Subgroups — C₃×C₅²×C₁₁×C₂₇ #72199722

Order 22275.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₉×C₁₁	C₃	1	1	yes	maximal, characteristic
2	C₅²×C₁₁×C₂₇	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₅×C₁₁×C₂₇	C₅	1	6	yes	maximal, direct product factor
4	C₃×C₅²×C₂₇	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃²×C₅²×C₁₁	C₉	1	1	yes	characteristic
6	C₅²×C₉×C₁₁	C₉	1	2	yes	—
7	C₅²×C₉×C₁₁	C₃²	1	1	yes	characteristic
8	C₃×C₅×C₉×C₁₁	C₃×C₅	1	6	yes	—
9	C₅×C₁₁×C₂₇	C₃×C₅	1	18	yes	direct product factor
10	C₃×C₁₁×C₂₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₅²×C₁₁	C₂₇	1	3	yes	direct product factor
12	C₃×C₅²×C₁₁	C₃×C₉	1	1	yes	characteristic
13	C₃×C₅²×C₉	C₃×C₁₁	1	1	yes	characteristic
14	C₅²×C₂₇	C₃×C₁₁	1	3	yes	direct product factor
15	C₃²×C₅×C₁₁	C₅×C₉	1	6	yes	—
16	C₅×C₉×C₁₁	C₅×C₉	1	12	yes	—
17	C₅×C₉×C₁₁	C₃²×C₅	1	6	yes	—
18	C₃×C₅×C₂₇	C₅×C₁₁	1	6	yes	direct product factor
19	C₃×C₉×C₁₁	C₃×C₅²	1	1	yes	characteristic
20	C₁₁×C₂₇	C₃×C₅²	1	3	yes	direct product factor
21	C₅²×C₁₁	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₅²	C₉×C₁₁	1	1	yes	characteristic
23	C₅²×C₉	C₉×C₁₁	1	2	yes	—
24	C₅²×C₉	C₃²×C₁₁	1	1	yes	characteristic
25	C₃×C₅×C₁₁	C₅×C₂₇	1	18	yes	direct product factor
26	C₃×C₅×C₁₁	C₃×C₅×C₉	1	6	yes	—
27	C₃×C₅×C₉	C₃×C₅×C₁₁	1	6	yes	—
28	C₅×C₂₇	C₃×C₅×C₁₁	1	18	yes	direct product factor
29	C₃²×C₁₁	C₅²×C₉	1	1	yes	characteristic
30	C₉×C₁₁	C₅²×C₉	1	2	yes	—
31	C₉×C₁₁	C₃²×C₅²	1	1	yes	characteristic
32	C₃×C₂₇	C₅²×C₁₁	1	1	yes	characteristic, direct product factor, Sylow
33	C₃×C₅²	C₁₁×C₂₇	1	3	yes	direct product factor
34	C₃×C₅²	C₃×C₉×C₁₁	1	1	yes	characteristic
35	C₅×C₁₁	C₃×C₅×C₂₇	1	6	yes	direct product factor
36	C₃²×C₅	C₅×C₉×C₁₁	1	6	yes	—
37	C₅×C₉	C₅×C₉×C₁₁	1	12	yes	—
38	C₅×C₉	C₃²×C₅×C₁₁	1	6	yes	—
39	C₃×C₁₁	C₅²×C₂₇	1	3	yes	direct product factor
40	C₃×C₁₁	C₃×C₅²×C₉	1	1	yes	characteristic
41	C₃×C₉	C₃×C₅²×C₁₁	1	1	yes	characteristic
42	C₂₇	C₃×C₅²×C₁₁	1	3	yes	direct product factor
43	C₅²	C₃×C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₃×C₅	C₅×C₁₁×C₂₇	1	18	yes	direct product factor
45	C₃×C₅	C₃×C₅×C₉×C₁₁	1	6	yes	—
46	C₁₁	C₃×C₅²×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
47	C₃²	C₅²×C₉×C₁₁	1	1	yes	characteristic
48	C₉	C₅²×C₉×C₁₁	1	2	yes	—
49	C₉	C₃²×C₅²×C₁₁	1	1	yes	characteristic
50	C₅	C₃×C₅×C₁₁×C₂₇	1	6	yes	direct product factor
51	C₃	C₅²×C₁₁×C₂₇	1	3	yes	direct product factor
52	C₃	C₃×C₅²×C₉×C₁₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₃²×C₅²×C₁₁
socle, prime radical	C₃²×C₅²×C₁₁	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅²×C₁₁×C₂₇	C₃	direct (N × H)
3. C₃×C₅×C₁₁×C₂₇	C₅	direct (N × H)
4. C₃×C₅²×C₂₇	C₁₁	direct (N × H)
9. C₅×C₁₁×C₂₇	C₃×C₅	direct (N × H)
10. C₃×C₁₁×C₂₇	C₅²	direct (N × H)
11. C₃×C₅²×C₁₁	C₂₇	direct (N × H)
14. C₅²×C₂₇	C₃×C₁₁	direct (N × H)
18. C₃×C₅×C₂₇	C₅×C₁₁	direct (N × H)
20. C₁₁×C₂₇	C₃×C₅²	direct (N × H)
21. C₅²×C₁₁	C₃×C₂₇	direct (N × H)
25. C₃×C₅×C₁₁	C₅×C₂₇	direct (N × H)
28. C₅×C₂₇	C₃×C₅×C₁₁	direct (N × H)
32. C₃×C₂₇	C₅²×C₁₁	direct (N × H)
33. C₃×C₅²	C₁₁×C₂₇	direct (N × H)
35. C₅×C₁₁	C₃×C₅×C₂₇	direct (N × H)
39. C₃×C₁₁	C₅²×C₂₇	direct (N × H)
42. C₂₇	C₃×C₅²×C₁₁	direct (N × H)
43. C₅²	C₃×C₁₁×C₂₇	direct (N × H)
44. C₃×C₅	C₅×C₁₁×C₂₇	direct (N × H)
46. C₁₁	C₃×C₅²×C₂₇	direct (N × H)
50. C₅	C₃×C₅×C₁₁×C₂₇	direct (N × H)
51. C₃	C₅²×C₁₁×C₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₉×C₁₁	3
C₅²×C₁₁×C₂₇	3
C₃×C₅×C₁₁×C₂₇	5
C₃×C₅²×C₂₇	11

Subgroups — C3×C52×C11×C27 #72199722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₁₁×C₂₇ #72199722