Group #14280305 — subgroups

Subgroups — C₈²×D₁₅ #14280305

Order 1920.

560 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 251 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	C₄×C₈×D₁₅	C₂	1	3	yes	maximal
2	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₃⋊#168	C₂	1	3	yes	maximal, complemented
4	C₈²×D₅	—	3	1	no	maximal, Hall
5	C₂×C₈×D₁₅	C₄	1	6	yes	—
6	C₄²×D₁₅	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₅×C₈	C₂²	1	3	yes	—
8	C₈×Dic₁₅	C₄	1	6	yes	—
9	C₄×C₃⋊#168	C₂²	1	3	yes	—
10	D₃×C₈²	—	5	1	no	maximal, Hall
11	C₄×C₈×D₅	—	3	3	no	—
12	C₅×C₈²	D₃	1	1	yes	characteristic, complemented
13	C₈×#168	—	3	3	no	—
14	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
15	C₂×C₃×C₅×C₈	C₂×C₄	1	6	yes	—
16	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
17	C₄×Dic₁₅	C₂×C₄	1	3	yes	—
18	C₂×C₃⋊#168	C₂×C₄	1	6	yes	—
19	C₈×D₁₅	C₈	1	24	yes	direct product factor
20	C₄×D₃×C₈	—	5	3	no	—
21	C₃×C₈²	D₅	1	1	yes	characteristic, complemented
22	C₈×C₃⋊C₈	—	5	3	no	—
23	C₂×C₈×D₅	—	3	6	no	—
24	C₄²×D₅	—	3	1	no	—
25	C₄×C₅×C₈	C₂×D₃	1	3	yes	—
26	C₈×Dic₅	—	3	6	no	—
27	C₄×#168	—	3	3	no	—
28	C₂×C₈²	—	15	1	no	Sylow
29	C₂²×D₁₅	C₄²	1	1	yes	characteristic
30	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
31	C₂×Dic₁₅	C₄²	1	3	yes	—
32	C₄×D₁₅	C₂×C₈	1	12	yes	—
33	C₃×C₅×C₈	C₂×C₈	1	12	yes	complemented
34	C₃⋊#168	C₂×C₈	1	12	yes	complemented
35	C₂×D₃×C₈	—	5	6	no	—
36	C₄²×D₃	—	5	1	no	—
37	C₃×C₄×C₈	C₂×D₅	1	3	yes	—
38	C₈×Dic₃	—	5	6	no	—
39	C₄×C₃⋊C₈	—	5	3	no	—
40	C₂×C₄×D₅	—	3	3	no	—
41	C₂×C₅×C₈	C₄×D₃	1	6	yes	—
42	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
43	C₄×Dic₅	—	3	3	no	—
44	C₂×#168	—	3	6	no	—
45	C₈×D₅	—	3	24	no	—
46	C₂×C₄×C₈	—	15	3	no	—
47	C₈²	—	15	3	no	—
48	C₈²	D₁₅	1	1	yes	characteristic, direct product factor
49	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
50	C₂×D₁₅	C₄×C₈	1	6	yes	—
51	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
52	Dic₁₅	C₄×C₈	1	6	yes	—
53	C₂×C₄×D₃	—	5	3	no	—
54	C₂×C₃×C₈	C₄×D₅	1	6	yes	—
55	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
56	C₄×Dic₃	—	5	3	no	—
57	C₂×C₃⋊C₈	—	5	6	no	—
58	D₃×C₈	—	5	24	no	—
59	C₂²×D₅	—	3	1	no	—
60	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
61	C₂×Dic₅	—	3	3	no	—
62	C₄×D₅	—	3	12	no	—
63	C₅×C₈	D₃×C₈	1	12	yes	complemented
64	#168	—	3	12	no	—
65	C₂²×C₈	—	15	6	no	—
66	C₂×C₄²	—	15	1	no	—
67	C₄×C₈	—	15	6	no	—
68	C₄×C₈	—	15	3	no	—
69	C₄×C₈	C₂×D₁₅	1	3	yes	—
70	C₂×C₃×C₅	C₂×C₄×C₈	1	3	yes	—
71	D₁₅	C₈²	1	4	yes	direct product factor
72	C₂²×D₃	—	5	1	no	—
73	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
74	C₂×Dic₃	—	5	3	no	—
75	C₄×D₃	—	5	12	no	—
76	C₃×C₈	C₈×D₅	1	12	yes	complemented
77	C₃⋊C₈	—	5	12	no	—
78	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
79	C₂×D₅	—	3	6	no	—
80	C₄×C₅	C₂×D₃×C₈	1	6	yes	—
81	Dic₅	—	3	6	no	—
82	C₂²×C₄	—	15	3	no	—
83	C₂×C₈	—	15	6	no	—
84	C₂×C₈	—	15	24	no	—
85	C₂×C₈	C₄×D₁₅	1	6	yes	—
86	C₄²	—	15	3	no	—
87	C₄²	C₂²×D₁₅	1	1	yes	characteristic
88	C₃×C₅	C₂×C₈²	1	1	yes	characteristic, complemented, Hall
89	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
90	C₂×D₃	—	5	6	no	—
91	C₃×C₄	C₂×C₈×D₅	1	6	yes	—
92	Dic₃	—	5	6	no	—
93	C₂×C₅	C₄×D₃×C₈	1	3	yes	—
94	D₅	—	3	4	no	—
95	C₂³	—	15	1	no	—
96	C₂×C₄	—	15	3	no	—
97	C₂×C₄	—	15	12	no	—
98	C₂×C₄	C₂×C₄×D₁₅	1	3	yes	—
99	C₈	—	15	12	no	—
100	C₈	C₈×D₁₅	1	12	yes	direct product factor
101	C₂×C₃	C₄×C₈×D₅	1	3	yes	—
102	D₃	—	5	4	no	—
103	C₅	D₃×C₈²	1	1	yes	characteristic, complemented, Sylow
104	C₂²	—	15	6	no	—
105	C₂²	C₄²×D₁₅	1	1	yes	characteristic
106	C₄	—	15	6	no	—
107	C₄	C₂×C₈×D₁₅	1	6	yes	—
108	C₃	C₈²×D₅	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	15	4	no	—
110	C₂	C₄×C₈×D₁₅	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₁₅

C₂

yes

maximal

C₃×C₅×C₈²

C₂

yes

maximal, characteristic, complemented

C₈×C₃⋊#168

C₂

yes

maximal, complemented

C₈²×D₅

—

maximal, Hall

C₂×C₈×D₁₅

C₄

yes

—

C₄²×D₁₅

C₂²

yes

characteristic

C₃×C₄×C₅×C₈

C₂²

yes

—

C₈×Dic₁₅

C₄

yes

—

C₄×C₃⋊#168

C₂²

yes

—

D₃×C₈²

—

maximal, Hall

C₄×C₈×D₅

—

C₅×C₈²

D₃

yes

characteristic, complemented

C₈×#168

—

C₂×C₄×D₁₅

C₂×C₄

yes

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄×Dic₁₅

C₂×C₄

yes

—

C₂×C₃⋊#168

C₂×C₄

yes

—

C₈×D₁₅

C₈

yes

direct product factor

C₄×D₃×C₈

—

C₃×C₈²

D₅

yes

characteristic, complemented

C₈×C₃⋊C₈

—

C₂×C₈×D₅

—

C₄²×D₅

—

C₄×C₅×C₈

C₂×D₃

yes

—

C₈×Dic₅

—

C₄×#168

—

C₂×C₈²

—

Sylow

C₂²×D₁₅

C₄²

yes

characteristic

C₂×C₃×C₄×C₅

C₂²×C₄

yes

—

C₂×Dic₁₅

C₄²

yes

—

C₄×D₁₅

C₂×C₈

yes

—

C₃×C₅×C₈

C₂×C₈

yes

complemented

C₃⋊#168

C₂×C₈

yes

complemented

C₂×D₃×C₈

—

C₄²×D₃

—

C₃×C₄×C₈

C₂×D₅

yes

—

C₈×Dic₃

—

C₄×C₃⋊C₈

—

C₂×C₄×D₅

—

C₂×C₅×C₈

C₄×D₃

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄×Dic₅

—

C₂×#168

—

C₈×D₅

—

C₂×C₄×C₈

—

C₈²

—

C₈²

D₁₅

yes

characteristic, direct product factor

C₂²×C₃×C₅

C₂×C₄²

yes

characteristic

C₂×D₁₅

C₄×C₈

yes

—

C₃×C₄×C₅

C₂²×C₈

yes

—

Dic₁₅

C₄×C₈

yes

—

C₂×C₄×D₃

—

C₂×C₃×C₈

C₄×D₅

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄×Dic₃

—

C₂×C₃⋊C₈

—

D₃×C₈

—

C₂²×D₅

—

C₂×C₄×C₅

C₂×C₄×D₃

yes

—

C₂×Dic₅

—

C₄×D₅

—

C₅×C₈

D₃×C₈

yes

complemented

#168

—

C₂²×C₈

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₅

yes

—

C₂×C₃×C₅

C₂×C₄×C₈

yes

—

D₁₅

C₈²

yes

direct product factor

C₂²×D₃

—

C₂×C₃×C₄

C₂×C₄×D₅

yes

—

C₂×Dic₃

—

C₄×D₃

—

C₃×C₈

C₈×D₅

yes

complemented

C₃⋊C₈

—

C₂²×C₅

C₄²×D₃

yes

characteristic

C₂×D₅

—

C₄×C₅

C₂×D₃×C₈

yes

—

Dic₅

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₅

yes

—

C₄²

—

C₄²

C₂²×D₁₅

yes

characteristic

C₃×C₅

C₂×C₈²

yes

characteristic, complemented, Hall

C₂²×C₃

C₄²×D₅

yes

characteristic

C₂×D₃

—

C₃×C₄

C₂×C₈×D₅

yes

—

Dic₃

—

C₂×C₅

C₄×D₃×C₈

yes

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₅

yes

—

C₈

—

100

C₈

C₈×D₁₅

yes

direct product factor

101

C₂×C₃

C₄×C₈×D₅

yes

—

102

D₃

—

103

C₅

D₃×C₈²

yes

characteristic, complemented, Sylow

104

C₂²

—

105

C₂²

C₄²×D₁₅

yes

characteristic

106

C₄

—

107

C₄

C₂×C₈×D₁₅

yes

—

108

C₃

C₈²×D₅

yes

characteristic, complemented, Sylow

109

C₂

—

110

C₂

C₄×C₈×D₁₅

yes

—

	subgroup	G/H
center, hypercenter	C₈²	D₁₅
commutator, hypocenter	C₃×C₅	C₂×C₈²
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×D₁₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₈²

subgroup

G/H

center, hypercenter

C₈²

D₁₅

commutator, hypocenter

C₃×C₅

C₂×C₈²

Frattini

C₄²

C₂²×D₁₅

Fitting

C₃×C₅×C₈²

C₂

socle

C₂²×C₃×C₅

C₂×C₄²

prime radical

C₂²×D₁₅

C₄²

cosocle

C₃×C₄²×C₅

C₂³

LCS

G ▷ C₃×C₅

UCS

C₁ ◁ C₈²

N	H ≅ G/N	type
2. C₃×C₅×C₈²	C₂	semidirect
3. C₈×C₃⋊#168	C₂	semidirect
12. C₅×C₈²	D₃	semidirect
19. C₈×D₁₅	C₈	direct (N × H)
21. C₃×C₈²	D₅	semidirect
33. C₃×C₅×C₈	C₂×C₈	semidirect
34. C₃⋊#168	C₂×C₈	semidirect
48. C₈²	D₁₅	direct (N × H)
63. C₅×C₈	D₃×C₈	semidirect
71. D₁₅	C₈²	direct (N × H)
76. C₃×C₈	C₈×D₅	semidirect
88. C₃×C₅	C₂×C₈²	semidirect
100. C₈	C₈×D₁₅	direct (N × H)
103. C₅	D₃×C₈²	semidirect
108. C₃	C₈²×D₅	semidirect

H ≅ G/N

type

2. C₃×C₅×C₈²

C₂

semidirect

3. C₈×C₃⋊#168

C₂

semidirect

12. C₅×C₈²

D₃

semidirect

19. C₈×D₁₅

C₈

direct (N × H)

21. C₃×C₈²

D₅

semidirect

33. C₃×C₅×C₈

C₂×C₈

semidirect

34. C₃⋊#168

C₂×C₈

semidirect

48. C₈²

D₁₅

direct (N × H)

63. C₅×C₈

D₃×C₈

semidirect

71. D₁₅

C₈²

direct (N × H)

76. C₃×C₈

C₈×D₅

semidirect

88. C₃×C₅

C₂×C₈²

semidirect

100. C₈

C₈×D₁₅

direct (N × H)

103. C₅

D₃×C₈²

semidirect

108. C₃

C₈²×D₅

semidirect

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

subgroup	index
C₄×C₈×D₁₅	2
C₃×C₅×C₈²	2
C₈×C₃⋊#168	2
C₈²×D₅	3
D₃×C₈²	5

subgroup

index

C₄×C₈×D₁₅

C₃×C₅×C₈²

C₈×C₃⋊#168

C₈²×D₅

D₃×C₈²

Subgroups — C₈²×D₁₅ #14280305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C82×D15 #14280305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×D₁₅ #14280305