Subgroups — C₅₃×#7530 #27215042

Order 13568.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1408	C₂	1	2	yes	maximal
3	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
4	C₅₃×#493	C₂²	1	1	yes	characteristic
5	C₅₃×#486	—	2	2	no	—
6	C₅₃×#353	—	2	2	no	—
7	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₁₆×C₅₃	—	4	1	no	—
9	C₂×D₈×C₅₃	—	4	1	no	—
10	C₄⋊Q₈×C₅₃	—	2	1	no	—
11	#127×C₅₃	D₄	1	2	yes	—
12	#102×C₅₃	—	4	4	no	—
13	#99×C₅₃	—	4	2	no	—
14	C₄×C₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	4	2	no	—
19	Q₁₆×C₅₃	—	8	2	no	—
20	D₈×C₅₃	—	8	2	no	—
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	4	2	no	—
24	C₂²⋊C₄×C₅₃	—	8	2	no	—
25	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂³×C₅₃	—	4	2	no	—
27	Q₈×C₅₃	—	8	2	no	—
28	D₄×C₅₃	—	8	2	no	—
29	D₄×C₅₃	—	8	4	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	8	1	no	—
32	C₂×C₄×C₅₃	—	16	2	no	—
33	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
34	C₈×C₅₃	—	2	2	no	—
35	C₈×C₅₃	—	4	2	no	—
36	#7530	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₅₃	—	8	2	no	—
38	C₂²×C₅₃	—	16	2	no	—
39	C₂²×C₅₃	#353	1	1	yes	characteristic
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	16	1	no	—
43	C₄×C₅₃	—	32	2	no	—
44	#1709	C₂×C₅₃	1	1	yes	characteristic
45	#1408	C₂×C₅₃	1	2	yes	—
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	—	16	2	no	—
48	C₂×C₅₃	#1408	1	1	yes	characteristic
49	#494	C₄×C₅₃	1	1	yes	characteristic, complemented
50	#493	C₂²×C₅₃	1	1	yes	characteristic
51	#486	—	2	2	no	—
52	#353	—	2	2	no	—
53	C₈²	C₄×C₅₃	1	1	yes	characteristic, complemented
54	C₅₃	#7530	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×D₈	—	4	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	D₄×C₅₃	1	2	yes	—
59	#102	—	4	4	no	—
60	#99	—	4	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
63	C₂×Q₈	—	4	1	no	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	—	4	2	no	—
66	Q₁₆	—	8	2	no	—
67	D₈	—	8	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₂²⋊C₄	—	8	2	no	—
72	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
73	C₂³	—	4	2	no	—
74	Q₈	—	8	2	no	—
75	D₄	—	8	2	no	—
76	D₄	—	8	4	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	—	16	2	no	—
80	C₂×C₄	#99×C₅₃	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	16	2	no	—
85	C₂²	C₅₃×#353	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	—	16	1	no	—
89	C₄	—	32	2	no	—
90	C₂	—	2	1	no	—
91	C₂	—	16	2	no	—
92	C₂	C₅₃×#1408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1408
commutator	C₄×C₈	C₂×C₄×C₅₃
Frattini, cosocle	#493	C₂²×C₅₃
prime radical	C₅₃×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄²×C₅₃ ◁ C₅₃×#493 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#494	C₄	semidirect
7. C₈²×C₅₃	C₄	semidirect
36. #7530	C₅₃	direct (N × H)
49. #494	C₄×C₅₃	semidirect
53. C₈²	C₄×C₅₃	semidirect
54. C₅₃	#7530	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×#1709	2
C₅₃×#1408	2
#7530	53

Subgroups — C53×#7530 #27215042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7530 #27215042