Group #13015256 — subgroups

Subgroups — #13015256

Order 1280.

526 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 228 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×#170⋊C₈	C₂	1	4	yes	maximal
2	#10604173	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
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	C₄	1	8	yes	—
11	#170⋊C₈	C₄	1	16	yes	complemented
12	#7843	—	5	1	no	maximal, Sylow
13	C₂²×#170	C₂×C₄	1	8	yes	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
17	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
19	C₂²×#168	C₂×C₄	1	2	yes	—
20	C₄×#170	C₂×C₄	1	16	yes	—
21	C₄×C₅×C₈	—	2	2	no	—
22	C₈×Dic₅	—	2	4	no	—
23	C₈×Dic₅	C₂×C₄	1	8	yes	—
24	C₄×#168	—	2	2	no	—
25	#1744	—	5	2	no	—
26	C₄²×C₈	—	5	1	no	—
27	C₂×#322	—	5	4	no	—
28	C₂²×C₄×C₅	C₄²	1	2	yes	—
29	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₅	C₄²	1	2	yes	—
31	C₂²×Dic₅	C₂²×C₄	1	2	yes	—
32	C₂×#170	—	2	16	no	—
33	C₂×#170	C₄²	1	16	yes	—
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	—	2	2	no	—
36	C₂×C₅×C₈	C₄²	1	4	yes	—
37	C₄²×C₅	M₁₆	1	2	yes	—
38	C₄²×C₅	M₁₆	1	2	yes	—
39	C₄×Dic₅	—	2	8	no	—
40	C₄×Dic₅	M₁₆	1	4	yes	—
41	C₄×Dic₅	C₂²×C₄	1	4	yes	—
42	C₂×#168	—	2	2	no	—
43	C₂×#168	—	2	2	no	—
44	C₂×#168	C₄²	1	4	yes	—
45	C₂×C₄×C₈	—	5	4	no	—
46	C₂×C₄×C₈	—	5	2	no	—
47	C₂×C₄×C₈	—	5	2	no	—
48	C₂×C₄×C₈	—	5	1	no	—
49	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄³	—	5	1	no	—
51	#336	—	5	8	no	—
52	#322	—	5	16	no	—
53	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅	—	2	4	no	—
55	C₂×C₄×C₅	#97	1	4	yes	—
56	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
60	C₂×Dic₅	—	2	4	no	—
61	C₂×Dic₅	#97	1	4	yes	—
62	C₂×Dic₅	C₂×C₄²	1	4	yes	—
63	C₂×Dic₅	#131	1	8	yes	—
64	#170	—	2	16	no	—
65	C₅×C₈	—	2	4	no	—
66	#168	—	2	4	no	—
67	C₂²×C₈	—	5	8	no	—
68	C₂²×C₈	—	5	2	no	—
69	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
70	C₂×C₄²	—	5	1	no	—
71	C₂×C₄²	—	5	1	no	—
72	C₂×C₄²	—	5	4	no	—
73	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	5	16	no	—
76	C₄×C₈	—	5	8	no	—
77	C₄×C₈	—	10	4	no	—
78	C₄×C₈	—	10	2	no	—
79	C₂²×C₅	C₄³	1	1	yes	characteristic
80	C₂²×C₅	C₂×#97	1	1	yes	characteristic
81	C₂²×C₅	C₂×#97	1	1	yes	characteristic
82	C₂²×C₅	#405	1	4	yes	—
83	C₄×C₅	—	2	4	no	—
84	C₄×C₅	C₄×M₁₆	1	2	yes	—
85	C₄×C₅	C₄×M₁₆	1	2	yes	—
86	Dic₅	—	2	4	no	—
87	Dic₅	#405	1	8	yes	—
88	C₂²×C₄	—	5	2	no	—
89	C₂²×C₄	—	5	2	no	—
90	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
91	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	5	16	no	—
95	C₂×C₈	—	5	4	no	—
96	C₂×C₈	—	10	16	no	—
97	C₂×C₈	—	10	2	no	—
98	C₂×C₈	—	10	2	no	—
99	C₂×C₈	C₄×AGL(1,𝔽₅)	1	4	yes	—
100	C₄²	—	5	4	no	—
101	C₄²	—	5	4	no	—
102	C₄²	—	10	8	no	—
103	C₄²	#700	1	2	yes	—
104	C₄²	#704	1	2	yes	—
105	C₂×C₅	#1452	1	2	yes	—
106	C₂×C₅	#1452	1	2	yes	—
107	C₂×C₅	C₄×#97	1	1	yes	characteristic
108	C₂×C₅	#1726	1	2	yes	—
109	C₂³	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	5	4	no	—
112	C₂×C₄	—	5	8	no	—
113	C₂×C₄	—	5	4	no	—
114	C₂×C₄	—	10	4	no	—
115	C₂×C₄	#4002	1	4	yes	—
116	C₂×C₄	C₂×#700	1	1	yes	characteristic
117	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
118	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
119	C₂×C₄	C₂×#704	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	10	16	no	—
122	C₈	—	10	4	no	—
123	C₅	#7843	1	1	yes	characteristic, complemented, Sylow
124	C₂²	#65676	1	1	yes	characteristic
125	C₂²	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂²	#65718	1	4	yes	—
127	C₂²	C₂×#4002	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	—	5	8	no	—
130	C₄	—	10	4	no	—
131	C₄	C₄×#700	1	2	yes	—
132	C₄	C₄×#704	1	2	yes	—
133	C₂	#10603811	1	2	yes	—
134	C₂	#10603838	1	2	yes	—
135	C₂	C₄×#4002	1	1	yes	characteristic
136	C₂	#10604689	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#170⋊C₈

C₂

yes

maximal

#10604173

C₂

yes

maximal

C₄×C₈×Dic₅

C₂

yes

maximal, characteristic

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

characteristic

#64918

C₄

yes

—

#170⋊C₈

C₄

yes

complemented

#7843

—

maximal, Sylow

C₂²×#170

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂×C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₄×#170

C₂×C₄

yes

—

C₄×C₅×C₈

—

C₈×Dic₅

—

C₈×Dic₅

C₂×C₄

yes

—

C₄×#168

—

#1744

—

C₄²×C₈

—

C₂×#322

—

C₂²×C₄×C₅

C₄²

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₄²

yes

—

C₂²×Dic₅

C₂²×C₄

yes

—

C₂×#170

—

C₂×#170

C₄²

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

—

C₄²×C₅

M₁₆

yes

—

C₄²×C₅

M₁₆

yes

—

C₄×Dic₅

—

C₄×Dic₅

M₁₆

yes

—

C₄×Dic₅

C₂²×C₄

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₄²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄³

—

#336

—

#322

—

C₂³×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#97

yes

—

C₂×C₄×C₅

C₂×C₄²

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

#97

yes

—

C₂×Dic₅

C₂×C₄²

yes

—

C₂×Dic₅

#131

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×AGL(1,𝔽₅)

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₄³

yes

characteristic

C₂²×C₅

C₂×#97

yes

characteristic

C₂²×C₅

C₂×#97

yes

characteristic

C₂²×C₅

#405

yes

—

C₄×C₅

—

C₄×C₅

C₄×M₁₆

yes

—

C₄×C₅

C₄×M₁₆

yes

—

Dic₅

—

Dic₅

#405

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×AGL(1,𝔽₅)

yes

—

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

#700

yes

—

104

C₄²

#704

yes

—

105

C₂×C₅

#1452

yes

—

106

C₂×C₅

#1452

yes

—

107

C₂×C₅

C₄×#97

yes

characteristic

108

C₂×C₅

#1726

yes

—

109

C₂³

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#4002

yes

—

116

C₂×C₄

C₂×#700

yes

characteristic

117

C₂×C₄

D₅⋊M₁₆

yes

—

118

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

—

119

C₂×C₄

C₂×#704

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₅

#7843

yes

characteristic, complemented, Sylow

124

C₂²

#65676

yes

characteristic

125

C₂²

C₄²×AGL(1,𝔽₅)

yes

characteristic

126

C₂²

#65718

yes

—

127

C₂²

C₂×#4002

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₄×#700

yes

—

132

C₄

C₄×#704

yes

—

133

C₂

#10603811

yes

—

134

C₂

#10603838

yes

—

135

C₂

C₄×#4002

yes

characteristic

136

C₂

#10604689

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₄³
Fitting	C₂×C₄×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂×C₄²
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#7843
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

C₄³

Fitting

C₂×C₄×C₅×C₈

C₄

socle, prime radical

C₂³×C₅

C₂×C₄²

cosocle

C₂×C₄×Dic₅

C₂³

C₂×C₄×C₈

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. #170⋊C₈	C₄	semidirect
123. C₅	#7843	semidirect

H ≅ G/N

type

11. #170⋊C₈

C₄

semidirect

123. C₅

#7843

semidirect

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

subgroup	index
C₂×#170⋊C₈	2
#10604173	2
C₄×C₈×Dic₅	2
#7843	5

subgroup

index

C₂×#170⋊C₈

#10604173

C₄×C₈×Dic₅

#7843

Subgroups — #13015256

Conjugacy classes of subgroups

Complements

Maximal subgroups