Group #13013761 — subgroups

Subgroups — #13013761

Order 1280.

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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	#10603887	C₂	1	1	yes	maximal, characteristic
2	#10603864	C₂	1	1	yes	maximal, characteristic
3	#10599881	C₂	1	1	yes	maximal, characteristic
4	C₄×#700	—	2	1	no	—
5	#64996	C₂²	1	1	yes	characteristic
6	#64897	C₄	1	1	yes	characteristic
7	#64895	C₄	1	1	yes	characteristic
8	#64894	—	2	1	no	—
9	#170⋊C₈	—	2	1	no	—
10	#64879	C₄	1	1	yes	characteristic, complemented
11	C₈×#170	C₄	1	1	yes	characteristic, complemented
12	#64862	—	2	1	no	—
13	C₅×#347	C₄	1	1	yes	characteristic, complemented
14	#64734	—	2	1	no	—
15	#64731	—	2	1	no	—
16	#64686	C₄	1	1	yes	characteristic, complemented
17	#7225	—	5	1	no	maximal, Sylow
18	#4130	—	2	1	no	—
19	D₅⋊M₁₆	—	2	1	no	—
20	C₂×#700	—	2	2	no	—
21	C₂×#676	C₂×C₄	1	1	yes	characteristic
22	C₂×#676	D₄	1	1	yes	characteristic
23	C₄²×D₅	D₄	1	1	yes	characteristic
24	#4002	—	2	2	no	—
25	C₄×#170	—	2	1	no	—
26	C₄×#170	—	2	1	no	—
27	C₅×M₃₂	—	2	1	no	—
28	C₅×#97	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅	D₄	1	1	yes	characteristic
31	#3944	—	2	1	no	—
32	#3935	Q₈	1	1	yes	characteristic
33	#2165	—	5	1	no	—
34	#1841	—	5	1	no	—
35	#1455	—	5	1	no	—
36	C₂×C₄×D₅	—	2	1	no	—
37	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²⋊AGL(1,𝔽₅)	—	4	1	no	—
39	#704	—	4	1	no	—
40	C₂×#170	—	2	2	no	—
41	C₂×#170	—	2	2	no	—
42	C₄⋊AGL(1,𝔽₅)	—	4	1	no	—
43	C₄×AGL(1,𝔽₅)	—	4	1	no	—
44	#700	—	4	4	no	—
45	#700	—	4	1	no	—
46	D₅⋊C₈	—	4	1	no	—
47	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₅	—	2	1	no	—
51	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
54	#676	—	4	1	no	—
55	#676	—	4	1	no	—
56	C₅×C₁₆	—	4	1	no	—
57	#672	—	4	1	no	—
58	C₄×M₁₆	—	5	1	no	—
59	C₄×M₁₆	—	10	1	no	—
60	#364	—	10	1	no	—
61	#349	—	10	1	no	—
62	#347	—	5	1	no	—
63	#347	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
64	#344	—	10	1	no	—
65	#339	—	10	1	no	—
66	#338	—	5	1	no	—
67	H(3,ℤ/4)	—	5	1	no	—
68	#322	—	5	1	no	—
69	#322	—	10	1	no	—
70	C₈²	—	5	1	no	—
71	C₂²×D₅	#104	1	1	yes	characteristic
72	C₂×AGL(1,𝔽₅)	—	8	1	no	—
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	#95	1	1	yes	characteristic
75	C₂×Dic₅	—	2	1	no	—
76	C₂×Dic₅	#104	1	1	yes	characteristic
77	C₄×D₅	—	2	1	no	—
78	C₄×D₅	—	2	1	no	—
79	C₄×D₅	—	4	2	no	—
80	#170	—	2	4	no	—
81	#170	—	4	2	no	—
82	#170	—	8	1	no	—
83	C₅×C₈	—	2	1	no	—
84	C₅×C₈	—	2	1	no	—
85	#168	—	2	1	no	—
86	#168	—	2	1	no	—
87	C₂×M₁₆	—	5	1	no	—
88	C₂×M₁₆	—	5	1	no	—
89	C₂×M₁₆	—	10	2	no	—
90	C₂×M₁₆	—	10	1	no	—
91	#117	—	10	1	no	—
92	C₂×C₄²	—	5	1	no	—
93	M₃₂	—	2	1	no	—
94	M₃₂	—	10	1	no	—
95	#97	—	5	1	no	—
96	#97	—	10	2	no	—
97	#97	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₄×C₈	—	5	1	no	—
99	C₄×C₈	—	5	1	no	—
100	C₄×C₈	—	10	1	no	—
101	C₄×C₈	—	10	1	no	—
102	C₂²×C₅	#339	1	1	yes	characteristic
103	C₂×D₅	—	2	1	no	—
104	C₂×D₅	—	4	1	no	—
105	AGL(1,𝔽₅)	—	16	1	no	—
106	C₄×C₅	—	4	1	no	—
107	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
108	C₄×C₅	#344	1	1	yes	characteristic
109	Dic₅	—	2	2	no	—
110	Dic₅	—	2	1	no	—
111	C₂²×C₄	—	5	1	no	—
112	C₂²×C₄	—	10	1	no	—
113	M₁₆	—	20	1	no	—
114	M₁₆	—	20	1	no	—
115	M₁₆	—	20	4	no	—
116	M₁₆	—	20	1	no	—
117	M₁₆	—	20	1	no	—
118	C₂×C₈	—	5	1	no	—
119	C₂×C₈	—	5	1	no	—
120	C₂×C₈	—	10	2	no	—
121	C₂×C₈	—	10	2	no	—
122	C₂×C₈	—	20	1	no	—
123	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₄⋊C₄	—	20	1	no	—
126	C₂²⋊C₄	—	20	1	no	—
127	C₄²	—	5	1	no	—
128	C₄²	—	10	1	no	—
129	C₄²	—	20	1	no	—
130	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₁₆	—	4	1	no	—
132	C₁₆	—	20	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	#1288	1	1	yes	characteristic
135	D₅	—	4	1	no	—
136	C₂³	—	5	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	5	1	no	—
139	C₂×C₄	—	10	1	no	—
140	C₂×C₄	—	10	1	no	—
141	C₂×C₄	—	10	1	no	—
142	C₂×C₄	—	20	2	no	—
143	C₂×C₄	—	40	1	no	—
144	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	10	4	no	—
148	C₈	—	10	1	no	—
149	C₈	—	10	1	no	—
150	C₈	—	20	2	no	—
151	C₈	—	40	1	no	—
152	C₅	#7225	1	1	yes	characteristic, complemented, Sylow
153	C₂²	—	10	1	no	—
154	C₂²	—	20	1	no	—
155	C₂²	#64900	1	1	yes	characteristic
156	C₄	—	4	1	no	—
157	C₄	—	10	2	no	—
158	C₄	—	10	1	no	—
159	C₄	—	80	1	no	—
160	C₄	#64863	1	1	yes	characteristic
161	C₄	#64902	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	20	1	no	—
164	C₂	#10598546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603887

C₂

yes

maximal, characteristic

#10603864

C₂

yes

maximal, characteristic

#10599881

C₂

yes

maximal, characteristic

C₄×#700

—

#64996

C₂²

yes

characteristic

#64897

C₄

yes

characteristic

#64895

C₄

yes

—

—

C₄

yes

characteristic, complemented

C₈×#170

C₄

yes

characteristic, complemented

#64862

—

C₅×#347

C₄

yes

characteristic, complemented

#64734

—

#64731

—

#64686

C₄

yes

characteristic, complemented

#7225

—

maximal, Sylow

#4130

—

D₅⋊M₁₆

—

C₂×#700

—

C₂×#676

C₂×C₄

yes

characteristic

C₂×#676

D₄

yes

characteristic

C₄²×D₅

D₄

yes

—

—

—

C₅×M₃₂

—

C₅×#97

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

characteristic

#3944

—

#3935

Q₈

yes

—

—

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂²⋊C₄

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

—

—

—

—

—

—

—

—

D₅⋊C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄×Dic₅

—

C₄×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

#676

—

#676

—

C₅×C₁₆

—

#672

—

C₄×M₁₆

—

C₄×M₁₆

—

—

—

—

yes

characteristic, complemented

—

—

—

—

—

—

C₈²

—

C₂²×D₅

#104

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#95

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

#104

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#117

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

—

—

yes

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂²×C₅

#339

yes

characteristic

103

C₂×D₅

—

104

C₂×D₅

—

105

AGL(1,𝔽₅)

—

106

C₄×C₅

—

107

C₄×C₅

H(3,ℤ/4)

yes

characteristic

108

C₄×C₅

#344

yes

characteristic

109

Dic₅

—

110

Dic₅

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

124

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₄⋊AGL(1,𝔽₅)

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₂×C₅

—

134

C₂×C₅

#1288

yes

characteristic

135

D₅

—

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₅

#7225

yes

characteristic, complemented, Sylow

153

C₂²

—

154

C₂²

—

155

C₂²

#64900

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#64863

yes

characteristic

161

C₄

#64902

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

#10598546

yes

characteristic

	subgroup	G/H
center	C₄	#64863
commutator	C₂×C₅×C₈	C₄²
Frattini	#97	C₂×AGL(1,𝔽₅)
Fitting	C₅×#347	C₄
socle	C₂×C₅	#1288
prime radical	C₂²×D₅	#104
cosocle	#64996	C₂²
hypercenter	#347	AGL(1,𝔽₅)
hypocenter	C₅	#7225
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

subgroup

G/H

center

C₄

#64863

commutator

C₂×C₅×C₈

C₄²

C₄

C₂×C₅

C₂²×D₅

C₂²

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

N	H ≅ G/N	type
10. #64879	C₄	semidirect
11. C₈×#170	C₄	semidirect
13. C₅×#347	C₄	semidirect
16. #64686	C₄	semidirect
63. #347	AGL(1,𝔽₅)	semidirect
152. C₅	#7225	semidirect

H ≅ G/N

type

10. #64879

C₄

semidirect

11. C₈×#170

C₄

semidirect

13. C₅×#347

C₄

semidirect

16. #64686

C₄

semidirect

63. #347

AGL(1,𝔽₅)

semidirect

152. C₅

#7225

semidirect

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

subgroup	index
#10603887	2
#10603864	2
#10599881	2
#7225	5

subgroup

index

Subgroups — #13013761

Conjugacy classes of subgroups

Complements

Maximal subgroups