Subgroups — C₂₇×H(3,𝔽₃)×C₅₃ #50311599

Order 38637.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇×C₅₃	C₃	1	4	yes	maximal, complemented
2	C₉×H(3,𝔽₃)×C₅₃	C₃	1	1	yes	maximal, characteristic
3	C₅₃×C₃²⋊C₂₇	C₃	1	8	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₅₃	C₉	1	1	yes	characteristic
5	C₃²×C₉×C₅₃	C₃²	1	4	yes	—
6	C₃×C₂₇×C₅₃	—	3	12	no	—
7	C₃×C₂₇×C₅₃	C₃²	1	8	yes	—
8	C₃×C₂₇×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×C₃²⋊C₉	C₉	1	8	yes	—
10	C₃³×C₅₃	C₃×C₉	1	4	yes	—
11	H(3,𝔽₃)×C₅₃	C₂₇	1	9	yes	direct product factor
12	C₃×C₉×C₅₃	—	3	4	no	—
13	C₃×C₉×C₅₃	—	3	8	no	—
14	C₃×C₉×C₅₃	C₃×C₉	1	8	yes	—
15	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
16	C₂₇×C₅₃	—	3	8	no	—
17	C₂₇×C₅₃	H(3,𝔽₃)	1	3	yes	direct product factor
18	C₂₇×H(3,𝔽₃)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃²×C₅₃	—	3	4	no	—
20	C₃²×C₅₃	—	3	8	no	—
21	C₃²×C₅₃	C₃×C₂₇	1	12	yes	complemented
22	C₃²×C₅₃	C₃²×C₉	1	1	yes	characteristic
23	C₉×C₅₃	—	3	8	no	—
24	C₉×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₉×C₅₃	#737	1	2	yes	—
26	C₃²×C₂₇	C₃×C₅₃	1	4	yes	complemented
27	C₉×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic
28	C₃²⋊C₂₇	C₃×C₅₃	1	8	yes	complemented
29	C₃×C₅₃	—	3	12	no	—
30	C₃×C₅₃	C₉×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₅₃	C₃²×C₂₇	1	1	yes	characteristic
32	C₃×C₅₃	#6903	1	2	yes	—
33	C₃×H(3,𝔽₃)	C₉×C₅₃	1	1	yes	characteristic
34	C₃²×C₉	C₃²×C₅₃	1	4	yes	—
35	C₃×C₂₇	—	3	12	no	—
36	C₃×C₂₇	C₃²×C₅₃	1	8	yes	—
37	C₃×C₂₇	C₃²×C₅₃	1	1	yes	characteristic
38	C₃²⋊C₉	C₉×C₅₃	1	8	yes	—
39	C₅₃	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
40	C₃³	C₃×C₉×C₅₃	1	4	yes	—
41	H(3,𝔽₃)	C₂₇×C₅₃	1	9	yes	direct product factor
42	C₃×C₉	—	3	4	no	—
43	C₃×C₉	—	3	8	no	—
44	C₃×C₉	C₃×C₉×C₅₃	1	8	yes	—
45	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
46	C₂₇	—	3	8	no	—
47	C₂₇	H(3,𝔽₃)×C₅₃	1	3	yes	direct product factor
48	C₃²	—	3	4	no	—
49	C₃²	—	3	8	no	—
50	C₃²	C₃×C₂₇×C₅₃	1	12	yes	complemented
51	C₃²	C₃²×C₉×C₅₃	1	1	yes	characteristic
52	C₉	—	3	8	no	—
53	C₉	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
54	C₉	C₅₃×#737	1	2	yes	—
55	C₃	—	3	12	no	—
56	C₃	C₉×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
57	C₃	C₃²×C₂₇×C₅₃	1	1	yes	characteristic
58	C₃	C₅₃×#6903	1	2	yes	—

	subgroup	G/H
center	C₃×C₂₇×C₅₃	C₃²
commutator	C₃	C₃²×C₂₇×C₅₃
Frattini, cosocle	C₃×C₉	C₃³×C₅₃
socle	C₃²×C₅₃	C₃²×C₉
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₂₇×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂₇×C₅₃	C₃	semidirect
3. C₅₃×C₃²⋊C₂₇	C₃	semidirect
11. H(3,𝔽₃)×C₅₃	C₂₇	direct (N × H)
17. C₂₇×C₅₃	H(3,𝔽₃)	direct (N × H)
18. C₂₇×H(3,𝔽₃)	C₅₃	direct (N × H)
21. C₃²×C₅₃	C₃×C₂₇	semidirect
26. C₃²×C₂₇	C₃×C₅₃	semidirect
28. C₃²⋊C₂₇	C₃×C₅₃	semidirect
39. C₅₃	C₂₇×H(3,𝔽₃)	direct (N × H)
41. H(3,𝔽₃)	C₂₇×C₅₃	direct (N × H)
47. C₂₇	H(3,𝔽₃)×C₅₃	direct (N × H)
50. C₃²	C₃×C₂₇×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇×C₅₃	3
C₉×H(3,𝔽₃)×C₅₃	3
C₅₃×C₃²⋊C₂₇	3
C₂₇×H(3,𝔽₃)	53

Subgroups — C27×H(3,𝔽3)×C53 #50311599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×H(3,𝔽₃)×C₅₃ #50311599