Group #12013999 — subgroups

Subgroups — #12013999

Order 1280.

268 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 83 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	#10599858	C₂	1	2	yes	maximal, complemented
2	#10599851	C₂	1	2	yes	maximal
3	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598853	C₂	1	1	yes	maximal, characteristic
5	C₅×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#64988	—	2	1	no	—
7	#64988	C₂²	1	1	yes	characteristic
8	#64984	C₂²	1	1	yes	characteristic
9	#64983	—	2	1	no	—
10	#64983	—	2	1	no	—
11	#64971	—	2	1	no	—
12	#64971	C₂²	1	2	yes	—
13	C₄×C₅×C₁₆	C₂²	1	2	yes	—
14	C₅×C₈²	C₂²	1	1	yes	characteristic
15	#64729	—	2	4	no	—
16	#64723	—	2	4	no	—
17	#12512	—	5	1	no	maximal, Sylow
18	C₂×Dic₂₀	—	2	2	no	—
19	C₂×Dic₂₀	—	4	2	no	—
20	C₂×D₄₀	—	2	2	no	—
21	#4020	—	2	1	no	—
22	#4015	—	2	2	no	—
23	#4015	—	2	1	no	—
24	C₂×C₅×C₁₆	D₄	1	4	yes	—
25	C₄×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×C₈	D₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
28	#3953	—	4	2	no	—
29	#3953	—	4	2	no	—
30	#3950	—	2	2	no	—
31	#3950	—	2	2	no	—
32	#3950	—	4	2	no	—
33	#2241	—	5	2	no	—
34	#2240	—	5	2	no	—
35	#1713	—	5	1	no	—
36	#1709	—	5	1	no	—
37	#1311	D₅	1	1	yes	characteristic, complemented
38	C₂×D₂₀	—	4	1	no	—
39	C₂×D₂₀	—	4	1	no	—
40	C₂×D₂₀	—	4	1	no	—
41	C₂×Dic₁₀	—	4	1	no	—
42	C₂×Dic₁₀	—	4	2	no	—
43	C₂×C₅×C₈	D₈	1	2	yes	—
44	C₂×C₅×C₈	D₈	1	2	yes	—
45	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
46	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Dic₅	—	4	2	no	—
48	C₄⋊Dic₅	—	4	2	no	—
49	C₄⋊Dic₅	—	4	1	no	—
50	C₄⋊Dic₅	—	4	1	no	—
51	Dic₂₀	—	4	4	no	—
52	Dic₂₀	—	8	4	no	—
53	D₄₀	—	4	4	no	—
54	C₅×C₁₆	—	2	4	no	—
55	#500	—	5	2	no	—
56	#500	—	10	1	no	—
57	#494	—	5	1	no	—
58	#494	—	10	1	no	—
59	#493	—	5	1	no	—
60	#486	—	10	1	no	—
61	#486	—	10	1	no	—
62	#358	—	10	4	no	—
63	#357	—	10	4	no	—
64	C₄×C₁₆	C₂×D₅	1	2	yes	—
65	C₈²	C₂×D₅	1	1	yes	characteristic
66	C₂²×D₅	—	8	1	no	—
67	C₂×C₄×C₅	#127	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
70	C₂×Dic₅	—	8	1	no	—
71	C₂×Dic₅	—	8	2	no	—
72	D₂₀	—	8	2	no	—
73	D₂₀	—	8	2	no	—
74	D₂₀	—	8	2	no	—
75	Dic₁₀	—	8	2	no	—
76	Dic₁₀	—	8	4	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	#135	1	4	yes	—
80	C₂×Q₁₆	—	10	2	no	—
81	C₂×Q₁₆	—	20	2	no	—
82	C₂×D₈	—	10	2	no	—
83	C₄⋊Q₈	—	10	1	no	—
84	C₄⋊Q₈	—	10	2	no	—
85	#127	—	10	1	no	—
86	C₂×C₁₆	D₂₀	1	4	yes	—
87	#107	—	10	2	no	—
88	#107	—	10	2	no	—
89	#107	—	20	2	no	—
90	#102	—	20	2	no	—
91	#102	—	20	2	no	—
92	C₄×C₈	D₂₀	1	1	yes	characteristic
93	C₄×C₈	D₂₀	1	1	yes	characteristic
94	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
95	C₂²×C₅	#493	1	1	yes	characteristic
96	C₂×D₅	—	16	1	no	—
97	C₂×D₅	—	16	1	no	—
98	C₂×D₅	—	16	1	no	—
99	C₄×C₅	#495	1	2	yes	—
100	C₄×C₅	#509	1	2	yes	—
101	C₄×C₅	#510	1	2	yes	—
102	Dic₅	—	16	1	no	—
103	Dic₅	—	16	2	no	—
104	C₂×Q₈	—	20	1	no	—
105	C₂×Q₈	—	20	2	no	—
106	C₂×D₄	—	20	1	no	—
107	C₂×D₄	—	20	1	no	—
108	C₂×D₄	—	20	1	no	—
109	Q₁₆	—	20	4	no	—
110	Q₁₆	—	40	4	no	—
111	D₈	—	20	4	no	—
112	C₂×C₈	D₄₀	1	2	yes	—
113	C₂×C₈	D₄₀	1	2	yes	—
114	C₂×C₈	C₂×D₂₀	1	2	yes	—
115	C₄⋊C₄	—	20	2	no	—
116	C₄⋊C₄	—	20	1	no	—
117	C₄⋊C₄	—	20	1	no	—
118	C₄⋊C₄	—	20	2	no	—
119	C₄²	C₂×D₂₀	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₂×C₅	#1709	1	1	yes	characteristic
122	C₂×C₅	#2250	1	1	yes	characteristic
123	C₂×C₅	#2251	1	1	yes	characteristic
124	D₅	—	32	1	no	—
125	C₂³	—	40	1	no	—
126	Q₈	—	40	4	no	—
127	Q₈	—	40	2	no	—
128	D₄	—	40	2	no	—
129	D₄	—	40	2	no	—
130	D₄	—	40	2	no	—
131	C₂×C₄	—	40	2	no	—
132	C₂×C₄	—	40	1	no	—
133	C₂×C₄	#4020	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	#4050	1	4	yes	—
139	C₅	#12512	1	1	yes	characteristic, complemented, Sylow
140	C₂²	—	80	1	no	—
141	C₂²	—	80	1	no	—
142	C₂²	—	80	1	no	—
143	C₂²	#64984	1	1	yes	characteristic
144	C₄	—	80	2	no	—
145	C₄	—	80	1	no	—
146	C₄	#64985	1	2	yes	—
147	C₄	#65197	1	2	yes	—
148	C₄	#65198	1	2	yes	—
149	C₂	—	160	1	no	—
150	C₂	#10598853	1	1	yes	characteristic
151	C₂	#10599874	1	1	yes	characteristic
152	C₂	#10599877	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599858

C₂

yes

maximal, complemented

#10599851

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic, complemented

#10598853

C₂

yes

maximal, characteristic

C₅×#1311

C₂

yes

maximal, characteristic, complemented

#64988

—

#64988

C₂²

yes

characteristic

#64984

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×D₄₀

—

#4020

—

#4015

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

D₄₀

—

C₅×C₁₆

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₂₀

yes

—

—

—

—

—

—

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#495

yes

—

100

C₄×C₅

#509

yes

—

101

C₄×C₅

#510

yes

—

102

Dic₅

—

103

Dic₅

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

Q₁₆

—

110

Q₁₆

—

111

D₈

—

112

C₂×C₈

D₄₀

yes

—

113

C₂×C₈

D₄₀

yes

—

114

C₂×C₈

C₂×D₂₀

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂×D₂₀

yes

characteristic

120

C₁₆

—

121

C₂×C₅

#1709

yes

characteristic

122

C₂×C₅

#2250

yes

characteristic

123

C₂×C₅

#2251

yes

characteristic

124

D₅

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#4020

yes

characteristic

134

C₂×C₄

C₂×D₄₀

yes

characteristic

135

C₂×C₄

C₂×D₄₀

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

#4050

yes

—

139

C₅

#12512

yes

characteristic, complemented, Sylow

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#64984

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#64985

yes

—

147

C₄

#65197

yes

—

148

C₄

#65198

yes

—

149

C₂

—

160

—

150

C₂

#10598853

yes

characteristic

151

C₂

#10599874

yes

characteristic

152

C₂

#10599877

yes

characteristic

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1311	C₂
socle	C₂²×C₅	#493
prime radical	#64984	C₂²
hypercenter	#1311	D₅
hypocenter	C₅	#12512
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #10599858	C₂	semidirect
3. #10598858	C₂	semidirect
5. C₅×#1311	C₂	semidirect
37. #1311	D₅	semidirect
139. C₅	#12512	semidirect

H ≅ G/N

type

1. #10599858

C₂

semidirect

3. #10598858

C₂

semidirect

5. C₅×#1311

C₂

semidirect

37. #1311

D₅

semidirect

139. C₅

#12512

semidirect

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

subgroup	index
#10599858	2
#10599851	2
#10598858	2
#10598853	2
C₅×#1311	2
#12512	5

subgroup

index

Subgroups — #12013999

Conjugacy classes of subgroups

Complements

Maximal subgroups