131	1	2	yes	—
84	C₂×C₄×C₅	#131	1	2	yes	—
85	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₅	#137	1	2	yes	—
87	C₅×C₈	—	8	1	no	—
88	C₅×C₈	—	8	1	no	—
89	#168	—	2	8	no	—
90	#168	—	8	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	5	4	no	—
94	C₂²×C₈	—	10	2	no	—
95	C₂×C₄²	Dic₁₀	1	2	yes	—
96	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
97	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
98	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
99	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	10	2	no	—
103	C₄⋊C₈	—	10	2	no	—
104	C₄⋊C₈	—	20	4	no	—
105	C₄×C₈	—	5	4	no	—
106	C₄×C₈	—	5	4	no	—
107	C₂²×C₅	#384	1	1	yes	characteristic
108	C₂²×C₅	#424	1	1	yes	characteristic
109	C₂²×C₅	#424	1	1	yes	characteristic
110	C₂²×C₅	#428	1	2	yes	—
111	C₂²×C₅	C₂×#108	1	2	yes	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	4	2	no	—
115	C₄×C₅	#424	1	4	yes	—
116	C₄×C₅	#496	1	1	yes	characteristic
117	C₄×C₅	#497	1	1	yes	characteristic
118	C₄×C₅	#501	1	2	yes	—
119	C₂²×C₄	#683	1	2	yes	—
120	C₂²×C₄	#683	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	5	8	no	—
129	C₂×C₈	—	10	4	no	—
130	C₂×C₈	—	10	4	no	—
131	C₂×C₈	—	20	2	no	—
132	C₂×C₈	—	20	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	D₄×D₅	1	2	yes	—
139	C₄²	Q₈×D₅	1	2	yes	—
140	C₂×C₅	#1573	1	2	yes	—
141	C₂×C₅	#1574	1	2	yes	—
142	C₂×C₅	#1843	1	1	yes	characteristic
143	C₂×C₅	#1847	1	1	yes	characteristic
144	C₂×C₅	#1848	1	1	yes	characteristic
145	C₂³	C₂×#683	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	#3954	1	4	yes	—
153	C₂×C₄	#4034	1	2	yes	—
154	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
155	C₂×C₄	#4047	1	2	yes	—
156	C₂×C₄	#4053	1	2	yes	—
157	C₂×C₄	#4086	1	1	yes	characteristic
158	C₂×C₄	#4090	1	1	yes	characteristic
159	C₂×C₄	#4095	1	2	yes	—
160	C₂×C₄	#4097	1	2	yes	—
161	C₈	—	8	1	no	—
162	C₈	—	8	1	no	—
163	C₈	—	10	8	no	—
164	C₈	—	40	2	no	—
165	C₅	#8812	1	1	yes	characteristic, complemented, Sylow
166	C₂²	#65265	1	1	yes	characteristic
167	C₂²	#65284	1	2	yes	—
168	C₂²	#65391	1	1	yes	characteristic
169	C₂²	#65408	1	1	yes	characteristic
170	C₂²	C₂×#3954	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	#65128	1	4	yes	—
175	C₄	#65347	1	1	yes	characteristic
176	C₄	#65349	1	1	yes	characteristic
177	C₄	#65354	1	2	yes	—
178	C₂	#10600343	1	1	yes	characteristic
179	C₂	#10601239	1	1	yes	characteristic
180	C₂	#10601311	1	1	yes	characteristic
181	C₂	#10601541	1	2	yes	—
182	C₂	#10601590	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1831

C₂

yes

maximal, characteristic

C₂×#64699

C₂

yes

maximal

C₂×#64700

C₂

yes

maximal

#10600187

C₂

yes

maximal, characteristic

#10600181

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₈

C₂²

yes

characteristic

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

—

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

C₅×#336

—

#64744

C₄

yes

—

#64744

C₄

yes

—

—

—

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

Q₈

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₂²×#168

C₂×C₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

—

—

—

D₄

yes

—

C₄×#168

Q₈

yes

—

#1842

—

#1831

D₅

yes

—

—

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅

yes

characteristic

C₂×C₄⋊C₈

C₂×D₅

yes

characteristic

C₂×C₄×C₈

—

C₄³

C₂×D₅

yes

—

—

—

—

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#108

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₁₀

yes

—

C₂×C₄²

C₄×D₅

yes

characteristic

C₂×C₄²

C₄×D₅

yes

characteristic

C₂×C₄²

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₂²×C₅

#384

yes

characteristic

108

C₂²×C₅

#424

yes

characteristic

109

C₂²×C₅

#424

yes

characteristic

110

C₂²×C₅

#428

yes

—

111

C₂²×C₅

C₂×#108

yes

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

#424

yes

—

116

C₄×C₅

#496

yes

characteristic

117

C₄×C₅

#497

yes

characteristic

118

C₄×C₅

#501

yes

—

119

C₂²×C₄

#683

yes

—

120

C₂²×C₄

#683

yes

—

121

C₂²×C₄

C₂×Dic₁₀

yes

characteristic

122

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

123

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

D₄×D₅

yes

—

139

C₄²

Q₈×D₅

yes

—

140

C₂×C₅

#1573

yes

—

141

C₂×C₅

#1574

yes

—

142

C₂×C₅

#1843

yes

characteristic

143

C₂×C₅

#1847

yes

characteristic

144

C₂×C₅

#1848

yes

characteristic

145

C₂³

C₂×#683

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#3954

yes

—

153

C₂×C₄

#4034

yes

—

154

C₂×C₄

D₅×C₄⋊C₄

yes

—

155

C₂×C₄

#4047

yes

—

156

C₂×C₄

#4053

yes

—

157

C₂×C₄

#4086

yes

characteristic

158

C₂×C₄

#4090

yes

characteristic

159

C₂×C₄

#4095

yes

—

160

C₂×C₄

#4097

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₅

#8812

yes

characteristic, complemented, Sylow

166

C₂²

#65265

yes

characteristic

167

C₂²

#65284

yes

—

168

C₂²

#65391

yes

characteristic

169

C₂²

#65408

yes

characteristic

170

C₂²

C₂×#3954

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#65128

yes

—

175

C₄

#65347

yes

characteristic

176

C₄

#65349

yes

characteristic

177

C₄

#65354

yes

—

178

C₂

#10600343

yes

characteristic

179

C₂

#10601239

yes

characteristic

180

C₂

#10601311

yes

characteristic

181

C₂

#10601541

yes

—

182

C₂

#10601590

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1831	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1831	D₅
hypocenter	C₅	#8812
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1831

C₂

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
165. C₅	#8812	semidirect

H ≅ G/N

type

165. C₅

#8812

semidirect

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

subgroup	index
C₅×#1831	2
C₂×#64699	2
C₂×#64700	2
#10600187	2
#10600181	2
#8812	5

subgroup

index

Subgroups — #11993065

Conjugacy classes of subgroups

Complements

Maximal subgroups