Group #13013776 — subgroups

Subgroups — #13013776

Order 1280.

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

normal subgroups only

This group has 244 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	#10604733	C₂	1	2	yes	maximal
2	#10602225	C₂	1	1	yes	maximal, characteristic
3	C₂×#4047	C₂²	1	1	yes	characteristic
4	#65761	—	2	2	no	—
5	D₅⋊#108	—	2	2	no	—
6	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×#672	C₄	1	1	yes	characteristic
8	#64897	C₄	1	4	yes	—
9	#64727	—	2	2	no	—
10	#7229	—	5	1	no	maximal, Sylow
11	C₂×#709	D₄	1	1	yes	characteristic
12	D₅⋊M₁₆	—	2	4	no	—
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×#681	Q₈	1	1	yes	characteristic
15	#4047	—	4	1	no	—
16	#4047	—	4	1	no	—
17	C₂×#676	C₂×C₄	1	2	yes	—
18	C₂×C₈×D₅	D₄	1	2	yes	—
19	#4013	—	4	2	no	—
20	#4005	—	4	2	no	—
21	#3996	—	4	2	no	—
22	#3995	—	4	2	no	—
23	C₅×M₃₂	—	4	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×#672	—	2	1	no	—
26	C₂×#672	—	4	1	no	—
27	#2113	—	5	1	no	—
28	#1813	—	5	2	no	—
29	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂²⋊D₅	SD₁₆	1	2	yes	—
31	#709	—	4	1	no	—
32	#709	—	4	1	no	—
33	C₂×D₂₀	Q₁₆	1	1	yes	characteristic
34	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
35	C₂×Dic₁₀	D₈	1	1	yes	characteristic
36	#704	—	4	4	no	—
37	#700	—	4	4	no	—
38	D₅⋊C₈	—	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	#681	—	4	1	no	—
44	C₂×#168	C₄⋊C₄	1	2	yes	—
45	#676	—	4	2	no	—
46	C₈×D₅	—	2	2	no	—
47	C₈×D₅	—	2	2	no	—
48	C₅×C₁₆	—	4	1	no	—
49	#672	—	4	1	no	—
50	C₂×#131	—	5	1	no	—
51	C₂×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂²×C₁₆	—	5	1	no	—
53	C₂×#108	—	10	2	no	—
54	#413	—	10	2	no	—
55	C₂²⋊C₁₆	—	10	2	no	—
56	#338	—	5	4	no	—
57	C₂³×C₅	#106	1	1	yes	characteristic
58	C₂²×D₅	#103	1	2	yes	—
59	C₂×C₄×C₅	—	2	1	no	—
60	C₂×C₄×C₅	—	2	1	no	—
61	C₂×C₄×C₅	#95	1	1	yes	characteristic
62	C₂×C₄×C₅	#107	1	1	yes	characteristic
63	C₂²⋊D₅	—	4	2	no	—
64	C₂×Dic₅	#102	1	2	yes	—
65	D₂₀	—	4	1	no	—
66	C₄×D₅	—	2	2	no	—
67	C₄×D₅	—	2	2	no	—
68	Dic₁₀	—	4	1	no	—
69	#170	—	8	4	no	—
70	C₅×C₈	—	2	1	no	—
71	C₅×C₈	#99	1	1	yes	characteristic
72	C₅×C₈	#99	1	1	yes	characteristic
73	#168	—	2	2	no	—
74	C₂×#41	—	5	1	no	—
75	#131	—	20	1	no	—
76	#131	—	20	1	no	—
77	C₂×M₁₆	—	5	1	no	—
78	C₂×M₁₆	—	5	2	no	—
79	C₂×M₁₆	—	10	4	no	—
80	C₂²×C₈	—	5	2	no	—
81	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	M₃₂	—	4	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	10	1	no	—
85	C₂×C₁₆	—	20	1	no	—
86	#108	—	20	2	no	—
87	#108	—	20	2	no	—
88	#101	—	20	2	no	—
89	#100	—	20	2	no	—
90	C₂²×C₅	—	2	1	no	—
91	C₂²×C₅	—	4	1	no	—
92	C₂²×C₅	#340	1	1	yes	characteristic
93	C₂×D₅	—	2	2	no	—
94	C₂×D₅	—	4	2	no	—
95	C₄×C₅	—	2	1	no	—
96	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
97	C₄×C₅	#342	1	1	yes	characteristic
98	Dic₅	—	2	2	no	—
99	#41	—	20	1	no	—
100	#41	—	20	1	no	—
101	C₂×Q₈	—	5	1	no	—
102	C₂×D₄	—	5	2	no	—
103	C₂×D₄	—	5	1	no	—
104	C₂²×C₄	—	5	2	no	—
105	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
106	M₁₆	—	20	2	no	—
107	M₁₆	—	20	1	no	—
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	4	no	—
116	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₁₆	—	4	1	no	—
119	C₁₆	—	20	1	no	—
120	C₂×C₅	—	2	1	no	—
121	C₂×C₅	—	4	1	no	—
122	C₂×C₅	#1289	1	1	yes	characteristic
123	D₅	—	4	2	no	—
124	C₂³	—	5	2	no	—
125	C₂³	#3993	1	1	yes	characteristic
126	Q₈	—	20	1	no	—
127	D₄	—	20	1	no	—
128	D₄	—	20	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	5	2	no	—
132	C₂×C₄	—	10	2	no	—
133	C₂×C₄	—	10	2	no	—
134	C₂×C₄	#3994	1	1	yes	characteristic
135	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	10	2	no	—
138	C₈	—	40	4	no	—
139	C₈	#3999	1	1	yes	characteristic
140	C₈	#4011	1	1	yes	characteristic
141	C₅	#7229	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	10	2	no	—
145	C₂²	—	20	2	no	—
146	C₂²	#64893	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	10	2	no	—
149	C₄	#64895	1	1	yes	characteristic
150	C₄	#64917	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	20	2	no	—
154	C₂	#10598764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604733

C₂

yes

maximal

#10602225

C₂

yes

maximal, characteristic

C₂×#4047

C₂²

yes

characteristic

#65761

—

D₅⋊#108

—

C₂×C₅×M₃₂

C₄

yes

characteristic

C₂²×#672

C₄

yes

characteristic

#64897

C₄

yes

—

—

—

D₄

yes

characteristic

D₅⋊M₁₆

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×#681

Q₈

yes

—

—

C₂×C₄

yes

—

C₂×C₈×D₅

D₄

yes

—

—

—

—

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

—

—

—

—

C₂²×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₂²⋊D₅

SD₁₆

yes

—

#709

—

#709

—

C₂×D₂₀

Q₁₆

yes

characteristic

C₂×C₄×D₅

C₂²⋊C₄

yes

—

C₂×Dic₁₀

D₈

yes

characteristic

#704

—

#700

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

#681

—

C₂×#168

C₄⋊C₄

yes

—

#676

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

—

C₂×#131

—

C₂×M₃₂

AGL(1,𝔽₅)

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

#413

—

C₂²⋊C₁₆

—

#338

—

C₂³×C₅

#106

yes

characteristic

C₂²×D₅

#103

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂²⋊D₅

—

C₂×Dic₅

#102

yes

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#170

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#340

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

106

M₁₆

—

107

M₁₆

—

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

C₄×AGL(1,𝔽₅)

yes

characteristic

117

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

#1289

yes

characteristic

123

D₅

—

124

C₂³

—

125

C₂³

#3993

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#3994

yes

characteristic

135

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#3999

yes

characteristic

140

C₈

#4011

yes

characteristic

141

C₅

#7229

yes

characteristic, complemented, Sylow

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#64893

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#64895

yes

characteristic

150

C₄

#64917

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#10598764

yes

characteristic

	subgroup	G/H
center	C₄	#64895
commutator	C₂×C₅×C₈	C₄²
Frattini	C₂²×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₃₂	C₄
socle	C₂×C₅	#1289
prime radical	C₂×#709	D₄
cosocle	C₂×#4047	C₂²
hypercenter	C₂×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#7229
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#64895

commutator

C₂×C₅×C₈

C₄²

Frattini

C₂²×C₈

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₅×M₃₂

C₄

C₂×C₅

D₄

C₂²

C₂×M₃₂

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
141. C₅	#7229	semidirect

H ≅ G/N

type

141. C₅

#7229

semidirect

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

subgroup	index
#10604733	2
#10602225	2
#7229	5

subgroup

index

#10604733

#10602225

#7229

Subgroups — #13013776

Conjugacy classes of subgroups

Complements

Maximal subgroups