Subgroups — C₅₃×#488 #16810243

Order 3392.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#124×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#103×C₅₃	C₂	1	2	yes	maximal, complemented
4	#102×C₅₃	C₂	1	2	yes	maximal
5	C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₅₃	—	2	2	no	—
12	C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₅₃	—	2	1	no	—
14	Q₈×C₅₃	—	2	2	no	—
15	D₄×C₅₃	—	2	2	no	—
16	C₂×C₄×C₅₃	—	2	1	no	—
17	C₂×C₄×C₅₃	—	2	2	no	—
18	C₂×C₄×C₅₃	D₄	1	2	yes	—
19	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₈×C₅₃	—	2	2	no	—
21	C₂²×C₅₃	—	4	1	no	—
22	C₂²×C₅₃	—	4	2	no	—
23	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₅₃	—	2	2	no	—
25	C₄×C₅₃	—	4	1	no	—
26	C₄×C₅₃	—	4	2	no	—
27	C₄×C₅₃	#41	1	2	yes	—
28	C₂×C₅₃	—	8	1	no	—
29	C₂×C₅₃	#124	1	1	yes	characteristic
30	C₂×C₅₃	#135	1	2	yes	—
31	#488	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₅₃	#488	1	1	yes	characteristic, direct product factor, Sylow
33	#125	C₂×C₅₃	1	1	yes	characteristic, complemented
34	#124	C₂×C₅₃	1	1	yes	characteristic
35	#103	C₂×C₅₃	1	2	yes	complemented
36	#102	C₂×C₅₃	1	2	yes	—
37	C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
38	C₂×Q₈	C₂²×C₅₃	1	1	yes	characteristic
39	C₂×D₄	C₂²×C₅₃	1	1	yes	characteristic
40	C₂×C₈	C₂²×C₅₃	1	2	yes	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₄²	C₂²×C₅₃	1	1	yes	characteristic
45	C₂³	—	2	1	no	—
46	Q₈	—	2	2	no	—
47	D₄	—	2	2	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	D₄×C₅₃	1	2	yes	—
51	C₂×C₄	C₂³×C₅₃	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×D₄×C₅₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	1	no	—
58	C₄	—	4	2	no	—
59	C₄	#41×C₅₃	1	2	yes	—
60	C₂	—	8	1	no	—
61	C₂	#124×C₅₃	1	1	yes	characteristic
62	C₂	#135×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅₃
prime radical	C₂×D₄×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #125×C₅₃	C₂	semidirect
3. #103×C₅₃	C₂	semidirect
5. C₄×C₈×C₅₃	C₂	semidirect
31. #488	C₅₃	direct (N × H)
32. C₅₃	#488	direct (N × H)
33. #125	C₂×C₅₃	semidirect
35. #103	C₂×C₅₃	semidirect
37. C₄×C₈	C₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
#125×C₅₃	2
#124×C₅₃	2
#103×C₅₃	2
#102×C₅₃	2
C₄×C₈×C₅₃	2
#488	53

Subgroups — C53×#488 #16810243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#488 #16810243