Group #13085495 — subgroups

Subgroups — #13085495

Order 1440.

190 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 45 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×#67141	C₂	1	1	yes	maximal, characteristic
2	C₃×#6762	C₂	1	2	yes	maximal
3	#91175	—	3	2	no	maximal
4	C₂×C₃⋊Dic₁₅	C₄	1	1	yes	characteristic
5	C₂×C₅×C₃⋊Dic₃	C₄	1	1	yes	characteristic
6	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	#67141	—	2	2	no	—
8	C₃×C₃⋊#170	—	2	2	no	—
9	#63561	—	5	1	no	maximal, Hall
10	C₂×#1204	—	3	2	no	—
11	C₂×#1204	—	6	1	no	—
12	#6762	D₃	1	2	yes	—
13	C₃×#704	—	3	2	no	—
14	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₃⋊Dic₁₅	—	2	1	no	—
16	C₅×C₃⋊Dic₃	—	2	1	no	—
17	C₃²×Dic₅	D₄	1	2	yes	—
18	#4005	—	9	1	no	Hall
19	C₂×#634	—	5	1	no	—
20	C₃×#208	—	5	2	no	—
21	C₂×Dic₁₅	—	3	2	no	—
22	C₂×Dic₁₅	—	6	1	no	—
23	C₂×C₅×Dic₃	—	3	2	no	—
24	C₂×C₅×Dic₃	—	6	1	no	—
25	C₂×C₃×Dic₅	—	2	1	no	—
26	C₂×C₃×Dic₅	C₂×D₃	1	2	yes	—
27	#1204	—	6	4	no	—
28	#1204	—	6	2	no	—
29	#1204	—	6	2	no	—
30	C₃⋊#170	—	2	2	no	—
31	C₃×#170	—	6	2	no	—
32	#824	—	15	2	no	—
33	C₂×C₃²×C₅	—	2	1	no	—
34	C₂×C₃²×C₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀	—	9	1	no	—
36	#704	—	3	2	no	—
37	C₂×C₃²×C₄	—	5	1	no	—
38	C₂×C₃⋊Dic₃	—	5	1	no	—
39	C₂×C₃⋊Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
40	#634	—	10	2	no	—
41	C₃×C₃⋊C₈	—	10	2	no	—
42	C₂²×C₃×C₅	—	2	1	no	—
43	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
44	Dic₁₅	—	6	2	no	—
45	Dic₁₅	—	6	2	no	—
46	C₃×Dic₅	—	2	2	no	—
47	C₃×Dic₅	D₁₂	1	2	yes	—
48	C₃×Dic₅	#67	1	2	yes	—
49	C₅×Dic₃	—	6	2	no	—
50	C₅×Dic₃	—	6	2	no	—
51	C₂×Dic₆	—	15	2	no	—
52	C₂×Dic₆	—	30	1	no	—
53	C₃×M₁₆	—	15	2	no	—
54	#208	—	5	2	no	—
55	C₃²×C₅	#101	1	1	yes	characteristic, complemented, Hall
56	C₂×C₄×C₅	—	9	1	no	—
57	C₂×Dic₅	—	9	1	no	—
58	C₂×Dic₅	D₃²	1	1	yes	characteristic
59	Dic₁₀	—	18	2	no	—
60	#170	—	6	2	no	—
61	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₃²×C₄	—	5	2	no	—
63	C₃⋊Dic₃	—	2	1	no	—
64	C₃⋊Dic₃	—	10	1	no	—
65	#101	—	45	1	no	Sylow
66	C₂×C₃×C₅	—	2	2	no	—
67	C₂×C₃×C₅	—	2	2	no	—
68	C₂×C₃×C₅	—	2	1	no	—
69	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
70	C₂×C₃×C₄	—	5	2	no	—
71	C₂×C₃×C₄	—	10	1	no	—
72	C₂×Dic₃	—	3	2	no	—
73	C₂×Dic₃	—	6	1	no	—
74	C₂×Dic₃	—	15	2	no	—
75	C₂×Dic₃	—	30	1	no	—
76	Dic₆	—	30	2	no	—
77	Dic₆	—	30	2	no	—
78	Dic₆	—	30	4	no	—
79	C₃×C₈	—	30	2	no	—
80	C₃⋊C₈	—	10	2	no	—
81	C₂²×C₅	#624	1	1	yes	characteristic
82	C₄×C₅	—	18	1	no	—
83	Dic₅	—	18	1	no	—
84	Dic₅	#626	1	2	yes	—
85	C₂×C₃²	—	2	1	no	—
86	C₂×C₃²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂×Q₈	—	45	1	no	—
88	M₁₆	—	15	2	no	—
89	C₃×C₅	—	2	1	no	—
90	C₃×C₅	#824	1	2	yes	complemented
91	C₂²×C₃	—	2	1	no	—
92	C₂²×C₃	D₃×AGL(1,𝔽₅)	1	2	yes	—
93	C₃×C₄	—	5	2	no	—
94	C₃×C₄	—	5	2	no	—
95	C₃×C₄	—	10	2	no	—
96	Dic₃	—	6	2	no	—
97	Dic₃	—	6	2	no	—
98	Dic₃	—	30	2	no	—
99	Dic₃	—	30	2	no	—
100	C₂×C₅	—	2	1	no	—
101	C₂×C₅	#3722	1	1	yes	characteristic
102	C₃²	#4005	1	1	yes	characteristic, complemented, Sylow
103	Q₈	—	90	2	no	—
104	C₂×C₄	—	5	1	no	—
105	C₂×C₄	—	9	1	no	—
106	C₂×C₄	—	45	1	no	—
107	C₈	—	30	2	no	—
108	C₂×C₃	—	2	2	no	—
109	C₂×C₃	—	2	2	no	—
110	C₂×C₃	—	2	1	no	—
111	C₂×C₃	(C₂×D₃)⋊AGL(1,𝔽₅)	1	2	yes	—
112	C₅	#63561	1	1	yes	characteristic, complemented, Sylow
113	C₂²	ΓL(1,𝔽₁₆)⋊D₃	1	1	yes	characteristic
114	C₄	—	5	2	no	—
115	C₄	—	18	1	no	—
116	C₄	—	90	1	no	—
117	C₃	—	2	1	no	—
118	C₃	#91175	1	2	yes	complemented
119	C₂	—	2	1	no	—
120	C₂	#10623978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67141

C₂

yes

maximal, characteristic

C₃×#6762

C₂

yes

maximal

#91175

—

maximal

C₂×C₃⋊Dic₁₅

C₄

yes

characteristic

C₂×C₅×C₃⋊Dic₃

C₄

yes

characteristic

C₂×C₃²×Dic₅

C₂²

yes

characteristic

#67141

—

C₃×C₃⋊#170

—

—

—

—

D₃

yes

—

C₃×#704

—

C₂²×C₃²×C₅

C₂×C₄

yes

characteristic

C₃⋊Dic₁₅

—

C₅×C₃⋊Dic₃

—

C₃²×Dic₅

D₄

yes

—

—

—

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

C₂×D₃

yes

—

—

—

—

—

—

—

C₂×C₃²×C₅

—

C₂×C₃²×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀

—

#704

—

C₂×C₃²×C₄

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

AGL(1,𝔽₅)

yes

characteristic

#634

—

C₃×C₃⋊C₈

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₄×D₃

yes

—

Dic₁₅

—

Dic₁₅

—

C₃×Dic₅

—

C₃×Dic₅

D₁₂

yes

—

C₃×Dic₅

#67

yes

—

C₅×Dic₃

—

C₅×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

—

#208

—

C₃²×C₅

#101

yes

characteristic, complemented, Hall

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

D₃²

yes

characteristic

Dic₁₀

—

#170

—

C₂²×C₃²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#101

—

Sylow

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂×C₃×C₅

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

#624

yes

characteristic

C₄×C₅

—

Dic₅

—

Dic₅

#626

yes

—

C₂×C₃²

—

C₂×C₃²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₂×Q₈

—

M₁₆

—

C₃×C₅

—

C₃×C₅

#824

yes

complemented

C₂²×C₃

—

C₂²×C₃

D₃×AGL(1,𝔽₅)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

100

C₂×C₅

—

101

C₂×C₅

#3722

yes

characteristic

102

C₃²

#4005

yes

characteristic, complemented, Sylow

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₈

—

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

—

112

C₅

#63561

yes

characteristic, complemented, Sylow

113

C₂²

ΓL(1,𝔽₁₆)⋊D₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₃

—

118

C₃

#91175

yes

complemented

119

C₂

—

120

C₂

#10623978

yes

characteristic

	subgroup	G/H
center	C₂	#10623978
commutator, Fitting, prime radical	C₂²×C₃²×C₅	C₂×C₄
Frattini, hypercenter	C₂²	ΓL(1,𝔽₁₆)⋊D₃
socle	C₂×C₃²×C₅	C₂²⋊C₄
cosocle	C₂×C₃²×Dic₅	C₂²
hypocenter	C₃²×C₅	#101
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

subgroup

G/H

center

C₂

#10623978

commutator, Fitting, prime radical

C₂²×C₃²×C₅

C₂×C₄

Frattini, hypercenter

C₂²

ΓL(1,𝔽₁₆)⋊D₃

socle

C₂×C₃²×C₅

C₂²⋊C₄

cosocle

C₂×C₃²×Dic₅

C₂²

hypocenter

C₃²×C₅

#101

LCS

G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅

UCS

C₁ ◁ C₂ ◁ C₂²

N	H ≅ G/N	type
55. C₃²×C₅	#101	semidirect
90. C₃×C₅	#824	semidirect
102. C₃²	#4005	semidirect
112. C₅	#63561	semidirect
118. C₃	#91175	semidirect

H ≅ G/N

type

55. C₃²×C₅

#101

semidirect

90. C₃×C₅

#824

semidirect

102. C₃²

#4005

semidirect

112. C₅

#63561

semidirect

118. C₃

#91175

semidirect

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

subgroup	index
C₂×#67141	2
C₃×#6762	2
#91175	3
#63561	5

subgroup

index

Subgroups — #13085495

Conjugacy classes of subgroups

Complements

Maximal subgroups