Group #13013653 — subgroups

Subgroups — #13013653

Order 1280.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	#10604152	C₂	1	2	yes	maximal
2	C₂×#64674	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
4	C₂×C₄×#168	C₄	1	1	yes	characteristic
5	C₂×C₄×#168	C₂²	1	1	yes	characteristic
6	C₂²⋊#674	C₄	1	4	yes	—
7	#64859	—	2	2	no	—
8	C₄⋊#674	—	2	2	no	—
9	#64674	—	2	2	no	—
10	#7196	—	5	1	no	maximal, Sylow
11	C₂²×C₅×C₈	—	2	1	no	—
12	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×#168	—	2	1	no	—
14	C₂²×#168	C₂×C₄	1	2	yes	—
15	C₂×#674	—	2	4	no	—
16	C₄×C₅×C₈	—	2	1	no	—
17	C₄×C₅×C₈	C₈	1	2	yes	—
18	C₄×#168	—	2	1	no	—
19	C₄×#168	C₈	1	2	yes	—
20	C₄×#168	D₄	1	1	yes	characteristic
21	C₄×#168	D₄	1	2	yes	—
22	C₄×#168	Q₈	1	1	yes	characteristic
23	#2163	—	5	2	no	—
24	C₂×#322	—	5	1	no	—
25	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
27	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
28	C₂×C₅×C₈	—	2	2	no	—
29	C₂×C₅×C₈	—	4	2	no	—
30	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
32	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
34	C₂×#168	—	2	4	no	—
35	C₂×#168	—	2	2	no	—
36	C₂×#168	—	4	2	no	—
37	C₂×#168	C₂²⋊C₄	1	2	yes	—
38	C₂×#168	C₄⋊C₄	1	2	yes	—
39	#674	—	4	4	no	—
40	C₂×C₄×C₈	—	5	1	no	—
41	C₂×C₄×C₈	—	5	1	no	—
42	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₄⋊C₁₆	—	10	2	no	—
44	C₂²⋊C₁₆	—	5	4	no	—
45	#346	—	10	2	no	—
46	#322	—	10	2	no	—
47	C₂³×C₅	#97	1	1	yes	characteristic
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	—	2	2	no	—
50	C₂×C₄×C₅	—	2	2	no	—
51	C₂×C₄×C₅	#95	1	1	yes	characteristic
52	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
57	C₅×C₈	—	4	2	no	—
58	#168	—	2	4	no	—
59	#168	—	2	2	no	—
60	#168	—	2	2	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	5	2	no	—
63	C₂²×C₈	—	10	1	no	—
64	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₁₆	—	10	4	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	5	2	no	—
68	C₄×C₈	—	5	1	no	—
69	C₄×C₈	—	5	1	no	—
70	C₄×C₈	—	5	2	no	—
71	C₄×C₈	—	10	1	no	—
72	C₄×C₈	#170	1	2	yes	—
73	C₂²×C₅	—	2	1	no	—
74	C₂²×C₅	—	2	1	no	—
75	C₂²×C₅	#336	1	1	yes	characteristic
76	C₂²×C₅	#347	1	2	yes	—
77	C₄×C₅	—	2	1	no	—
78	C₄×C₅	—	2	1	no	—
79	C₄×C₅	—	2	1	no	—
80	C₄×C₅	—	2	1	no	—
81	C₄×C₅	#344	1	2	yes	—
82	C₄×C₅	#364	1	2	yes	—
83	C₂²×C₄	#700	1	1	yes	characteristic
84	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₂²×C₄	#704	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	5	2	no	—
89	C₂×C₈	—	5	2	no	—
90	C₂×C₈	—	10	4	no	—
91	C₂×C₈	—	10	2	no	—
92	C₂×C₈	—	20	2	no	—
93	C₄²	D₅⋊C₈	1	1	yes	characteristic
94	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₄²	C₂×#170	1	1	yes	characteristic
96	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₁₆	—	20	4	no	—
98	C₂×C₅	—	2	2	no	—
99	C₂×C₅	#1281	1	1	yes	characteristic
100	C₂×C₅	#1375	1	1	yes	characteristic
101	C₂×C₅	#1376	1	1	yes	characteristic
102	C₂³	#4002	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	C₄×#170	1	1	yes	characteristic
107	C₂×C₄	C₄⋊#170	1	1	yes	characteristic
108	C₂×C₄	#4003	1	1	yes	characteristic
109	C₂×C₄	#4004	1	1	yes	characteristic
110	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊#170	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	10	4	no	—
114	C₈	—	10	2	no	—
115	C₈	—	10	2	no	—
116	C₅	#7196	1	1	yes	characteristic, complemented, Sylow
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#64860	1	2	yes	—
120	C₂²	#64918	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#64889	1	2	yes	—
126	C₄	#64894	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#10598699	1	1	yes	characteristic
129	C₂	#10598774	1	1	yes	characteristic
130	C₂	#10598779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604152

C₂

yes

maximal

C₂×#64674

C₂

yes

maximal, characteristic

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₄×#168

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

C₂²⋊#674

C₄

yes

—

—

—

—

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂²×#168

—

C₂²×#168

C₂×C₄

yes

—

C₂×#674

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₈

yes

—

C₄×#168

—

C₄×#168

C₈

yes

—

C₄×#168

D₄

yes

characteristic

C₄×#168

D₄

yes

—

C₄×#168

Q₈

yes

characteristic

#2163

—

C₂×#322

—

C₂²×C₄×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

—

—

C₂²⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

#674

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₂³×C₅

#97

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#95

yes

characteristic

C₂×C₄×C₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₅×C₈

—

#168

—

#168

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#170

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#336

yes

characteristic

C₂²×C₅

#347

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#344

yes

—

C₄×C₅

#364

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₅⋊C₈

yes

C₄²

yes

C₄²

yes

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

#1281

yes

characteristic

100

C₂×C₅

#1375

yes

characteristic

101

C₂×C₅

#1376

yes

characteristic

102

C₂³

#4002

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×#170

yes

characteristic

107

C₂×C₄

C₄⋊#170

yes

characteristic

108

C₂×C₄

#4003

yes

characteristic

109

C₂×C₄

#4004

yes

characteristic

110

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

111

C₂×C₄

C₂²⋊#170

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₅

#7196

yes

characteristic, complemented, Sylow

117

C₂²

—

118

C₂²

—

119

C₂²

#64860

yes

—

120

C₂²

#64918

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#64889

yes

—

126

C₄

#64894

yes

—

127

C₂

—

128

C₂

#10598699

yes

characteristic

129

C₂

#10598774

yes

characteristic

130

C₂

#10598779

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄
commutator	C₂×C₄×C₅	C₄×C₈
Frattini	C₂×C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	#97
cosocle	C₂×C₄×#168	C₂²
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#7196
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

commutator

C₂×C₄×C₅

C₄×C₈

Frattini

C₂×C₄²

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅×C₈

C₄

C₂²×C₅

C₂³×C₅

C₂×C₄×#168

C₂²

C₂×C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
116. C₅	#7196	semidirect

H ≅ G/N

type

116. C₅

#7196

semidirect

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

subgroup	index
#10604152	2
C₂×#64674	2
#7196	5

subgroup

index

#10604152

C₂×#64674

#7196

Subgroups — #13013653

Conjugacy classes of subgroups

Complements

Maximal subgroups