Subgroups — C₅₃×#12522 #27220034

Order 13568.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2254	C₂	1	2	yes	maximal
2	C₅₃×#2252	C₂	1	2	yes	maximal
3	C₅₃×#1710	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#500	C₂²	1	4	yes	—
6	C₅₃×#487	—	2	4	no	—
7	C₅₃×#367	—	2	4	no	—
8	C₅₃×#366	—	2	4	no	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₅₃	—	2	4	no	—
13	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
14	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
15	#107×C₅₃	—	2	4	no	—
16	#107×C₅₃	—	2	4	no	—
17	#103×C₅₃	—	4	8	no	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	4	4	no	—
21	C₂×C₈×C₅₃	D₈	1	4	yes	—
22	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	4	4	no	—
25	C₄⋊C₄×C₅₃	—	4	4	no	—
26	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₅₃	Q₁₆	1	4	yes	—
28	C₁₆×C₅₃	Q₁₆	1	4	yes	—
29	Q₈×C₅₃	—	8	8	no	—
30	C₂×C₄×C₅₃	—	8	4	no	—
31	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
33	C₈×C₅₃	—	2	4	no	—
34	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
35	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
36	#12522	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₅₃	#493	1	1	yes	characteristic
38	C₄×C₅₃	—	16	4	no	—
39	C₄×C₅₃	#494	1	1	yes	characteristic
40	C₄×C₅₃	#494	1	1	yes	characteristic
41	C₄×C₅₃	#508	1	4	yes	—
42	#2254	C₂×C₅₃	1	2	yes	—
43	#2252	C₂×C₅₃	1	2	yes	—
44	#1710	C₂×C₅₃	1	2	yes	—
45	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
46	C₂×C₅₃	#1713	1	1	yes	characteristic
47	C₂×C₅₃	#2249	1	2	yes	—
48	#500	C₂²×C₅₃	1	4	yes	—
49	#487	—	2	4	no	—
50	#367	—	2	4	no	—
51	#366	—	2	4	no	—
52	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
53	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
54	C₈²	C₂²×C₅₃	1	1	yes	characteristic
55	C₅₃	#12522	1	1	yes	characteristic, direct product factor, Sylow
56	C₄⋊Q₈	—	2	4	no	—
57	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
58	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
59	#107	—	2	4	no	—
60	#107	—	2	4	no	—
61	#103	—	4	8	no	—
62	C₄×C₈	D₄×C₅₃	1	2	yes	—
63	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	4	4	no	—
65	C₂×C₈	D₈×C₅₃	1	4	yes	—
66	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
68	C₄⋊C₄	—	4	4	no	—
69	C₄⋊C₄	—	4	4	no	—
70	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
71	C₁₆	Q₁₆×C₅₃	1	4	yes	—
72	C₁₆	Q₁₆×C₅₃	1	4	yes	—
73	Q₈	—	8	8	no	—
74	C₂×C₄	—	8	4	no	—
75	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
76	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Q₁₆×C₅₃	1	2	yes	—
79	C₈	C₂×Q₁₆×C₅₃	1	2	yes	—
80	C₂²	C₅₃×#493	1	1	yes	characteristic
81	C₄	—	16	4	no	—
82	C₄	C₅₃×#494	1	1	yes	characteristic
83	C₄	C₅₃×#494	1	1	yes	characteristic
84	C₄	C₅₃×#508	1	4	yes	—
85	C₂	C₅₃×#1713	1	1	yes	characteristic
86	C₂	C₅₃×#2249	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
36. #12522	C₅₃	direct (N × H)
55. C₅₃	#12522	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×#2254	2
C₅₃×#2252	2
C₅₃×#1710	2
C₈×C₁₆×C₅₃	2
#12522	53

Subgroups — C53×#12522 #27220034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12522 #27220034