Subgroups — Q₈×D₁₀₅ #13132896

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₀₅	C₂	1	3	yes	maximal
3	Dic₂₁₀	C₂	1	3	yes	maximal, complemented
4	Q₈×D₃₅	—	3	1	no	maximal, Hall
5	C₂×D₁₀₅	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	3	yes	—
7	Dic₁₀₅	C₂²	1	3	yes	—
8	Q₈×D₂₁	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
10	C₄×D₃₅	—	3	3	no	—
11	Dic₇₀	—	3	3	no	—
12	Q₈×D₁₅	—	7	1	no	maximal, Hall
13	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
14	D₁₀₅	Q₈	1	2	yes	direct product factor
15	C₃×C₇×Q₈	D₅	1	1	yes	characteristic, complemented
16	C₄×D₂₁	—	5	3	no	—
17	Dic₄₂	—	5	3	no	—
18	C₂×D₃₅	—	3	1	no	—
19	C₄×C₅×C₇	C₂×D₃	1	3	yes	—
20	Dic₃₅	—	3	3	no	—
21	C₃×C₅×Q₈	D₇	1	1	yes	characteristic, complemented
22	C₄×D₁₅	—	7	3	no	—
23	Dic₃₀	—	7	3	no	—
24	Q₈×D₇	—	15	1	no	Hall
25	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
26	C₂×D₂₁	—	5	1	no	—
27	C₃×C₄×C₇	C₂×D₅	1	3	yes	—
28	Dic₂₁	—	5	3	no	—
29	Q₈×D₅	—	21	1	no	Hall
30	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
31	D₃₅	—	3	2	no	—
32	C₂×D₁₅	—	7	1	no	—
33	C₃×C₄×C₅	C₂×D₇	1	3	yes	—
34	Dic₁₅	—	7	3	no	—
35	C₇×Q₈	D₁₅	1	1	yes	characteristic, complemented
36	C₄×D₇	—	15	3	no	—
37	Dic₁₄	—	15	3	no	—
38	D₃×Q₈	—	35	1	no	Hall
39	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
40	D₂₁	—	5	2	no	—
41	C₅×Q₈	D₂₁	1	1	yes	characteristic, complemented
42	C₄×D₅	—	21	3	no	—
43	Dic₁₀	—	21	3	no	—
44	C₅×C₇	D₃×Q₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
46	D₁₅	—	7	2	no	—
47	C₂×D₇	—	15	1	no	—
48	C₄×C₇	C₂×D₁₅	1	3	yes	—
49	Dic₇	—	15	3	no	—
50	C₃×Q₈	D₃₅	1	1	yes	characteristic, complemented
51	C₄×D₃	—	35	3	no	—
52	Dic₆	—	35	3	no	—
53	C₃×C₇	Q₈×D₅	1	1	yes	characteristic, complemented, Hall
54	C₂×D₅	—	21	1	no	—
55	C₄×C₅	C₂×D₂₁	1	3	yes	—
56	Dic₅	—	21	3	no	—
57	C₂×Q₈	—	105	1	no	Sylow
58	C₃×C₅	Q₈×D₇	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
60	D₇	—	15	2	no	—
61	C₂×D₃	—	35	1	no	—
62	C₃×C₄	C₂×D₃₅	1	3	yes	—
63	Dic₃	—	35	3	no	—
64	C₂×C₅	C₂²×D₂₁	1	1	yes	characteristic
65	D₅	—	21	2	no	—
66	Q₈	—	105	3	no	—
67	Q₈	D₁₀₅	1	1	yes	characteristic, direct product factor
68	C₂×C₄	—	105	3	no	—
69	C₇	Q₈×D₁₅	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	C₂²×D₃₅	1	1	yes	characteristic
71	D₃	—	35	2	no	—
72	C₅	Q₈×D₂₁	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	105	1	no	—
74	C₄	—	105	3	no	—
75	C₄	C₂×D₁₀₅	1	3	yes	—
76	C₃	Q₈×D₃₅	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	105	2	no	—
78	C₂	C₂²×D₁₀₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₀₅
commutator, socle, cosocle	C₂×C₃×C₅×C₇	C₂³
Fitting	C₃×C₅×C₇×Q₈	C₂
prime radical	C₂×D₁₀₅	C₂²
hypercenter	Q₈	D₁₀₅
hypocenter	C₃×C₅×C₇	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×Q₈	C₂	semidirect
3. Dic₂₁₀	C₂	semidirect
9. C₅×C₇×Q₈	D₃	semidirect
14. D₁₀₅	Q₈	direct (N × H)
15. C₃×C₇×Q₈	D₅	semidirect
21. C₃×C₅×Q₈	D₇	semidirect
25. C₃×C₅×C₇	C₂×Q₈	semidirect
35. C₇×Q₈	D₁₅	semidirect
41. C₅×Q₈	D₂₁	semidirect
44. C₅×C₇	D₃×Q₈	semidirect
50. C₃×Q₈	D₃₅	semidirect
53. C₃×C₇	Q₈×D₅	semidirect
58. C₃×C₅	Q₈×D₇	semidirect
67. Q₈	D₁₀₅	direct (N × H)
69. C₇	Q₈×D₁₅	semidirect
72. C₅	Q₈×D₂₁	semidirect
76. C₃	Q₈×D₃₅	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×Q₈	2
C₄×D₁₀₅	2
Dic₂₁₀	2
Q₈×D₃₅	3
Q₈×D₂₁	5
Q₈×D₁₅	7

Subgroups — Q8×D105 #13132896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₀₅ #13132896