Group #12008647 — subgroups

Subgroups — #12008647

Order 1280.

296 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 127 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×#2960	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	2	yes	maximal
3	#10598209	C₂	1	4	yes	maximal
4	C₅×#533	C₄	1	1	yes	characteristic
5	C₅×#446	—	2	2	no	—
6	C₅×C₈×Q₈	—	2	2	no	—
7	C₅×#424	C₂²	1	2	yes	—
8	C₅×#423	C₄	1	1	yes	characteristic
9	C₅×#405	C₄	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²⋊#672	C₂²	1	4	yes	—
12	C₄⋊#672	—	2	2	no	—
13	C₄×#672	—	2	2	no	—
14	#11496	—	5	1	no	maximal, Sylow
15	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₅×C₄⋊Q₈	C₈	1	2	yes	—
19	C₅×#125	C₈	1	2	yes	—
20	C₅×C₂²⋊Q₈	—	2	2	no	—
21	C₄×C₅×Q₈	—	2	2	no	—
22	C₅×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×C₄⋊C₈	—	2	4	no	—
25	C₅×C₄⋊C₈	—	2	2	no	—
26	C₅×#97	—	2	2	no	—
27	C₄×C₅×C₈	—	2	2	no	—
28	C₄×C₅×C₈	D₄	1	2	yes	—
29	C₄×C₅×C₈	D₄	1	2	yes	—
30	C₂×#672	—	2	4	no	—
31	#2960	D₅	1	1	yes	characteristic
32	#2162	—	5	2	no	—
33	#1323	—	5	4	no	—
34	C₂×C₅×Q₈	—	2	2	no	—
35	C₂²×C₄×C₅	M₁₆	1	2	yes	—
36	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₅×M₁₆	—	4	2	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	—	2	2	no	—
40	C₂×C₅×C₈	—	4	1	no	—
41	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
44	C₅×C₄⋊C₄	—	2	4	no	—
45	C₅×C₄⋊C₄	—	2	2	no	—
46	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
47	C₅×C₂²⋊C₄	—	2	2	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
52	#672	—	4	4	no	—
53	#533	Dic₅	1	1	yes	characteristic
54	#446	—	2	2	no	—
55	C₈×Q₈	—	2	2	no	—
56	#424	C₂×D₅	1	2	yes	—
57	#423	Dic₅	1	1	yes	characteristic
58	#405	Dic₅	1	2	yes	—
59	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
60	C₄⋊C₁₆	—	10	2	no	—
61	C₂²⋊C₁₆	—	5	4	no	—
62	C₄×C₁₆	—	10	2	no	—
63	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
64	C₅×Q₈	—	4	4	no	—
65	C₂×C₄×C₅	—	2	4	no	—
66	C₂×C₄×C₅	—	2	1	no	—
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	—	2	4	no	—
69	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
73	C₅×C₈	—	2	2	no	—
74	C₅×C₈	—	2	1	no	—
75	C₅×C₈	—	4	2	no	—
76	C₅×C₈	#101	1	2	yes	—
77	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
78	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
80	C₄⋊Q₈	#168	1	2	yes	—
81	#125	#168	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×Dic₅	1	2	yes	—
85	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
86	C₂×C₁₆	—	10	4	no	—
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	2	2	no	—
89	#97	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₂²⋊D₅	1	2	yes	—
92	C₄×C₈	C₂²⋊D₅	1	2	yes	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	—	2	1	no	—
95	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	—	2	1	no	—
98	C₄×C₅	—	4	4	no	—
99	C₄×C₅	#350	1	4	yes	—
100	C₄×C₅	#410	1	1	yes	characteristic
101	C₄×C₅	C₂×#101	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	C₂²×C₄	#681	1	2	yes	—
104	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
105	M₁₆	—	4	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
111	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂×#168	1	4	yes	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₂×#168	1	1	yes	characteristic
118	C₄²	C₂×#168	1	1	yes	characteristic
119	C₄²	C₂²⋊Dic₅	1	2	yes	—
120	C₁₆	—	20	4	no	—
121	C₂×C₅	—	4	1	no	—
122	C₂×C₅	#1461	1	1	yes	characteristic
123	C₂×C₅	C₂×#350	1	2	yes	—
124	C₂³	C₂×#681	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₂²⋊#168	1	2	yes	—
131	C₂×C₄	C₂²⋊#168	1	2	yes	—
132	C₂×C₄	C₂²×#168	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	2	no	—
136	C₈	—	4	2	no	—
137	C₈	#3968	1	2	yes	—
138	C₅	#11496	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	#64781	1	4	yes	—
146	C₄	C₂×#3968	1	1	yes	characteristic
147	C₄	#65529	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#10602417	1	1	yes	characteristic
150	C₂	C₂×#64781	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2960

C₂

yes

maximal, characteristic

#10600537

C₂

yes

maximal

#10598209

C₂

yes

maximal

C₅×#533

C₄

yes

characteristic

C₅×#446

—

C₅×C₈×Q₈

—

C₅×#424

C₂²

yes

—

C₅×#423

C₄

yes

characteristic

C₅×#405

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂²⋊#672

C₂²

yes

—

—

—

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×C₄⋊Q₈

C₈

yes

—

C₅×#125

C₈

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂×#672

—

#2960

D₅

yes

characteristic

#2162

—

#1323

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₈

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

#672

—

#533

Dic₅

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂×D₅

yes

—

#423

Dic₅

yes

characteristic

#405

Dic₅

yes

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#101

yes

—

C₂×M₁₆

C₂×Dic₅

yes

—

C₂²×C₈

C₂×Dic₅

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

C₄⋊Q₈

#168

yes

—

#125

#168

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₅

yes

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²⋊D₅

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#350

yes

—

100

C₄×C₅

#410

yes

characteristic

101

C₄×C₅

C₂×#101

yes

characteristic

102

C₂×Q₈

—

103

C₂²×C₄

#681

yes

—

104

C₂²×C₄

C₂²×Dic₅

yes

characteristic

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊Dic₅

yes

characteristic

110

C₂×C₈

C₂²⋊Dic₅

yes

characteristic

111

C₂×C₈

C₂×C₂²⋊D₅

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×#168

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₂×#168

yes

characteristic

118

C₄²

C₂×#168

yes

characteristic

119

C₄²

C₂²⋊Dic₅

yes

—

120

C₁₆

—

121

C₂×C₅

—

122

C₂×C₅

#1461

yes

characteristic

123

C₂×C₅

C₂×#350

yes

—

124

C₂³

C₂×#681

yes

characteristic

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²⋊#168

yes

—

131

C₂×C₄

C₂²⋊#168

yes

—

132

C₂×C₄

C₂²×#168

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

#3968

yes

—

138

C₅

#11496

yes

characteristic, complemented, Sylow

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#168

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#64781

yes

—

146

C₄

C₂×#3968

yes

characteristic

147

C₄

#65529

yes

characteristic

148

C₂

—

149

C₂

#10602417

yes

characteristic

150

C₂

C₂×#64781

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2960	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2960	D₅
hypocenter	C₅	#11496
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₅

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₅×#2960

C₂

socle

C₂²×C₅

C₂×C₂²⋊C₈

prime radical

C₂³×C₅

C₂×M₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

N	H ≅ G/N	type
138. C₅	#11496	semidirect

H ≅ G/N

type

138. C₅

#11496

semidirect

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

subgroup	index
C₅×#2960	2
#10600537	2
#10598209	2
#11496	5

subgroup

index

Subgroups — #12008647

Conjugacy classes of subgroups

Complements

Maximal subgroups