Group #12014074 — subgroups

Subgroups — #12014074

Order 1280.

318 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 91 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	#10599854	C₂	1	2	yes	maximal, complemented
2	#10599851	C₂	1	2	yes	maximal, complemented
3	#10598856	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598855	C₂	1	1	yes	maximal, characteristic
5	C₅×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3932	—	2	4	no	—
7	#64988	C₂²	1	1	yes	characteristic
8	#64984	—	2	1	no	—
9	#64984	—	2	1	no	—
10	#64984	C₂²	1	2	yes	—
11	#64971	C₂²	1	1	yes	characteristic
12	#64970	—	2	1	no	—
13	#64969	—	2	1	no	—
14	C₄×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×C₈²	C₂²	1	1	yes	characteristic
16	#64729	—	2	4	no	—
17	#12529	—	5	1	no	maximal, Sylow
18	C₂×Dic₂₀	—	2	2	no	—
19	C₂×D₄₀	—	2	2	no	—
20	C₂×D₄₀	—	2	2	no	—
21	C₂×D₄₀	—	4	2	no	—
22	C₂×D₄₀	—	4	2	no	—
23	#4020	—	2	2	no	—
24	#4015	—	2	1	no	—
25	#4015	—	2	1	no	—
26	C₂×C₅×C₁₆	D₄	1	4	yes	—
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
30	#3950	—	2	2	no	—
31	#3949	—	4	2	no	—
32	#3948	—	4	2	no	—
33	#3932	—	4	8	no	—
34	#2248	—	5	2	no	—
35	#2240	—	5	2	no	—
36	#1712	—	5	1	no	—
37	#1707	—	5	1	no	—
38	#1312	D₅	1	1	yes	characteristic, complemented
39	C₂×D₂₀	—	4	2	no	—
40	C₂×D₂₀	—	4	2	no	—
41	C₂×D₂₀	—	4	2	no	—
42	C₂×Dic₁₀	—	4	1	no	—
43	C₂×Dic₁₀	—	4	1	no	—
44	C₂×C₅×C₈	D₈	1	2	yes	—
45	C₂×C₅×C₈	D₈	1	2	yes	—
46	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
47	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₅	—	4	1	no	—
49	C₄⋊Dic₅	—	4	1	no	—
50	C₄⋊Dic₅	—	4	1	no	—
51	C₄⋊Dic₅	—	4	1	no	—
52	Dic₂₀	—	4	4	no	—
53	D₄₀	—	4	4	no	—
54	D₄₀	—	4	4	no	—
55	D₄₀	—	8	4	no	—
56	D₄₀	—	8	4	no	—
57	C₅×C₁₆	—	2	4	no	—
58	C₂×SD₃₂	—	10	4	no	—
59	#500	—	5	1	no	—
60	#498	—	10	1	no	—
61	#494	—	5	1	no	—
62	#493	—	5	2	no	—
63	#493	—	10	1	no	—
64	#493	—	10	1	no	—
65	#487	—	10	1	no	—
66	#357	—	10	4	no	—
67	C₄×C₁₆	C₂×D₅	1	2	yes	—
68	C₈²	C₂×D₅	1	1	yes	characteristic
69	C₂²×D₅	—	8	2	no	—
70	C₂×C₄×C₅	#127	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
73	C₂×Dic₅	—	8	1	no	—
74	C₂×Dic₅	—	8	1	no	—
75	D₂₀	—	8	2	no	—
76	D₂₀	—	8	2	no	—
77	D₂₀	—	8	4	no	—
78	D₂₀	—	8	4	no	—
79	Dic₁₀	—	8	2	no	—
80	Dic₁₀	—	8	2	no	—
81	C₅×C₈	—	2	2	no	—
82	C₅×C₈	SD₃₂	1	4	yes	complemented
83	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
84	C₂×Q₁₆	—	10	2	no	—
85	C₂×D₈	—	10	2	no	—
86	C₂×D₈	—	10	2	no	—
87	C₂×D₈	—	20	2	no	—
88	C₂×D₈	—	20	2	no	—
89	C₄⋊Q₈	—	10	1	no	—
90	C₄⋊Q₈	—	10	1	no	—
91	#127	—	10	2	no	—
92	SD₃₂	—	20	8	no	—
93	C₂×C₁₆	D₂₀	1	4	yes	—
94	#107	—	10	2	no	—
95	#106	—	20	2	no	—
96	#103	—	20	2	no	—
97	C₄×C₈	D₂₀	1	1	yes	characteristic
98	C₄×C₈	D₂₀	1	1	yes	characteristic
99	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
100	C₂²×C₅	#493	1	1	yes	characteristic
101	C₂×D₅	—	16	2	no	—
102	C₂×D₅	—	16	2	no	—
103	C₂×D₅	—	16	2	no	—
104	C₄×C₅	#496	1	2	yes	—
105	C₄×C₅	C₂×SD₃₂	1	2	yes	—
106	C₄×C₅	#509	1	2	yes	—
107	Dic₅	—	16	1	no	—
108	Dic₅	—	16	1	no	—
109	C₂×Q₈	—	20	1	no	—
110	C₂×Q₈	—	20	1	no	—
111	C₂×D₄	—	20	2	no	—
112	C₂×D₄	—	20	2	no	—
113	C₂×D₄	—	20	2	no	—
114	Q₁₆	—	20	4	no	—
115	D₈	—	20	4	no	—
116	D₈	—	20	4	no	—
117	D₈	—	40	4	no	—
118	D₈	—	40	4	no	—
119	C₂×C₈	D₄₀	1	2	yes	—
120	C₂×C₈	D₄₀	1	2	yes	—
121	C₂×C₈	C₂×D₂₀	1	2	yes	—
122	C₄⋊C₄	—	20	1	no	—
123	C₄⋊C₄	—	20	1	no	—
124	C₄⋊C₄	—	20	1	no	—
125	C₄⋊C₄	—	20	1	no	—
126	C₄²	C₂×D₂₀	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₂×C₅	#1721	1	1	yes	characteristic
129	C₂×C₅	#2248	1	1	yes	characteristic
130	C₂×C₅	#2250	1	1	yes	characteristic
131	D₅	—	32	2	no	—
132	C₂³	—	40	2	no	—
133	Q₈	—	40	2	no	—
134	Q₈	—	40	2	no	—
135	D₄	—	40	4	no	—
136	D₄	—	40	4	no	—
137	D₄	—	40	2	no	—
138	D₄	—	40	2	no	—
139	C₂×C₄	—	40	1	no	—
140	C₂×C₄	—	40	1	no	—
141	C₂×C₄	#4020	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
143	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	#3932	1	4	yes	complemented
146	C₈	#4054	1	4	yes	—
147	C₅	#12529	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	80	2	no	—
149	C₂²	—	80	2	no	—
150	C₂²	—	80	2	no	—
151	C₂²	#64984	1	1	yes	characteristic
152	C₄	—	80	1	no	—
153	C₄	—	80	1	no	—
154	C₄	#65001	1	2	yes	—
155	C₄	C₂×#3932	1	2	yes	—
156	C₄	#65197	1	2	yes	—
157	C₂	—	160	2	no	—
158	C₂	#10598887	1	1	yes	characteristic
159	C₂	#10599854	1	1	yes	characteristic
160	C₂	#10599874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599854

C₂

yes

maximal, complemented

#10599851

C₂

yes

maximal, complemented

#10598856

C₂

yes

maximal, characteristic, complemented

#10598855

C₂

yes

maximal, characteristic

C₅×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#3932

—

#64988

C₂²

yes

—

—

C₂²

yes

—

#64971

C₂²

yes

characteristic

#64970

—

#64969

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂×Dic₂₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

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

—

D₄₀

—

D₄₀

—

D₄₀

—

D₄₀

—

C₅×C₁₆

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

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

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

SD₃₂

yes

complemented

C₅×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₂₀

yes

—

#107

—

#106

—

#103

—

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

100

C₂²×C₅

#493

yes

characteristic

101

C₂×D₅

—

102

C₂×D₅

—

103

C₂×D₅

—

104

C₄×C₅

#496

yes

—

105

C₄×C₅

C₂×SD₃₂

yes

—

106

C₄×C₅

#509

yes

—

107

Dic₅

—

108

Dic₅

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

D₈

—

116

D₈

—

117

D₈

—

118

D₈

—

119

C₂×C₈

D₄₀

yes

—

120

C₂×C₈

D₄₀

yes

—

121

C₂×C₈

C₂×D₂₀

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

C₂×D₂₀

yes

characteristic

127

C₁₆

—

128

C₂×C₅

#1721

yes

characteristic

129

C₂×C₅

#2248

yes

characteristic

130

C₂×C₅

#2250

yes

characteristic

131

D₅

—

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#4020

yes

characteristic

142

C₂×C₄

C₂×D₄₀

yes

characteristic

143

C₂×C₄

C₂×D₄₀

yes

characteristic

144

C₈

—

145

C₈

#3932

yes

complemented

146

C₈

#4054

yes

—

147

C₅

#12529

yes

characteristic, complemented, Sylow

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#64984

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

#65001

yes

—

155

C₄

C₂×#3932

yes

—

156

C₄

#65197

yes

—

157

C₂

—

160

—

158

C₂

#10598887

yes

characteristic

159

C₂

#10599854

yes

characteristic

160

C₂

#10599874

yes

characteristic

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1312	C₂
socle	C₂²×C₅	#493
prime radical	#10598855	C₂
hypercenter	#1312	D₅
hypocenter	C₅	#12529
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #10599854	C₂	semidirect
2. #10599851	C₂	semidirect
3. #10598856	C₂	semidirect
5. C₅×#1312	C₂	semidirect
38. #1312	D₅	semidirect
82. C₅×C₈	SD₃₂	semidirect
145. C₈	#3932	semidirect
147. C₅	#12529	semidirect

H ≅ G/N

type

1. #10599854

C₂

semidirect

2. #10599851

C₂

semidirect

3. #10598856

C₂

semidirect

5. C₅×#1312

C₂

semidirect

38. #1312

D₅

semidirect

82. C₅×C₈

SD₃₂

semidirect

145. C₈

#3932

semidirect

147. C₅

#12529

semidirect

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

subgroup	index
#10599854	2
#10599851	2
#10598856	2
#10598855	2
C₅×#1312	2
#12529	5

subgroup

index

Subgroups — #12014074

Conjugacy classes of subgroups

Complements

Maximal subgroups