Subgroups — C₅₃×#2247 #20271184

Order 6784.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₅₃	C₂	1	4	yes	maximal
2	C₅₃×#494	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₅₃	—	2	2	no	—
5	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
6	C₄⋊Q₈×C₅₃	—	2	2	no	—
7	Q₃₂×C₅₃	—	2	8	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Q₈×C₅₃	—	2	4	no	—
11	Q₁₆×C₅₃	—	2	8	no	—
12	Q₁₆×C₅₃	—	4	4	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	—	4	4	no	—
16	C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	D₄	1	4	yes	—
18	Q₈×C₅₃	—	4	4	no	—
19	Q₈×C₅₃	—	4	4	no	—
20	C₂×C₄×C₅₃	—	4	4	no	—
21	C₂×C₄×C₅₃	D₈	1	2	yes	—
22	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₈×C₅₃	D₈	1	2	yes	—
24	C₈×C₅₃	C₂×D₄	1	2	yes	—
25	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
26	C₄×C₅₃	—	8	4	no	—
27	C₄×C₅₃	Q₃₂	1	4	yes	complemented
28	C₄×C₅₃	#127	1	1	yes	characteristic
29	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
30	#2247	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅₃	#493	1	1	yes	characteristic
32	C₂×C₅₃	C₂×Q₃₂	1	2	yes	—
33	C₂×Q₃₂	C₂×C₅₃	1	4	yes	—
34	#494	C₂×C₅₃	1	2	yes	—
35	C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
36	C₅₃	#2247	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×Q₁₆	—	2	2	no	—
38	C₂×Q₁₆	C₂²×C₅₃	1	4	yes	—
39	C₄⋊Q₈	—	2	2	no	—
40	Q₃₂	—	2	8	no	—
41	C₂×C₁₆	C₂²×C₅₃	1	2	yes	—
42	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
43	C₂×Q₈	—	2	4	no	—
44	Q₁₆	—	2	8	no	—
45	Q₁₆	—	4	4	no	—
46	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
47	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
48	C₄⋊C₄	—	4	4	no	—
49	C₄²	D₄×C₅₃	1	1	yes	characteristic
50	C₁₆	D₄×C₅₃	1	4	yes	—
51	Q₈	—	4	4	no	—
52	Q₈	—	4	4	no	—
53	C₂×C₄	—	4	4	no	—
54	C₂×C₄	D₈×C₅₃	1	2	yes	—
55	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
56	C₈	D₈×C₅₃	1	2	yes	—
57	C₈	C₂×D₄×C₅₃	1	2	yes	—
58	C₂²	C₂×D₈×C₅₃	1	1	yes	characteristic
59	C₄	—	8	4	no	—
60	C₄	Q₃₂×C₅₃	1	4	yes	complemented
61	C₄	#127×C₅₃	1	1	yes	characteristic
62	C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
63	C₂	C₅₃×#493	1	1	yes	characteristic
64	C₂	C₂×Q₃₂×C₅₃	1	2	yes	—

	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
27. C₄×C₅₃	Q₃₂	semidirect
30. #2247	C₅₃	direct (N × H)
36. C₅₃	#2247	direct (N × H)
60. C₄	Q₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×C₅₃	2
C₅₃×#494	2
C₄×C₁₆×C₅₃	2
#2247	53

Subgroups — C53×#2247 #20271184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2247 #20271184