Subgroups — Dic₁₅₂₀₀ #72292736

Order 60800.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₇₆₀₀	C₂	1	2	yes	maximal
2	Dic₃₈₀₀	—	2	2	no	—
3	Dic₃₀₄₀	—	5	1	no	maximal
4	Dic₁₉₀₀	—	4	2	no	—
5	Dic₁₅₂₀	—	5	2	no	—
6	Dic₉₅₀	—	8	2	no	—
7	Dic₈₀₀	—	19	1	no	maximal, Hall
8	Dic₇₆₀	—	10	2	no	—
9	C₁₉×C₂₅×C₆₄	C₂	1	1	yes	maximal, characteristic
10	C₁₉×C₂₅×C₃₂	C₂²	1	1	yes	characteristic
11	C₁₆×C₁₉×C₂₅	D₄	1	1	yes	characteristic
12	C₅×C₁₉×C₆₄	D₅	1	1	yes	characteristic
13	C₈×C₁₉×C₂₅	D₈	1	1	yes	characteristic
14	C₅×C₁₉×C₃₂	C₂×D₅	1	1	yes	characteristic
15	Dic₆₀₈	—	25	1	no	Hall
16	C₄×C₁₉×C₂₅	D₁₆	1	1	yes	characteristic
17	Dic₄₇₅	—	16	2	no	—
18	Dic₄₀₀	—	19	2	no	—
19	C₂₅×C₆₄	D₁₉	1	1	yes	characteristic
20	Dic₃₈₀	—	20	2	no	—
21	C₅×C₁₆×C₁₉	D₂₀	1	1	yes	characteristic
22	Dic₃₀₄	—	25	2	no	—
23	C₁₉×C₆₄	D₂₅	1	1	yes	characteristic
24	C₂×C₁₉×C₂₅	D₃₂	1	1	yes	characteristic
25	Dic₂₀₀	—	38	2	no	—
26	C₂₅×C₃₂	C₂×D₁₉	1	1	yes	characteristic
27	Dic₁₉₀	—	40	2	no	—
28	C₅×C₈×C₁₉	D₄₀	1	1	yes	characteristic
29	Dic₁₆₀	—	95	1	no	—
30	Dic₁₅₂	—	50	2	no	—
31	C₁₉×C₃₂	C₂×D₂₅	1	1	yes	characteristic
32	C₁₉×C₂₅	Q₁₂₈	1	1	yes	characteristic, complemented, Hall
33	Dic₁₀₀	—	76	2	no	—
34	C₁₆×C₂₅	D₇₆	1	1	yes	characteristic
35	C₄×C₅×C₁₉	D₈₀	1	1	yes	characteristic
36	Dic₉₅	—	80	2	no	—
37	Dic₈₀	—	95	2	no	—
38	C₅×C₆₄	D₉₅	1	1	yes	characteristic
39	Dic₇₆	—	100	2	no	—
40	C₁₆×C₁₉	D₁₀₀	1	1	yes	characteristic
41	Dic₅₀	—	152	2	no	—
42	C₈×C₂₅	D₁₅₂	1	1	yes	characteristic
43	C₂×C₅×C₁₉	D₁₆₀	1	1	yes	characteristic
44	Dic₄₀	—	190	2	no	—
45	C₅×C₃₂	C₂×D₉₅	1	1	yes	characteristic
46	Dic₃₈	—	200	2	no	—
47	C₈×C₁₉	D₂₀₀	1	1	yes	characteristic
48	Q₁₂₈	—	475	1	no	Sylow
49	C₄×C₂₅	D₃₀₄	1	1	yes	characteristic
50	Dic₂₅	—	304	2	no	—
51	C₅×C₁₉	Dic₁₆₀	1	1	yes	characteristic
52	Dic₂₀	—	380	2	no	—
53	C₅×C₁₆	D₃₈₀	1	1	yes	characteristic
54	C₄×C₁₉	D₄₀₀	1	1	yes	characteristic
55	Dic₁₉	—	400	2	no	—
56	Q₆₄	—	475	2	no	—
57	C₆₄	D₄₇₅	1	1	yes	characteristic
58	C₂×C₂₅	D₆₀₈	1	1	yes	characteristic
59	Dic₁₀	—	760	2	no	—
60	C₅×C₈	D₇₆₀	1	1	yes	characteristic
61	C₂×C₁₉	D₈₀₀	1	1	yes	characteristic
62	Q₃₂	—	950	2	no	—
63	C₃₂	C₂×D₄₇₅	1	1	yes	characteristic
64	C₂₅	Dic₆₀₈	1	1	yes	characteristic, complemented, Sylow
65	C₄×C₅	D₁₅₂₀	1	1	yes	characteristic
66	Dic₅	—	1520	2	no	—
67	C₁₉	Dic₈₀₀	1	1	yes	characteristic, complemented, Sylow
68	Q₁₆	—	1900	2	no	—
69	C₁₆	D₁₉₀₀	1	1	yes	characteristic
70	C₂×C₅	D₃₀₄₀	1	1	yes	characteristic
71	Q₈	—	3800	2	no	—
72	C₈	D₃₈₀₀	1	1	yes	characteristic
73	C₅	Dic₃₀₄₀	1	1	yes	characteristic
74	C₄	—	7600	2	no	—
75	C₄	D₇₆₀₀	1	1	yes	characteristic
76	C₂	D₁₅₂₀₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₅₂₀₀
commutator, cosocle	C₁₉×C₂₅×C₃₂	C₂²
Frattini	C₅×C₃₂	C₂×D₉₅
Fitting	C₁₉×C₂₅×C₆₄	C₂
socle, prime radical	C₂×C₅×C₁₉	D₁₆₀
hypercenter	C₆₄	D₄₇₅
hypocenter	C₁₉×C₂₅	Q₁₂₈
LCS	G ▷ C₁₉×C₂₅×C₃₂ ▷ C₁₆×C₁₉×C₂₅ ▷ C₈×C₁₉×C₂₅ ▷ C₄×C₁₉×C₂₅ ▷ C₂×C₁₉×C₂₅ ▷ C₁₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
32. C₁₉×C₂₅	Q₁₂₈	semidirect
64. C₂₅	Dic₆₀₈	semidirect
67. C₁₉	Dic₈₀₀	semidirect

Maximal subgroups

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

subgroup	index
Dic₇₆₀₀	2
Dic₃₀₄₀	5
Dic₈₀₀	19
C₁₉×C₂₅×C₆₄	2

Subgroups — Dic15200 #72292736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₅₂₀₀ #72292736