Group #14278613 — subgroups

Subgroups — #14278613

Order 1920.

120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11746284	C₂	1	1	yes	maximal, characteristic
2	C₅×#4463	C₂	1	1	yes	maximal, characteristic
3	C₃×#64681	C₂	1	1	yes	maximal, characteristic
4	#10598058	—	3	1	no	maximal, Hall
5	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	#91096	—	2	1	no	—
7	C₅×#812	—	2	1	no	—
8	C₂×C₃×#672	C₄	1	1	yes	characteristic
9	C₂×C₃×#672	C₄	1	1	yes	characteristic
10	#67514	—	5	1	no	maximal, Hall
11	C₅×#364	—	3	1	no	—
12	#64683	—	3	1	no	—
13	#64681	D₃	1	1	yes	characteristic
14	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
16	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
17	C₃⋊#672	—	4	1	no	—
18	C₃×#672	—	2	1	no	—
19	C₃×#672	C₈	1	2	yes	—
20	C₅×C₃⋊C₁₆	—	4	1	no	—
21	C₃×#346	—	5	1	no	—
22	#4463	—	5	1	no	—
23	#4463	D₅	1	1	yes	characteristic
24	C₅×M₃₂	—	6	1	no	—
25	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
26	#3944	—	6	1	no	—
27	C₂×#672	Dic₃	1	1	yes	characteristic
28	C₂×#672	Dic₃	1	1	yes	characteristic
29	#1369	—	15	1	no	Sylow
30	C₂×C₃×C₄×C₅	—	2	1	no	—
31	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
32	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
33	C₃×C₅×C₈	M₁₆	1	1	yes	characteristic
34	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
35	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
36	C₂×C₃×C₁₆	—	5	1	no	—
37	C₂×C₃×C₁₆	—	5	1	no	—
38	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
39	#812	—	2	1	no	—
40	#812	—	10	1	no	—
41	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₄²×C₅	Dic₆	1	1	yes	characteristic
44	C₅×C₁₆	—	12	1	no	—
45	#672	—	2	1	no	—
46	#672	—	12	1	no	—
47	#672	C₃⋊C₈	1	2	yes	—
48	#364	—	3	1	no	—
49	#364	—	15	1	no	—
50	#346	—	5	1	no	—
51	C₂²×C₃×C₅	#108	1	1	yes	characteristic
52	C₃×C₄×C₅	—	4	1	no	—
53	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
54	C₃×C₄×C₅	#106	1	1	yes	characteristic
55	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
57	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
58	C₃×C₁₆	—	5	2	no	—
59	C₃×C₁₆	—	10	1	no	—
60	C₃⋊C₁₆	—	4	1	no	—
61	C₃⋊C₁₆	—	20	1	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
64	C₅×C₈	#204	1	1	yes	characteristic
65	C₅×C₈	#204	1	1	yes	characteristic
66	C₅×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
67	C₅×C₈	#208	1	1	yes	characteristic
68	M₃₂	—	6	1	no	—
69	M₃₂	—	30	1	no	—
70	C₂×C₁₆	—	5	1	no	—
71	C₂×C₁₆	—	5	1	no	—
72	C₄×C₈	D₃×D₅	1	1	yes	characteristic
73	C₂×C₃×C₅	—	2	1	no	—
74	C₂×C₃×C₅	#334	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	1	no	—
76	C₂×C₃×C₄	#683	1	1	yes	characteristic
77	C₃×C₈	C₈×D₅	1	1	yes	characteristic
78	C₃×C₈	#676	1	1	yes	characteristic
79	C₃×C₈	#687	1	1	yes	characteristic
80	C₃×C₈	#688	1	1	yes	characteristic
81	C₂²×C₅	#819	1	1	yes	characteristic
82	C₄×C₅	—	4	1	no	—
83	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
84	C₄×C₅	#817	1	1	yes	characteristic
85	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
86	C₂×C₈	#1203	1	1	yes	characteristic
87	C₄²	#1204	1	1	yes	characteristic
88	C₁₆	—	5	2	no	—
89	C₁₆	—	10	1	no	—
90	C₁₆	—	12	1	no	—
91	C₁₆	—	60	1	no	—
92	C₃×C₅	#1369	1	1	yes	characteristic, complemented, Hall
93	C₂²×C₃	#3954	1	1	yes	characteristic
94	C₃×C₄	—	4	1	no	—
95	C₃×C₄	#3940	1	1	yes	characteristic
96	C₃×C₄	#3946	1	1	yes	characteristic
97	C₂×C₅	—	2	1	no	—
98	C₂×C₅	#4457	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	#6668	1	1	yes	characteristic
101	C₈	D₅×C₃⋊C₈	1	1	yes	characteristic
102	C₈	#6649	1	1	yes	characteristic
103	C₈	#6659	1	1	yes	characteristic
104	C₈	#6660	1	1	yes	characteristic
105	C₂×C₃	—	2	1	no	—
106	C₂×C₃	#64700	1	1	yes	characteristic
107	C₅	#67514	1	1	yes	characteristic, complemented, Sylow
108	C₂²	#90991	1	1	yes	characteristic
109	C₄	—	4	1	no	—
110	C₄	#90982	1	1	yes	characteristic
111	C₄	#90985	1	1	yes	characteristic
112	C₃	#10598058	1	1	yes	characteristic, complemented, Sylow
113	C₂	—	2	1	no	—
114	C₂	#11745864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11746284

C₂

yes

maximal, characteristic

C₅×#4463

C₂

yes

maximal, characteristic

C₃×#64681

C₂

yes

maximal, characteristic

#10598058

—

maximal, Hall

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

#91096

—

C₅×#812

—

C₂×C₃×#672

C₄

yes

characteristic

C₂×C₃×#672

C₄

yes

—

—

—

D₃

yes

characteristic

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₃×C₄²×C₅

Q₈

yes

—

—

C₈

yes

—

C₅×C₃⋊C₁₆

—

C₃×#346

—

#4463

—

#4463

D₅

yes

characteristic

C₅×M₃₂

—

C₄×C₅×C₈

C₂×D₃

yes

characteristic

#3944

—

C₂×#672

Dic₃

yes

characteristic

C₂×#672

Dic₃

yes

characteristic

#1369

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

characteristic

C₃×C₅×C₈

C₂×C₈

yes

characteristic

C₃×C₅×C₈

M₁₆

yes

characteristic

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#812

—

#812

—

C₂×C₅×C₈

D₁₂

yes

characteristic

C₂×C₅×C₈

C₂×Dic₃

yes

characteristic

C₄²×C₅

Dic₆

yes

characteristic

C₅×C₁₆

—

#672

—

#672

—

#672

C₃⋊C₈

yes

—

#364

—

#364

—

#346

—

C₂²×C₃×C₅

#108

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₄⋊C₈

yes

characteristic

C₃×C₄×C₅

#106

yes

characteristic

C₂×C₃×C₈

C₄×D₅

yes

characteristic

C₂×C₃×C₈

C₂²⋊D₅

yes

characteristic

C₃×C₄²

Dic₁₀

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊Dic₃

yes

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

C₂×C₃⋊C₈

yes

characteristic

C₅×C₈

#208

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

D₃×D₅

yes

characteristic

C₂×C₃×C₅

—

C₂×C₃×C₅

#334

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

#683

yes

characteristic

C₃×C₈

C₈×D₅

yes

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

C₂²×C₅

yes

C₄×C₅

—

C₄×C₅

C₄⋊(C₃⋊C₈)

yes

characteristic

C₄×C₅

#817

yes

characteristic

C₂×C₈

D₅×Dic₃

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₃×C₅

#1369

yes

characteristic, complemented, Hall

C₂²×C₃

#3954

yes

characteristic

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₂×C₅

—

C₂×C₅

#4457

yes

characteristic

C₂×C₄

—

100

C₂×C₄

#6668

yes

characteristic

101

C₈

D₅×C₃⋊C₈

yes

characteristic

102

C₈

#6649

yes

characteristic

103

C₈

#6659

yes

characteristic

104

C₈

#6660

yes

characteristic

105

C₂×C₃

—

106

C₂×C₃

#64700

yes

characteristic

107

C₅

#67514

yes

characteristic, complemented, Sylow

108

C₂²

#90991

yes

characteristic

109

C₄

—

110

C₄

#90982

yes

characteristic

111

C₄

#90985

yes

characteristic

112

C₃

#10598058

yes

characteristic, complemented, Sylow

113

C₂

—

114

C₂

#11745864

yes

characteristic

	subgroup	G/H
center	C₄	#90985
commutator	C₃×C₅×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₃×D₅
Fitting, cosocle	C₃×C₄×C₅×C₈	C₂²
socle	C₂×C₃×C₅	#334
prime radical	C₂²×C₃×C₅	#108
hypocenter	C₃×C₅	#1369
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#90985

commutator

C₃×C₅×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₃×D₅

Fitting, cosocle

C₃×C₄×C₅×C₈

C₂²

C₂×C₃×C₅

C₂²×C₃×C₅

C₃×C₅

G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
92. C₃×C₅	#1369	semidirect
107. C₅	#67514	semidirect
112. C₃	#10598058	semidirect

H ≅ G/N

type

92. C₃×C₅

#1369

semidirect

107. C₅

#67514

semidirect

112. C₃

#10598058

semidirect

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

subgroup	index
#11746284	2
C₅×#4463	2
C₃×#64681	2
#10598058	3
#67514	5

subgroup

index

Subgroups — #14278613

Conjugacy classes of subgroups

Complements

Maximal subgroups