Subgroups — C₂²×C₅₃×#564 #27270204

Order 13568.

6374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#564	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#125×C₅₃	C₂	1	15	yes	maximal
3	C₅₃×#564	C₂²	1	256	yes	direct product factor
4	C₂×#125×C₅₃	C₂²	1	360	yes	—
5	C₂²×C₄⋊C₄×C₅₃	C₂²	1	30	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	5	yes	—
7	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
8	#125×C₅₃	C₂³	1	960	yes	—
9	C₂×C₄⋊C₄×C₅₃	C₂³	1	360	yes	—
10	C₂×C₄²×C₅₃	C₂³	1	60	yes	—
11	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	60	no	—
13	C₂²×C₄×C₅₃	C₂⁴	1	90	yes	—
14	C₄⋊C₄×C₅₃	C₂⁴	1	480	yes	—
15	C₄²×C₅₃	C₂⁴	1	80	yes	—
16	C₂³×C₅₃	2₋¹⁺⁴	1	12	yes	—
17	C₂³×C₅₃	C₂⁵	1	3	yes	—
18	C₂×C₄×C₅₃	—	2	180	no	—
19	C₂×C₄×C₅₃	C₂⁵	1	60	yes	—
20	C₂²×#564	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅₃	#564	1	16	yes	direct product factor
22	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	18	yes	—
23	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
24	C₄×C₅₃	—	2	60	no	—
25	C₂×#564	C₂×C₅₃	1	48	yes	direct product factor
26	C₂²×#125	C₂×C₅₃	1	15	yes	—
27	C₂×C₅₃	C₂×#564	1	12	yes	direct product factor
28	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	3	yes	—
29	#564	C₂²×C₅₃	1	256	yes	direct product factor
30	C₂×#125	C₂²×C₅₃	1	360	yes	—
31	C₂²×C₄⋊C₄	C₂²×C₅₃	1	30	yes	—
32	C₂²×C₄²	C₂²×C₅₃	1	5	yes	—
33	C₅₃	C₂²×#564	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₄	C₂³×C₅₃	1	15	yes	—
35	#125	C₂³×C₅₃	1	960	yes	—
36	C₂×C₄⋊C₄	C₂³×C₅₃	1	360	yes	—
37	C₂×C₄²	C₂³×C₅₃	1	60	yes	—
38	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
39	C₂²×C₄	—	2	60	no	—
40	C₂²×C₄	C₂⁴×C₅₃	1	90	yes	—
41	C₄⋊C₄	C₂⁴×C₅₃	1	480	yes	—
42	C₄²	C₂⁴×C₅₃	1	80	yes	—
43	C₂³	2₋¹⁺⁴×C₅₃	1	12	yes	—
44	C₂³	C₂⁵×C₅₃	1	3	yes	—
45	C₂×C₄	—	2	180	no	—
46	C₂×C₄	C₂⁵×C₅₃	1	60	yes	—
47	C₂²	C₅₃×#564	1	16	yes	direct product factor
48	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	18	yes	—
49	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
50	C₄	—	2	60	no	—
51	C₂	C₂×C₅₃×#564	1	12	yes	direct product factor
52	C₂	C₂²×2₋¹⁺⁴×C₅₃	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#564	C₂	direct (N × H)
3. C₅₃×#564	C₂²	direct (N × H)
20. C₂²×#564	C₅₃	direct (N × H)
21. C₂²×C₅₃	#564	direct (N × H)
25. C₂×#564	C₂×C₅₃	direct (N × H)
27. C₂×C₅₃	C₂×#564	direct (N × H)
29. #564	C₂²×C₅₃	direct (N × H)
33. C₅₃	C₂²×#564	direct (N × H)
47. C₂²	C₅₃×#564	direct (N × H)
51. C₂	C₂×C₅₃×#564	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#564	2
C₂²×#125×C₅₃	2
C₂²×#564	53

Subgroups — C22×C53×#564 #27270204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#564 #27270204