Group #13013755 — subgroups

Subgroups — #13013755

Order 1280.

184 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 58 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	#10603856	C₂	1	1	yes	maximal, characteristic
2	#10603821	C₂	1	1	yes	maximal, characteristic
3	#10599866	C₂	1	1	yes	maximal, characteristic
4	#65683	—	2	1	no	—
5	#65681	—	2	1	no	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64896	—	2	1	no	—
8	#64896	C₄	1	2	yes	—
9	#64884	—	2	1	no	—
10	#64883	C₄	1	2	yes	—
11	C₅×#346	C₄	1	1	yes	characteristic
12	#64727	—	2	1	no	—
13	#64721	—	2	1	no	—
14	#64681	C₄	1	1	yes	characteristic
15	#7224	—	5	1	no	maximal, Sylow
16	C₂×#700	—	2	2	no	—
17	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₅	D₄	1	1	yes	characteristic
19	C₄²×D₅	D₄	1	1	yes	characteristic
20	#4003	—	4	1	no	—
21	#4002	—	4	1	no	—
22	C₄⋊#170	—	2	2	no	—
23	C₄⋊#170	—	4	1	no	—
24	C₄⋊#170	—	4	1	no	—
25	C₂×C₅×C₁₆	—	2	1	no	—
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅	D₄	1	1	yes	characteristic
29	C₂×#672	—	2	1	no	—
30	C₄×#168	Q₈	1	1	yes	characteristic
31	#2163	—	5	1	no	—
32	#1837	—	5	1	no	—
33	#1568	—	5	1	no	—
34	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₅	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
37	C₂×#170	—	4	2	no	—
38	C₂×#170	—	4	2	no	—
39	#700	—	4	2	no	—
40	#700	—	4	2	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅	D₈	1	1	yes	characteristic
46	C₄×Dic₅	SD₁₆	1	1	yes	characteristic
47	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₅	—	4	1	no	—
50	C₈×D₅	—	4	1	no	—
51	C₅×C₁₆	—	4	1	no	—
52	#672	—	4	1	no	—
53	#432	—	10	1	no	—
54	#423	—	10	1	no	—
55	C₂×C₄×C₈	—	5	1	no	—
56	C₄⋊C₁₆	—	10	1	no	—
57	C₂²⋊C₁₆	—	10	1	no	—
58	#346	—	5	1	no	—
59	#346	AGL(1,𝔽₅)	1	1	yes	characteristic
60	#341	—	5	2	no	—
61	#341	—	10	1	no	—
62	#335	—	10	1	no	—
63	#334	—	5	2	no	—
64	C₂²×D₅	#103	1	1	yes	characteristic
65	C₂×C₄×C₅	#95	1	1	yes	characteristic
66	C₂×C₄×C₅	#106	1	1	yes	characteristic
67	C₂×C₄×C₅	#107	1	1	yes	characteristic
68	C₂×Dic₅	#102	1	1	yes	characteristic
69	C₂×Dic₅	#102	1	1	yes	characteristic
70	C₂×Dic₅	#103	1	1	yes	characteristic
71	C₄×D₅	—	2	1	no	—
72	C₄×D₅	—	2	1	no	—
73	C₄×D₅	—	4	1	no	—
74	C₄×D₅	—	4	1	no	—
75	#170	—	8	2	no	—
76	#170	—	8	2	no	—
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	#168	—	2	1	no	—
80	#168	—	2	1	no	—
81	C₂×M₁₆	—	10	2	no	—
82	C₂²×C₈	—	5	1	no	—
83	C₂²×C₈	—	5	1	no	—
84	C₂×C₄²	—	5	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	10	1	no	—
87	C₄⋊C₈	—	10	2	no	—
88	C₄⋊C₈	—	20	1	no	—
89	C₄⋊C₈	—	20	1	no	—
90	C₂²⋊C₈	—	20	1	no	—
91	#97	—	20	1	no	—
92	C₄×C₈	—	5	1	no	—
93	C₄×C₈	—	5	1	no	—
94	C₄×C₈	—	5	1	no	—
95	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₂²×C₅	#340	1	1	yes	characteristic
97	C₂×D₅	—	2	1	no	—
98	C₂×D₅	—	2	1	no	—
99	C₂×D₅	—	2	1	no	—
100	C₄×C₅	—	2	1	no	—
101	C₄×C₅	—	2	1	no	—
102	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
103	C₄×C₅	#344	1	1	yes	characteristic
104	Dic₅	—	2	1	no	—
105	Dic₅	—	2	1	no	—
106	Dic₅	—	2	1	no	—
107	C₂²×C₄	—	5	1	no	—
108	C₂²×C₄	—	5	1	no	—
109	C₂²×C₄	—	5	1	no	—
110	M₁₆	—	20	2	no	—
111	M₁₆	—	20	2	no	—
112	C₂×C₈	—	5	1	no	—
113	C₂×C₈	—	5	1	no	—
114	C₂×C₈	—	20	1	no	—
115	C₂×C₈	—	20	1	no	—
116	C₂×C₈	—	20	2	no	—
117	C₂×C₈	—	20	2	no	—
118	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₄²	—	5	1	no	—
121	C₄²	—	5	1	no	—
122	C₄²	—	5	1	no	—
123	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₁₆	—	4	1	no	—
125	C₁₆	—	20	1	no	—
126	C₂×C₅	#1286	1	1	yes	characteristic
127	C₂×C₅	#1388	1	1	yes	characteristic
128	C₂×C₅	#1389	1	1	yes	characteristic
129	D₅	—	4	1	no	—
130	C₂³	—	5	1	no	—
131	C₂×C₄	—	5	1	no	—
132	C₂×C₄	—	5	1	no	—
133	C₂×C₄	—	5	1	no	—
134	C₂×C₄	—	10	1	no	—
135	C₂×C₄	—	10	1	no	—
136	C₂×C₄	—	20	1	no	—
137	C₂×C₄	—	20	1	no	—
138	C₂×C₄	#3993	1	1	yes	characteristic
139	C₂×C₄	#3994	1	1	yes	characteristic
140	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	10	1	no	—
144	C₈	—	10	1	no	—
145	C₈	—	40	2	no	—
146	C₈	—	40	2	no	—
147	C₅	#7224	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	10	1	no	—
149	C₂²	—	10	1	no	—
150	C₂²	—	10	1	no	—
151	C₂²	#64893	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	10	1	no	—
155	C₄	—	10	1	no	—
156	C₄	—	10	1	no	—
157	C₄	#64863	1	1	yes	characteristic
158	C₄	#64894	1	1	yes	characteristic
159	C₂	—	20	1	no	—
160	C₂	#10598527	1	1	yes	characteristic
161	C₂	#10598676	1	1	yes	characteristic
162	C₂	#10598677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603856

C₂

yes

maximal, characteristic

#10603821

C₂

yes

maximal, characteristic

#10599866

C₂

yes

maximal, characteristic

#65683

—

#65681

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64896

—

#64896

C₄

yes

—

#64884

—

#64883

C₄

yes

—

C₅×#346

C₄

yes

—

—

C₄

yes

—

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

D₄

yes

characteristic

C₄²×D₅

D₄

yes

—

—

—

—

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

characteristic

C₂×#672

—

C₄×#168

Q₈

yes

—

—

—

C₂×C₄×D₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅

SD₁₆

yes

characteristic

C₂×C₄×D₅

Q₁₆

yes

—

—

—

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄×Dic₅

C₂²⋊C₄

yes

characteristic

C₄×Dic₅

D₈

yes

characteristic

C₄×Dic₅

SD₁₆

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

—

#432

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

yes

—

—

—

—

C₂²×D₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

H(3,ℤ/4)

yes

characteristic

103

C₄×C₅

#344

yes

characteristic

104

Dic₅

—

105

Dic₅

—

106

Dic₅

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

119

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄⋊AGL(1,𝔽₅)

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₂×C₅

#1286

yes

characteristic

127

C₂×C₅

#1388

yes

characteristic

128

C₂×C₅

#1389

yes

characteristic

129

D₅

—

130

C₂³

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#3993

yes

characteristic

139

C₂×C₄

#3994

yes

characteristic

140

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₅

#7224

yes

characteristic, complemented, Sylow

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#64893

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#64863

yes

characteristic

158

C₄

#64894

yes

characteristic

159

C₂

—

160

C₂

#10598527

yes

characteristic

161

C₂

#10598676

yes

characteristic

162

C₂

#10598677

yes

characteristic

	subgroup	G/H
center	C₂²	#64893
commutator	C₂×C₅×C₈	C₄²
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#346	C₄
socle	C₂²×C₅	#340
prime radical	C₂²×D₅	#103
cosocle	C₄×C₈×D₅	C₂²
hypercenter	#346	AGL(1,𝔽₅)
hypocenter	C₅	#7224
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

#64893

commutator

C₂×C₅×C₈

C₄²

C₄×C₈

C₄

C₂²×C₅

C₂²×D₅

C₄×C₈×D₅

C₂²

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
147. C₅	#7224	semidirect

H ≅ G/N

type

147. C₅

#7224

semidirect

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

subgroup	index
#10603856	2
#10603821	2
#10599866	2
#7224	5

subgroup

index

Subgroups — #13013755

Conjugacy classes of subgroups

Complements

Maximal subgroups