Subgroups — C₅₃×#10624833 #50311718

Order 38637.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6903	C₃	1	1	yes	maximal, characteristic
2	C₅₃×C₈₁⋊C₃	C₃	1	8	yes	maximal, complemented
3	C₃×C₅₃×C₈₁	C₃	1	4	yes	maximal, complemented
4	C₅₃×#737	C₉	1	1	yes	characteristic
5	C₅₃×C₂₇⋊C₃	C₉	1	8	yes	—
6	C₃×C₂₇×C₅₃	C₃²	1	4	yes	—
7	C₅₃×C₈₁	C₃²	1	8	yes	—
8	C₅₃×C₈₁	C₃²	1	1	yes	characteristic
9	3₋¹⁺²×C₅₃	C₂₇	1	8	yes	—
10	H(3,𝔽₃)×C₅₃	C₂₇	1	1	yes	characteristic
11	C₃×C₉×C₅₃	C₃×C₉	1	4	yes	—
12	C₂₇×C₅₃	C₃×C₉	1	8	yes	—
13	C₂₇×C₅₃	C₃³	1	1	yes	characteristic
14	#10624833	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₃²×C₅₃	C₃×C₂₇	1	4	yes	—
16	C₉×C₅₃	C₃×C₂₇	1	8	yes	—
17	C₉×C₅₃	C₃²×C₉	1	1	yes	characteristic
18	#6903	C₃×C₅₃	1	1	yes	characteristic
19	C₈₁⋊C₃	C₃×C₅₃	1	8	yes	complemented
20	C₃×C₈₁	C₃×C₅₃	1	4	yes	complemented
21	C₃×C₅₃	—	3	4	no	—
22	C₃×C₅₃	C₃²×C₂₇	1	1	yes	characteristic
23	#737	C₉×C₅₃	1	1	yes	characteristic
24	C₂₇⋊C₃	C₉×C₅₃	1	8	yes	—
25	C₃×C₂₇	C₃²×C₅₃	1	4	yes	—
26	C₈₁	C₃²×C₅₃	1	8	yes	—
27	C₈₁	C₃²×C₅₃	1	1	yes	characteristic
28	C₅₃	#10624833	1	1	yes	characteristic, direct product factor, Sylow
29	3₋¹⁺²	C₂₇×C₅₃	1	8	yes	—
30	H(3,𝔽₃)	C₂₇×C₅₃	1	1	yes	characteristic
31	C₃×C₉	C₃×C₉×C₅₃	1	4	yes	—
32	C₂₇	C₃×C₉×C₅₃	1	8	yes	—
33	C₂₇	C₃³×C₅₃	1	1	yes	characteristic
34	C₃²	C₃×C₂₇×C₅₃	1	4	yes	—
35	C₉	C₃×C₂₇×C₅₃	1	8	yes	—
36	C₉	C₃²×C₉×C₅₃	1	1	yes	characteristic
37	C₃	—	3	4	no	—
38	C₃	C₃²×C₂₇×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₅₃×C₈₁	C₃²
commutator	C₃	C₃²×C₂₇×C₅₃
Frattini, cosocle	C₂₇	C₃³×C₅₃
socle	C₃×C₅₃	C₃²×C₂₇
prime radical	H(3,𝔽₃)×C₅₃	C₂₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₅₃×C₈₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×C₈₁⋊C₃	C₃	semidirect
3. C₃×C₅₃×C₈₁	C₃	semidirect
14. #10624833	C₅₃	direct (N × H)
19. C₈₁⋊C₃	C₃×C₅₃	semidirect
20. C₃×C₈₁	C₃×C₅₃	semidirect
28. C₅₃	#10624833	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#6903	3
C₅₃×C₈₁⋊C₃	3
C₃×C₅₃×C₈₁	3
#10624833	53

Subgroups — C53×#10624833 #50311718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624833 #50311718