Group #13013818 — subgroups

Subgroups — #13013818

Order 1280.

200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 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	#10604697	C₂	1	2	yes	maximal
2	#10602213	C₂	1	1	yes	maximal, characteristic
3	#65758	—	2	2	no	—
4	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
5	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
6	C₂×#3944	C₄	1	1	yes	characteristic
7	#64884	C₄	1	4	yes	—
8	#64882	—	2	2	no	—
9	#64721	—	2	2	no	—
10	#7245	—	5	1	no	maximal, Sylow
11	C₂×#704	—	4	2	no	—
12	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
14	C₂²×#168	Q₈	1	1	yes	characteristic
15	#4004	—	4	2	no	—
16	C₄⋊#170	—	2	4	no	—
17	C₅×M₃₂	—	4	1	no	—
18	C₂×C₅×C₁₆	—	2	1	no	—
19	C₈×Dic₅	C₂×C₄	1	2	yes	—
20	C₈×Dic₅	D₄	1	2	yes	—
21	#3944	—	4	1	no	—
22	C₂×#672	—	2	1	no	—
23	#2150	—	5	1	no	—
24	#1567	—	5	2	no	—
25	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×Dic₅	SD₁₆	1	2	yes	—
27	#704	—	8	4	no	—
28	C₂×#170	—	4	4	no	—
29	C₂×C₅×C₈	—	2	1	no	—
30	C₂×C₅×C₈	—	2	1	no	—
31	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
32	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
34	C₄×Dic₅	D₈	1	1	yes	characteristic
35	C₄×Dic₅	Q₁₆	1	1	yes	characteristic
36	C₂×#168	—	4	1	no	—
37	C₂×#168	C₄⋊C₄	1	2	yes	—
38	C₅×C₁₆	—	4	1	no	—
39	#672	—	4	1	no	—
40	C₂×M₃₂	—	5	1	no	—
41	C₂×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
42	#423	—	10	2	no	—
43	C₂×C₄×C₈	—	5	1	no	—
44	C₄⋊C₁₆	—	10	2	no	—
45	#335	—	5	4	no	—
46	#335	—	10	2	no	—
47	C₂³×C₅	#106	1	1	yes	characteristic
48	C₂×C₄×C₅	—	2	1	no	—
49	C₂×C₄×C₅	—	2	1	no	—
50	C₂×C₄×C₅	#95	1	1	yes	characteristic
51	C₂×C₄×C₅	#107	1	1	yes	characteristic
52	C₂×Dic₅	—	2	2	no	—
53	C₂×Dic₅	—	2	2	no	—
54	C₂×Dic₅	#102	1	2	yes	—
55	C₂×Dic₅	#103	1	2	yes	—
56	#170	—	8	4	no	—
57	C₅×C₈	—	2	1	no	—
58	C₅×C₈	#100	1	1	yes	characteristic
59	C₅×C₈	#101	1	1	yes	characteristic
60	#168	—	2	2	no	—
61	C₂×M₁₆	—	20	2	no	—
62	C₂²×C₈	—	5	1	no	—
63	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂×C₄²	—	5	1	no	—
65	M₃₂	—	4	1	no	—
66	M₃₂	—	20	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	—	10	1	no	—
69	C₄⋊C₈	—	10	4	no	—
70	C₄⋊C₈	—	20	2	no	—
71	C₄×C₈	—	5	2	no	—
72	C₄×C₈	—	5	2	no	—
73	C₂²×C₅	—	2	1	no	—
74	C₂²×C₅	—	2	2	no	—
75	C₂²×C₅	#340	1	1	yes	characteristic
76	C₄×C₅	—	2	1	no	—
77	C₄×C₅	#338	1	1	yes	characteristic
78	C₄×C₅	#343	1	1	yes	characteristic
79	Dic₅	—	2	2	no	—
80	Dic₅	#361	1	2	yes	—
81	Dic₅	#362	1	2	yes	—
82	C₂²×C₄	—	5	2	no	—
83	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
84	M₁₆	—	40	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	5	2	no	—
88	C₂×C₈	—	20	1	no	—
89	C₂×C₈	—	20	4	no	—
90	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₄²	—	5	1	no	—
93	C₄²	—	5	1	no	—
94	C₄²	—	5	2	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	—	20	1	no	—
97	C₂×C₅	—	4	1	no	—
98	C₂×C₅	#1292	1	1	yes	characteristic
99	C₂×C₅	#1387	1	2	yes	—
100	C₂³	#3993	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	—	5	2	no	—
105	C₂×C₄	—	10	2	no	—
106	C₂×C₄	—	10	2	no	—
107	C₂×C₄	#3994	1	1	yes	characteristic
108	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	10	2	no	—
111	C₈	—	40	4	no	—
112	C₈	#4005	1	1	yes	characteristic
113	C₈	#4013	1	1	yes	characteristic
114	C₅	#7245	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	#64893	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	5	2	no	—
120	C₄	—	5	2	no	—
121	C₄	—	10	2	no	—
122	C₄	#64897	1	1	yes	characteristic
123	C₄	#64919	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	#10598674	1	2	yes	—
126	C₂	#10598772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604697

C₂

yes

maximal

#10602213

C₂

yes

maximal, characteristic

#65758

—

C₂×C₅×M₃₂

C₄

yes

characteristic

C₂×C₈×Dic₅

C₂²

yes

characteristic

C₂×#3944

C₄

yes

characteristic

#64884

C₄

yes

—

—

—

—

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

D₄

yes

characteristic

C₂²×#168

Q₈

yes

characteristic

#4004

—

C₄⋊#170

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₈×Dic₅

C₂×C₄

yes

—

C₈×Dic₅

D₄

yes

—

—

—

—

—

C₂²×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂²×Dic₅

SD₁₆

yes

—

#704

—

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₄×Dic₅

C₂²⋊C₄

yes

—

C₄×Dic₅

D₈

yes

characteristic

C₄×Dic₅

Q₁₆

yes

characteristic

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂×M₃₂

AGL(1,𝔽₅)

yes

characteristic

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

#335

—

#335

—

C₂³×C₅

#106

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#102

yes

—

C₂×Dic₅

#103

yes

—

#170

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#340

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

#361

yes

—

Dic₅

#362

yes

—

C₂²×C₄

—

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

#1292

yes

characteristic

C₂×C₅

#1387

yes

—

100

C₂³

#3993

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#3994

yes

characteristic

108

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

#4005

yes

characteristic

113

C₈

#4013

yes

characteristic

114

C₅

#7245

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

—

117

C₂²

#64893

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#64897

yes

characteristic

123

C₄

#64919

yes

characteristic

124

C₂

—

125

C₂

#10598674

yes

—

126

C₂

#10598772

yes

characteristic

	subgroup	G/H
center	C₂²	#64893
commutator	C₂×C₅×C₈	C₄²
Frattini	C₂²×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₃₂	C₄
socle	C₂²×C₅	#340
prime radical	C₂³×C₅	#106
cosocle	C₂×C₈×Dic₅	C₂²
hypercenter	C₂×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#7245
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#64893

commutator

C₂×C₅×C₈

C₄²

Frattini

C₂²×C₈

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₅×M₃₂

C₄

C₂²×C₅

C₂³×C₅

C₂×C₈×Dic₅

C₂²

C₂×M₃₂

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
114. C₅	#7245	semidirect

H ≅ G/N

type

114. C₅

#7245

semidirect

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

subgroup	index
#10604697	2
#10602213	2
#7245	5

subgroup

index

#10604697

#10602213

#7245

Subgroups — #13013818

Conjugacy classes of subgroups

Complements

Maximal subgroups