Subgroups — C₅₃×#2864 #20271801

Order 6784.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#389	C₂	1	4	yes	maximal
2	C₅₃×#387	C₂	1	8	yes	maximal
3	C₅₃×#382	C₂	1	2	yes	maximal
4	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
6	C₂×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	#95×C₅₃	C₂²	1	8	yes	—
11	#95×C₅₃	C₂²	1	8	yes	—
12	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	—	2	16	no	—
17	C₄⋊C₄×C₅₃	—	2	8	no	—
18	C₄²×C₅₃	—	2	8	no	—
19	C₄²×C₅₃	—	2	4	no	—
20	C₄²×C₅₃	Q₈	1	4	yes	—
21	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
22	C₂×C₄×C₅₃	—	2	4	no	—
23	C₂×C₄×C₅₃	—	2	4	no	—
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	—	2	8	no	—
26	C₂×C₄×C₅₃	—	2	16	no	—
27	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
28	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
29	C₂×C₄×C₅₃	#41	1	8	yes	—
30	C₂×C₄×C₅₃	#41	1	4	yes	—
31	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₅₃	C₂×#41	1	2	yes	—
33	C₂²×C₅₃	C₂×#41	1	4	yes	—
34	C₄×C₅₃	—	2	8	no	—
35	C₄×C₅₃	—	2	4	no	—
36	C₄×C₅₃	—	4	8	no	—
37	C₄×C₅₃	#125	1	4	yes	—
38	#2864	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₅₃	C₂×#125	1	1	yes	characteristic
40	C₂×C₅₃	#529	1	4	yes	—
41	C₂×C₅₃	#533	1	2	yes	—
42	#389	C₂×C₅₃	1	4	yes	—
43	#387	C₂×C₅₃	1	8	yes	—
44	#382	C₂×C₅₃	1	2	yes	—
45	C₄³	C₂×C₅₃	1	1	yes	characteristic
46	C₅₃	#2864	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
48	C₂×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
49	C₂×C₄²	C₂²×C₅₃	1	4	yes	—
50	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
51	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
52	#95	C₂²×C₅₃	1	8	yes	—
53	#95	C₂²×C₅₃	1	8	yes	—
54	C₂²×C₄	C₂³×C₅₃	1	8	yes	—
55	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
56	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
57	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
58	C₄⋊C₄	—	2	16	no	—
59	C₄⋊C₄	—	2	8	no	—
60	C₄²	—	2	8	no	—
61	C₄²	—	2	4	no	—
62	C₄²	Q₈×C₅₃	1	4	yes	—
63	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	2	16	no	—
69	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
70	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
71	C₂×C₄	#41×C₅₃	1	8	yes	—
72	C₂×C₄	#41×C₅₃	1	4	yes	—
73	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
74	C₂²	C₂×#41×C₅₃	1	2	yes	—
75	C₂²	C₂×#41×C₅₃	1	4	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	4	no	—
78	C₄	—	4	8	no	—
79	C₄	#125×C₅₃	1	4	yes	—
80	C₂	C₂×#125×C₅₃	1	1	yes	characteristic
81	C₂	C₅₃×#529	1	4	yes	—
82	C₂	C₅₃×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
38. #2864	C₅₃	direct (N × H)
46. C₅₃	#2864	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×#389	2
C₅₃×#387	2
C₅₃×#382	2
C₄³×C₅₃	2
#2864	53

Subgroups — C53×#2864 #20271801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2864 #20271801