Group #13016693 — subgroups

Subgroups — #13016693

Order 1280.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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	#10603840	C₂	1	1	yes	maximal, characteristic
2	#10603811	C₂	1	1	yes	maximal, characteristic
3	#10598861	C₂	1	1	yes	maximal, characteristic
4	#10598529	C₂	1	2	yes	maximal
5	#10598526	C₂	1	2	yes	maximal
6	#65686	C₂²	1	1	yes	characteristic
7	#65678	C₂²	1	2	yes	—
8	#65676	C₂²	1	1	yes	characteristic
9	#64974	C₂²	1	2	yes	—
10	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
11	(C₂×AGL(1,𝔽₅))⋊C₈	C₄	1	4	yes	—
12	#170⋊C₈	C₄	1	4	yes	complemented
13	C₅×#322	C₄	1	1	yes	characteristic, complemented
14	#64675	C₄	1	1	yes	characteristic, complemented
15	C₈×#168	C₄	1	2	yes	complemented
16	#8142	—	5	1	no	maximal, Sylow
17	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
18	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
20	C₄²×D₅	C₂³	1	1	yes	characteristic
21	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
22	#4004	—	2	2	no	—
23	#4003	—	2	2	no	—
24	#4002	—	2	2	no	—
25	C₄×#170	C₂×C₄	1	4	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	#3952	—	2	2	no	—
29	#3946	—	2	2	no	—
30	C₈×Dic₅	C₂×C₄	1	2	yes	—
31	C₄×#168	C₂×C₄	1	2	yes	—
32	C₄×#168	C₂×C₄	1	1	yes	characteristic
33	#1746	—	5	1	no	—
34	#1452	—	5	1	no	—
35	#1450	—	5	1	no	—
36	#1275	—	5	2	no	—
37	#1274	—	5	2	no	—
38	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
39	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×D₅	#41	1	2	yes	—
41	C₂×#170	—	2	2	no	—
42	C₂×#170	—	2	4	no	—
43	C₄×AGL(1,𝔽₅)	—	2	4	no	—
44	D₅⋊C₈	—	2	4	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	C₄²	1	2	yes	—
47	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
49	C₄×Dic₅	#41	1	2	yes	—
50	C₂×#168	—	2	2	no	—
51	C₂×#168	C₄²	1	2	yes	—
52	C₈×D₅	—	2	4	no	—
53	#431	—	5	2	no	—
54	#431	—	5	2	no	—
55	C₂×#97	—	5	1	no	—
56	C₂×C₄×C₈	—	5	1	no	—
57	#376	—	5	1	no	—
58	#336	—	5	4	no	—
59	#322	—	5	4	no	—
60	#322	—	5	1	no	—
61	#322	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
62	C₈²	—	5	2	no	—
63	C₂²×D₅	#117	1	1	yes	characteristic
64	C₂×AGL(1,𝔽₅)	—	4	4	no	—
65	C₂×AGL(1,𝔽₅)	—	4	2	no	—
66	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅	#117	1	2	yes	—
68	C₂×Dic₅	#117	1	2	yes	—
69	C₂×Dic₅	#117	1	1	yes	characteristic
70	C₄×D₅	—	4	2	no	—
71	C₄×D₅	#131	1	4	yes	—
72	#170	—	4	4	no	—
73	#170	—	4	2	no	—
74	C₅×C₈	—	2	2	no	—
75	C₅×C₈	—	4	2	no	—
76	#168	—	2	2	no	—
77	#168	—	4	2	no	—
78	C₂²×C₈	—	5	2	no	—
79	C₂²×C₈	—	5	2	no	—
80	C₂×C₄²	—	5	1	no	—
81	C₂×C₄²	—	5	2	no	—
82	C₄⋊C₈	—	10	2	no	—
83	C₄⋊C₈	—	10	2	no	—
84	C₂²⋊C₈	—	10	2	no	—
85	C₂²⋊C₈	—	10	2	no	—
86	#97	—	10	2	no	—
87	C₄×C₈	—	5	2	no	—
88	C₄×C₈	—	5	4	no	—
89	C₄×C₈	—	5	1	no	—
90	C₄×C₈	—	5	2	no	—
91	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
92	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	#95	—	10	2	no	—
94	C₂²×C₅	#376	1	1	yes	characteristic
95	C₂×D₅	—	2	2	no	—
96	C₂×D₅	#433	1	2	yes	—
97	AGL(1,𝔽₅)	—	8	2	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	#405	1	2	yes	—
100	Dic₅	—	2	2	no	—
101	Dic₅	#433	1	2	yes	—
102	C₂²×C₄	—	5	1	no	—
103	C₂²×C₄	—	5	2	no	—
104	C₂²×C₄	—	10	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	5	2	no	—
107	C₂×C₈	—	10	4	no	—
108	C₂×C₈	—	10	4	no	—
109	C₂×C₈	—	10	2	no	—
110	C₂×C₈	—	10	4	no	—
111	C₂×C₈	—	10	2	no	—
112	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
113	C₄²	—	5	1	no	—
114	C₄²	—	5	2	no	—
115	C₄²	—	10	4	no	—
116	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₅	#1746	1	1	yes	characteristic
118	C₂×C₅	#1747	1	2	yes	—
119	D₅	—	2	2	no	—
120	C₂³	—	5	1	no	—
121	C₂×C₄	—	5	1	no	—
122	C₂×C₄	—	5	4	no	—
123	C₂×C₄	—	5	2	no	—
124	C₂×C₄	—	20	2	no	—
125	C₂×C₄	—	20	2	no	—
126	C₂×C₄	—	20	4	no	—
127	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₂×C₄	#4130	1	2	yes	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₈	—	10	2	no	—
132	C₈	—	20	2	no	—
133	C₈	—	20	2	no	—
134	C₈	—	20	4	no	—
135	C₅	#8142	1	1	yes	characteristic, complemented, Sylow
136	C₂²	—	5	2	no	—
137	C₂²	—	10	2	no	—
138	C₂²	#65686	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	5	2	no	—
141	C₄	—	10	2	no	—
142	C₄	—	40	2	no	—
143	C₄	#65718	1	2	yes	—
144	C₂	—	10	2	no	—
145	C₂	#10603840	1	1	yes	characteristic
146	C₂	#10603884	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603840

C₂

yes

maximal, characteristic

#10603811

C₂

yes

maximal, characteristic

#10598861

C₂

yes

maximal, characteristic

#10598529

C₂

yes

maximal

#10598526

C₂

yes

maximal

#65686

C₂²

yes

characteristic

#65678

C₂²

yes

—

#65676

C₂²

yes

characteristic

#64974

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₈

C₄

yes

—

#170⋊C₈

C₄

yes

complemented

C₅×#322

C₄

yes

characteristic, complemented

#64675

C₄

yes

characteristic, complemented

C₈×#168

C₄

yes

complemented

#8142

—

maximal, Sylow

C₂×C₄×AGL(1,𝔽₅)

C₂×C₄

yes

—

C₂×D₅⋊C₈

C₂×C₄

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

C₄²×D₅

C₂³

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

—

—

—

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

—

—

—

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂²×C₄

yes

characteristic

C₂×C₄×D₅

#41

yes

—

C₂×#170

—

C₂×#170

—

C₄×AGL(1,𝔽₅)

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

C₄²

yes

—

C₈×D₅

—

#431

—

#431

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

yes

characteristic, complemented

C₈²

—

C₂²×D₅

yes

—

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×Dic₅

#117

yes

—

C₂×Dic₅

#117

yes

characteristic

C₄×D₅

—

C₄×D₅

#131

yes

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

#95

—

C₂²×C₅

#376

yes

characteristic

C₂×D₅

—

C₂×D₅

#433

yes

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

#405

yes

—

100

Dic₅

—

101

Dic₅

#433

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

117

C₂×C₅

#1746

yes

characteristic

118

C₂×C₅

#1747

yes

—

119

D₅

—

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

128

C₂×C₄

#4130

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₅

#8142

yes

characteristic, complemented, Sylow

136

C₂²

—

137

C₂²

—

138

C₂²

#65686

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#65718

yes

—

144

C₂

—

145

C₂

#10603840

yes

characteristic

146

C₂

#10603884

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂²×C₅	#376
prime radical	C₂²×D₅	#117
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#8142
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂²×D₅

C₄²×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ #322

N	H ≅ G/N	type
12. #170⋊C₈	C₄	semidirect
13. C₅×#322	C₄	semidirect
14. #64675	C₄	semidirect
15. C₈×#168	C₄	semidirect
61. #322	AGL(1,𝔽₅)	semidirect
135. C₅	#8142	semidirect

H ≅ G/N

type

12. #170⋊C₈

C₄

semidirect

13. C₅×#322

C₄

semidirect

14. #64675

C₄

semidirect

15. C₈×#168

C₄

semidirect

61. #322

AGL(1,𝔽₅)

semidirect

135. C₅

#8142

semidirect

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

subgroup	index
#10603840	2
#10603811	2
#10598861	2
#10598529	2
#10598526	2
#8142	5

subgroup

index

Subgroups — #13016693

Conjugacy classes of subgroups

Complements

Maximal subgroups