Group #12013579 — subgroups

Subgroups — #12013579

Order 1280.

212 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 71 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×#2183	C₂	1	1	yes	maximal, characteristic
2	#10601735	C₂	1	2	yes	maximal
3	#10599859	C₂	1	1	yes	maximal, characteristic
4	#10598052	C₂	1	2	yes	maximal
5	#10597992	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
7	#65309	—	2	2	no	—
8	C₂×Dic₄₀	—	2	1	no	—
9	C₂×Dic₄₀	—	2	1	no	—
10	#64988	C₂²	1	2	yes	—
11	C₅×C₄⋊C₁₆	—	2	1	no	—
12	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₅×#346	—	2	1	no	—
14	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	#64702	—	2	2	no	—
16	#64678	C₂²	1	2	yes	—
17	#12443	—	5	1	no	maximal, Sylow
18	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
19	C₂×Dic₂₀	—	2	2	no	—
20	C₂×Dic₂₀	—	2	2	no	—
21	#4015	—	2	2	no	—
22	C₂×C₅×C₁₆	—	2	1	no	—
23	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
24	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
25	C₅×C₄⋊C₈	—	2	1	no	—
26	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
27	C₄×C₅×C₈	—	2	1	no	—
28	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
29	#3967	—	4	2	no	—
30	#3940	—	4	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	Dic₄₀	—	4	2	no	—
33	Dic₄₀	—	4	2	no	—
34	#2247	—	5	1	no	—
35	#2183	D₅	1	1	yes	characteristic
36	#1670	—	5	2	no	—
37	#1372	—	5	1	no	—
38	#1362	—	5	2	no	—
39	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
40	C₂×Dic₁₀	—	4	2	no	—
41	C₂×C₅×C₈	—	2	1	no	—
42	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₅×C₈	#41	1	1	yes	characteristic
44	C₅×C₄⋊C₄	—	2	1	no	—
45	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
46	C₄²×C₅	—	2	1	no	—
47	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₅	—	4	2	no	—
49	C₄⋊Dic₅	—	4	2	no	—
50	C₂×#168	—	4	2	no	—
51	Dic₂₀	—	4	2	no	—
52	Dic₂₀	—	4	2	no	—
53	Dic₂₀	—	8	2	no	—
54	C₅×C₁₆	—	2	1	no	—
55	C₅×C₁₆	—	4	1	no	—
56	C₅×C₁₆	SD₁₆	1	2	yes	—
57	C₂×Q₃₂	—	10	1	no	—
58	C₂×Q₃₂	—	10	1	no	—
59	#494	—	5	2	no	—
60	#475	—	10	2	no	—
61	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	1	no	—
63	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
64	#346	—	2	1	no	—
65	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
66	#335	—	5	2	no	—
67	#332	—	10	2	no	—
68	C₅×Q₈	—	2	2	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	#121	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅	#135	1	1	yes	characteristic
74	C₂×Dic₅	—	8	2	no	—
75	Dic₁₀	—	8	2	no	—
76	Dic₁₀	—	8	2	no	—
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	4	1	no	—
79	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
80	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
81	#168	—	8	2	no	—
82	C₂×Q₁₆	—	10	2	no	—
83	C₂×Q₁₆	—	10	2	no	—
84	C₄⋊Q₈	—	10	2	no	—
85	C₄×Q₈	D₂₀	1	1	yes	characteristic
86	Q₃₂	—	20	2	no	—
87	Q₃₂	—	20	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
90	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
91	#106	—	20	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	10	2	no	—
94	C₄⋊C₈	D₂₀	1	1	yes	characteristic
95	#103	—	20	2	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
98	C₂²×C₅	#466	1	1	yes	characteristic
99	C₄×C₅	—	2	1	no	—
100	C₄×C₅	—	4	1	no	—
101	C₄×C₅	—	4	1	no	—
102	C₄×C₅	#460	1	1	yes	characteristic
103	C₄×C₅	#472	1	1	yes	characteristic
104	C₄×C₅	#510	1	2	yes	—
105	Dic₅	—	16	2	no	—
106	C₂×Q₈	—	20	2	no	—
107	C₂×Q₈	D₄₀	1	1	yes	characteristic
108	Q₁₆	—	20	2	no	—
109	Q₁₆	—	20	2	no	—
110	Q₁₆	—	40	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	20	2	no	—
113	C₂×C₈	#709	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	20	2	no	—
117	C₄⋊C₄	—	20	2	no	—
118	C₄⋊C₄	D₄₀	1	1	yes	characteristic
119	C₄²	—	2	1	no	—
120	C₄²	C₂×D₂₀	1	1	yes	characteristic
121	C₁₆	—	2	1	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	#688	1	2	yes	—
124	C₂×C₅	#1668	1	1	yes	characteristic
125	C₂×C₅	#2219	1	1	yes	characteristic
126	C₂×C₅	#2222	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	40	2	no	—
129	Q₈	—	40	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	40	2	no	—
133	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
134	C₂×C₄	#4050	1	1	yes	characteristic
135	C₂×C₄	#4076	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	40	2	no	—
139	C₈	C₂×#688	1	1	yes	characteristic
140	C₈	#4095	1	1	yes	characteristic
141	C₅	#12443	1	1	yes	characteristic, complemented, Sylow
142	C₂²	#65404	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	80	2	no	—
147	C₄	#65198	1	2	yes	—
148	C₄	#65316	1	1	yes	characteristic
149	C₄	#65431	1	1	yes	characteristic
150	C₂	#10601732	1	1	yes	characteristic
151	C₂	#10602240	1	1	yes	characteristic
152	C₂	#10602249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2183

C₂

yes

maximal, characteristic

#10601735

C₂

yes

maximal

#10599859

C₂

yes

maximal, characteristic

#10598052

C₂

yes

maximal

#10597992

C₂

yes

maximal, characteristic

C₅×C₈×Q₈

C₂²

yes

characteristic

#65309

—

C₂×Dic₄₀

—

C₂×Dic₄₀

—

#64988

C₂²

yes

—

C₅×C₄⋊C₁₆

—

C₅×C₄⋊C₁₆

C₂²

yes

characteristic

C₅×#346

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

#64702

—

#64678

C₂²

yes

—

#12443

—

maximal, Sylow

C₄×C₅×Q₈

D₄

yes

characteristic

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

D₄

yes

characteristic

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂³

yes

—

—

—

Dic₄₀

—

Dic₄₀

—

#2247

—

#2183

D₅

yes

—

—

—

C₂×C₅×Q₈

D₈

yes

characteristic

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

D₈

yes

characteristic

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

Dic₂₀

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

C₅×C₁₆

—

C₅×C₁₆

SD₁₆

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₅

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₅

yes

characteristic

#335

—

#332

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×SD₁₆

yes

C₅×C₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₂₀

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

characteristic

C₂×C₁₆

C₂²⋊D₅

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₂₀

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#466

yes

characteristic

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

#460

yes

characteristic

103

C₄×C₅

#472

yes

characteristic

104

C₄×C₅

#510

yes

—

105

Dic₅

—

106

C₂×Q₈

—

107

C₂×Q₈

D₄₀

yes

characteristic

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

#709

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

D₄₀

yes

characteristic

119

C₄²

—

120

C₄²

C₂×D₂₀

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

#688

yes

—

124

C₂×C₅

#1668

yes

characteristic

125

C₂×C₅

#2219

yes

characteristic

126

C₂×C₅

#2222

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×D₄₀

yes

characteristic

134

C₂×C₄

#4050

yes

characteristic

135

C₂×C₄

#4076

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

C₂×#688

yes

characteristic

140

C₈

#4095

yes

characteristic

141

C₅

#12443

yes

characteristic, complemented, Sylow

142

C₂²

#65404

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#65198

yes

—

148

C₄

#65316

yes

characteristic

149

C₄

#65431

yes

characteristic

150

C₂

#10601732

yes

characteristic

151

C₂

#10602240

yes

characteristic

152

C₂

#10602249

yes

characteristic

	subgroup	G/H
center	C₂²	#65404
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2183	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#2183	D₅
hypocenter	C₅	#12443
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#65404

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
141. C₅	#12443	semidirect

H ≅ G/N

type

141. C₅

#12443

semidirect

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

subgroup	index
C₅×#2183	2
#10601735	2
#10599859	2
#10598052	2
#10597992	2
#12443	5

subgroup

index

Subgroups — #12013579

Conjugacy classes of subgroups

Complements

Maximal subgroups