Group #13043395 — subgroups

Subgroups — #13043395

Order 1280.

684 conjugacy classes of subgroups; only the 380 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 380 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	#10617530	C₂	1	2	yes	maximal
2	Dic₅×#131	C₂	1	1	yes	maximal, characteristic
3	#10604751	C₂	1	6	yes	maximal, complemented
4	C₂×#64889	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#131	C₄	1	1	yes	characteristic
6	#41×Dic₅	C₄	1	1	yes	characteristic
7	C₂×#4094	C₂²	1	1	yes	characteristic
8	#65791	—	2	8	no	—
9	C₂×#3998	C₂²	1	12	yes	—
10	#65411	C₂²	1	3	yes	—
11	M₁₆×Dic₅	C₄	1	3	yes	—
12	#65390	C₄	1	3	yes	—
13	C₂×C₈×Dic₅	C₂²	1	3	yes	—
14	#64889	C₂²	1	4	yes	—
15	#64889	C₂²	1	12	yes	—
16	#33401	—	5	1	no	maximal, Sylow
17	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
18	C₅×#131	C₈	1	8	yes	—
19	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
21	#4094	D₄	1	4	yes	—
22	#4094	Q₈	1	4	yes	—
23	Q₈×Dic₅	—	2	1	no	—
24	D₄×Dic₅	—	2	3	no	—
25	#4072	C₂×C₄	1	3	yes	—
26	C₂×C₄×Dic₅	C₂×C₄	1	3	yes	—
27	C₂×#681	C₂×C₄	1	3	yes	—
28	C₂²×#168	C₂³	1	3	yes	—
29	#3998	—	2	12	no	—
30	#3998	C₂³	1	8	yes	—
31	C₂×#674	—	2	12	no	—
32	#3947	C₂×C₄	1	6	yes	—
33	C₈×Dic₅	C₂×C₄	1	3	yes	—
34	C₈×Dic₅	C₂×C₄	1	3	yes	—
35	C₈×Dic₅	C₂³	1	1	yes	characteristic
36	C₈×Dic₅	C₂³	1	3	yes	—
37	#3407	—	5	2	no	—
38	C₄×#131	—	5	1	no	—
39	#2153	—	5	6	no	—
40	C₂×#364	—	5	6	no	—
41	C₅×#41	C₄⋊C₄	1	4	yes	—
42	C₅×#41	C₂×C₈	1	4	yes	—
43	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
44	C₂×C₅×D₄	M₁₆	1	3	yes	—
45	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
46	C₂²×Dic₅	—	2	3	no	—
47	C₅×M₁₆	C₂×C₈	1	12	yes	—
48	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
51	C₂²⋊Dic₅	—	2	3	no	—
52	C₄⋊Dic₅	—	2	3	no	—
53	C₄×Dic₅	—	2	3	no	—
54	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₅	C₂²×C₄	1	3	yes	—
56	#681	C₄⋊C₄	1	12	yes	—
57	C₂×#168	C₂²×C₄	1	3	yes	—
58	C₂×#168	C₂×D₄	1	6	yes	—
59	C₂×#168	C₂×Q₈	1	6	yes	—
60	C₂×#168	C₂⁴	1	1	yes	characteristic
61	#674	—	2	8	no	—
62	C₂×#131	—	5	1	no	—
63	C₂×#131	AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₄×#41	—	5	1	no	—
65	#504	—	10	8	no	—
66	C₂×M₃₂	—	5	12	no	—
67	#405	—	5	3	no	—
68	#405	—	5	3	no	—
69	C₄×M₁₆	—	5	3	no	—
70	C₂×C₄×C₈	—	5	3	no	—
71	#364	—	5	4	no	—
72	#364	—	5	12	no	—
73	C₂³×C₅	C₂×M₁₆	1	3	yes	—
74	C₅×Q₈	C₄⋊C₈	1	4	yes	—
75	C₅×D₄	C₄⋊C₈	1	12	yes	—
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
77	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
78	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
79	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
80	C₂×Dic₅	—	2	1	no	—
81	C₂×Dic₅	—	2	3	no	—
82	C₂×Dic₅	—	4	6	no	—
83	C₅×C₈	C₂²×C₈	1	2	yes	—
84	C₅×C₈	C₂²×C₈	1	6	yes	—
85	#168	C₂×C₄⋊C₄	1	6	yes	—
86	#168	C₂²×D₄	1	1	yes	characteristic
87	#168	C₂²×Q₈	1	1	yes	characteristic
88	C₂×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
89	#131	—	5	4	no	—
90	#131	—	5	4	no	—
91	#131	#170	1	8	yes	—
92	C₂×M₁₆	—	5	3	no	—
93	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	3	yes	—
94	C₂²×C₈	—	5	3	no	—
95	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
96	C₄×Q₈	—	10	1	no	—
97	C₄×D₄	—	10	3	no	—
98	#117	—	5	3	no	—
99	C₂×C₄²	—	5	3	no	—
100	M₃₂	—	5	8	no	—
101	M₃₂	—	10	12	no	—
102	C₂×C₁₆	—	10	12	no	—
103	#97	—	5	6	no	—
104	C₄×C₈	—	5	3	no	—
105	C₄×C₈	—	5	3	no	—
106	C₄×C₈	—	5	1	no	—
107	C₄×C₈	—	5	3	no	—
108	C₂²×C₅	—	4	3	no	—
109	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
110	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
111	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
112	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
114	Dic₅	—	2	2	no	—
115	Dic₅	—	4	3	no	—
116	#41	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
117	#41	C₂×#170	1	4	yes	—
118	C₂×Q₈	#700	1	1	yes	characteristic
119	C₂×D₄	#700	1	3	yes	—
120	C₂²×C₄	—	10	3	no	—
121	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
122	M₁₆	—	5	12	no	—
123	M₁₆	C₂×#170	1	12	yes	—
124	C₂×C₈	—	5	6	no	—
125	C₂×C₈	—	5	6	no	—
126	C₂×C₈	—	5	1	no	—
127	C₂×C₈	—	5	3	no	—
128	C₂×C₈	C₂×#170	1	12	yes	—
129	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
131	C₄⋊C₄	—	10	3	no	—
132	C₂²⋊C₄	—	10	3	no	—
133	C₄²	—	5	1	no	—
134	C₄²	—	5	3	no	—
135	C₄²	—	10	3	no	—
136	C₁₆	—	10	8	no	—
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	—	2	6	no	—
139	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	C₂³	C₂×#700	1	3	yes	—
141	Q₈	C₄⋊#170	1	4	yes	—
142	D₄	C₄⋊#170	1	12	yes	—
143	C₂×C₄	—	10	1	no	—
144	C₂×C₄	—	10	3	no	—
145	C₂×C₄	—	20	6	no	—
146	C₂×C₄	C₂×#700	1	3	yes	—
147	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	6	yes	—
148	C₂×C₄	C₂²×#170	1	6	yes	—
149	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₈	—	5	1	no	—
151	C₈	—	5	1	no	—
152	C₈	—	5	6	no	—
153	C₈	C₂²×#170	1	2	yes	—
154	C₈	C₂²×#170	1	6	yes	—
155	C₅	#33401	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊#170	1	6	yes	—
158	C₂²	C₂²×#700	1	1	yes	characteristic
159	C₄	—	10	2	no	—
160	C₄	—	20	3	no	—
161	C₄	C₂×C₄⋊#170	1	6	yes	—
162	C₄	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₄	C₂³×#170	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊#170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10617530

C₂

yes

maximal

Dic₅×#131

C₂

yes

maximal, characteristic

#10604751

C₂

yes

maximal, complemented

C₂×#64889

C₂

yes

maximal, complemented

C₂×C₅×#131

C₄

yes

characteristic

#41×Dic₅

C₄

yes

characteristic

C₂×#4094

C₂²

yes

characteristic

#65791

—

C₂×#3998

C₂²

yes

—

#65411

C₂²

yes

—

M₁₆×Dic₅

C₄

yes

—

#65390

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

#64889

C₂²

yes

—

#64889

C₂²

yes

—

#33401

—

maximal, Sylow

C₂×C₅×#41

C₂×C₄

yes

characteristic

C₅×#131

C₈

yes

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

—

#4094

D₄

yes

—

#4094

Q₈

yes

—

Q₈×Dic₅

—

D₄×Dic₅

—

#4072

C₂×C₄

yes

—

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂×#681

C₂×C₄

yes

—

C₂²×#168

C₂³

yes

—

#3998

—

#3998

C₂³

yes

—

C₂×#674

—

#3947

C₂×C₄

yes

—

C₈×Dic₅

C₂×C₄

yes

—

C₈×Dic₅

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₈×Dic₅

C₂³

yes

—

—

—

—

—

C₅×#41

C₄⋊C₄

yes

—

C₅×#41

C₂×C₈

yes

—

C₂×C₅×Q₈

M₁₆

yes

characteristic

C₂×C₅×D₄

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

—

C₂²×Dic₅

—

C₅×M₁₆

C₂×C₈

yes

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂²×C₄

yes

—

#681

C₄⋊C₄

yes

—

C₂×#168

C₂²×C₄

yes

—

C₂×#168

C₂×D₄

yes

—

C₂×#168

C₂×Q₈

yes

—

C₂×#168

C₂⁴

yes

—

—

yes

C₄×#41

—

#504

—

C₂×M₃₂

—

#405

—

#405

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#364

—

#364

—

C₂³×C₅

C₂×M₁₆

yes

—

C₅×Q₈

C₄⋊C₈

yes

—

C₅×D₄

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂³×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

C₂²×C₈

yes

—

C₅×C₈

C₂²×C₈

yes

—

#168

C₂×C₄⋊C₄

yes

—

#168

C₂²×D₄

yes

characteristic

#168

C₂²×Q₈

yes

C₂×#41

yes

—

—

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

#97

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂²×C₅

—

109

C₂²×C₅

C₂×C₄⋊C₈

yes

—

110

C₂²×C₅

C₂²×M₁₆

yes

characteristic

111

C₄×C₅

C₂×C₄⋊C₈

yes

—

112

C₄×C₅

C₂²×C₄⋊C₄

yes

characteristic

113

C₄×C₅

C₂³×C₈

yes

characteristic

114

Dic₅

—

115

Dic₅

—

116

#41

C₄⋊AGL(1,𝔽₅)

yes

—

117

#41

C₂×#170

yes

—

118

C₂×Q₈

#700

yes

characteristic

119

C₂×D₄

#700

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

—

122

M₁₆

—

123

M₁₆

C₂×#170

yes

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×#170

yes

—

129

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

130

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₁₆

—

137

C₂×C₅

—

138

C₂×C₅

—

139

C₂×C₅

C₂²×C₄⋊C₈

yes

characteristic

140

C₂³

C₂×#700

yes

—

141

Q₈

C₄⋊#170

yes

—

142

D₄

C₄⋊#170

yes

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×#700

yes

—

147

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

148

C₂×C₄

C₂²×#170

yes

—

149

C₂×C₄

C₂³×AGL(1,𝔽₅)

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

C₂²×#170

yes

—

154

C₈

C₂²×#170

yes

—

155

C₅

#33401

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

C₂×C₄⋊#170

yes

—

158

C₂²

C₂²×#700

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

C₂×C₄⋊#170

yes

—

162

C₄

C₂²×C₄⋊AGL(1,𝔽₅)

yes

characteristic

163

C₄

C₂³×#170

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

C₂²×C₄⋊#170

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×#131	C₄
socle	C₂×C₅	C₂²×C₄⋊C₈
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×#131	AGL(1,𝔽₅)
hypocenter	C₅	#33401
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₄

C₂²×C₄⋊AGL(1,𝔽₅)

commutator

C₄×C₅

C₂³×C₈

Frattini

C₂×C₄

C₂³×AGL(1,𝔽₅)

Fitting

C₂×C₅×#131

C₄

socle

C₂×C₅

C₂²×C₄⋊C₈

prime radical

C₂×C₅×#41

C₂×C₄

C₂⁴

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
3. #10604751	C₂	semidirect
4. C₂×#64889	C₂	semidirect
155. C₅	#33401	semidirect

H ≅ G/N

type

3. #10604751

C₂

semidirect

4. C₂×#64889

C₂

semidirect

155. C₅

#33401

semidirect

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

subgroup	index
#10617530	2
Dic₅×#131	2
#10604751	2
C₂×#64889	2
#33401	5

subgroup

index

Subgroups — #13043395

Conjugacy classes of subgroups

Complements

Maximal subgroups