Subgroups — D₂₇₄₀₀ #72290708

Order 54800.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇₀₀	C₂	1	2	yes	maximal, complemented
2	D₅₄₈₀	—	5	1	no	maximal
3	D₂₇₄₀	—	5	2	no	—
4	C₈×C₂₅×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×C₈×C₁₃₇	D₅	1	1	yes	characteristic
6	C₂×D₃₄₂₅	—	2	2	no	—
7	C₄×C₂₅×C₁₃₇	C₂²	1	1	yes	characteristic
8	D₃₄₂₅	—	4	2	no	—
9	C₂×C₂₅×C₁₃₇	D₄	1	1	yes	characteristic
10	C₂₅×C₁₃₇	D₈	1	1	yes	characteristic, complemented, Hall
11	C₂×D₆₈₅	—	10	2	no	—
12	C₄×C₅×C₁₃₇	C₂×D₅	1	1	yes	characteristic
13	D₁₀₉₆	—	25	1	no	Hall
14	C₂×C₅×C₁₃₇	D₂₀	1	1	yes	characteristic
15	D₆₈₅	—	20	2	no	—
16	D₅₄₈	—	25	2	no	—
17	C₈×C₁₃₇	D₂₅	1	1	yes	characteristic, complemented
18	C₅×C₁₃₇	D₄₀	1	1	yes	characteristic
19	C₂×D₁₃₇	—	50	2	no	—
20	C₄×C₁₃₇	C₂×D₂₅	1	1	yes	characteristic
21	D₂₀₀	—	137	1	no	maximal, Hall
22	C₂×C₁₃₇	D₁₀₀	1	1	yes	characteristic
23	D₁₃₇	—	100	2	no	—
24	D₁₀₀	—	137	2	no	—
25	C₈×C₂₅	D₁₃₇	1	1	yes	characteristic, complemented
26	C₁₃₇	D₂₀₀	1	1	yes	characteristic, complemented, Sylow
27	C₂×D₂₅	—	274	2	no	—
28	C₄×C₂₅	C₂×D₁₃₇	1	1	yes	characteristic
29	D₄₀	—	685	1	no	—
30	C₂×C₂₅	D₅₄₈	1	1	yes	characteristic
31	D₂₅	—	548	2	no	—
32	D₂₀	—	685	2	no	—
33	C₅×C₈	D₆₈₅	1	1	yes	characteristic
34	C₂₅	D₁₀₉₆	1	1	yes	characteristic, complemented, Sylow
35	C₂×D₅	—	1370	2	no	—
36	C₄×C₅	C₂×D₆₈₅	1	1	yes	characteristic
37	D₈	—	3425	1	no	Sylow
38	C₂×C₅	D₂₇₄₀	1	1	yes	characteristic
39	D₅	—	2740	2	no	—
40	D₄	—	3425	2	no	—
41	C₈	D₃₄₂₅	1	1	yes	characteristic, complemented
42	C₅	D₅₄₈₀	1	1	yes	characteristic
43	C₂²	—	6850	2	no	—
44	C₄	C₂×D₃₄₂₅	1	1	yes	characteristic
45	C₂	—	13700	2	no	—
46	C₂	D₁₃₇₀₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₃₇₀₀
commutator, cosocle	C₄×C₂₅×C₁₃₇	C₂²
Frattini	C₄×C₅	C₂×D₆₈₅
Fitting	C₈×C₂₅×C₁₃₇	C₂
socle	C₂×C₅×C₁₃₇	D₂₀
hypercenter	C₈	D₃₄₂₅
hypocenter	C₂₅×C₁₃₇	D₈
LCS	G ▷ C₄×C₂₅×C₁₃₇ ▷ C₂×C₂₅×C₁₃₇ ▷ C₂₅×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₁₃₇₀₀	C₂	semidirect
4. C₈×C₂₅×C₁₃₇	C₂	semidirect
10. C₂₅×C₁₃₇	D₈	semidirect
17. C₈×C₁₃₇	D₂₅	semidirect
25. C₈×C₂₅	D₁₃₇	semidirect
26. C₁₃₇	D₂₀₀	semidirect
34. C₂₅	D₁₀₉₆	semidirect
41. C₈	D₃₄₂₅	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇₀₀	2
D₅₄₈₀	5
C₈×C₂₅×C₁₃₇	2
D₂₀₀	137

Subgroups — D27400 #72290708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₇₄₀₀ #72290708