Subgroups — C₂⁵×C₅×C₂₇ #72183866

Order 4320.

2992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₂₇	C₂	1	31	yes	maximal, direct product factor
2	C₂⁵×C₅×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₅×C₂₇	C₂²	1	155	yes	direct product factor
4	C₂⁵×C₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁴×C₅×C₉	C₂×C₃	1	31	yes	—
6	C₂²×C₅×C₂₇	C₂³	1	155	yes	direct product factor
7	C₂⁵×C₃×C₅	C₉	1	1	yes	characteristic
8	C₂⁴×C₂₇	C₂×C₅	1	31	yes	direct product factor
9	C₂³×C₅×C₉	C₂²×C₃	1	155	yes	—
10	C₂⁵×C₉	C₃×C₅	1	1	yes	characteristic
11	C₂×C₅×C₂₇	C₂⁴	1	31	yes	direct product factor
12	C₂⁴×C₃×C₅	C₂×C₉	1	31	yes	—
13	C₂³×C₂₇	C₂²×C₅	1	155	yes	direct product factor
14	C₂²×C₅×C₉	C₂³×C₃	1	155	yes	—
15	C₂⁵×C₅	C₂₇	1	1	yes	characteristic, direct product factor, Hall
16	C₂⁴×C₉	C₂×C₃×C₅	1	31	yes	—
17	C₅×C₂₇	C₂⁵	1	1	yes	characteristic, direct product factor, Hall
18	C₂³×C₃×C₅	C₂²×C₉	1	155	yes	—
19	C₂²×C₂₇	C₂³×C₅	1	155	yes	direct product factor
20	C₂⁵×C₃	C₅×C₉	1	1	yes	characteristic
21	C₂×C₅×C₉	C₂⁴×C₃	1	31	yes	—
22	C₂⁴×C₅	C₂×C₂₇	1	31	yes	direct product factor
23	C₂³×C₉	C₂²×C₃×C₅	1	155	yes	—
24	C₂²×C₃×C₅	C₂³×C₉	1	155	yes	—
25	C₂×C₂₇	C₂⁴×C₅	1	31	yes	direct product factor
26	C₂⁴×C₃	C₂×C₅×C₉	1	31	yes	—
27	C₅×C₉	C₂⁵×C₃	1	1	yes	characteristic
28	C₂³×C₅	C₂²×C₂₇	1	155	yes	direct product factor
29	C₂²×C₉	C₂³×C₃×C₅	1	155	yes	—
30	C₂⁵	C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃×C₅	C₂⁴×C₉	1	31	yes	—
32	C₂₇	C₂⁵×C₅	1	1	yes	characteristic, direct product factor, Sylow
33	C₂³×C₃	C₂²×C₅×C₉	1	155	yes	—
34	C₂²×C₅	C₂³×C₂₇	1	155	yes	direct product factor
35	C₂×C₉	C₂⁴×C₃×C₅	1	31	yes	—
36	C₂⁴	C₂×C₅×C₂₇	1	31	yes	direct product factor
37	C₃×C₅	C₂⁵×C₉	1	1	yes	characteristic
38	C₂²×C₃	C₂³×C₅×C₉	1	155	yes	—
39	C₂×C₅	C₂⁴×C₂₇	1	31	yes	direct product factor
40	C₉	C₂⁵×C₃×C₅	1	1	yes	characteristic
41	C₂³	C₂²×C₅×C₂₇	1	155	yes	direct product factor
42	C₂×C₃	C₂⁴×C₅×C₉	1	31	yes	—
43	C₅	C₂⁵×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²	C₂³×C₅×C₂₇	1	155	yes	direct product factor
45	C₃	C₂⁵×C₅×C₉	1	1	yes	characteristic
46	C₂	C₂⁴×C₅×C₂₇	1	31	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂⁵×C₃×C₅
socle, prime radical	C₂⁵×C₃×C₅	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×C₂₇	C₂	direct (N × H)
3. C₂³×C₅×C₂₇	C₂²	direct (N × H)
4. C₂⁵×C₂₇	C₅	direct (N × H)
6. C₂²×C₅×C₂₇	C₂³	direct (N × H)
8. C₂⁴×C₂₇	C₂×C₅	direct (N × H)
11. C₂×C₅×C₂₇	C₂⁴	direct (N × H)
13. C₂³×C₂₇	C₂²×C₅	direct (N × H)
15. C₂⁵×C₅	C₂₇	direct (N × H)
17. C₅×C₂₇	C₂⁵	direct (N × H)
19. C₂²×C₂₇	C₂³×C₅	direct (N × H)
22. C₂⁴×C₅	C₂×C₂₇	direct (N × H)
25. C₂×C₂₇	C₂⁴×C₅	direct (N × H)
28. C₂³×C₅	C₂²×C₂₇	direct (N × H)
30. C₂⁵	C₅×C₂₇	direct (N × H)
32. C₂₇	C₂⁵×C₅	direct (N × H)
34. C₂²×C₅	C₂³×C₂₇	direct (N × H)
36. C₂⁴	C₂×C₅×C₂₇	direct (N × H)
39. C₂×C₅	C₂⁴×C₂₇	direct (N × H)
41. C₂³	C₂²×C₅×C₂₇	direct (N × H)
43. C₅	C₂⁵×C₂₇	direct (N × H)
44. C₂²	C₂³×C₅×C₂₇	direct (N × H)
46. C₂	C₂⁴×C₅×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
C₂⁴×C₅×C₂₇	2
C₂⁵×C₅×C₉	3
C₂⁵×C₂₇	5

Subgroups — C25×C5×C27 #72183866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₅×C₂₇ #72183866