Subgroups — D₂₇₁₀₈ #72290439

Order 54216.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₇₇₇	C₂	1	2	yes	maximal, complemented
2	D₉₀₃₆	—	3	1	no	maximal
3	D₆₇₇₇	—	2	2	no	—
4	D₃₀₁₂	—	9	1	no	—
5	C₄×C₂₇×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₇×C₂₅₁	C₂²	1	1	yes	characteristic
7	C₂×D₂₂₅₉	—	3	2	no	—
8	C₄×C₉×C₂₅₁	D₃	1	1	yes	characteristic
9	C₂₇×C₂₅₁	D₄	1	1	yes	characteristic, complemented, Hall
10	D₂₂₅₉	—	6	2	no	—
11	C₂×C₉×C₂₅₁	C₂×D₃	1	1	yes	characteristic
12	C₂×D₇₅₃	—	9	2	no	—
13	C₃×C₄×C₂₅₁	D₉	1	1	yes	characteristic
14	C₉×C₂₅₁	D₁₂	1	1	yes	characteristic
15	D₁₀₀₄	—	27	1	no	Hall
16	C₂×C₃×C₂₅₁	C₂×D₉	1	1	yes	characteristic
17	D₇₅₃	—	18	2	no	—
18	C₂×D₂₅₁	—	27	2	no	—
19	C₄×C₂₅₁	D₂₇	1	1	yes	characteristic, complemented
20	C₃×C₂₅₁	D₃₆	1	1	yes	characteristic
21	C₂×C₂₅₁	C₂×D₂₇	1	1	yes	characteristic
22	D₂₅₁	—	54	2	no	—
23	C₂₅₁	D₁₀₈	1	1	yes	characteristic, complemented, Sylow
24	D₁₀₈	—	251	1	no	maximal, Hall
25	C₂×D₂₇	—	251	2	no	—
26	C₄×C₂₇	D₂₅₁	1	1	yes	characteristic, complemented
27	D₃₆	—	753	1	no	—
28	C₂×C₂₇	C₂×D₂₅₁	1	1	yes	characteristic
29	D₂₇	—	502	2	no	—
30	C₂×D₉	—	753	2	no	—
31	C₄×C₉	D₇₅₃	1	1	yes	characteristic
32	C₂₇	D₁₀₀₄	1	1	yes	characteristic, complemented, Sylow
33	D₁₂	—	2259	1	no	—
34	C₂×C₉	C₂×D₇₅₃	1	1	yes	characteristic
35	D₉	—	1506	2	no	—
36	C₂×D₃	—	2259	2	no	—
37	C₃×C₄	D₂₂₅₉	1	1	yes	characteristic
38	C₉	D₃₀₁₂	1	1	yes	characteristic
39	D₄	—	6777	1	no	Sylow
40	C₂×C₃	C₂×D₂₂₅₉	1	1	yes	characteristic
41	D₃	—	4518	2	no	—
42	C₂²	—	6777	2	no	—
43	C₄	D₆₇₇₇	1	1	yes	characteristic, complemented
44	C₃	D₉₀₃₆	1	1	yes	characteristic
45	C₂	—	13554	2	no	—
46	C₂	C₂×D₆₇₇₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₆₇₇₇
commutator, cosocle	C₂×C₂₇×C₂₅₁	C₂²
Frattini	C₂×C₉	C₂×D₇₅₃
Fitting	C₄×C₂₇×C₂₅₁	C₂
socle	C₂×C₃×C₂₅₁	C₂×D₉
hypercenter	C₄	D₆₇₇₇
hypocenter	C₂₇×C₂₅₁	D₄
LCS	G ▷ C₂×C₂₇×C₂₅₁ ▷ C₂₇×C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₆₇₇₇	C₂	semidirect
5. C₄×C₂₇×C₂₅₁	C₂	semidirect
9. C₂₇×C₂₅₁	D₄	semidirect
19. C₄×C₂₅₁	D₂₇	semidirect
23. C₂₅₁	D₁₀₈	semidirect
26. C₄×C₂₇	D₂₅₁	semidirect
32. C₂₇	D₁₀₀₄	semidirect
43. C₄	D₆₇₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₆₇₇₇	2
D₉₀₃₆	3
C₄×C₂₇×C₂₅₁	2
D₁₀₈	251

Subgroups — D27108 #72290439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₇₁₀₈ #72290439