Group #12004085 — subgroups

Subgroups — #12004085

Order 1280.

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

normal subgroups only

This group has 272 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	#10601569	C₂	1	2	yes	maximal
2	#10600234	C₂	1	1	yes	maximal, characteristic
3	#10600230	C₂	1	1	yes	maximal, characteristic
4	C₅×#1274	C₂	1	1	yes	maximal, characteristic
5	#10598236	C₂	1	2	yes	maximal
6	#65804	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	C₅×#376	C₂²	1	1	yes	characteristic
9	#65213	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#168	C₂²	1	2	yes	—
11	#64744	—	2	2	no	—
12	#64743	—	2	2	no	—
13	#64702	—	2	2	no	—
14	#64701	—	2	2	no	—
15	#10630	—	5	1	no	maximal, Sylow
16	C₂×C₄²×C₅	D₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
18	#4072	—	2	1	no	—
19	#4070	—	2	2	no	—
20	C₂×#681	—	2	1	no	—
21	C₂²×#168	—	2	2	no	—
22	#4016	—	2	1	no	—
23	#4015	D₄	1	2	yes	—
24	C₄×Dic₁₀	—	2	2	no	—
25	C₅×C₄⋊C₈	—	2	2	no	—
26	C₅×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₅×#95	—	2	2	no	—
29	C₄⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	#1995	—	5	1	no	—
33	#1832	—	5	1	no	—
34	#1542	—	5	2	no	—
35	#1301	—	5	2	no	—
36	#1274	D₅	1	1	yes	characteristic
37	C₂²×C₄×C₅	—	2	2	no	—
38	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	#41	1	2	yes	—
40	C₂×Dic₁₀	—	2	2	no	—
41	C₂×C₅×C₈	—	4	2	no	—
42	C₄²×C₅	—	2	2	no	—
43	C₄²×C₅	—	2	2	no	—
44	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅	#41	1	2	yes	—
46	C₄²×C₅	#41	1	1	yes	characteristic
47	C₂²⋊Dic₅	—	4	1	no	—
48	C₄⋊Dic₅	—	2	2	no	—
49	C₄⋊Dic₅	—	2	2	no	—
50	#683	—	4	2	no	—
51	C₄×Dic₅	—	4	1	no	—
52	#681	—	4	2	no	—
53	C₂×#168	—	4	2	no	—
54	C₂×#168	—	4	1	no	—
55	C₂×#168	—	4	2	no	—
56	C₂×#168	—	4	2	no	—
57	#533	—	5	1	no	—
58	#431	C₂×D₅	1	2	yes	—
59	#423	—	5	1	no	—
60	C₂×C₄⋊C₈	—	5	2	no	—
61	#376	C₂×D₅	1	1	yes	characteristic
62	#339	—	10	2	no	—
63	#336	—	10	2	no	—
64	#333	—	10	2	no	—
65	#332	—	10	2	no	—
66	C₂³×C₅	#123	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	2	no	—
68	C₂×C₄×C₅	—	4	4	no	—
69	C₂×C₄×C₅	—	4	2	no	—
70	C₂×C₄×C₅	#121	1	1	yes	characteristic
71	C₂×C₄×C₅	#123	1	2	yes	—
72	C₂×C₄×C₅	#124	1	1	yes	characteristic
73	C₂×C₄×C₅	#126	1	2	yes	—
74	C₂×C₄×C₅	#135	1	2	yes	—
75	C₂×C₄×C₅	#137	1	2	yes	—
76	C₂×Dic₅	—	4	2	no	—
77	Dic₁₀	—	8	2	no	—
78	C₅×C₈	—	8	2	no	—
79	#168	—	8	2	no	—
80	#168	—	8	1	no	—
81	C₂×M₁₆	—	10	1	no	—
82	C₂²×C₈	—	10	2	no	—
83	C₄⋊Q₈	—	5	2	no	—
84	#125	—	10	1	no	—
85	C₂²⋊Q₈	—	10	2	no	—
86	C₄×Q₈	—	10	2	no	—
87	#117	—	10	1	no	—
88	C₂×C₄²	D₂₀	1	2	yes	—
89	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	10	2	no	—
92	C₄⋊C₈	—	10	2	no	—
93	C₄⋊C₈	—	10	2	no	—
94	C₂²⋊C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	#95	—	2	2	no	—
97	C₂²×C₅	—	2	2	no	—
98	C₂²×C₅	#397	1	1	yes	characteristic
99	C₂²×C₅	#485	1	2	yes	—
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	—	8	2	no	—
103	C₄×C₅	#464	1	2	yes	—
104	C₄×C₅	#488	1	1	yes	characteristic
105	C₄×C₅	#491	1	1	yes	characteristic
106	Dic₅	—	8	2	no	—
107	C₂×Q₈	—	10	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
110	C₂²×C₄	#711	1	2	yes	—
111	M₁₆	—	20	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	20	1	no	—
114	C₂×C₈	—	20	2	no	—
115	C₂×C₈	—	20	2	no	—
116	C₂×C₈	—	20	2	no	—
117	C₄⋊C₄	—	10	2	no	—
118	C₄⋊C₄	—	10	2	no	—
119	C₄⋊C₄	—	20	2	no	—
120	C₂²⋊C₄	—	20	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	20	1	no	—
124	C₄²	#709	1	2	yes	—
125	C₄²	D₄×D₅	1	1	yes	characteristic
126	C₄²	#713	1	1	yes	characteristic
127	C₂×C₅	—	2	2	no	—
128	C₂×C₅	#2074	1	1	yes	characteristic
129	C₂×C₅	#2080	1	2	yes	—
130	C₂³	#4032	1	1	yes	characteristic
131	Q₈	—	40	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	20	2	no	—
136	C₂×C₄	#4021	1	1	yes	characteristic
137	C₂×C₄	#4029	1	2	yes	—
138	C₂×C₄	#4041	1	1	yes	characteristic
139	C₂×C₄	#4044	1	2	yes	—
140	C₂×C₄	#4050	1	2	yes	—
141	C₂×C₄	#4055	1	2	yes	—
142	C₈	—	8	2	no	—
143	C₈	—	40	2	no	—
144	C₈	—	40	1	no	—
145	C₅	#10630	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	2	2	no	—
147	C₂²	#64963	1	1	yes	characteristic
148	C₂²	#65012	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	40	2	no	—
153	C₄	#64986	1	1	yes	characteristic
154	C₄	#64992	1	1	yes	characteristic
155	C₄	#65134	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#10599431	1	1	yes	characteristic
158	C₂	#10599493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10601569

C₂

yes

maximal

#10600234

C₂

yes

maximal, characteristic

#10600230

C₂

yes

maximal, characteristic

C₅×#1274

C₂

yes

maximal, characteristic

#10598236

C₂

yes

maximal

#65804

C₂²

yes

characteristic

C₅×#431

C₂²

yes

—

C₅×#376

C₂²

yes

characteristic

#65213

C₂²

yes

characteristic

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

—

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₂²×#168

—

#4016

—

#4015

D₄

yes

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₅×#95

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

—

—

—

C₂×D₅

yes

—

#423

—

C₂×C₄⋊C₈

—

#376

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#123

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

#126

yes

—

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₂₀

yes

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₅

—

C₂²×C₅

#397

yes

characteristic

C₂²×C₅

#485

yes

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

#464

yes

—

104

C₄×C₅

#488

yes

characteristic

105

C₄×C₅

#491

yes

characteristic

106

Dic₅

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×D₂₀

yes

characteristic

110

C₂²×C₄

#711

yes

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

#709

yes

—

125

C₄²

D₄×D₅

yes

characteristic

126

C₄²

#713

yes

characteristic

127

C₂×C₅

—

128

C₂×C₅

#2074

yes

characteristic

129

C₂×C₅

#2080

yes

—

130

C₂³

#4032

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#4021

yes

characteristic

137

C₂×C₄

#4029

yes

—

138

C₂×C₄

#4041

yes

characteristic

139

C₂×C₄

#4044

yes

—

140

C₂×C₄

#4050

yes

—

141

C₂×C₄

#4055

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₅

#10630

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

#64963

yes

characteristic

148

C₂²

#65012

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#64986

yes

characteristic

154

C₄

#64992

yes

characteristic

155

C₄

#65134

yes

—

156

C₂

—

157

C₂

#10599431

yes

characteristic

158

C₂

#10599493

yes

—

	subgroup	G/H
center	C₂×C₄	#4021
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1274	C₂
socle	C₂²×C₅	#397
prime radical	C₂³×C₅	#123
hypercenter	#1274	D₅
hypocenter	C₅	#10630
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#4021

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 ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
145. C₅	#10630	semidirect

H ≅ G/N

type

145. C₅

#10630

semidirect

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

subgroup	index
#10601569	2
#10600234	2
#10600230	2
C₅×#1274	2
#10598236	2
#10630	5

subgroup

index

Subgroups — #12004085

Conjugacy classes of subgroups

Complements

Maximal subgroups