Group #13018739 — subgroups

Subgroups — #13018739

Order 1280.

382 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 152 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	#10604185	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#10598771	C₂	1	4	yes	maximal
4	C₂×C₄⋊#170	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	C₂²	1	1	yes	characteristic
9	C₂×C₄×#168	C₄	1	1	yes	characteristic
10	#64918	—	2	4	no	—
11	#8785	—	5	1	no	maximal, Sylow
12	C₂²×#170	—	2	4	no	—
13	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅	D₄	1	2	yes	—
17	C₂×C₄×Dic₅	Q₈	1	2	yes	—
18	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
19	C₂²×#168	C₂×C₄	1	2	yes	—
20	C₄⋊#170	—	2	8	no	—
21	C₄×C₅×C₈	C₈	1	4	yes	complemented
22	C₈×Dic₅	—	2	4	no	—
23	C₈×Dic₅	—	2	4	no	—
24	C₄×#168	—	2	2	no	—
25	#1831	—	5	2	no	—
26	C₄²×C₈	—	5	1	no	—
27	#1277	—	5	4	no	—
28	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
31	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
33	C₂×#170	—	4	4	no	—
34	C₂×#170	—	4	8	no	—
35	C₂×C₅×C₈	—	2	2	no	—
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
38	C₄²×C₅	C₂×C₈	1	2	yes	—
39	C₄²×C₅	M₁₆	1	2	yes	—
40	C₄×Dic₅	—	2	2	no	—
41	C₄×Dic₅	—	2	4	no	—
42	C₄×Dic₅	—	2	4	no	—
43	C₄×Dic₅	#41	1	4	yes	—
44	C₂×#168	—	2	2	no	—
45	C₂×#168	—	2	2	no	—
46	C₂×#168	M₁₆	1	4	yes	—
47	C₂×C₄⋊C₈	—	5	4	no	—
48	C₂×C₄×C₈	—	5	2	no	—
49	C₂×C₄×C₈	—	5	2	no	—
50	C₂×C₄×C₈	—	5	1	no	—
51	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₄³	—	5	1	no	—
53	#336	—	10	4	no	—
54	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₅	#117	1	2	yes	—
59	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
62	C₂×Dic₅	—	2	2	no	—
63	C₂×Dic₅	—	2	2	no	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	#108	1	4	yes	—
66	C₂×Dic₅	#124	1	2	yes	—
67	C₂×Dic₅	#125	1	2	yes	—
68	C₂×Dic₅	#135	1	4	yes	—
69	#170	—	8	4	no	—
70	C₅×C₈	—	2	4	no	—
71	#168	—	2	4	no	—
72	C₂²×C₈	—	5	2	no	—
73	C₂²×C₈	—	10	4	no	—
74	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
75	C₂×C₄²	—	5	1	no	—
76	C₂×C₄²	—	5	2	no	—
77	C₂×C₄²	—	5	2	no	—
78	C₂×C₄²	—	5	1	no	—
79	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₄⋊C₈	—	10	8	no	—
81	C₄×C₈	—	10	4	no	—
82	C₄×C₈	—	10	4	no	—
83	C₄×C₈	—	10	2	no	—
84	C₄×C₈	#170	1	4	yes	complemented
85	C₂²×C₅	#382	1	1	yes	characteristic
86	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₅	#423	1	1	yes	characteristic
88	C₂²×C₅	#427	1	2	yes	—
89	C₂²×C₅	C₂×#108	1	2	yes	—
90	C₄×C₅	—	2	4	no	—
91	C₄×C₅	#431	1	2	yes	—
92	C₄×C₅	#432	1	2	yes	—
93	Dic₅	—	2	4	no	—
94	Dic₅	—	2	2	no	—
95	Dic₅	#488	1	4	yes	—
96	C₂²×C₄	—	5	1	no	—
97	C₂²×C₄	—	5	2	no	—
98	C₂²×C₄	—	5	1	no	—
99	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
100	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	5	4	no	—
104	C₂×C₈	—	10	2	no	—
105	C₂×C₈	—	10	2	no	—
106	C₂×C₈	—	20	4	no	—
107	C₂×C₈	—	20	8	no	—
108	C₂×C₈	C₂×#170	1	4	yes	—
109	C₄²	—	5	4	no	—
110	C₄²	—	10	4	no	—
111	C₄²	—	10	4	no	—
112	C₄²	—	10	2	no	—
113	C₄²	#700	1	2	yes	—
114	C₄²	C₂×#170	1	2	yes	—
115	C₂×C₅	#1564	1	2	yes	—
116	C₂×C₅	#1568	1	2	yes	—
117	C₂×C₅	#1831	1	1	yes	characteristic
118	C₂×C₅	#1835	1	1	yes	characteristic
119	C₂×C₅	#1837	1	1	yes	characteristic
120	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	5	2	no	—
124	C₂×C₄	—	5	4	no	—
125	C₂×C₄	—	5	2	no	—
126	C₂×C₄	—	5	4	no	—
127	C₂×C₄	—	10	2	no	—
128	C₂×C₄	—	10	2	no	—
129	C₂×C₄	—	10	2	no	—
130	C₂×C₄	C₄⋊#170	1	4	yes	—
131	C₂×C₄	C₂×#700	1	1	yes	characteristic
132	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
133	C₂×C₄	#4130	1	2	yes	—
134	C₂×C₄	C₂²×#170	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	10	4	no	—
137	C₈	—	40	4	no	—
138	C₅	#8785	1	1	yes	characteristic, complemented, Sylow
139	C₂²	#65682	1	1	yes	characteristic
140	C₂²	#65689	1	1	yes	characteristic
141	C₂²	#65721	1	2	yes	—
142	C₂²	D₅⋊#108	1	2	yes	—
143	C₂²	C₂×C₄⋊#170	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	—	5	4	no	—
146	C₄	—	10	4	no	—
147	C₄	—	10	2	no	—
148	C₄	#65683	1	2	yes	—
149	C₄	#65748	1	2	yes	—
150	C₂	#10603821	1	2	yes	—
151	C₂	#10603846	1	1	yes	characteristic
152	C₂	#10603857	1	1	yes	characteristic
153	C₂	#10604185	1	1	yes	characteristic
154	C₂	#10604694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10604185

C₂

yes

maximal

C₄×C₈×Dic₅

C₂

yes

maximal, characteristic

#10598771

C₂

yes

maximal

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₈×Dic₅

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₄²×Dic₅

C₂²

yes

characteristic

C₂×C₄×#168

C₄

yes

—

—

C₂²×#170

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

D₄

yes

—

C₂×C₄×Dic₅

Q₈

yes

—

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₄⋊#170

—

C₄×C₅×C₈

C₈

yes

complemented

C₈×Dic₅

—

C₈×Dic₅

—

C₄×#168

—

#1831

—

C₄²×C₈

—

#1277

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₄⋊C₄

yes

—

C₂²×Dic₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

C₂×Q₈

yes

characteristic

C₂×#170

—

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₄²×C₅

C₂×C₈

yes

—

C₄²×C₅

M₁₆

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

M₁₆

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄³

—

#336

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#108

yes

—

C₂×Dic₅

#124

yes

—

C₂×Dic₅

#125

yes

—

C₂×Dic₅

#135

yes

—

#170

—

C₅×C₈

—

#168

—

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

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂²×C₅

yes

C₂²×C₅

C₂×C₄⋊C₈

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

C₂×#108

yes

—

C₄×C₅

—

C₄×C₅

#431

yes

—

C₄×C₅

#432

yes

—

Dic₅

—

Dic₅

—

Dic₅

#488

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

100

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×#170

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

#700

yes

—

114

C₄²

C₂×#170

yes

—

115

C₂×C₅

#1564

yes

—

116

C₂×C₅

#1568

yes

—

117

C₂×C₅

#1831

yes

characteristic

118

C₂×C₅

#1835

yes

characteristic

119

C₂×C₅

#1837

yes

characteristic

120

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₄⋊#170

yes

—

131

C₂×C₄

C₂×#700

yes

characteristic

132

C₂×C₄

D₅⋊M₁₆

yes

—

133

C₂×C₄

#4130

yes

—

134

C₂×C₄

C₂²×#170

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₅

#8785

yes

characteristic, complemented, Sylow

139

C₂²

#65682

yes

characteristic

140

C₂²

#65689

yes

characteristic

141

C₂²

#65721

yes

—

142

C₂²

D₅⋊#108

yes

—

143

C₂²

C₂×C₄⋊#170

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#65683

yes

—

149

C₄

#65748

yes

—

150

C₂

#10603821

yes

—

151

C₂

#10603846

yes

characteristic

152

C₂

#10603857

yes

characteristic

153

C₂

#10604185

yes

characteristic

154

C₂

#10604694

yes

—

	subgroup	G/H
center	C₂³	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#8785
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅×C₈

C₄

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅

C₂³

C₂×C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
21. C₄×C₅×C₈	C₈	semidirect
84. C₄×C₈	#170	semidirect
138. C₅	#8785	semidirect

H ≅ G/N

type

21. C₄×C₅×C₈

C₈

semidirect

84. C₄×C₈

#170

semidirect

138. C₅

#8785

semidirect

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

subgroup	index
#10604185	2
C₄×C₈×Dic₅	2
#10598771	2
#8785	5

subgroup

index

#10604185

C₄×C₈×Dic₅

#10598771

#8785

Subgroups — #13018739

Conjugacy classes of subgroups

Complements

Maximal subgroups