Group #12013660 — subgroups

Subgroups — #12013660

Order 1280.

204 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	#10599837	C₂	1	1	yes	maximal, characteristic
3	#10599381	C₂	1	2	yes	maximal
4	C₅×#1371	C₂	1	1	yes	maximal, characteristic
5	#10597966	C₂	1	2	yes	maximal
6	#65138	—	2	2	no	—
7	#64971	C₂²	1	2	yes	—
8	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₅×#335	C₂²	1	2	yes	—
11	#64724	—	2	1	no	—
12	#64724	—	2	1	no	—
13	#64721	—	2	1	no	—
14	C₁₆×Dic₅	—	2	1	no	—
15	#64702	—	2	2	no	—
16	C₄⋊#672	C₂²	1	1	yes	characteristic
17	#12457	—	5	1	no	maximal, Sylow
18	C₂×Dic₂₀	—	4	2	no	—
19	#4015	—	2	2	no	—
20	C₄×Dic₁₀	D₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
23	C₅×C₄⋊C₈	—	2	2	no	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3950	—	2	2	no	—
26	#3950	—	2	2	no	—
27	#3948	—	4	2	no	—
28	#3946	—	2	1	no	—
29	C₈×Dic₅	—	2	1	no	—
30	C₂×#672	—	2	1	no	—
31	C₄⋊#168	D₄	1	1	yes	characteristic
32	#2252	—	5	1	no	—
33	Q₈×C₁₆	—	5	1	no	—
34	#1674	—	5	2	no	—
35	#1371	D₅	1	1	yes	characteristic
36	#1364	—	5	2	no	—
37	C₂×Dic₁₀	—	4	2	no	—
38	C₂×Dic₁₀	D₈	1	1	yes	characteristic
39	C₂×C₅×C₈	—	4	2	no	—
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₅×C₈	#41	1	1	yes	characteristic
42	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅	—	4	2	no	—
44	C₄⋊Dic₅	—	4	2	no	—
45	C₄⋊Dic₅	D₈	1	1	yes	characteristic
46	#683	—	2	1	no	—
47	C₄×Dic₅	—	2	1	no	—
48	C₂×#168	—	2	1	no	—
49	Dic₂₀	—	8	4	no	—
50	C₅×C₁₆	—	2	1	no	—
51	C₅×C₁₆	Q₁₆	1	2	yes	complemented
52	#672	—	4	1	no	—
53	#500	—	5	2	no	—
54	#462	—	10	2	no	—
55	C₈×Q₈	—	5	1	no	—
56	#366	—	10	1	no	—
57	#366	—	10	1	no	—
58	C₄⋊C₁₆	—	5	1	no	—
59	C₄⋊C₁₆	—	10	1	no	—
60	C₄×C₁₆	—	10	1	no	—
61	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
62	#335	C₂×D₅	1	2	yes	—
63	#332	—	10	2	no	—
64	C₂×C₄×C₅	#121	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅	#135	1	1	yes	characteristic
67	C₂×Dic₅	—	2	1	no	—
68	C₂×Dic₅	—	2	1	no	—
69	C₂×Dic₅	—	8	2	no	—
70	Dic₁₀	—	2	1	no	—
71	Dic₁₀	—	8	2	no	—
72	Dic₁₀	—	8	2	no	—
73	Dic₁₀	D₁₆	1	2	yes	—
74	C₅×C₈	—	2	1	no	—
75	C₅×C₈	—	8	2	no	—
76	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
77	C₅×C₈	#137	1	1	yes	characteristic
78	#168	—	4	1	no	—
79	C₂×Q₁₆	—	20	2	no	—
80	C₄⋊Q₈	—	10	2	no	—
81	C₄×Q₈	—	5	1	no	—
82	C₂×C₁₆	—	10	1	no	—
83	C₂×C₁₆	D₂₀	1	1	yes	characteristic
84	C₂×C₁₆	D₂₀	1	1	yes	characteristic
85	#107	—	10	2	no	—
86	#107	—	10	2	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	5	1	no	—
89	C₄⋊C₈	—	10	1	no	—
90	#103	—	20	2	no	—
91	C₄×C₈	—	10	1	no	—
92	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
93	C₂²×C₅	#466	1	1	yes	characteristic
94	C₄×C₅	—	2	1	no	—
95	C₄×C₅	#462	1	1	yes	characteristic
96	C₄×C₅	#473	1	1	yes	characteristic
97	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
98	C₄×C₅	#508	1	1	yes	characteristic
99	Dic₅	—	2	2	no	—
100	Dic₅	—	4	1	no	—
101	Dic₅	—	16	2	no	—
102	C₂×Q₈	—	5	1	no	—
103	C₂×Q₈	—	20	2	no	—
104	Q₁₆	—	40	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	10	1	no	—
107	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
108	C₂×C₈	#713	1	1	yes	characteristic
109	C₄⋊C₄	—	5	1	no	—
110	C₄⋊C₄	—	10	1	no	—
111	C₄⋊C₄	—	20	2	no	—
112	C₄⋊C₄	—	20	2	no	—
113	C₄²	—	10	1	no	—
114	C₄²	D₄×D₅	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	20	1	no	—
117	C₁₆	Dic₂₀	1	2	yes	complemented
118	C₂×C₅	#1674	1	1	yes	characteristic
119	C₂×C₅	#2215	1	1	yes	characteristic
120	C₂×C₅	#2222	1	1	yes	characteristic
121	Q₈	—	5	2	no	—
122	Q₈	—	10	1	no	—
123	Q₈	—	40	2	no	—
124	Q₈	—	40	2	no	—
125	C₂×C₄	—	10	1	no	—
126	C₂×C₄	—	10	1	no	—
127	C₂×C₄	—	40	2	no	—
128	C₂×C₄	#4041	1	1	yes	characteristic
129	C₂×C₄	D₅×D₈	1	1	yes	characteristic
130	C₂×C₄	#4065	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	8	2	no	—
133	C₈	—	20	1	no	—
134	C₈	C₂×Dic₂₀	1	1	yes	characteristic
135	C₈	#4055	1	1	yes	characteristic
136	C₅	#12457	1	1	yes	characteristic, complemented, Sylow
137	C₂²	#65172	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	10	2	no	—
140	C₄	—	20	1	no	—
141	C₄	—	80	2	no	—
142	C₄	#65138	1	1	yes	characteristic
143	C₄	#65185	1	1	yes	characteristic
144	C₄	D₅×D₁₆	1	1	yes	characteristic
145	C₄	#65208	1	1	yes	characteristic
146	C₂	#10599381	1	1	yes	characteristic
147	C₂	#10600120	1	1	yes	characteristic
148	C₂	#10600155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₀

C₂

yes

maximal, characteristic

#10599837

C₂

yes

maximal, characteristic

#10599381

C₂

yes

maximal

C₅×#1371

C₂

yes

maximal, characteristic

#10597966

C₂

yes

maximal

#65138

—

#64971

C₂²

yes

—

C₈×Dic₁₀

C₂²

yes

characteristic

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₅×#335

C₂²

yes

—

#64724

—

#64724

—

#64721

—

C₁₆×Dic₅

—

#64702

—

C₄⋊#672

C₂²

yes

characteristic

#12457

—

maximal, Sylow

C₂×Dic₂₀

—

#4015

—

C₄×Dic₁₀

D₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

characteristic

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂³

yes

—

—

—

—

C₈×Dic₅

—

C₂×#672

—

C₄⋊#168

D₄

yes

characteristic

#2252

—

Q₈×C₁₆

—

#1674

—

#1371

D₅

yes

characteristic

#1364

—

C₂×Dic₁₀

—

C₂×Dic₁₀

D₈

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

D₈

yes

characteristic

#683

—

C₄×Dic₅

—

C₂×#168

—

Dic₂₀

—

C₅×C₁₆

—

C₅×C₁₆

Q₁₆

yes

—

—

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

#335

C₂×D₅

yes

—

#332

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

D₁₆

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×Q₁₆

yes

C₅×C₈

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

D₂₀

yes

characteristic

C₂×C₁₆

D₂₀

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#466

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

C₂×D₁₆

yes

characteristic

C₄×C₅

#508

yes

characteristic

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂×D₂₀

yes

characteristic

108

C₂×C₈

#713

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

D₄×D₅

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

Dic₂₀

yes

complemented

118

C₂×C₅

#1674

yes

characteristic

119

C₂×C₅

#2215

yes

characteristic

120

C₂×C₅

#2222

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#4041

yes

characteristic

129

C₂×C₄

D₅×D₈

yes

characteristic

130

C₂×C₄

#4065

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₂×Dic₂₀

yes

characteristic

135

C₈

#4055

yes

characteristic

136

C₅

#12457

yes

characteristic, complemented, Sylow

137

C₂²

#65172

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#65138

yes

characteristic

143

C₄

#65185

yes

characteristic

144

C₄

D₅×D₁₆

yes

characteristic

145

C₄

#65208

yes

characteristic

146

C₂

#10599381

yes

characteristic

147

C₂

#10600120

yes

characteristic

148

C₂

#10600155

yes

characteristic

	subgroup	G/H
center	C₂²	#65172
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1371	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#1371	D₅
hypocenter	C₅	#12457
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#65172

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 ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
51. C₅×C₁₆	Q₁₆	semidirect
117. C₁₆	Dic₂₀	semidirect
136. C₅	#12457	semidirect

H ≅ G/N

type

51. C₅×C₁₆

Q₁₆

semidirect

117. C₁₆

Dic₂₀

semidirect

136. C₅

#12457

semidirect

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

subgroup	index
C₁₆×Dic₁₀	2
#10599837	2
#10599381	2
C₅×#1371	2
#10597966	2
#12457	5

subgroup

index

C₁₆×Dic₁₀

Subgroups — #12013660

Conjugacy classes of subgroups

Complements

Maximal subgroups