Group #13013787 — subgroups

Subgroups — #13013787

Order 1280.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10604731	C₂	1	2	yes	maximal
2	#10602224	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
4	#65756	—	2	2	no	—
5	C₂×#3995	—	2	2	no	—
6	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
7	C₂×#3944	C₄	1	1	yes	characteristic
8	#64896	C₄	1	4	yes	—
9	#64727	—	2	2	no	—
10	#7234	—	5	1	no	maximal, Sylow
11	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
12	C₂×#700	—	2	4	no	—
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×#168	Q₈	1	1	yes	characteristic
15	C₂×C₈×D₅	—	4	1	no	—
16	C₂×C₈×D₅	—	4	1	no	—
17	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
18	C₂×C₈×D₅	D₄	1	2	yes	—
19	C₂²⋊#170	—	4	2	no	—
20	#4003	—	4	2	no	—
21	#3995	—	4	2	no	—
22	#3995	—	4	2	no	—
23	C₅×M₃₂	—	4	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	#3944	—	4	1	no	—
26	C₂×#672	—	2	1	no	—
27	#2112	—	5	1	no	—
28	#1810	—	5	2	no	—
29	C₂³×D₅	Q₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₅	D₈	1	1	yes	characteristic
32	C₂×C₄×D₅	—	4	1	no	—
33	C₂×C₄×D₅	—	4	1	no	—
34	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×D₅	SD₁₆	1	2	yes	—
36	C₂×#170	—	4	4	no	—
37	#700	—	4	4	no	—
38	#700	—	4	4	no	—
39	C₂×C₅×C₈	—	2	1	no	—
40	C₂×C₅×C₈	—	2	1	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#168	—	4	1	no	—
44	C₂×#168	C₄⋊C₄	1	2	yes	—
45	C₈×D₅	—	2	2	no	—
46	C₈×D₅	—	2	2	no	—
47	C₈×D₅	—	4	2	no	—
48	C₅×C₁₆	—	4	1	no	—
49	#672	—	4	1	no	—
50	C₂³×C₈	—	5	1	no	—
51	C₂×M₃₂	—	5	1	no	—
52	C₂×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×#108	—	10	2	no	—
54	#407	—	10	2	no	—
55	C₂²⋊C₁₆	—	10	2	no	—
56	#341	—	5	4	no	—
57	C₂³×C₅	#107	1	1	yes	characteristic
58	C₂²×D₅	—	4	1	no	—
59	C₂²×D₅	—	4	2	no	—
60	C₂²×D₅	#103	1	2	yes	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	#95	1	1	yes	characteristic
64	C₂×C₄×C₅	#106	1	1	yes	characteristic
65	C₂×Dic₅	—	4	1	no	—
66	C₂×Dic₅	#102	1	2	yes	—
67	C₄×D₅	—	2	2	no	—
68	C₄×D₅	—	2	2	no	—
69	C₄×D₅	—	4	2	no	—
70	#170	—	8	4	no	—
71	C₅×C₈	—	2	1	no	—
72	C₅×C₈	#99	1	1	yes	characteristic
73	C₅×C₈	#99	1	1	yes	characteristic
74	#168	—	2	2	no	—
75	C₂³×C₄	—	5	1	no	—
76	C₂×M₁₆	—	10	4	no	—
77	C₂²×C₈	—	5	1	no	—
78	C₂²×C₈	—	5	2	no	—
79	C₂²×C₈	—	5	2	no	—
80	C₂²×C₈	—	20	1	no	—
81	C₂²×C₈	—	20	1	no	—
82	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	M₃₂	—	4	1	no	—
84	M₃₂	—	20	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	10	1	no	—
87	#108	—	20	2	no	—
88	#108	—	20	2	no	—
89	C₂²⋊C₈	—	20	2	no	—
90	C₂²⋊C₈	—	20	2	no	—
91	C₂²×C₅	—	2	1	no	—
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	#340	1	1	yes	characteristic
94	C₂×D₅	—	2	2	no	—
95	C₂×D₅	—	2	4	no	—
96	C₂×D₅	—	4	4	no	—
97	C₄×C₅	—	2	1	no	—
98	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
99	C₄×C₅	#342	1	1	yes	characteristic
100	Dic₅	—	2	2	no	—
101	C₂⁴	—	5	1	no	—
102	C₂²×C₄	—	5	1	no	—
103	C₂²×C₄	—	5	2	no	—
104	C₂²×C₄	—	5	2	no	—
105	C₂²×C₄	—	20	1	no	—
106	C₂²×C₄	—	20	1	no	—
107	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
108	M₁₆	—	20	4	no	—
109	M₁₆	—	20	4	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	5	2	no	—
113	C₂×C₈	—	10	2	no	—
114	C₂×C₈	—	10	2	no	—
115	C₂×C₈	—	20	2	no	—
116	C₂×C₈	—	20	1	no	—
117	C₂×C₈	—	20	4	no	—
118	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₁₆	—	4	1	no	—
121	C₁₆	—	20	1	no	—
122	C₂×C₅	—	4	1	no	—
123	C₂×C₅	#1289	1	1	yes	characteristic
124	C₂×C₅	#1388	1	2	yes	—
125	D₅	—	4	2	no	—
126	C₂³	—	5	2	no	—
127	C₂³	—	20	1	no	—
128	C₂³	—	20	2	no	—
129	C₂³	#3994	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	5	2	no	—
133	C₂×C₄	—	10	2	no	—
134	C₂×C₄	—	10	2	no	—
135	C₂×C₄	—	20	2	no	—
136	C₂×C₄	—	20	1	no	—
137	C₂×C₄	#3993	1	1	yes	characteristic
138	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	10	2	no	—
141	C₈	—	40	4	no	—
142	C₈	#3999	1	1	yes	characteristic
143	C₈	#4011	1	1	yes	characteristic
144	C₅	#7234	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	2	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	10	4	no	—
148	C₂²	—	10	2	no	—
149	C₂²	—	20	4	no	—
150	C₂²	#64893	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	10	2	no	—
153	C₄	#64895	1	1	yes	characteristic
154	C₄	#64917	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	—	20	2	no	—
157	C₂	#10598676	1	2	yes	—
158	C₂	#10598764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604731

C₂

yes

maximal

#10602224

C₂

yes

maximal, characteristic

C₂²×C₈×D₅

C₂²

yes

characteristic

#65756

—

C₂×#3995

—

C₂×C₅×M₃₂

C₄

yes

characteristic

C₂×#3944

C₄

yes

characteristic

#64896

C₄

yes

—

#64727

—

#7234

—

maximal, Sylow

C₂²×C₄×D₅

D₄

yes

characteristic

C₂×#700

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×#168

Q₈

yes

characteristic

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

C₂×C₄

yes

—

C₂×C₈×D₅

D₄

yes

—

C₂²⋊#170

—

#4003

—

#3995

—

#3995

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

—

—

—

—

C₂³×D₅

Q₁₆

yes

characteristic

C₂²×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂²×Dic₅

D₈

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂²⋊C₄

yes

—

C₂×C₄×D₅

SD₁₆

yes

—

C₂×#170

—

#700

—

#700

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

yes

—

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₅

#107

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

#103

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

—

C₂×Dic₅

#102

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#340

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

100

Dic₅

—

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

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

C₁₆

—

121

C₁₆

—

122

C₂×C₅

—

123

C₂×C₅

#1289

yes

characteristic

124

C₂×C₅

#1388

yes

—

125

D₅

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#3994

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#3993

yes

characteristic

138

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#3999

yes

characteristic

143

C₈

#4011

yes

characteristic

144

C₅

#7234

yes

characteristic, complemented, Sylow

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#64893

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

#64895

yes

characteristic

154

C₄

#64917

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#10598676

yes

—

158

C₂

#10598764

yes

characteristic

	subgroup	G/H
center	C₂²	#64893
commutator	C₂×C₅×C₈	C₄²
Frattini	C₂²×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₃₂	C₄
socle	C₂²×C₅	#340
prime radical	C₂³×D₅	Q₁₆
cosocle	C₂²×C₈×D₅	C₂²
hypercenter	C₂×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#7234
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#64893

commutator

C₂×C₅×C₈

C₄²

Frattini

C₂²×C₈

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₅×M₃₂

C₄

socle

C₂²×C₅

#340

prime radical

C₂³×D₅

Q₁₆

cosocle

C₂²×C₈×D₅

C₂²

C₂×M₃₂

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
144. C₅	#7234	semidirect

H ≅ G/N

type

144. C₅

#7234

semidirect

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

subgroup	index
#10604731	2
#10602224	2
#7234	5

subgroup

index

#10604731

#10602224

#7234

Subgroups — #13013787

Conjugacy classes of subgroups

Complements

Maximal subgroups