Subgroups — #11761663

Order 1080.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₅	C₂	1	1	yes	maximal, characteristic
2	Dic₅×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#67079	—	3	1	no	maximal
5	C₂×C₅×C₂₇	C₂²	1	1	yes	characteristic
6	D₁₃₅	C₄	1	2	yes	complemented
7	C₄×D₂₇	—	5	1	no	maximal, Hall
8	C₂×D₄₅	—	3	1	no	—
9	C₉×Dic₅	D₃	1	1	yes	characteristic
10	C₅×Dic₉	—	3	1	no	—
11	C₅×C₂₇	C₂×C₄	1	1	yes	characteristic, complemented, Hall
12	#1200	—	9	1	no	—
13	C₂×D₂₇	—	5	1	no	—
14	C₄×C₂₇	—	5	1	no	—
15	Dic₂₇	D₅	1	1	yes	characteristic, complemented
16	C₂×C₅×C₉	C₂×D₃	1	1	yes	characteristic
17	D₄₅	—	3	2	no	—
18	C₄×D₉	—	15	1	no	—
19	C₂×D₁₅	—	9	1	no	—
20	C₃×Dic₅	D₉	1	1	yes	characteristic
21	C₅×Dic₃	—	9	1	no	—
22	C₂×C₂₇	C₂×D₅	1	1	yes	characteristic
23	D₂₇	—	5	2	no	—
24	C₅×C₉	C₄×D₃	1	1	yes	characteristic
25	C₄×D₅	—	27	1	no	Hall
26	C₂×D₉	—	15	1	no	—
27	C₄×C₉	—	5	1	no	—
28	Dic₉	—	3	1	no	—
29	C₂×C₃×C₅	C₂×D₉	1	1	yes	characteristic
30	D₁₅	—	9	2	no	—
31	C₂₇	C₄×D₅	1	1	yes	characteristic, complemented, Sylow
32	C₄×D₃	—	45	1	no	—
33	C₂×D₅	—	27	1	no	—
34	C₄×C₅	—	27	1	no	—
35	Dic₅	D₂₇	1	1	yes	characteristic, complemented
36	C₂×C₉	D₃×D₅	1	1	yes	characteristic
37	D₉	—	15	2	no	—
38	C₃×C₅	C₄×D₉	1	1	yes	characteristic
39	C₂×D₃	—	45	1	no	—
40	C₃×C₄	—	5	1	no	—
41	Dic₃	—	9	1	no	—
42	C₂×C₅	C₂×D₂₇	1	1	yes	characteristic
43	D₅	—	27	2	no	—
44	C₉	#1200	1	1	yes	characteristic
45	C₂×C₄	—	135	1	no	Sylow
46	C₂×C₃	D₅×D₉	1	1	yes	characteristic
47	D₃	—	45	2	no	—
48	C₅	C₄×D₂₇	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	135	1	no	—
50	C₄	—	5	1	no	—
51	C₄	—	27	1	no	—
52	C₃	#67079	1	1	yes	characteristic
53	C₂	—	135	2	no	—
54	C₂	D₅×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₅×D₂₇
commutator, hypocenter	C₅×C₂₇	C₂×C₄
Frattini	C₂×C₉	D₃×D₅
Fitting, cosocle	C₂×C₅×C₂₇	C₂²
socle	C₂×C₃×C₅	C₂×D₉
prime radical	C₂×D₁₃₅	C₂
LCS	G ▷ C₅×C₂₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. Dic₅×C₂₇	C₂	semidirect
3. C₅×Dic₂₇	C₂	semidirect
6. D₁₃₅	C₄	semidirect
11. C₅×C₂₇	C₂×C₄	semidirect
15. Dic₂₇	D₅	semidirect
31. C₂₇	C₄×D₅	semidirect
35. Dic₅	D₂₇	semidirect
48. C₅	C₄×D₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃₅	2
Dic₅×C₂₇	2
C₅×Dic₂₇	2
#67079	3
C₄×D₂₇	5