Group #12004112 — subgroups

Subgroups — #12004112

Order 1280.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#10601571	C₂	1	2	yes	maximal
2	#10600634	C₂	1	1	yes	maximal, characteristic
3	#10600615	C₂	1	1	yes	maximal, characteristic
4	C₅×#1281	C₂	1	1	yes	maximal, characteristic
5	#10598229	C₂	1	2	yes	maximal
6	C₂×#4015	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₅×#97	C₂²	1	1	yes	characteristic
9	C₂×#3948	—	2	2	no	—
10	#65225	C₂²	1	1	yes	characteristic
11	#65213	C₂²	1	2	yes	—
12	#64772	—	2	2	no	—
13	#64706	—	2	4	no	—
14	#10635	—	5	1	no	maximal, Sylow
15	C₂²×Dic₁₀	—	2	1	no	—
16	C₂²×C₅×C₈	D₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅	—	2	2	no	—
19	C₂×C₄⋊Dic₅	—	2	1	no	—
20	C₂×#681	—	2	2	no	—
21	#4015	—	2	2	no	—
22	#4015	—	2	1	no	—
23	#4015	D₄	1	2	yes	—
24	C₅×C₄⋊C₈	—	2	2	no	—
25	C₅×C₂²⋊C₈	—	2	2	no	—
26	C₅×#97	—	2	2	no	—
27	C₄×C₅×C₈	—	2	2	no	—
28	#3963	—	2	2	no	—
29	#3948	—	4	4	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	#1960	—	5	1	no	—
33	#1838	—	5	1	no	—
34	#1546	—	5	2	no	—
35	#1286	—	5	2	no	—
36	#1281	D₅	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	#41	1	2	yes	—
39	C₂²×Dic₅	—	4	1	no	—
40	C₂²×Dic₅	—	4	1	no	—
41	C₂×Dic₁₀	—	2	2	no	—
42	C₂×Dic₁₀	—	4	2	no	—
43	C₂×Dic₁₀	—	4	2	no	—
44	C₂×Dic₁₀	—	4	1	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	4	2	no	—
49	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅	#41	1	2	yes	—
51	C₄²×C₅	#41	1	1	yes	characteristic
52	C₄⋊Dic₅	—	2	4	no	—
53	C₄⋊Dic₅	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	#681	—	4	2	no	—
56	#681	—	4	2	no	—
57	C₂×#168	—	4	2	no	—
58	C₂×C₄⋊Q₈	—	5	1	no	—
59	#431	C₂×D₅	1	2	yes	—
60	#423	—	5	2	no	—
61	C₂×#103	—	10	2	no	—
62	C₂×#97	C₂×D₅	1	1	yes	characteristic
63	#382	—	5	1	no	—
64	#341	—	10	2	no	—
65	#332	—	10	4	no	—
66	C₂³×C₅	#124	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	2	no	—
68	C₂×C₄×C₅	#121	1	1	yes	characteristic
69	C₂×C₄×C₅	#123	1	2	yes	—
70	C₂×C₄×C₅	#123	1	1	yes	characteristic
71	C₂×C₄×C₅	#126	1	2	yes	—
72	C₂×C₄×C₅	#135	1	2	yes	—
73	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₅	—	4	2	no	—
75	C₂×Dic₅	—	8	1	no	—
76	C₂×Dic₅	—	8	1	no	—
77	C₂×Dic₅	—	8	2	no	—
78	C₂×Dic₅	—	8	1	no	—
79	Dic₁₀	—	4	4	no	—
80	Dic₁₀	—	8	2	no	—
81	C₅×C₈	—	4	2	no	—
82	C₅×C₈	—	8	2	no	—
83	#168	—	8	2	no	—
84	C₂²×Q₈	—	10	1	no	—
85	C₂×M₁₆	—	10	2	no	—
86	C₂²×C₈	D₂₀	1	2	yes	—
87	C₄⋊Q₈	—	5	2	no	—
88	C₄⋊Q₈	—	10	2	no	—
89	C₄⋊Q₈	—	10	1	no	—
90	C₂×C₄⋊C₄	—	10	2	no	—
91	C₂×C₄⋊C₄	—	10	1	no	—
92	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	10	2	no	—
95	C₄⋊C₈	—	10	2	no	—
96	#103	—	20	4	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	#97	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	#95	—	10	2	no	—
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	#397	1	1	yes	characteristic
103	C₂²×C₅	#488	1	1	yes	characteristic
104	C₂²×C₅	#490	1	1	yes	characteristic
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	2	2	no	—
107	C₄×C₅	#463	1	2	yes	—
108	C₄×C₅	#482	1	2	yes	—
109	Dic₅	—	8	2	no	—
110	Dic₅	—	16	1	no	—
111	C₂×Q₈	—	10	2	no	—
112	C₂×Q₈	—	20	1	no	—
113	C₂×Q₈	—	20	2	no	—
114	C₂×Q₈	—	20	2	no	—
115	C₂²×C₄	—	20	1	no	—
116	C₂²×C₄	—	20	1	no	—
117	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
118	C₂²×C₄	#709	1	2	yes	—
119	M₁₆	—	20	2	no	—
120	M₁₆	—	20	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	20	2	no	—
126	C₄⋊C₄	—	10	4	no	—
127	C₄⋊C₄	—	20	2	no	—
128	C₄⋊C₄	—	20	2	no	—
129	C₄²	D₄×D₅	1	1	yes	characteristic
130	C₄²	#711	1	2	yes	—
131	C₄²	#713	1	1	yes	characteristic
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	#2073	1	1	yes	characteristic
135	C₂×C₅	#2082	1	2	yes	—
136	C₂³	#4021	1	1	yes	characteristic
137	Q₈	—	20	4	no	—
138	Q₈	—	40	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	20	2	no	—
141	C₂×C₄	—	40	1	no	—
142	C₂×C₄	—	40	1	no	—
143	C₂×C₄	—	40	2	no	—
144	C₂×C₄	—	40	1	no	—
145	C₂×C₄	#4029	1	2	yes	—
146	C₂×C₄	#4032	1	1	yes	characteristic
147	C₂×C₄	#4041	1	1	yes	characteristic
148	C₂×C₄	#4044	1	2	yes	—
149	C₂×C₄	#4050	1	2	yes	—
150	C₂×C₄	#4054	1	2	yes	—
151	C₈	—	4	2	no	—
152	C₈	—	8	2	no	—
153	C₈	—	40	2	no	—
154	C₅	#10635	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	2	no	—
156	C₂²	#64963	1	1	yes	characteristic
157	C₂²	#64986	1	1	yes	characteristic
158	C₂²	#65002	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	40	2	no	—
162	C₄	—	80	1	no	—
163	C₄	#65026	1	2	yes	—
164	C₄	#65133	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#10599430	1	1	yes	characteristic
168	C₂	#10599815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10601571

C₂

yes

maximal

#10600634

C₂

yes

maximal, characteristic

#10600615

C₂

yes

maximal, characteristic

C₅×#1281

C₂

yes

maximal, characteristic

#10598229

C₂

yes

maximal

C₂×#4015

C₂²

yes

characteristic

C₅×#431

C₂²

yes

—

C₂×C₅×#97

C₂²

yes

characteristic

C₂×#3948

—

#65225

C₂²

yes

characteristic

#65213

C₂²

yes

—

—

—

—

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

—

—

—

D₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

—

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#681

—

#681

—

C₂×#168

—

C₂×C₄⋊Q₈

—

#431

C₂×D₅

yes

—

#423

—

C₂×#103

—

C₂×#97

C₂×D₅

yes

—

—

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#123

yes

characteristic

C₂×C₄×C₅

#126

yes

—

C₂×C₄×C₅

#135

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

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

D₂₀

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

#95

—

101

C₂²×C₅

—

102

C₂²×C₅

#397

yes

characteristic

103

C₂²×C₅

#488

yes

characteristic

104

C₂²×C₅

#490

yes

characteristic

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

#463

yes

—

108

C₄×C₅

#482

yes

—

109

Dic₅

—

110

Dic₅

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×D₂₀

yes

characteristic

118

C₂²×C₄

#709

yes

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

D₄×D₅

yes

characteristic

130

C₄²

#711

yes

—

131

C₄²

#713

yes

characteristic

132

C₂×C₅

—

133

C₂×C₅

—

134

C₂×C₅

#2073

yes

characteristic

135

C₂×C₅

#2082

yes

—

136

C₂³

#4021

yes

characteristic

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#4029

yes

—

146

C₂×C₄

#4032

yes

characteristic

147

C₂×C₄

#4041

yes

characteristic

148

C₂×C₄

#4044

yes

—

149

C₂×C₄

#4050

yes

—

150

C₂×C₄

#4054

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₅

#10635

yes

characteristic, complemented, Sylow

155

C₂²

—

156

C₂²

#64963

yes

characteristic

157

C₂²

#64986

yes

characteristic

158

C₂²

#65002

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#65026

yes

—

164

C₄

#65133

yes

—

165

C₂

—

166

C₂

—

167

C₂

#10599430

yes

characteristic

168

C₂

#10599815

yes

—

	subgroup	G/H
center	C₂²	#64963
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1281	C₂
socle	C₂²×C₅	#397
prime radical	C₂³×C₅	#124
hypercenter	#1281	D₅
hypocenter	C₅	#10635
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#64963

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
154. C₅	#10635	semidirect

H ≅ G/N

type

154. C₅

#10635

semidirect

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

subgroup	index
#10601571	2
#10600634	2
#10600615	2
C₅×#1281	2
#10598229	2
#10635	5

subgroup

index

Subgroups — #12004112

Conjugacy classes of subgroups

Complements

Maximal subgroups