Subgroups — D₅×D₅₆ #11764701

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×D₂₈	C₂	1	2	yes	maximal, complemented
2	D₂₈₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#10587842	C₂	1	2	yes	maximal, complemented
6	C₂×D₅×D₇	—	2	2	no	—
7	D₁₄₀	C₂²	1	2	yes	complemented
8	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
9	C₅×D₂₈	C₂²	1	2	yes	complemented
10	#63306	—	2	2	no	—
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
12	C₇×#168	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₅₆	—	5	1	no	maximal, Hall
14	D₅×D₈	—	7	1	no	maximal, Hall
15	C₂×D₃₅	—	2	2	no	—
16	C₂×C₇×D₅	D₄	1	1	yes	characteristic
17	C₂×C₅×D₇	—	2	2	no	—
18	D₅×D₇	—	4	4	no	—
19	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
20	C₇×Dic₅	D₄	1	1	yes	characteristic
21	C₂×D₂₈	—	5	2	no	—
22	C₂×C₇×C₈	—	5	1	no	—
23	D₅₆	—	5	2	no	—
24	D₅₆	—	5	1	no	—
25	D₅₆	D₅	1	1	yes	characteristic, direct product factor
26	D₄×D₅	—	7	2	no	—
27	C₅×D₈	—	7	1	no	—
28	#686	—	7	2	no	—
29	D₄₀	—	7	1	no	—
30	C₈×D₅	D₇	1	1	yes	characteristic, complemented
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	D₃₅	—	4	2	no	—
33	C₅×D₇	—	4	2	no	—
34	C₇×D₅	D₈	1	2	yes	complemented
35	C₂²×D₇	—	10	2	no	—
36	C₂×C₄×C₇	—	5	1	no	—
37	D₂₈	—	5	2	no	—
38	D₂₈	—	10	2	no	—
39	D₂₈	C₂×D₅	1	2	yes	complemented
40	C₇×C₈	—	5	1	no	—
41	C₇×C₈	C₂×D₅	1	1	yes	characteristic, complemented
42	C₂²×D₅	—	14	2	no	—
43	C₅×D₄	—	7	2	no	—
44	C₂²⋊D₅	—	14	2	no	—
45	D₂₀	—	7	2	no	—
46	C₄×D₅	C₂×D₇	1	1	yes	characteristic
47	C₅×C₈	C₂×D₇	1	1	yes	characteristic, complemented
48	#168	C₂×D₇	1	1	yes	characteristic, complemented
49	C₅×C₇	C₂×D₈	1	1	yes	characteristic, complemented, Hall
50	C₂×D₈	—	35	1	no	Sylow
51	C₂²×C₇	—	5	1	no	—
52	C₂×D₇	—	2	2	no	—
53	C₂×D₇	—	10	2	no	—
54	C₂×D₇	—	20	4	no	—
55	C₄×C₇	—	5	1	no	—
56	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
57	C₂²×C₅	—	14	2	no	—
58	C₂×D₅	—	14	2	no	—
59	C₂×D₅	—	28	4	no	—
60	C₂×D₅	D₂₈	1	1	yes	characteristic
61	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
62	Dic₅	D₂₈	1	1	yes	characteristic
63	C₂×D₄	—	35	2	no	—
64	D₈	—	7	1	no	—
65	D₈	—	35	2	no	—
66	D₈	—	35	1	no	—
67	C₂×C₈	—	5	1	no	—
68	C₂×C₇	—	5	2	no	—
69	C₂×C₇	D₄×D₅	1	1	yes	characteristic
70	D₇	—	4	2	no	—
71	D₇	—	20	2	no	—
72	C₂×C₅	—	28	2	no	—
73	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
74	D₅	—	28	2	no	—
75	D₅	D₅₆	1	2	yes	direct product factor
76	C₂³	—	70	2	no	—
77	D₄	—	7	2	no	—
78	D₄	—	35	2	no	—
79	D₄	—	70	2	no	—
80	C₂×C₄	—	5	1	no	—
81	C₈	—	5	1	no	—
82	C₈	D₅×D₇	1	1	yes	characteristic, complemented
83	C₇	D₅×D₈	1	1	yes	characteristic, complemented, Sylow
84	C₅	C₂×D₅₆	1	1	yes	characteristic, complemented, Sylow
85	C₂²	—	5	1	no	—
86	C₂²	—	14	2	no	—
87	C₂²	—	70	2	no	—
88	C₂²	—	140	4	no	—
89	C₄	—	5	1	no	—
90	C₄	C₂×D₅×D₇	1	1	yes	characteristic
91	C₂	—	5	2	no	—
92	C₂	—	28	2	no	—
93	C₂	—	140	2	no	—
94	C₂	D₅×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅×D₂₈
commutator, cosocle	C₄×C₅×C₇	C₂³
Frattini	C₄	C₂×D₅×D₇
Fitting	C₅×C₇×C₈	C₂²
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₈	D₅×D₇
hypocenter	C₅×C₇	C₂×D₈
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₅×D₂₈	C₂	semidirect
2. D₂₈₀	C₂	semidirect
3. C₇×C₈×D₅	C₂	semidirect
4. C₅×D₅₆	C₂	semidirect
5. #10587842	C₂	semidirect
7. D₁₄₀	C₂²	semidirect
9. C₅×D₂₈	C₂²	semidirect
11. C₅×C₇×C₈	C₂²	semidirect
12. C₇×#168	C₂²	semidirect
25. D₅₆	D₅	direct (N × H)
30. C₈×D₅	D₇	semidirect
34. C₇×D₅	D₈	semidirect
39. D₂₈	C₂×D₅	semidirect
41. C₇×C₈	C₂×D₅	semidirect
47. C₅×C₈	C₂×D₇	semidirect
48. #168	C₂×D₇	semidirect
49. C₅×C₇	C₂×D₈	semidirect
75. D₅	D₅₆	direct (N × H)
82. C₈	D₅×D₇	semidirect
83. C₇	D₅×D₈	semidirect
84. C₅	C₂×D₅₆	semidirect

Maximal subgroups

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

subgroup	index
D₅×D₂₈	2
D₂₈₀	2
C₇×C₈×D₅	2
C₅×D₅₆	2
#10587842	2
C₂×D₅₆	5
D₅×D₈	7

Subgroups — D5×D56 #11764701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×D₅₆ #11764701