Group #13085439 — subgroups

Subgroups — #13085439

Order 1440.

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

normal subgroups only

This group has 228 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	C₂×#67141	C₂	1	1	yes	maximal, characteristic
2	#10624097	C₂	1	1	yes	maximal, characteristic
3	C₃²×#704	C₂	1	1	yes	maximal, characteristic
4	#91175	—	3	4	no	maximal
5	C₂×C₃⋊Dic₁₅	C₄	1	1	yes	characteristic
6	C₂×C₅×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
8	#67141	—	2	1	no	—
9	#67141	—	2	1	no	—
10	C₃⋊(C₃⋊#170)	—	2	1	no	—
11	C₃²×#170	—	2	1	no	—
12	#63651	—	5	1	no	maximal, Hall
13	C₂×#1204	—	3	4	no	—
14	#6762	—	3	4	no	—
15	C₃×#704	D₃	1	4	yes	—
16	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃⋊Dic₁₅	—	2	1	no	—
18	C₅×C₃⋊Dic₃	—	2	1	no	—
19	C₃²×Dic₅	D₄	1	1	yes	characteristic
20	C₃²×Dic₅	D₄	1	1	yes	characteristic
21	#4005	—	9	1	no	Hall
22	C₂×#634	—	5	1	no	—
23	C₃²×M₁₆	—	5	1	no	—
24	#3748	—	5	1	no	—
25	C₂×Dic₁₅	—	3	4	no	—
26	C₂×C₅×Dic₃	—	3	4	no	—
27	C₂×C₃×Dic₅	C₂×D₃	1	4	yes	—
28	#1204	—	6	4	no	—
29	#1204	—	6	4	no	—
30	C₃⋊#170	—	6	4	no	—
31	C₃×#170	—	2	4	no	—
32	#824	—	15	4	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	—	9	1	no	—
37	#704	C₃⋊D₃	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	5	1	no	—
39	C₂×C₃⋊Dic₃	—	5	1	no	—
40	C₂×C₃⋊Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
41	#634	—	10	1	no	—
42	#634	—	10	1	no	—
43	C₃²×C₈	—	10	1	no	—
44	C₃⋊(C₃⋊C₈)	—	10	1	no	—
45	C₂²×C₃×C₅	C₄×D₃	1	4	yes	—
46	Dic₁₅	—	6	4	no	—
47	C₃×Dic₅	D₁₂	1	4	yes	—
48	C₃×Dic₅	#67	1	4	yes	—
49	C₅×Dic₃	—	6	4	no	—
50	C₂×Dic₆	—	15	4	no	—
51	C₃×M₁₆	—	5	4	no	—
52	#208	—	15	4	no	—
53	C₃²×C₅	#101	1	1	yes	characteristic, complemented, Hall
54	C₂×C₄×C₅	—	9	1	no	—
55	C₂×Dic₅	—	9	1	no	—
56	C₂×Dic₅	C₂×C₃⋊D₃	1	1	yes	characteristic
57	Dic₁₀	—	18	1	no	—
58	Dic₁₀	—	18	1	no	—
59	#170	—	2	1	no	—
60	#170	—	18	1	no	—
61	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₃²×C₄	—	5	1	no	—
63	C₃²×C₄	—	5	1	no	—
64	C₃⋊Dic₃	—	2	1	no	—
65	C₃⋊Dic₃	—	10	1	no	—
66	#101	—	45	1	no	Sylow
67	C₂×C₃×C₅	—	2	4	no	—
68	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
69	C₂×C₃×C₄	—	5	4	no	—
70	C₂×Dic₃	—	3	4	no	—
71	C₂×Dic₃	—	15	4	no	—
72	Dic₆	—	30	4	no	—
73	Dic₆	—	30	4	no	—
74	C₃×C₈	—	10	4	no	—
75	C₃⋊C₈	—	30	4	no	—
76	C₂²×C₅	C₄×C₃⋊D₃	1	1	yes	characteristic
77	C₄×C₅	—	18	1	no	—
78	Dic₅	—	18	1	no	—
79	Dic₅	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
80	Dic₅	#638	1	1	yes	characteristic
81	C₂×C₃²	—	2	1	no	—
82	C₂×C₃²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×Q₈	—	45	1	no	—
84	M₁₆	—	5	1	no	—
85	M₁₆	—	45	1	no	—
86	C₃×C₅	#824	1	4	yes	complemented
87	C₂²×C₃	D₃×AGL(1,𝔽₅)	1	4	yes	—
88	C₃×C₄	—	5	4	no	—
89	C₃×C₄	—	5	4	no	—
90	Dic₃	—	6	4	no	—
91	Dic₃	—	30	4	no	—
92	C₂×C₅	—	2	1	no	—
93	C₂×C₅	#3752	1	1	yes	characteristic
94	C₃²	#4005	1	1	yes	characteristic, complemented, Sylow
95	Q₈	—	90	1	no	—
96	Q₈	—	90	1	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	—	9	1	no	—
99	C₂×C₄	—	45	1	no	—
100	C₈	—	10	1	no	—
101	C₈	—	90	1	no	—
102	C₂×C₃	—	2	4	no	—
103	C₂×C₃	(C₂×D₃)⋊AGL(1,𝔽₅)	1	4	yes	—
104	C₅	#63651	1	1	yes	characteristic, complemented, Sylow
105	C₂²	C₃⋊D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₄	—	5	1	no	—
107	C₄	—	5	1	no	—
108	C₄	—	18	1	no	—
109	C₄	—	90	1	no	—
110	C₃	#91175	1	4	yes	complemented
111	C₂	—	2	1	no	—
112	C₂	(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67141

C₂

yes

maximal, characteristic

#10624097

C₂

yes

maximal, characteristic

C₃²×#704

C₂

yes

maximal, characteristic

#91175

—

maximal

C₂×C₃⋊Dic₁₅

C₄

yes

characteristic

C₂×C₅×C₃⋊Dic₃

C₄

yes

characteristic

C₂×C₃²×Dic₅

C₂²

yes

—

—

—

C₃²×#170

—

—

—

—

D₃

yes

—

C₂²×C₃²×C₅

C₂×C₄

yes

characteristic

C₃⋊Dic₁₅

—

C₅×C₃⋊Dic₃

—

C₃²×Dic₅

D₄

yes

characteristic

C₃²×Dic₅

D₄

yes

—

—

C₃²×M₁₆

—

#3748

—

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₂×C₃×Dic₅

C₂×D₃

yes

—

—

—

—

—

—

C₂×C₃²×C₅

—

C₂×C₃²×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀

—

#704

—

#704

C₃⋊D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

yes

—

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₂²×C₃×C₅

C₄×D₃

yes

—

Dic₁₅

—

C₃×Dic₅

D₁₂

yes

—

C₃×Dic₅

#67

yes

—

C₅×Dic₃

—

C₂×Dic₆

—

C₃×M₁₆

—

#208

—

C₃²×C₅

#101

yes

characteristic, complemented, Hall

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×C₃⋊D₃

yes

characteristic

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

C₂²×C₃²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#101

—

Sylow

C₂×C₃×C₅

—

C₂×C₃×C₅

(C₂×D₃)⋊C₄

yes

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

C₄×C₃⋊D₃

yes

characteristic

C₄×C₅

—

Dic₅

—

Dic₅

C₄⋊(C₃⋊D₃)

yes

characteristic

Dic₅

#638

yes

characteristic

C₂×C₃²

—

C₂×C₃²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₂×Q₈

—

M₁₆

—

M₁₆

—

C₃×C₅

#824

yes

complemented

C₂²×C₃

D₃×AGL(1,𝔽₅)

yes

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂×C₅

—

C₂×C₅

#3752

yes

characteristic

C₃²

#4005

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₈

—

101

C₈

—

102

C₂×C₃

—

103

C₂×C₃

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

—

104

C₅

#63651

yes

characteristic, complemented, Sylow

105

C₂²

C₃⋊D₃×AGL(1,𝔽₅)

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₃

#91175

yes

complemented

111

C₂

—

112

C₂

(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)

yes

characteristic

	subgroup	G/H
center	C₂	(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)
commutator, Fitting, prime radical	C₂²×C₃²×C₅	C₂×C₄
Frattini, hypercenter	C₂²	C₃⋊D₃×AGL(1,𝔽₅)
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

(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)

commutator, Fitting, prime radical

C₂²×C₃²×C₅

C₂×C₄

Frattini, hypercenter

C₂²

C₃⋊D₃×AGL(1,𝔽₅)

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
53. C₃²×C₅	#101	semidirect
86. C₃×C₅	#824	semidirect
94. C₃²	#4005	semidirect
104. C₅	#63651	semidirect
110. C₃	#91175	semidirect

H ≅ G/N

type

53. C₃²×C₅

#101

semidirect

86. C₃×C₅

#824

semidirect

94. C₃²

#4005

semidirect

104. C₅

#63651

semidirect

110. 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
#10624097	2
C₃²×#704	2
#91175	3
#63651	5

subgroup

index

C₃²×#704

Subgroups — #13085439

Conjugacy classes of subgroups

Complements

Maximal subgroups