Subgroups — C₅₃×#10624725 #50311610

Order 38637.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6908	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6903	C₃	1	3	yes	maximal, complemented
3	C₃×C₅₃×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#6873	C₃	1	3	yes	maximal, complemented
5	C₅₃×#6872	C₃	1	3	yes	maximal, complemented
6	C₅₃×#6869	C₃	1	2	yes	maximal, complemented
7	C₅₃×#737	C₃²	1	3	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×#733	C₉	1	2	yes	—
10	C₅₃×#732	C₉	1	1	yes	characteristic
11	C₅₃×#731	C₉	1	3	yes	—
12	C₅₃×AΣL(1,𝔽₂₇)	C₉	1	3	yes	—
13	C₅₃×C₂₇⋊C₃	—	3	3	no	—
14	C₅₃×C₂₇⋊C₃	—	3	6	no	—
15	C₅₃×C₂₇⋊C₃	C₃²	1	6	yes	—
16	C₃×C₂₇×C₅₃	—	3	3	no	—
17	C₃×C₂₇×C₅₃	C₃²	1	1	yes	characteristic
18	C₃×C₂₇×C₅₃	C₃²	1	1	yes	characteristic
19	C₃×C₂₇×C₅₃	C₃²	1	1	yes	characteristic
20	C₃³×C₅₃	C₃×C₉	1	1	yes	characteristic
21	3₋¹⁺²×C₅₃	—	3	6	no	—
22	3₋¹⁺²×C₅₃	C₃×C₉	1	6	yes	—
23	H(3,𝔽₃)×C₅₃	C₃×C₉	1	3	yes	—
24	C₃×C₉×C₅₃	—	3	3	no	—
25	C₃×C₉×C₅₃	—	3	1	no	—
26	C₃×C₉×C₅₃	—	3	2	no	—
27	C₃×C₉×C₅₃	C₃×C₉	1	1	yes	characteristic
28	C₃×C₉×C₅₃	C₃×C₉	1	1	yes	characteristic
29	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
30	C₂₇×C₅₃	—	3	6	no	—
31	C₂₇×C₅₃	—	3	1	no	—
32	C₂₇×C₅₃	—	3	1	no	—
33	C₂₇×C₅₃	H(3,𝔽₃)	1	3	yes	—
34	#10624725	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₃²×C₅₃	—	3	1	no	—
36	C₃²×C₅₃	—	3	3	no	—
37	C₃²×C₅₃	—	9	1	no	—
38	C₃²×C₅₃	C₃²×C₉	1	1	yes	characteristic
39	C₉×C₅₃	—	3	1	no	—
40	C₉×C₅₃	—	3	1	no	—
41	C₉×C₅₃	—	3	6	no	—
42	C₉×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₉×C₅₃	#737	1	2	yes	—
44	#6908	C₃×C₅₃	1	1	yes	characteristic
45	#6903	C₃×C₅₃	1	3	yes	complemented
46	C₃×C₂₇⋊C₃	C₃×C₅₃	1	1	yes	characteristic, complemented
47	#6873	C₃×C₅₃	1	3	yes	complemented
48	#6872	C₃×C₅₃	1	3	yes	complemented
49	#6869	C₃×C₅₃	1	2	yes	complemented
50	C₃×C₅₃	—	3	1	no	—
51	C₃×C₅₃	—	9	3	no	—
52	C₃×C₅₃	—	9	1	no	—
53	C₃×C₅₃	C₉×H(3,𝔽₃)	1	1	yes	characteristic
54	#737	C₃²×C₅₃	1	3	yes	—
55	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
56	#733	C₉×C₅₃	1	2	yes	—
57	#732	C₉×C₅₃	1	1	yes	characteristic
58	#731	C₉×C₅₃	1	3	yes	—
59	AΣL(1,𝔽₂₇)	C₉×C₅₃	1	3	yes	—
60	C₂₇⋊C₃	—	3	3	no	—
61	C₂₇⋊C₃	—	3	6	no	—
62	C₂₇⋊C₃	C₃²×C₅₃	1	6	yes	—
63	C₃×C₂₇	—	3	3	no	—
64	C₃×C₂₇	C₃²×C₅₃	1	1	yes	characteristic
65	C₃×C₂₇	C₃²×C₅₃	1	1	yes	characteristic
66	C₃×C₂₇	C₃²×C₅₃	1	1	yes	characteristic
67	C₅₃	#10624725	1	1	yes	characteristic, direct product factor, Sylow
68	C₃³	C₃×C₉×C₅₃	1	1	yes	characteristic
69	3₋¹⁺²	—	3	6	no	—
70	3₋¹⁺²	C₃×C₉×C₅₃	1	6	yes	—
71	H(3,𝔽₃)	C₃×C₉×C₅₃	1	3	yes	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	2	no	—
75	C₃×C₉	C₃×C₉×C₅₃	1	1	yes	characteristic
76	C₃×C₉	C₃×C₉×C₅₃	1	1	yes	characteristic
77	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
78	C₂₇	—	3	6	no	—
79	C₂₇	—	3	1	no	—
80	C₂₇	—	3	1	no	—
81	C₂₇	H(3,𝔽₃)×C₅₃	1	3	yes	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	3	no	—
84	C₃²	—	9	1	no	—
85	C₃²	C₃²×C₉×C₅₃	1	1	yes	characteristic
86	C₉	—	3	1	no	—
87	C₉	—	3	1	no	—
88	C₉	—	3	6	no	—
89	C₉	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
90	C₉	C₅₃×#737	1	2	yes	—
91	C₃	—	3	1	no	—
92	C₃	—	9	1	no	—
93	C₃	—	9	3	no	—
94	C₃	C₉×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₅₃	C₃×H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₅₃
Frattini, cosocle	C₃×C₉	C₃³×C₅₃
socle	C₃×C₅₃	C₉×H(3,𝔽₃)
prime radical	C₅₃×#6908	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₅₃ ◁ C₃×C₂₇×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#6903	C₃	semidirect
3. C₃×C₅₃×C₂₇⋊C₃	C₃	semidirect
4. C₅₃×#6873	C₃	semidirect
5. C₅₃×#6872	C₃	semidirect
6. C₅₃×#6869	C₃	semidirect
34. #10624725	C₅₃	direct (N × H)
45. #6903	C₃×C₅₃	semidirect
46. C₃×C₂₇⋊C₃	C₃×C₅₃	semidirect
47. #6873	C₃×C₅₃	semidirect
48. #6872	C₃×C₅₃	semidirect
49. #6869	C₃×C₅₃	semidirect
67. C₅₃	#10624725	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×#6908	3
C₅₃×#6903	3
C₃×C₅₃×C₂₇⋊C₃	3
C₅₃×#6873	3
C₅₃×#6872	3
C₅₃×#6869	3
#10624725	53

Subgroups — C53×#10624725 #50311610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624725 #50311610