Subgroups — C₃×C₅×C₂₃×C₂₇ #72188113

Order 9315.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₉×C₂₃	C₃	1	1	yes	maximal, characteristic
2	C₅×C₂₃×C₂₇	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₂₃×C₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃²×C₅×C₂₃	C₉	1	1	yes	characteristic
5	C₅×C₉×C₂₃	C₉	1	2	yes	—
6	C₅×C₉×C₂₃	C₃²	1	1	yes	characteristic
7	C₃×C₉×C₂₃	C₃×C₅	1	1	yes	characteristic
8	C₂₃×C₂₇	C₃×C₅	1	3	yes	direct product factor
9	C₃×C₅×C₂₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₅×C₂₃	C₂₇	1	3	yes	direct product factor
11	C₃×C₅×C₂₃	C₃×C₉	1	1	yes	characteristic
12	C₃²×C₂₃	C₅×C₉	1	1	yes	characteristic
13	C₉×C₂₃	C₅×C₉	1	2	yes	—
14	C₉×C₂₃	C₃²×C₅	1	1	yes	characteristic
15	C₃×C₅×C₉	C₃×C₂₃	1	1	yes	characteristic
16	C₅×C₂₇	C₃×C₂₃	1	3	yes	direct product factor
17	C₅×C₂₃	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₂₇	C₅×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×C₂₃	C₅×C₂₇	1	3	yes	direct product factor
20	C₃×C₂₃	C₃×C₅×C₉	1	1	yes	characteristic
21	C₃²×C₅	C₉×C₂₃	1	1	yes	characteristic
22	C₅×C₉	C₉×C₂₃	1	2	yes	—
23	C₅×C₉	C₃²×C₂₃	1	1	yes	characteristic
24	C₃×C₉	C₃×C₅×C₂₃	1	1	yes	characteristic
25	C₂₇	C₃×C₅×C₂₃	1	3	yes	direct product factor
26	C₂₃	C₃×C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
27	C₃×C₅	C₂₃×C₂₇	1	3	yes	direct product factor
28	C₃×C₅	C₃×C₉×C₂₃	1	1	yes	characteristic
29	C₃²	C₅×C₉×C₂₃	1	1	yes	characteristic
30	C₉	C₅×C₉×C₂₃	1	2	yes	—
31	C₉	C₃²×C₅×C₂₃	1	1	yes	characteristic
32	C₅	C₃×C₂₃×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₅×C₂₃×C₂₇	1	3	yes	direct product factor
34	C₃	C₃×C₅×C₉×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₃²×C₅×C₂₃
socle, prime radical	C₃²×C₅×C₂₃	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₂₃×C₂₇	C₃	direct (N × H)
3. C₃×C₂₃×C₂₇	C₅	direct (N × H)
8. C₂₃×C₂₇	C₃×C₅	direct (N × H)
9. C₃×C₅×C₂₇	C₂₃	direct (N × H)
10. C₃×C₅×C₂₃	C₂₇	direct (N × H)
16. C₅×C₂₇	C₃×C₂₃	direct (N × H)
17. C₅×C₂₃	C₃×C₂₇	direct (N × H)
18. C₃×C₂₇	C₅×C₂₃	direct (N × H)
19. C₃×C₂₃	C₅×C₂₇	direct (N × H)
25. C₂₇	C₃×C₅×C₂₃	direct (N × H)
26. C₂₃	C₃×C₅×C₂₇	direct (N × H)
27. C₃×C₅	C₂₃×C₂₇	direct (N × H)
32. C₅	C₃×C₂₃×C₂₇	direct (N × H)
33. C₃	C₅×C₂₃×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×C₅×C₉×C₂₃	3
C₅×C₂₃×C₂₇	3
C₃×C₂₃×C₂₇	5
C₃×C₅×C₂₇	23

Subgroups — C3×C5×C23×C27 #72188113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₂₃×C₂₇ #72188113