Subgroups — C₂⁵×#88324 #28330814

Order 13568.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88324	C₂	1	62	yes	maximal, direct product factor
2	C₂⁵×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×#88324	C₂²	1	620	yes	direct product factor
4	C₂⁴×Dic₅₃	C₄	1	31	yes	—
5	C₂⁴×Dic₅₃	C₂²	1	31	yes	—
6	C₂⁶×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×#88324	C₂³	1	1240	yes	direct product factor
8	C₂³×Dic₅₃	C₂×C₄	1	465	yes	—
9	C₂³×Dic₅₃	C₂³	1	155	yes	—
10	C₂⁵×C₅₃	C₈	1	32	yes	complemented
11	C₂⁵×C₅₃	C₂×C₄	1	31	yes	—
12	C₂⁴×C₅₃	C₂×C₈	1	496	yes	complemented
13	C₂⁴×C₅₃	C₂²×C₄	1	155	yes	—
14	C₂²×Dic₅₃	C₂²×C₄	1	1085	yes	—
15	C₂²×Dic₅₃	C₂⁴	1	155	yes	—
16	C₂×#88324	C₂⁴	1	496	yes	direct product factor
17	C₂³×C₅₃	C₂²×C₈	1	1240	yes	complemented
18	C₂³×C₅₃	C₂³×C₄	1	155	yes	—
19	C₂×Dic₅₃	C₂³×C₄	1	465	yes	—
20	C₂×Dic₅₃	C₂⁵	1	31	yes	—
21	#88324	C₂⁵	1	32	yes	direct product factor
22	C₂⁵×C₈	—	53	1	no	maximal, Sylow
23	C₂²×C₅₃	C₂³×C₈	1	620	yes	complemented
24	C₂²×C₅₃	C₂⁴×C₄	1	31	yes	—
25	Dic₅₃	C₂⁴×C₄	1	31	yes	—
26	Dic₅₃	C₂⁶	1	1	yes	characteristic
27	C₂⁵×C₄	—	53	1	no	—
28	C₂⁴×C₈	—	53	62	no	—
29	C₂×C₅₃	C₂⁴×C₈	1	62	yes	complemented
30	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
31	C₂⁶	#6157	1	1	yes	characteristic
32	C₂⁴×C₄	—	53	31	no	—
33	C₂⁴×C₄	—	53	31	no	—
34	C₂³×C₈	—	53	620	no	—
35	C₅₃	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
36	C₂⁵	#88324	1	32	yes	direct product factor
37	C₂⁵	C₂×#6157	1	31	yes	—
38	C₂³×C₄	—	53	155	no	—
39	C₂³×C₄	—	53	465	no	—
40	C₂²×C₈	—	53	1240	no	—
41	C₂⁴	C₂×#88324	1	496	yes	direct product factor
42	C₂⁴	C₂²×#6157	1	155	yes	—
43	C₂²×C₄	—	53	1085	no	—
44	C₂²×C₄	—	53	155	no	—
45	C₂×C₈	—	53	496	no	—
46	C₂³	C₂²×#88324	1	1240	yes	direct product factor
47	C₂³	C₂³×#6157	1	155	yes	—
48	C₂×C₄	—	53	465	no	—
49	C₂×C₄	—	53	31	no	—
50	C₈	—	53	32	no	—
51	C₂²	C₂³×#88324	1	620	yes	direct product factor
52	C₂²	C₂⁴×#6157	1	31	yes	—
53	C₄	—	53	1	no	—
54	C₄	—	53	31	no	—
55	C₂	C₂⁴×#88324	1	62	yes	direct product factor
56	C₂	C₂⁵×#6157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁶	#6157
commutator, hypocenter	C₅₃	C₂⁵×C₈
Frattini	C₂	C₂⁵×#6157
Fitting, socle, prime radical	C₂⁶×C₅₃	C₄
cosocle	Dic₅₃	C₂⁶
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#88324	C₂	direct (N × H)
3. C₂³×#88324	C₂²	direct (N × H)
7. C₂²×#88324	C₂³	direct (N × H)
10. C₂⁵×C₅₃	C₈	semidirect
12. C₂⁴×C₅₃	C₂×C₈	semidirect
16. C₂×#88324	C₂⁴	direct (N × H)
17. C₂³×C₅₃	C₂²×C₈	semidirect
21. #88324	C₂⁵	direct (N × H)
23. C₂²×C₅₃	C₂³×C₈	semidirect
29. C₂×C₅₃	C₂⁴×C₈	semidirect
35. C₅₃	C₂⁵×C₈	semidirect
36. C₂⁵	#88324	direct (N × H)
41. C₂⁴	C₂×#88324	direct (N × H)
46. C₂³	C₂²×#88324	direct (N × H)
51. C₂²	C₂³×#88324	direct (N × H)
55. C₂	C₂⁴×#88324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#88324	2
C₂⁵×Dic₅₃	2
C₂⁵×C₈	53

Subgroups — C25×#88324 #28330814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×#88324 #28330814