Subgroups — C₇×C₈×D₁₅ #13132337

Order 1680.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₈×D₅	—	3	1	no	maximal, Hall
5	C₂×C₇×D₁₅	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₁₅	C₄	1	1	yes	characteristic
8	D₃×C₇×C₈	—	5	1	no	maximal, Hall
9	C₄×C₇×D₅	—	3	1	no	—
10	C₅×C₇×C₈	D₃	1	1	yes	characteristic, complemented
11	C₇×#168	—	3	1	no	—
12	C₈×D₁₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×D₁₅	C₈	1	2	yes	direct product factor
15	C₄×D₃×C₇	—	5	1	no	—
16	C₃×C₇×C₈	D₅	1	1	yes	characteristic, complemented
17	C₇×C₃⋊C₈	—	5	1	no	—
18	C₂×C₇×D₅	—	3	1	no	—
19	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
20	C₇×Dic₅	—	3	1	no	—
21	C₄×D₁₅	C₂×C₇	1	1	yes	characteristic
22	C₃×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
23	C₃⋊#168	C₂×C₇	1	1	yes	characteristic, complemented
24	C₂×C₇×C₈	—	15	1	no	Hall
25	C₃×C₅×C₇	C₂×C₈	1	1	yes	characteristic, complemented, Hall
26	C₂×D₃×C₇	—	5	1	no	—
27	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
28	C₇×Dic₃	—	5	1	no	—
29	C₈×D₅	—	3	1	no	Hall
30	C₂×C₅×C₇	C₄×D₃	1	1	yes	characteristic
31	C₇×D₅	—	3	2	no	—
32	C₂×D₁₅	C₄×C₇	1	1	yes	characteristic
33	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
34	Dic₁₅	C₄×C₇	1	1	yes	characteristic
35	C₂×C₄×C₇	—	15	1	no	—
36	C₇×C₈	—	15	1	no	—
37	C₇×C₈	D₁₅	1	1	yes	characteristic, direct product factor
38	D₃×C₈	—	5	1	no	Hall
39	C₂×C₃×C₇	C₄×D₅	1	1	yes	characteristic
40	D₃×C₇	—	5	2	no	—
41	C₄×D₅	—	3	1	no	—
42	C₅×C₈	D₃×C₇	1	1	yes	characteristic, complemented
43	#168	—	3	1	no	—
44	C₅×C₇	D₃×C₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	C₂×C₄×C₇	1	1	yes	characteristic
46	D₁₅	C₇×C₈	1	2	yes	direct product factor
47	C₂²×C₇	—	15	1	no	—
48	C₄×C₇	—	15	1	no	—
49	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
50	C₄×D₃	—	5	1	no	—
51	C₃×C₈	C₇×D₅	1	1	yes	characteristic, complemented
52	C₃⋊C₈	—	5	1	no	—
53	C₃×C₇	C₈×D₅	1	1	yes	characteristic, complemented, Hall
54	C₂×D₅	—	3	1	no	—
55	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
56	Dic₅	—	3	1	no	—
57	C₂×C₈	—	15	1	no	Sylow
58	C₃×C₅	C₂×C₇×C₈	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	—	15	2	no	—
60	C₂×C₇	C₄×D₁₅	1	1	yes	characteristic
61	C₂×D₃	—	5	1	no	—
62	C₃×C₄	C₂×C₇×D₅	1	1	yes	characteristic
63	Dic₃	—	5	1	no	—
64	C₂×C₅	C₄×D₃×C₇	1	1	yes	characteristic
65	D₅	—	3	2	no	—
66	C₂×C₄	—	15	1	no	—
67	C₈	—	15	1	no	—
68	C₈	C₇×D₁₅	1	1	yes	characteristic, direct product factor
69	C₇	C₈×D₁₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	C₄×C₇×D₅	1	1	yes	characteristic
71	D₃	—	5	2	no	—
72	C₅	D₃×C₇×C₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	15	1	no	—
74	C₄	—	15	1	no	—
75	C₄	C₂×C₇×D₁₅	1	1	yes	characteristic
76	C₃	C₇×C₈×D₅	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	15	2	no	—
78	C₂	C₄×C₇×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₇×C₈	D₁₅
commutator, hypocenter	C₃×C₅	C₂×C₇×C₈
Frattini	C₄	C₂×C₇×D₁₅
Fitting	C₃×C₅×C₇×C₈	C₂
socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical	C₂×C₇×D₁₅	C₄
cosocle	C₃×C₄×C₅	C₂²×C₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₇×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×C₅×C₇×C₈	C₂	semidirect
3. C₇×C₃⋊#168	C₂	semidirect
10. C₅×C₇×C₈	D₃	semidirect
12. C₈×D₁₅	C₇	direct (N × H)
14. C₇×D₁₅	C₈	direct (N × H)
16. C₃×C₇×C₈	D₅	semidirect
22. C₃×C₅×C₈	C₂×C₇	semidirect
23. C₃⋊#168	C₂×C₇	semidirect
25. C₃×C₅×C₇	C₂×C₈	semidirect
37. C₇×C₈	D₁₅	direct (N × H)
42. C₅×C₈	D₃×C₇	semidirect
44. C₅×C₇	D₃×C₈	semidirect
46. D₁₅	C₇×C₈	direct (N × H)
51. C₃×C₈	C₇×D₅	semidirect
53. C₃×C₇	C₈×D₅	semidirect
58. C₃×C₅	C₂×C₇×C₈	semidirect
68. C₈	C₇×D₁₅	direct (N × H)
69. C₇	C₈×D₁₅	direct (N × H)
72. C₅	D₃×C₇×C₈	semidirect
76. C₃	C₇×C₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×D₁₅	2
C₃×C₅×C₇×C₈	2
C₇×C₃⋊#168	2
C₇×C₈×D₅	3
D₃×C₇×C₈	5
C₈×D₁₅	7

Subgroups — C7×C8×D15 #13132337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×D₁₅ #13132337