Subgroups — C₅₃×#2255 #20271192

Order 6784.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#367	C₂	1	2	yes	maximal
4	C₅₃×#366	C₂	1	2	yes	maximal
5	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₄⋊Q₈×C₅₃	—	2	1	no	—
7	#125×C₅₃	—	2	1	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	#107×C₅₃	—	2	1	no	—
10	#107×C₅₃	C₂²	1	1	yes	characteristic
11	#107×C₅₃	C₂²	1	1	yes	characteristic
12	#107×C₅₃	C₂²	1	2	yes	—
13	#106×C₅₃	—	2	1	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	—	4	1	no	—
22	C₄⋊C₄×C₅₃	—	4	1	no	—
23	C₄⋊C₄×C₅₃	—	4	1	no	—
24	C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₁₆×C₅₃	Q₈	1	4	yes	complemented
26	Q₈×C₅₃	—	8	2	no	—
27	C₂×C₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₈×C₅₃	Q₁₆	1	2	yes	—
34	C₈×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	—	8	1	no	—
39	C₄×C₅₃	—	8	2	no	—
40	C₄×C₅₃	—	8	1	no	—
41	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
43	#2255	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅₃	#500	1	1	yes	characteristic
45	C₂×C₅₃	#509	1	1	yes	characteristic
46	C₂×C₅₃	#510	1	1	yes	characteristic
47	#500	C₂×C₅₃	1	1	yes	characteristic
48	#499	C₂×C₅₃	1	1	yes	characteristic
49	#367	C₂×C₅₃	1	2	yes	—
50	#366	C₂×C₅₃	1	2	yes	—
51	#346	C₂×C₅₃	1	1	yes	characteristic
52	C₅₃	#2255	1	1	yes	characteristic, direct product factor, Sylow
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	2	1	no	—
55	C₂×C₁₆	C₂²×C₅₃	1	2	yes	—
56	#107	—	2	1	no	—
57	#107	C₂²×C₅₃	1	1	yes	characteristic
58	#107	C₂²×C₅₃	1	2	yes	—
59	#107	C₂²×C₅₃	1	1	yes	characteristic
60	#106	—	2	1	no	—
61	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
64	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₄⋊C₄	—	4	1	no	—
71	C₄²	D₄×C₅₃	1	1	yes	characteristic
72	C₁₆	Q₈×C₅₃	1	4	yes	complemented
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
78	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
79	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
80	C₈	Q₁₆×C₅₃	1	2	yes	—
81	C₈	C₂×Q₈×C₅₃	1	2	yes	—
82	C₂²	C₂×D₈×C₅₃	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	8	2	no	—
87	C₄	—	8	1	no	—
88	C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
89	C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
90	C₂	C₅₃×#500	1	1	yes	characteristic
91	C₂	C₅₃×#509	1	1	yes	characteristic
92	C₂	C₅₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₁₆×C₅₃	Q₈	semidirect
43. #2255	C₅₃	direct (N × H)
52. C₅₃	#2255	direct (N × H)
72. C₁₆	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#500	2
C₅₃×#499	2
C₅₃×#367	2
C₅₃×#366	2
C₅₃×#346	2
#2255	53

Subgroups — C53×#2255 #20271192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2255 #20271192