Group #12004289 — subgroups

Subgroups — #12004289

Order 1280.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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×#1849	C₂	1	1	yes	maximal, characteristic
2	#10601293	C₂	1	2	yes	maximal
3	#10601287	C₂	1	2	yes	maximal
4	#10600631	C₂	1	1	yes	maximal, characteristic
5	#10598151	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	—	2	2	no	—
8	C₅×#423	C₂²	1	2	yes	—
9	C₂×C₅×#97	C₂²	1	1	yes	characteristic
10	#65214	C₂²	1	2	yes	—
11	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
12	#64770	—	2	2	no	—
13	#64704	—	2	4	no	—
14	#64699	—	2	4	no	—
15	#10662	—	5	1	no	maximal, Sylow
16	C₂²×Dic₁₀	Q₈	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	D₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	Q₈	1	1	yes	characteristic
21	C₂×#683	—	2	1	no	—
22	C₂×C₄×Dic₅	—	2	1	no	—
23	C₂²×#168	—	2	1	no	—
24	C₄×Dic₁₀	—	2	2	no	—
25	C₄×Dic₁₀	—	2	2	no	—
26	C₅×#108	—	4	4	no	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×#97	—	2	2	no	—
30	C₂²⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄×#168	D₄	1	4	yes	—
34	#1985	—	5	1	no	—
35	#1849	D₅	1	1	yes	characteristic
36	#1571	—	5	2	no	—
37	#1487	—	5	2	no	—
38	#1275	—	5	1	no	—
39	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅	#41	1	1	yes	characteristic
42	C₂²×Dic₅	—	2	1	no	—
43	C₂²×Dic₅	—	2	1	no	—
44	C₂×Dic₁₀	—	2	2	no	—
45	C₂×Dic₁₀	—	4	2	no	—
46	C₅×M₁₆	—	4	4	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	4	2	no	—
50	C₂×C₅×C₈	Q₁₆	1	4	yes	—
51	C₄²×C₅	C₂×D₄	1	2	yes	—
52	C₄²×C₅	#41	1	2	yes	—
53	C₄⋊Dic₅	—	2	2	no	—
54	#683	—	4	2	no	—
55	C₄×Dic₅	—	4	1	no	—
56	C₄×Dic₅	—	4	1	no	—
57	C₂×#168	—	2	4	no	—
58	C₂×#168	—	4	1	no	—
59	C₂×#168	—	4	2	no	—
60	C₂×C₄×Q₈	—	5	1	no	—
61	#431	—	5	2	no	—
62	C₂×#108	—	2	2	no	—
63	#423	C₂×D₅	1	2	yes	—
64	C₂×C₄⋊C₈	—	5	1	no	—
65	C₂×#97	C₂×D₅	1	1	yes	characteristic
66	#336	—	10	2	no	—
67	#335	—	10	4	no	—
68	Q₈⋊C₈	—	10	4	no	—
69	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅	—	4	1	no	—
71	C₂×C₄×C₅	—	4	1	no	—
72	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅	#123	1	2	yes	—
74	C₂×C₄×C₅	#123	1	1	yes	characteristic
75	C₂×C₄×C₅	#127	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₅	—	2	2	no	—
79	C₂×Dic₅	—	4	1	no	—
80	C₂×Dic₅	—	4	1	no	—
81	C₂×Dic₅	—	4	2	no	—
82	C₂×Dic₅	—	4	1	no	—
83	Dic₁₀	—	2	4	no	—
84	Dic₁₀	—	4	2	no	—
85	C₅×C₈	—	2	4	no	—
86	C₅×C₈	—	8	2	no	—
87	#168	—	4	4	no	—
88	#168	—	8	1	no	—
89	C₂²×Q₈	—	5	1	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂²×C₈	—	10	1	no	—
92	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
93	C₄×Q₈	—	10	2	no	—
94	C₄×Q₈	—	10	2	no	—
95	C₂×C₄⋊C₄	—	5	1	no	—
96	C₂×C₄⋊C₄	—	10	1	no	—
97	C₂×C₄²	—	10	1	no	—
98	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
99	#108	—	4	4	no	—
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	C₂²⋊C₈	—	10	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	—	5	4	no	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	#399	1	1	yes	characteristic
110	C₂²×C₅	#477	1	2	yes	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	#484	1	2	yes	—
114	C₄×C₅	#494	1	1	yes	characteristic
115	C₄×C₅	#496	1	1	yes	characteristic
116	Dic₅	—	4	2	no	—
117	Dic₅	—	8	1	no	—
118	C₂×Q₈	—	10	2	no	—
119	C₂×Q₈	—	20	2	no	—
120	C₂²×C₄	—	10	1	no	—
121	C₂²×C₄	—	10	1	no	—
122	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
123	C₂²×C₄	#713	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
125	M₁₆	—	4	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	10	4	no	—
130	C₂×C₈	—	20	2	no	—
131	C₂×C₈	—	20	1	no	—
132	C₂×C₈	#689	1	4	yes	—
133	C₄⋊C₄	—	10	2	no	—
134	C₄⋊C₄	—	20	2	no	—
135	C₄²	—	20	1	no	—
136	C₄²	—	20	1	no	—
137	C₄²	#709	1	2	yes	—
138	C₄²	D₄×D₅	1	2	yes	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	#2091	1	1	yes	characteristic
141	C₂×C₅	#2095	1	1	yes	characteristic
142	C₂×C₅	#2098	1	1	yes	characteristic
143	C₂³	#4092	1	1	yes	characteristic
144	Q₈	—	10	4	no	—
145	Q₈	—	20	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	10	2	no	—
149	C₂×C₄	—	20	1	no	—
150	C₂×C₄	—	20	1	no	—
151	C₂×C₄	—	20	2	no	—
152	C₂×C₄	—	20	1	no	—
153	C₂×C₄	#4040	1	2	yes	—
154	C₂×C₄	#4042	1	2	yes	—
155	C₂×C₄	#4075	1	1	yes	characteristic
156	C₂×C₄	#4078	1	2	yes	—
157	C₂×C₄	#4086	1	1	yes	characteristic
158	C₂×C₄	C₂×#689	1	2	yes	—
159	C₈	—	2	4	no	—
160	C₈	—	8	2	no	—
161	C₈	—	20	4	no	—
162	C₈	—	40	1	no	—
163	C₅	#10662	1	1	yes	characteristic, complemented, Sylow
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#65281	1	1	yes	characteristic
167	C₂²	#65380	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	20	2	no	—
171	C₄	—	40	1	no	—
172	C₄	#65259	1	2	yes	—
173	C₄	#65363	1	1	yes	characteristic
174	C₄	#65381	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#10600383	1	1	yes	characteristic
177	C₂	#10601522	1	1	yes	characteristic
178	C₂	#10601531	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1849

C₂

yes

maximal, characteristic

#10601293

C₂

yes

maximal

#10601287

C₂

yes

maximal

#10600631

C₂

yes

maximal, characteristic

#10598151

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀

C₂²

yes

characteristic

C₂×C₅×#108

—

C₅×#423

C₂²

yes

—

C₂×C₅×#97

C₂²

yes

characteristic

#65214

C₂²

yes

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

C₂²×Dic₁₀

Q₈

yes

characteristic

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

Q₈

yes

characteristic

C₂×#683

—

C₂×C₄×Dic₅

—

C₂²×#168

—

C₄×Dic₁₀

—

C₄×Dic₁₀

—

C₅×#108

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#97

—

C₂²⋊#168

—

C₄⋊#168

—

C₄⋊#168

—

C₄×#168

D₄

yes

—

#1985

—

#1849

D₅

yes

—

—

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅

#41

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

Q₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×C₄×Q₈

—

#431

—

C₂×#108

—

#423

C₂×D₅

yes

—

C₂×C₄⋊C₈

—

C₂×#97

C₂×D₅

yes

characteristic

#336

—

#335

—

Q₈⋊C₈

—

C₂³×C₅

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#108

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

—

105

#97

—

106

C₄×C₈

—

107

C₂²×C₅

—

108

C₂²×C₅

—

109

C₂²×C₅

#399

yes

characteristic

110

C₂²×C₅

#477

yes

—

111

C₄×C₅

—

112

C₄×C₅

—

113

C₄×C₅

#484

yes

—

114

C₄×C₅

#494

yes

characteristic

115

C₄×C₅

#496

yes

characteristic

116

Dic₅

—

117

Dic₅

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

Q₈×D₅

yes

characteristic

123

C₂²×C₄

#713

yes

characteristic

124

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

#689

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

#709

yes

—

138

C₄²

D₄×D₅

yes

—

139

C₂×C₅

—

140

C₂×C₅

#2091

yes

characteristic

141

C₂×C₅

#2095

yes

characteristic

142

C₂×C₅

#2098

yes

characteristic

143

C₂³

#4092

yes

characteristic

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#4040

yes

—

154

C₂×C₄

#4042

yes

—

155

C₂×C₄

#4075

yes

characteristic

156

C₂×C₄

#4078

yes

—

157

C₂×C₄

#4086

yes

characteristic

158

C₂×C₄

C₂×#689

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₅

#10662

yes

characteristic, complemented, Sylow

164

C₂²

—

165

C₂²

—

166

C₂²

#65281

yes

characteristic

167

C₂²

#65380

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#65259

yes

—

173

C₄

#65363

yes

characteristic

174

C₄

#65381

yes

characteristic

175

C₂

—

176

C₂

#10600383

yes

characteristic

177

C₂

#10601522

yes

characteristic

178

C₂

#10601531

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#4086
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1849	C₂
socle	C₂²×C₅	#399
prime radical	C₂³×C₅	C₂²⋊Q₈
hypercenter	#1849	D₅
hypocenter	C₅	#10662
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

#4086

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂²⋊Q₈

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
163. C₅	#10662	semidirect

H ≅ G/N

type

163. C₅

#10662

semidirect

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

subgroup	index
C₅×#1849	2
#10601293	2
#10601287	2
#10600631	2
#10598151	2
#10662	5

subgroup

index

Subgroups — #12004289

Conjugacy classes of subgroups

Complements

Maximal subgroups