Subgroups — C₂⁴×#88324 #20291698

Order 6784.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#88324	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88324	C₂²	1	140	yes	direct product factor
4	C₂³×Dic₅₃	C₄	1	15	yes	—
5	C₂³×Dic₅₃	C₂²	1	15	yes	—
6	C₂⁵×C₅₃	C₄	1	1	yes	characteristic
7	C₂⁴×C₅₃	C₈	1	16	yes	complemented
8	C₂⁴×C₅₃	C₂×C₄	1	15	yes	—
9	C₂²×Dic₅₃	C₂×C₄	1	105	yes	—
10	C₂²×Dic₅₃	C₂³	1	35	yes	—
11	C₂×#88324	C₂³	1	120	yes	direct product factor
12	C₂³×C₅₃	C₂×C₈	1	120	yes	complemented
13	C₂³×C₅₃	C₂²×C₄	1	35	yes	—
14	C₂×Dic₅₃	C₂²×C₄	1	105	yes	—
15	C₂×Dic₅₃	C₂⁴	1	15	yes	—
16	#88324	C₂⁴	1	16	yes	direct product factor
17	C₂²×C₅₃	C₂²×C₈	1	140	yes	complemented
18	C₂²×C₅₃	C₂³×C₄	1	15	yes	—
19	Dic₅₃	C₂³×C₄	1	15	yes	—
20	Dic₅₃	C₂⁵	1	1	yes	characteristic
21	C₂⁴×C₈	—	53	1	no	maximal, Sylow
22	C₂×C₅₃	C₂³×C₈	1	30	yes	complemented
23	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
24	C₂⁴×C₄	—	53	1	no	—
25	C₂³×C₈	—	53	30	no	—
26	C₅₃	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
27	C₂⁵	#6157	1	1	yes	characteristic
28	C₂³×C₄	—	53	15	no	—
29	C₂³×C₄	—	53	15	no	—
30	C₂²×C₈	—	53	140	no	—
31	C₂⁴	#88324	1	16	yes	direct product factor
32	C₂⁴	C₂×#6157	1	15	yes	—
33	C₂²×C₄	—	53	105	no	—
34	C₂²×C₄	—	53	35	no	—
35	C₂×C₈	—	53	120	no	—
36	C₂³	C₂×#88324	1	120	yes	direct product factor
37	C₂³	C₂²×#6157	1	35	yes	—
38	C₂×C₄	—	53	105	no	—
39	C₂×C₄	—	53	15	no	—
40	C₈	—	53	16	no	—
41	C₂²	C₂²×#88324	1	140	yes	direct product factor
42	C₂²	C₂³×#6157	1	15	yes	—
43	C₄	—	53	1	no	—
44	C₄	—	53	15	no	—
45	C₂	C₂³×#88324	1	30	yes	direct product factor
46	C₂	C₂⁴×#6157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵	#6157
commutator, hypocenter	C₅₃	C₂⁴×C₈
Frattini	C₂	C₂⁴×#6157
Fitting, socle	C₂⁵×C₅₃	C₄
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂³×#88324	C₂	direct (N × H)
3. C₂²×#88324	C₂²	direct (N × H)
7. C₂⁴×C₅₃	C₈	semidirect
11. C₂×#88324	C₂³	direct (N × H)
12. C₂³×C₅₃	C₂×C₈	semidirect
16. #88324	C₂⁴	direct (N × H)
17. C₂²×C₅₃	C₂²×C₈	semidirect
22. C₂×C₅₃	C₂³×C₈	semidirect
26. C₅₃	C₂⁴×C₈	semidirect
31. C₂⁴	#88324	direct (N × H)
36. C₂³	C₂×#88324	direct (N × H)
41. C₂²	C₂²×#88324	direct (N × H)
45. C₂	C₂³×#88324	direct (N × H)

Subgroups — C24×#88324 #20291698

Conjugacy classes of subgroups

Complements

Subgroups — C₂⁴×#88324 #20291698