Group #12013677 — subgroups

Subgroups — #12013677

Order 1280.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	#10599841	C₂	1	1	yes	maximal, characteristic
2	#10599837	C₂	1	1	yes	maximal, characteristic
3	#10599381	C₂	1	2	yes	maximal
4	C₅×#1372	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	#64722	—	2	1	no	—
14	#64721	—	2	1	no	—
15	#64702	—	2	2	no	—
16	C₄⋊#672	C₂²	1	1	yes	characteristic
17	#12460	—	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	#2183	—	5	1	no	—
34	#1674	—	5	2	no	—
35	#1372	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	#346	—	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	2	no	—
71	Dic₁₀	—	8	2	no	—
72	Dic₁₀	—	8	2	no	—
73	C₅×C₈	—	2	1	no	—
74	C₅×C₈	—	8	2	no	—
75	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
76	C₅×C₈	#137	1	1	yes	characteristic
77	#168	—	4	1	no	—
78	C₂×Q₁₆	—	20	2	no	—
79	C₄⋊Q₈	—	10	2	no	—
80	C₄×Q₈	—	5	1	no	—
81	C₂×C₁₆	—	10	1	no	—
82	C₂×C₁₆	D₂₀	1	1	yes	characteristic
83	C₂×C₁₆	D₂₀	1	1	yes	characteristic
84	#107	—	10	2	no	—
85	#107	—	10	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	5	1	no	—
88	C₄⋊C₈	—	10	1	no	—
89	#103	—	20	2	no	—
90	C₄×C₈	—	10	1	no	—
91	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
92	C₂²×C₅	#466	1	1	yes	characteristic
93	C₄×C₅	—	2	1	no	—
94	C₄×C₅	#462	1	1	yes	characteristic
95	C₄×C₅	#473	1	1	yes	characteristic
96	C₄×C₅	#509	1	2	yes	—
97	Dic₅	—	4	1	no	—
98	Dic₅	—	4	1	no	—
99	Dic₅	—	16	2	no	—
100	C₂×Q₈	—	5	1	no	—
101	C₂×Q₈	—	20	2	no	—
102	Q₁₆	—	40	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	10	1	no	—
105	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
106	C₂×C₈	#713	1	1	yes	characteristic
107	C₄⋊C₄	—	5	1	no	—
108	C₄⋊C₄	—	10	1	no	—
109	C₄⋊C₄	—	20	2	no	—
110	C₄⋊C₄	—	20	2	no	—
111	C₄²	—	10	1	no	—
112	C₄²	D₄×D₅	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	20	1	no	—
115	C₁₆	Dic₂₀	1	2	yes	complemented
116	C₂×C₅	#1674	1	1	yes	characteristic
117	C₂×C₅	#2218	1	1	yes	characteristic
118	C₂×C₅	#2222	1	1	yes	characteristic
119	Q₈	—	10	2	no	—
120	Q₈	—	40	2	no	—
121	Q₈	—	40	2	no	—
122	C₂×C₄	—	10	1	no	—
123	C₂×C₄	—	10	1	no	—
124	C₂×C₄	—	40	2	no	—
125	C₂×C₄	#4041	1	1	yes	characteristic
126	C₂×C₄	D₅×D₈	1	1	yes	characteristic
127	C₂×C₄	#4065	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	8	2	no	—
130	C₈	—	20	1	no	—
131	C₈	C₂×Dic₂₀	1	1	yes	characteristic
132	C₈	#4055	1	1	yes	characteristic
133	C₅	#12460	1	1	yes	characteristic, complemented, Sylow
134	C₂²	#65172	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	20	1	no	—
137	C₄	—	20	1	no	—
138	C₄	—	80	2	no	—
139	C₄	#65138	1	1	yes	characteristic
140	C₄	#65185	1	1	yes	characteristic
141	C₄	#65203	1	2	yes	—
142	C₂	#10599381	1	1	yes	characteristic
143	C₂	#10600141	1	1	yes	characteristic
144	C₂	#10600155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599841

C₂

yes

maximal, characteristic

#10599837

C₂

yes

maximal, characteristic

#10599381

C₂

yes

maximal

C₅×#1372

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

—

—

—

—

—

—

C₂²

yes

characteristic

#12460

—

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

—

—

—

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

—

#346

—

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

—

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

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×D₂₀

yes

characteristic

106

C₂×C₈

#713

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

D₄×D₅

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

Dic₂₀

yes

complemented

116

C₂×C₅

#1674

yes

characteristic

117

C₂×C₅

#2218

yes

characteristic

118

C₂×C₅

#2222

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#4041

yes

characteristic

126

C₂×C₄

D₅×D₈

yes

characteristic

127

C₂×C₄

#4065

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

C₂×Dic₂₀

yes

characteristic

132

C₈

#4055

yes

characteristic

133

C₅

#12460

yes

characteristic, complemented, Sylow

134

C₂²

#65172

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#65138

yes

characteristic

140

C₄

#65185

yes

characteristic

141

C₄

#65203

yes

—

142

C₂

#10599381

yes

characteristic

143

C₂

#10600141

yes

characteristic

144

C₂

#10600155

yes

characteristic

	subgroup	G/H
center	C₂²	#65172
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1372	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#1372	D₅
hypocenter	C₅	#12460
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

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 ◁ #1372

N	H ≅ G/N	type
51. C₅×C₁₆	Q₁₆	semidirect
115. C₁₆	Dic₂₀	semidirect
133. C₅	#12460	semidirect

H ≅ G/N

type

51. C₅×C₁₆

Q₁₆

semidirect

115. C₁₆

Dic₂₀

semidirect

133. C₅

#12460

semidirect

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

subgroup	index
#10599841	2
#10599837	2
#10599381	2
C₅×#1372	2
#10597966	2
#12460	5

subgroup

index

Subgroups — #12013677

Conjugacy classes of subgroups

Complements

Maximal subgroups