Group #13018815 — subgroups

Subgroups — #13018815

Order 1280.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#10603859	C₂	1	2	yes	maximal
2	#10603817	C₂	1	2	yes	maximal
3	#10603816	C₂	1	2	yes	maximal
4	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#3996	—	2	4	no	—
6	#65681	C₂²	1	4	yes	—
7	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
10	#64883	—	2	4	no	—
11	#64881	—	2	4	no	—
12	C₅×C₈²	C₄	1	1	yes	characteristic
13	C₈×#168	C₄	1	1	yes	characteristic
14	C₈×#168	C₄	1	1	yes	characteristic
15	C₈×#168	C₄	1	1	yes	characteristic
16	#8822	—	5	1	no	maximal, Sylow
17	C₂×#700	—	2	4	no	—
18	C₂×C₈×D₅	D₄	1	2	yes	—
19	C₂×C₈×D₅	Q₈	1	2	yes	—
20	C₂×C₈×D₅	Q₈	1	2	yes	—
21	C₄²×D₅	C₂³	1	1	yes	characteristic
22	C₄⋊#170	—	2	4	no	—
23	C₄⋊#170	—	2	4	no	—
24	#3996	—	4	8	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅	D₄	1	2	yes	—
29	C₈×Dic₅	D₄	1	2	yes	—
30	C₈×Dic₅	Q₈	1	2	yes	—
31	C₄×#168	C₂×C₄	1	1	yes	characteristic
32	C₄×#168	C₂×C₄	1	1	yes	characteristic
33	C₄×#168	C₂×C₄	1	1	yes	characteristic
34	#1848	—	5	2	no	—
35	#1566	—	5	2	no	—
36	#1566	—	5	2	no	—
37	C₂×C₈²	—	5	1	no	—
38	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
41	C₂×#170	—	4	4	no	—
42	#700	—	4	8	no	—
43	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
48	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
49	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
50	C₂×#168	C₄⋊C₄	1	2	yes	—
51	C₂×#168	C₄⋊C₄	1	2	yes	—
52	C₂×#168	C₄⋊C₄	1	2	yes	—
53	C₈×D₅	—	2	4	no	—
54	C₈×D₅	—	2	4	no	—
55	C₈×D₅	SD₁₆	1	8	yes	—
56	C₂×#108	—	10	4	no	—
57	#423	—	5	4	no	—
58	C₂×C₄×C₈	—	5	1	no	—
59	C₂×C₄×C₈	—	5	1	no	—
60	C₂×C₄×C₈	—	5	1	no	—
61	#334	—	10	4	no	—
62	#334	—	10	4	no	—
63	C₈²	—	5	1	no	—
64	C₈²	—	5	1	no	—
65	C₈²	—	5	1	no	—
66	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂²×D₅	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×Dic₅	#127	1	1	yes	characteristic
72	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
74	C₄×D₅	—	2	2	no	—
75	C₄×D₅	—	2	2	no	—
76	C₄×D₅	C₂×SD₁₆	1	4	yes	—
77	#170	—	8	4	no	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	#106	1	4	yes	—
80	C₅×C₈	#108	1	4	yes	complemented
81	#168	—	2	2	no	—
82	#168	#106	1	4	yes	—
83	#168	#108	1	4	yes	complemented
84	C₂×M₁₆	—	10	4	no	—
85	C₂²×C₈	—	5	2	no	—
86	C₂²×C₈	—	5	2	no	—
87	C₂²×C₈	—	5	2	no	—
88	C₂×C₄²	—	5	1	no	—
89	#108	—	20	8	no	—
90	C₄⋊C₈	—	10	4	no	—
91	C₄⋊C₈	—	10	4	no	—
92	C₄×C₈	—	5	2	no	—
93	C₄×C₈	—	5	2	no	—
94	C₄×C₈	—	5	2	no	—
95	C₄×C₈	—	5	1	no	—
96	C₄×C₈	—	5	1	no	—
97	C₄×C₈	—	5	1	no	—
98	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₂²×C₅	#384	1	1	yes	characteristic
102	C₂×D₅	—	2	1	no	—
103	C₂×D₅	—	2	1	no	—
104	C₂×D₅	#498	1	2	yes	—
105	C₄×C₅	C₂×#106	1	2	yes	—
106	C₄×C₅	C₂×#108	1	2	yes	—
107	C₄×C₅	C₂×#108	1	2	yes	—
108	Dic₅	—	2	1	no	—
109	Dic₅	—	2	1	no	—
110	Dic₅	#492	1	2	yes	—
111	C₂²×C₄	—	5	1	no	—
112	C₂²×C₄	—	5	1	no	—
113	C₂²×C₄	—	5	1	no	—
114	M₁₆	—	20	8	no	—
115	C₂×C₈	—	5	2	no	—
116	C₂×C₈	—	5	2	no	—
117	C₂×C₈	—	5	8	no	—
118	C₂×C₈	—	5	2	no	—
119	C₂×C₈	—	10	4	no	—
120	C₂×C₈	—	10	4	no	—
121	C₂×C₈	—	20	4	no	—
122	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
123	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
124	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
125	C₄²	—	5	1	no	—
126	C₄²	—	5	1	no	—
127	C₄²	—	5	1	no	—
128	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₂×C₅	#1844	1	1	yes	characteristic
130	C₂×C₅	#1848	1	1	yes	characteristic
131	C₂×C₅	#1848	1	1	yes	characteristic
132	D₅	—	2	2	no	—
133	C₂³	—	5	1	no	—
134	C₂×C₄	—	5	1	no	—
135	C₂×C₄	—	5	4	no	—
136	C₂×C₄	—	5	1	no	—
137	C₂×C₄	—	5	1	no	—
138	C₂×C₄	—	10	2	no	—
139	C₂×C₄	—	10	2	no	—
140	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	5	4	no	—
145	C₈	—	5	4	no	—
146	C₈	—	10	2	no	—
147	C₈	—	40	4	no	—
148	C₈	#3993	1	4	yes	—
149	C₈	#3996	1	4	yes	complemented
150	C₅	#8822	1	1	yes	characteristic, complemented, Sylow
151	C₂²	—	5	2	no	—
152	C₂²	—	10	1	no	—
153	C₂²	—	10	1	no	—
154	C₂²	#65688	1	1	yes	characteristic
155	C₄	—	5	2	no	—
156	C₄	—	10	1	no	—
157	C₄	—	10	1	no	—
158	C₄	C₂×#3993	1	2	yes	—
159	C₄	C₂×#3996	1	2	yes	—
160	C₄	D₅⋊#108	1	2	yes	—
161	C₂	—	10	2	no	—
162	C₂	#10603847	1	1	yes	characteristic
163	C₂	#10603859	1	1	yes	characteristic
164	C₂	#10603860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603859

C₂

yes

maximal

#10603817

C₂

yes

maximal

#10603816

C₂

yes

maximal

C₈²×D₅

C₂

yes

maximal, characteristic

C₂×#3996

—

#65681

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

C₄×C₈×D₅

C₂²

yes

characteristic

C₄×C₈×D₅

C₂²

yes

characteristic

#64883

—

#64881

—

C₅×C₈²

C₄

yes

characteristic

C₈×#168

C₄

yes

characteristic

C₈×#168

C₄

yes

characteristic

C₈×#168

C₄

yes

—

—

C₂×C₈×D₅

D₄

yes

—

C₂×C₈×D₅

Q₈

yes

—

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

—

—

C₂×C₈²

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

characteristic

C₂×#170

—

#700

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

SD₁₆

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

#334

—

C₈²

—

C₈²

—

C₈²

—

C₈²

AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅

#127

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×SD₁₆

yes

—

#170

—

C₅×C₈

—

C₅×C₈

#106

yes

—

C₅×C₈

yes

—

yes

—

#168

#108

yes

complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₄×C₈

yes

100

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

101

C₂²×C₅

#384

yes

characteristic

102

C₂×D₅

—

103

C₂×D₅

—

104

C₂×D₅

#498

yes

—

105

C₄×C₅

C₂×#106

yes

—

106

C₄×C₅

C₂×#108

yes

—

107

C₄×C₅

C₂×#108

yes

—

108

Dic₅

—

109

Dic₅

—

110

Dic₅

#492

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

123

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

124

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

129

C₂×C₅

#1844

yes

characteristic

130

C₂×C₅

#1848

yes

characteristic

131

C₂×C₅

#1848

yes

characteristic

132

D₅

—

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

141

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

142

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#3993

yes

—

149

C₈

#3996

yes

complemented

150

C₅

#8822

yes

characteristic, complemented, Sylow

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#65688

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂×#3993

yes

—

159

C₄

C₂×#3996

yes

—

160

C₄

D₅⋊#108

yes

—

161

C₂

—

162

C₂

#10603847

yes

characteristic

163

C₂

#10603859

yes

characteristic

164

C₂

#10603860

yes

characteristic

	subgroup	G/H
center	C₂²	#65688
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#384
prime radical	C₂²×D₅	C₄⋊Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8822
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#65688

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

socle

C₂²×C₅

#384

prime radical

C₂²×D₅

C₄⋊Q₈

cosocle

C₄²×D₅

C₂³

C₈²

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
80. C₅×C₈	#108	semidirect
83. #168	#108	semidirect
149. C₈	#3996	semidirect
150. C₅	#8822	semidirect

H ≅ G/N

type

80. C₅×C₈

#108

semidirect

83. #168

#108

semidirect

149. C₈

#3996

semidirect

150. C₅

#8822

semidirect

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

subgroup	index
#10603859	2
#10603817	2
#10603816	2
C₈²×D₅	2
#8822	5

subgroup

index

C₈²×D₅

Subgroups — #13018815

Conjugacy classes of subgroups

Complements

Maximal subgroups