Subgroups — C₅₃×#2380 #20271317

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#386	C₂	1	12	yes	maximal, complemented
3	C₅₃×#383	C₂	1	2	yes	maximal, complemented
4	C₂²×D₄×C₅₃	C₂²	1	6	yes	—
5	C₂³×C₄×C₅₃	C₂²	1	4	yes	—
6	#127×C₅₃	C₄	1	8	yes	complemented
7	C₂×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#95×C₅₃	C₂²	1	12	yes	—
10	C₂⁴×C₅₃	C₂³	1	4	yes	—
11	C₂×D₄×C₅₃	—	4	6	no	—
12	C₂×D₄×C₅₃	—	4	6	no	—
13	C₂×D₄×C₅₃	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₅₃	—	4	12	no	—
15	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂²⋊C₄×C₅₃	—	4	12	no	—
18	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
19	C₂³×C₅₃	—	2	12	no	—
20	C₂³×C₅₃	—	2	12	no	—
21	C₂³×C₅₃	C₂²×C₄	1	8	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	D₄×C₅₃	—	4	24	no	—
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	—	2	24	no	—
26	C₂×C₄×C₅₃	—	4	16	no	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
28	C₂²×C₅₃	—	2	24	no	—
29	C₂²×C₅₃	—	4	4	no	—
30	C₂²×C₅₃	—	4	12	no	—
31	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₅₃	—	2	6	no	—
34	C₄×C₅₃	—	4	8	no	—
35	#2380	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅₃	—	4	8	no	—
37	C₂×C₅₃	#518	1	3	yes	—
38	C₂×C₅₃	#560	1	4	yes	—
39	C₂×#127	C₂×C₅₃	1	1	yes	characteristic
40	#386	C₂×C₅₃	1	12	yes	complemented
41	#383	C₂×C₅₃	1	2	yes	complemented
42	C₅₃	#2380	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×D₄	C₂²×C₅₃	1	6	yes	—
44	C₂³×C₄	C₂²×C₅₃	1	4	yes	—
45	#127	C₄×C₅₃	1	8	yes	complemented
46	C₂×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
47	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
48	#95	C₂²×C₅₃	1	12	yes	—
49	C₂⁴	C₂³×C₅₃	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×C₄×C₅₃	1	24	yes	complemented
53	C₂²×C₄	—	4	12	no	—
54	C₂²×C₄	C₂³×C₅₃	1	8	yes	—
55	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
56	C₂²⋊C₄	—	4	12	no	—
57	C₄²	C₂×C₄×C₅₃	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×C₄×C₅₃	1	8	yes	—
61	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	4	16	no	—
66	C₂×C₄	C₂²×C₄×C₅₃	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
71	C₂²	H(5,𝔽₂)×C₅₃	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	4	8	no	—
74	C₂	—	4	8	no	—
75	C₂	C₅₃×#518	1	3	yes	—
76	C₂	C₅₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#386	C₂	semidirect
3. C₅₃×#383	C₂	semidirect
6. #127×C₅₃	C₄	semidirect
13. C₂×D₄×C₅₃	C₂×C₄	semidirect
18. C₄²×C₅₃	C₂×C₄	semidirect
35. #2380	C₅₃	direct (N × H)
40. #386	C₂×C₅₃	semidirect
41. #383	C₂×C₅₃	semidirect
42. C₅₃	#2380	direct (N × H)
45. #127	C₄×C₅₃	semidirect
52. C₂×D₄	C₂×C₄×C₅₃	semidirect
57. C₄²	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₅₃	2
C₅₃×#386	2
C₅₃×#383	2
#2380	53

Subgroups — C53×#2380 #20271317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2380 #20271317