Subgroups — C₂⁵×#10596831 #33999738

Order 18688.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#10596831	C₂	1	62	yes	maximal, direct product factor
2	C₂⁵×C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×#10596831	C₂²	1	620	yes	direct product factor
4	C₂⁶×C₇₃	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₇₃	C₄	1	31	yes	—
6	C₂⁴×C₄×C₇₃	C₂²	1	31	yes	—
7	C₂²×#10596831	C₂³	1	1240	yes	direct product factor
8	C₂⁵×C₇₃	C₈	1	32	yes	complemented
9	C₂⁵×C₇₃	C₂×C₄	1	31	yes	—
10	C₂³×C₄×C₇₃	C₂×C₄	1	465	yes	—
11	C₂³×C₄×C₇₃	C₂³	1	155	yes	—
12	C₂×#10596831	C₂⁴	1	496	yes	direct product factor
13	C₂⁴×C₇₃	C₂×C₈	1	496	yes	complemented
14	C₂⁴×C₇₃	C₂²×C₄	1	155	yes	—
15	C₂²×C₄×C₇₃	C₂²×C₄	1	1085	yes	—
16	C₂²×C₄×C₇₃	C₂⁴	1	155	yes	—
17	C₂³×C₇₃	C₂²×C₈	1	1240	yes	complemented
18	C₂³×C₇₃	C₂³×C₄	1	155	yes	—
19	C₂×C₄×C₇₃	C₂³×C₄	1	465	yes	—
20	C₂×C₄×C₇₃	C₂⁵	1	31	yes	—
21	#10596831	C₂⁵	1	32	yes	direct product factor
22	C₂²×C₇₃	C₂³×C₈	1	620	yes	complemented
23	C₂²×C₇₃	C₂⁴×C₄	1	31	yes	—
24	C₄×C₇₃	C₂⁴×C₄	1	31	yes	—
25	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
26	C₂⁵×C₈	—	73	1	no	maximal, Sylow
27	C₂×C₇₃	C₂⁴×C₈	1	62	yes	complemented
28	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
29	C₂⁵×C₄	D₇₃	1	1	yes	characteristic
30	C₂⁴×C₈	—	73	62	no	—
31	C₇₃	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
32	C₂⁶	Dic₇₃	1	1	yes	characteristic
33	C₂⁴×C₄	Dic₇₃	1	31	yes	—
34	C₂⁴×C₄	C₂×D₇₃	1	31	yes	—
35	C₂³×C₈	—	73	620	no	—
36	C₂⁵	#10596831	1	32	yes	direct product factor
37	C₂⁵	C₂×Dic₇₃	1	31	yes	—
38	C₂³×C₄	C₂×Dic₇₃	1	465	yes	—
39	C₂³×C₄	C₂²×D₇₃	1	155	yes	—
40	C₂²×C₈	—	73	1240	no	—
41	C₂⁴	C₂×#10596831	1	496	yes	direct product factor
42	C₂⁴	C₂²×Dic₇₃	1	155	yes	—
43	C₂²×C₄	C₂²×Dic₇₃	1	1085	yes	—
44	C₂²×C₄	C₂³×D₇₃	1	155	yes	—
45	C₂×C₈	—	73	496	no	—
46	C₂³	C₂²×#10596831	1	1240	yes	direct product factor
47	C₂³	C₂³×Dic₇₃	1	155	yes	—
48	C₂×C₄	C₂³×Dic₇₃	1	465	yes	—
49	C₂×C₄	C₂⁴×D₇₃	1	31	yes	—
50	C₈	—	73	32	no	—
51	C₂²	C₂³×#10596831	1	620	yes	direct product factor
52	C₂²	C₂⁴×Dic₇₃	1	31	yes	—
53	C₄	C₂⁴×Dic₇₃	1	31	yes	—
54	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
55	C₂	C₂⁴×#10596831	1	62	yes	direct product factor
56	C₂	C₂⁵×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵×C₄	D₇₃
commutator, hypocenter	C₇₃	C₂⁵×C₈
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂⁵×C₄×C₇₃	C₂
socle, prime radical	C₂⁶×C₇₃	C₄
cosocle	C₄×C₇₃	C₂⁶
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#10596831	C₂	direct (N × H)
3. C₂³×#10596831	C₂²	direct (N × H)
7. C₂²×#10596831	C₂³	direct (N × H)
8. C₂⁵×C₇₃	C₈	semidirect
12. C₂×#10596831	C₂⁴	direct (N × H)
13. C₂⁴×C₇₃	C₂×C₈	semidirect
17. C₂³×C₇₃	C₂²×C₈	semidirect
21. #10596831	C₂⁵	direct (N × H)
22. C₂²×C₇₃	C₂³×C₈	semidirect
27. C₂×C₇₃	C₂⁴×C₈	semidirect
31. C₇₃	C₂⁵×C₈	semidirect
36. C₂⁵	#10596831	direct (N × H)
41. C₂⁴	C₂×#10596831	direct (N × H)
46. C₂³	C₂²×#10596831	direct (N × H)
51. C₂²	C₂³×#10596831	direct (N × H)
55. C₂	C₂⁴×#10596831	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×#10596831	2
C₂⁵×C₄×C₇₃	2
C₂⁵×C₈	73

Subgroups — C25×#10596831 #33999738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×#10596831 #33999738