Group #13026585 — subgroups

Subgroups — #13026585

Order 1280.

366 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 164 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	C₂×C₄⋊#674	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#10598777	C₂	1	4	yes	maximal
4	C₂²×#674	C₂²	1	4	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₅	C₄	1	2	yes	—
7	C₂×C₈×Dic₅	C₂²	1	2	yes	—
8	C₄²×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	C₄⋊#674	—	2	4	no	—
11	#12879	—	5	1	no	maximal, Sylow
12	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₈	1	2	yes	—
16	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
18	C₂²×#168	C₂×C₄	1	1	yes	characteristic
19	C₂²×#168	C₂³	1	1	yes	characteristic
20	C₂×#674	—	2	4	no	—
21	C₂×#674	—	2	8	no	—
22	C₄×C₅×C₈	—	2	2	no	—
23	C₈×Dic₅	—	2	4	no	—
24	C₈×Dic₅	—	2	4	no	—
25	C₄×#168	D₄	1	2	yes	—
26	C₄×#168	Q₈	1	2	yes	—
27	C₂×C₄⋊C₁₆	—	5	2	no	—
28	C₄²×C₈	—	5	1	no	—
29	#1374	—	5	4	no	—
30	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅	C₂×C₈	1	2	yes	—
34	C₂²×Dic₅	C₂×C₈	1	2	yes	—
35	C₂×C₅×C₈	—	2	2	no	—
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	M₁₆	1	4	yes	—
38	C₄²×C₅	C₄⋊C₄	1	2	yes	—
39	C₄²×C₅	M₁₆	1	2	yes	—
40	C₄×Dic₅	—	2	2	no	—
41	C₄×Dic₅	—	2	2	no	—
42	C₄×Dic₅	—	2	2	no	—
43	C₄×Dic₅	—	2	4	no	—
44	C₄×Dic₅	C₁₆	1	4	yes	complemented
45	C₂×#168	—	2	2	no	—
46	C₂×#168	C₄⋊C₄	1	2	yes	—
47	C₂×#168	C₂×D₄	1	1	yes	characteristic
48	C₂×#168	C₂×Q₈	1	1	yes	characteristic
49	C₂×#168	#41	1	4	yes	—
50	#674	—	4	4	no	—
51	C₂²×C₁₆	—	5	4	no	—
52	C₂×C₄×C₈	—	5	1	no	—
53	C₂×C₄×C₈	—	5	2	no	—
54	C₂×C₄×C₈	—	5	2	no	—
55	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₄³	—	5	1	no	—
57	C₄⋊C₁₆	—	10	4	no	—
58	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₅	M₃₂	1	2	yes	—
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅	#117	1	2	yes	—
66	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₅	—	2	2	no	—
69	C₂×Dic₅	—	2	2	no	—
70	C₂×Dic₅	—	2	2	no	—
71	C₂×Dic₅	—	2	2	no	—
72	C₂×Dic₅	C₂×C₁₆	1	4	yes	—
73	C₂×Dic₅	M₃₂	1	4	yes	—
74	C₅×C₈	—	2	4	no	—
75	#168	—	2	2	no	—
76	#168	#124	1	2	yes	—
77	#168	#125	1	2	yes	—
78	C₂²×C₈	—	5	1	no	—
79	C₂²×C₈	—	5	1	no	—
80	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
81	C₂×C₄²	—	5	1	no	—
82	C₂×C₄²	—	5	2	no	—
83	C₂×C₄²	—	5	1	no	—
84	C₂×C₄²	—	5	2	no	—
85	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₂×C₁₆	—	10	4	no	—
87	C₂×C₁₆	—	10	8	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	5	2	no	—
90	C₄×C₈	—	5	2	no	—
91	C₄×C₈	—	10	4	no	—
92	C₄×C₈	—	10	4	no	—
93	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₅	#431	1	2	yes	—
95	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
96	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
97	C₂²×C₅	C₂×M₃₂	1	2	yes	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	C₄⋊C₁₆	1	4	yes	—
100	C₄×C₅	#382	1	1	yes	characteristic
101	C₄×C₅	#423	1	1	yes	characteristic
102	C₄×C₅	#432	1	2	yes	—
103	Dic₅	—	2	4	no	—
104	Dic₅	—	2	4	no	—
105	C₂²×C₄	—	5	2	no	—
106	C₂²×C₄	—	5	2	no	—
107	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
109	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	5	1	no	—
113	C₂×C₈	—	5	1	no	—
114	C₂×C₈	—	5	4	no	—
115	C₂×C₈	—	5	2	no	—
116	C₂×C₈	—	10	2	no	—
117	C₂×C₈	#700	1	4	yes	—
118	C₄²	—	5	4	no	—
119	C₄²	—	10	4	no	—
120	C₄²	—	10	2	no	—
121	C₄²	—	10	2	no	—
122	C₄²	—	10	2	no	—
123	C₄²	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
124	C₄²	#704	1	2	yes	—
125	C₁₆	—	20	4	no	—
126	C₂×C₅	#1831	1	1	yes	characteristic
127	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
128	C₂×C₅	#2150	1	1	yes	characteristic
129	C₂×C₅	#2162	1	2	yes	—
130	C₂×C₅	#2163	1	2	yes	—
131	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	5	4	no	—
134	C₂×C₄	—	5	4	no	—
135	C₂×C₄	—	10	2	no	—
136	C₂×C₄	—	10	2	no	—
137	C₂×C₄	—	10	2	no	—
138	C₂×C₄	—	10	2	no	—
139	C₂×C₄	D₅⋊C₁₆	1	2	yes	—
140	C₂×C₄	#3990	1	2	yes	—
141	C₂×C₄	#4004	1	2	yes	—
142	C₂×C₄	#4004	1	2	yes	—
143	C₂×C₄	C₂×#700	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×C₄	#4130	1	2	yes	—
146	C₂×C₄	C₂×#704	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	5	2	no	—
149	C₈	—	5	2	no	—
150	C₈	—	10	2	no	—
151	C₅	#12879	1	1	yes	characteristic, complemented, Sylow
152	C₂²	#65680	1	2	yes	—
153	C₂²	C₂×D₅⋊C₁₆	1	1	yes	characteristic
154	C₂²	C₂×#3990	1	1	yes	characteristic
155	C₂²	D₅⋊M₃₂	1	2	yes	—
156	C₂²	C₂×#4004	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	10	4	no	—
159	C₄	—	10	4	no	—
160	C₄	#64888	1	4	yes	—
161	C₄	#65681	1	1	yes	characteristic
162	C₄	#65689	1	1	yes	characteristic
163	C₄	#65759	1	2	yes	—
164	C₂	#10603829	1	2	yes	—
165	C₂	#10603831	1	2	yes	—
166	C₂	#10604187	1	1	yes	characteristic
167	C₂	C₂×#64888	1	1	yes	characteristic
168	C₂	#10604708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#674

C₂

yes

maximal

C₄×C₈×Dic₅

C₂

yes

maximal, characteristic

#10598777

C₂

yes

maximal

C₂²×#674

C₂²

yes

—

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₈×Dic₅

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₄²×Dic₅

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

—

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₈

yes

characteristic

C₂×C₄×Dic₅

C₈

yes

—

C₂×C₄×Dic₅

C₈

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂³

yes

characteristic

C₂×#674

—

C₂×#674

—

C₄×C₅×C₈

—

C₈×Dic₅

—

C₈×Dic₅

—

C₄×#168

D₄

yes

—

C₄×#168

Q₈

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

—

#1374

—

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₈

yes

—

C₂²×Dic₅

C₂×C₈

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

M₁₆

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

M₁₆

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₁₆

yes

complemented

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

characteristic

C₂×#168

#41

yes

—

#674

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₂³×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

M₃₂

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×C₁₆

yes

—

C₂×Dic₅

M₃₂

yes

—

C₅×C₈

—

#168

—

#168

#124

yes

—

#168

#125

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅

#431

yes

—

C₂²×C₅

C₂²×C₁₆

yes

characteristic

C₂²×C₅

C₂×M₃₂

yes

characteristic

C₂²×C₅

C₂×M₃₂

yes

—

C₄×C₅

—

C₄×C₅

C₄⋊C₁₆

yes

—

100

C₄×C₅

#382

yes

characteristic

101

C₄×C₅

#423

yes

characteristic

102

C₄×C₅

#432

yes

—

103

Dic₅

—

104

Dic₅

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

D₅⋊C₈

yes

characteristic

108

C₂²×C₄

D₅⋊C₈

yes

characteristic

109

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

#700

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄⋊AGL(1,𝔽₅)

yes

—

124

C₄²

#704

yes

—

125

C₁₆

—

126

C₂×C₅

#1831

yes

characteristic

127

C₂×C₅

C₂×C₄⋊C₁₆

yes

characteristic

128

C₂×C₅

#2150

yes

characteristic

129

C₂×C₅

#2162

yes

—

130

C₂×C₅

#2163

yes

—

131

C₂³

C₂×D₅⋊C₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

D₅⋊C₁₆

yes

—

140

C₂×C₄

#3990

yes

—

141

C₂×C₄

#4004

yes

—

142

C₂×C₄

#4004

yes

—

143

C₂×C₄

C₂×#700

yes

—

144

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

145

C₂×C₄

#4130

yes

—

146

C₂×C₄

C₂×#704

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₅

#12879

yes

characteristic, complemented, Sylow

152

C₂²

#65680

yes

—

153

C₂²

C₂×D₅⋊C₁₆

yes

characteristic

154

C₂²

C₂×#3990

yes

characteristic

155

C₂²

D₅⋊M₃₂

yes

—

156

C₂²

C₂×#4004

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#64888

yes

—

161

C₄

#65681

yes

characteristic

162

C₄

#65689

yes

characteristic

163

C₄

#65759

yes

—

164

C₂

#10603829

yes

—

165

C₂

#10603831

yes

—

166

C₂

#10604187

yes

characteristic

167

C₂

C₂×#64888

yes

characteristic

168

C₂

#10604708

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₂×C₄×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×#168	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#12879
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

commutator

C₂²×C₅

C₂²×C₁₆

Fitting

C₂×C₄×C₅×C₈

C₄

socle, prime radical

C₂³×C₅

C₂²×C₈

cosocle

C₂²×#168

C₂³

C₂×C₄×C₈

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
44. C₄×Dic₅	C₁₆	semidirect
151. C₅	#12879	semidirect

H ≅ G/N

type

44. C₄×Dic₅

C₁₆

semidirect

151. C₅

#12879

semidirect

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

subgroup	index
C₂×C₄⋊#674	2
C₄×C₈×Dic₅	2
#10598777	2
#12879	5

subgroup

index

C₂×C₄⋊#674

C₄×C₈×Dic₅

#10598777

#12879

Subgroups — #13026585

Conjugacy classes of subgroups

Complements

Maximal subgroups