Subgroups — D₁₀₄₈₈ #72215403

Order 20976.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₂₄₄	C₂	1	2	yes	maximal, complemented
2	D₃₄₉₆	—	3	1	no	maximal, Hall
3	D₁₇₄₈	—	3	2	no	—
4	C₃×C₈×C₁₉×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₉×C₂₃	D₃	1	1	yes	characteristic, complemented
6	C₂×D₁₃₁₁	—	2	2	no	—
7	C₃×C₄×C₁₉×C₂₃	C₂²	1	1	yes	characteristic
8	D₁₃₁₁	—	4	2	no	—
9	C₂×C₃×C₁₉×C₂₃	D₄	1	1	yes	characteristic
10	C₂×D₄₃₇	—	6	2	no	—
11	C₄×C₁₉×C₂₃	C₂×D₃	1	1	yes	characteristic
12	C₃×C₁₉×C₂₃	D₈	1	1	yes	characteristic, complemented, Hall
13	D₅₅₂	—	19	1	no	maximal, Hall
14	D₄₅₆	—	23	1	no	maximal, Hall
15	C₂×C₁₉×C₂₃	D₁₂	1	1	yes	characteristic
16	D₄₃₇	—	12	2	no	—
17	D₂₇₆	—	19	2	no	—
18	C₃×C₈×C₂₃	D₁₉	1	1	yes	characteristic, complemented
19	D₂₂₈	—	23	2	no	—
20	C₃×C₈×C₁₉	D₂₃	1	1	yes	characteristic, complemented
21	C₁₉×C₂₃	D₂₄	1	1	yes	characteristic, complemented, Hall
22	D₁₈₄	—	57	1	no	Hall
23	D₁₅₂	—	69	1	no	Hall
24	C₂×D₆₉	—	38	2	no	—
25	C₃×C₄×C₂₃	C₂×D₁₉	1	1	yes	characteristic
26	C₂×D₅₇	—	46	2	no	—
27	C₃×C₄×C₁₉	C₂×D₂₃	1	1	yes	characteristic
28	D₉₂	—	57	2	no	—
29	C₈×C₂₃	D₅₇	1	1	yes	characteristic, complemented
30	D₇₆	—	69	2	no	—
31	C₈×C₁₉	D₆₉	1	1	yes	characteristic, complemented
32	C₂×C₃×C₂₃	D₇₆	1	1	yes	characteristic
33	D₆₉	—	76	2	no	—
34	C₂×C₃×C₁₉	D₉₂	1	1	yes	characteristic
35	D₅₇	—	92	2	no	—
36	C₂×D₂₃	—	114	2	no	—
37	C₄×C₂₃	C₂×D₅₇	1	1	yes	characteristic
38	C₂×D₁₉	—	138	2	no	—
39	C₄×C₁₉	C₂×D₆₉	1	1	yes	characteristic
40	C₃×C₂₃	D₁₅₂	1	1	yes	characteristic, complemented, Hall
41	C₃×C₁₉	D₁₈₄	1	1	yes	characteristic, complemented, Hall
42	D₂₄	—	437	1	no	Hall
43	C₂×C₂₃	D₂₂₈	1	1	yes	characteristic
44	D₂₃	—	228	2	no	—
45	C₂×C₁₉	D₂₇₆	1	1	yes	characteristic
46	D₁₉	—	276	2	no	—
47	D₁₂	—	437	2	no	—
48	C₃×C₈	D₄₃₇	1	1	yes	characteristic, complemented
49	C₂₃	D₄₅₆	1	1	yes	characteristic, complemented, Sylow
50	C₁₉	D₅₅₂	1	1	yes	characteristic, complemented, Sylow
51	D₈	—	1311	1	no	Sylow
52	C₂×D₃	—	874	2	no	—
53	C₃×C₄	C₂×D₄₃₇	1	1	yes	characteristic
54	D₄	—	1311	2	no	—
55	C₈	D₁₃₁₁	1	1	yes	characteristic, complemented
56	C₂×C₃	D₁₇₄₈	1	1	yes	characteristic
57	D₃	—	1748	2	no	—
58	C₂²	—	2622	2	no	—
59	C₄	C₂×D₁₃₁₁	1	1	yes	characteristic
60	C₃	D₃₄₉₆	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	5244	2	no	—
62	C₂	D₅₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅₂₄₄
commutator, cosocle	C₃×C₄×C₁₉×C₂₃	C₂²
Frattini	C₄	C₂×D₁₃₁₁
Fitting	C₃×C₈×C₁₉×C₂₃	C₂
socle	C₂×C₃×C₁₉×C₂₃	D₄
hypercenter	C₈	D₁₃₁₁
hypocenter	C₃×C₁₉×C₂₃	D₈
LCS	G ▷ C₃×C₄×C₁₉×C₂₃ ▷ C₂×C₃×C₁₉×C₂₃ ▷ C₃×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₅₂₄₄	C₂	semidirect
4. C₃×C₈×C₁₉×C₂₃	C₂	semidirect
5. C₈×C₁₉×C₂₃	D₃	semidirect
12. C₃×C₁₉×C₂₃	D₈	semidirect
18. C₃×C₈×C₂₃	D₁₉	semidirect
20. C₃×C₈×C₁₉	D₂₃	semidirect
21. C₁₉×C₂₃	D₂₄	semidirect
29. C₈×C₂₃	D₅₇	semidirect
31. C₈×C₁₉	D₆₉	semidirect
40. C₃×C₂₃	D₁₅₂	semidirect
41. C₃×C₁₉	D₁₈₄	semidirect
48. C₃×C₈	D₄₃₇	semidirect
49. C₂₃	D₄₅₆	semidirect
50. C₁₉	D₅₅₂	semidirect
55. C₈	D₁₃₁₁	semidirect
60. C₃	D₃₄₉₆	semidirect

Maximal subgroups

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

subgroup	index
D₅₂₄₄	2
D₃₄₉₆	3
C₃×C₈×C₁₉×C₂₃	2
D₅₅₂	19
D₄₅₆	23

Subgroups — D10488 #72215403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₀₄₈₈ #72215403