Subgroups — C₅₃×#2868 #20271805

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₅₃	C₂	1	6	yes	maximal
2	C₅₃×#384	C₂	1	8	yes	maximal
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₅₃	C₂²	1	4	yes	—
5	C₄⋊Q₈×C₅₃	—	2	24	no	—
6	C₂×C₄⋊C₄×C₅₃	C₂²	1	24	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
9	C₂×Q₈×C₅₃	—	2	24	no	—
10	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
13	C₄⋊C₄×C₅₃	—	2	48	no	—
14	C₄²×C₅₃	D₄	1	12	yes	—
15	C₄²×C₅₃	Q₈	1	16	yes	complemented
16	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
17	Q₈×C₅₃	—	4	16	no	—
18	C₂×C₄×C₅₃	—	2	24	no	—
19	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
20	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
21	C₂×C₄×C₅₃	C₂×Q₈	1	24	yes	—
22	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
23	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
24	C₄×C₅₃	—	4	8	no	—
25	C₄×C₅₃	#127	1	4	yes	—
26	C₄×C₅₃	C₄⋊Q₈	1	24	yes	complemented
27	#2868	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
29	C₂×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
30	C₂×C₄⋊Q₈	C₂×C₅₃	1	6	yes	—
31	#384	C₂×C₅₃	1	8	yes	—
32	C₄³	C₂×C₅₃	1	1	yes	characteristic
33	C₅₃	#2868	1	1	yes	characteristic, direct product factor, Sylow
34	C₂²×Q₈	C₂²×C₅₃	1	4	yes	—
35	C₄⋊Q₈	—	2	24	no	—
36	C₂×C₄⋊C₄	C₂²×C₅₃	1	24	yes	—
37	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
38	C₂×C₄²	C₂²×C₅₃	1	4	yes	—
39	C₂×Q₈	—	2	24	no	—
40	C₂²×C₄	C₂³×C₅₃	1	8	yes	—
41	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
42	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
43	C₄⋊C₄	—	2	48	no	—
44	C₄²	D₄×C₅₃	1	12	yes	—
45	C₄²	Q₈×C₅₃	1	16	yes	complemented
46	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
47	Q₈	—	4	16	no	—
48	C₂×C₄	—	2	24	no	—
49	C₂×C₄	C₂×D₄×C₅₃	1	6	yes	—
50	C₂×C₄	C₂×D₄×C₅₃	1	12	yes	—
51	C₂×C₄	C₂×Q₈×C₅₃	1	24	yes	—
52	C₂²	C₂²×D₄×C₅₃	1	3	yes	—
53	C₂²	C₂²×Q₈×C₅₃	1	4	yes	—
54	C₄	—	4	8	no	—
55	C₄	#127×C₅₃	1	4	yes	—
56	C₄	C₄⋊Q₈×C₅₃	1	24	yes	complemented
57	C₂	C₂×#127×C₅₃	1	1	yes	characteristic
58	C₂	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₄²×C₅₃	Q₈	semidirect
26. C₄×C₅₃	C₄⋊Q₈	semidirect
27. #2868	C₅₃	direct (N × H)
33. C₅₃	#2868	direct (N × H)
45. C₄²	Q₈×C₅₃	semidirect
56. C₄	C₄⋊Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₅₃	2
C₅₃×#384	2
C₄³×C₅₃	2
#2868	53

Subgroups — C53×#2868 #20271805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2868 #20271805