Subgroups — C₅₃×#13725 #27221237

Order 13568.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×D₃₂	C₂	1	4	yes	maximal, complemented
2	C₅₃×#2246	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₅₃	—	2	2	no	—
5	C₂×D₁₆×C₅₃	C₂²	1	4	yes	—
6	C₅₃×#493	—	2	2	no	—
7	C₅₃×D₃₂	—	2	8	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₅₃	—	2	4	no	—
11	C₂×D₈×C₅₃	—	4	2	no	—
12	#127×C₅₃	—	4	2	no	—
13	D₁₆×C₅₃	—	2	8	no	—
14	D₁₆×C₅₃	—	4	4	no	—
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	4	yes	complemented
19	C₂×D₄×C₅₃	—	4	4	no	—
20	C₂×D₄×C₅₃	—	8	4	no	—
21	D₈×C₅₃	—	4	4	no	—
22	D₈×C₅₃	—	4	4	no	—
23	D₈×C₅₃	—	8	4	no	—
24	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₅₃	D₈	1	1	yes	characteristic
27	C₁₆×C₅₃	D₈	1	2	yes	—
28	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
29	C₂³×C₅₃	—	8	4	no	—
30	D₄×C₅₃	—	8	4	no	—
31	D₄×C₅₃	—	8	4	no	—
32	D₄×C₅₃	—	16	8	no	—
33	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
34	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
35	C₈×C₅₃	D₁₆	1	2	yes	—
36	C₈×C₅₃	#127	1	1	yes	characteristic
37	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
38	#13725	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	—	16	4	no	—
40	C₂²×C₅₃	—	16	8	no	—
41	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
42	C₄×C₅₃	D₃₂	1	4	yes	complemented
43	C₄×C₅₃	#493	1	1	yes	characteristic
44	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
45	C₂×D₃₂	C₂×C₅₃	1	4	yes	complemented
46	#2246	C₂×C₅₃	1	2	yes	complemented
47	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	—	32	4	no	—
49	C₂×C₅₃	#2246	1	1	yes	characteristic
50	C₂×C₅₃	C₂×D₃₂	1	2	yes	—
51	C₂×D₁₆	—	2	2	no	—
52	C₂×D₁₆	C₂²×C₅₃	1	4	yes	—
53	#493	—	2	2	no	—
54	D₃₂	—	2	8	no	—
55	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
56	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
57	C₅₃	#13725	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×D₈	—	2	4	no	—
59	C₂×D₈	—	4	2	no	—
60	#127	—	4	2	no	—
61	D₁₆	—	2	8	no	—
62	D₁₆	—	4	4	no	—
63	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
64	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
65	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
66	C₃₂	D₄×C₅₃	1	4	yes	complemented
67	C₂×D₄	—	4	4	no	—
68	C₂×D₄	—	8	4	no	—
69	D₈	—	4	4	no	—
70	D₈	—	4	4	no	—
71	D₈	—	8	4	no	—
72	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
74	C₄²	D₈×C₅₃	1	1	yes	characteristic
75	C₁₆	D₈×C₅₃	1	2	yes	—
76	C₁₆	C₂×D₄×C₅₃	1	2	yes	—
77	C₂³	—	8	4	no	—
78	D₄	—	8	4	no	—
79	D₄	—	8	4	no	—
80	D₄	—	16	8	no	—
81	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
82	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
83	C₈	D₁₆×C₅₃	1	2	yes	—
84	C₈	#127×C₅₃	1	1	yes	characteristic
85	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
86	C₂²	—	16	4	no	—
87	C₂²	—	16	8	no	—
88	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
89	C₄	C₅₃×D₃₂	1	4	yes	complemented
90	C₄	C₅₃×#493	1	1	yes	characteristic
91	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
92	C₂	—	32	4	no	—
93	C₂	C₅₃×#2246	1	1	yes	characteristic
94	C₂	C₂×C₅₃×D₃₂	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×D₃₂	C₂	semidirect
2. C₅₃×#2246	C₂	semidirect
3. C₄×C₃₂×C₅₃	C₂	semidirect
18. C₃₂×C₅₃	D₄	semidirect
38. #13725	C₅₃	direct (N × H)
42. C₄×C₅₃	D₃₂	semidirect
45. C₂×D₃₂	C₂×C₅₃	semidirect
46. #2246	C₂×C₅₃	semidirect
47. C₄×C₃₂	C₂×C₅₃	semidirect
57. C₅₃	#13725	direct (N × H)
66. C₃₂	D₄×C₅₃	semidirect
89. C₄	C₅₃×D₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×D₃₂	2
C₅₃×#2246	2
C₄×C₃₂×C₅₃	2
#13725	53

Subgroups — C53×#13725 #27221237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13725 #27221237