Subgroups — C₂⁵×#10587739 #31797786

Order 17536.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#10587739	C₂	1	62	yes	maximal, direct product factor
2	C₂⁵×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×#10587739	C₂²	1	620	yes	direct product factor
4	C₂⁴×D₁₃₇	C₄	1	31	yes	complemented
5	C₂⁴×D₁₃₇	C₂²	1	31	yes	—
6	C₂⁵×C₁₃₇	C₄	1	1	yes	characteristic, complemented
7	C₂²×#10587739	C₂³	1	1240	yes	direct product factor
8	C₂³×D₁₃₇	C₂×C₄	1	465	yes	complemented
9	C₂³×D₁₃₇	C₂³	1	155	yes	—
10	C₂⁴×C₁₃₇	C₂×C₄	1	31	yes	complemented
11	C₂×#10587739	C₂⁴	1	496	yes	direct product factor
12	C₂²×D₁₃₇	C₂²×C₄	1	1085	yes	complemented
13	C₂²×D₁₃₇	C₂⁴	1	155	yes	—
14	C₂³×C₁₃₇	C₂²×C₄	1	155	yes	complemented
15	C₂²×C₁₃₇	C₂³×C₄	1	155	yes	complemented
16	C₂×D₁₃₇	C₂³×C₄	1	465	yes	complemented
17	C₂×D₁₃₇	C₂⁵	1	31	yes	—
18	#10587739	C₂⁵	1	32	yes	direct product factor
19	C₂×C₁₃₇	C₂⁴×C₄	1	31	yes	complemented
20	D₁₃₇	C₂⁴×C₄	1	31	yes	complemented
21	D₁₃₇	C₂⁶	1	1	yes	characteristic
22	C₁₃₇	C₂⁵×C₄	1	1	yes	characteristic, complemented, Sylow
23	C₂⁵×C₄	—	137	1	no	maximal, Sylow
24	C₂⁶	—	137	1	no	—
25	C₂⁴×C₄	—	137	62	no	—
26	C₂⁵	—	137	31	no	—
27	C₂⁵	—	137	31	no	—
28	C₂⁵	#10587739	1	1	yes	characteristic, direct product factor
29	C₂³×C₄	—	137	620	no	—
30	C₂⁴	—	137	465	no	—
31	C₂⁴	—	137	155	no	—
32	C₂⁴	C₂×#10587739	1	31	yes	direct product factor
33	C₂²×C₄	—	137	1240	no	—
34	C₂³	—	137	155	no	—
35	C₂³	—	137	1085	no	—
36	C₂³	C₂²×#10587739	1	155	yes	direct product factor
37	C₂×C₄	—	137	496	no	—
38	C₂²	—	137	31	no	—
39	C₂²	—	137	465	no	—
40	C₂²	C₂³×#10587739	1	155	yes	direct product factor
41	C₄	—	137	32	no	—
42	C₂	—	137	1	no	—
43	C₂	—	137	31	no	—
44	C₂	C₂⁴×#10587739	1	31	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁵	#10587739
commutator, hypocenter	C₁₃₇	C₂⁵×C₄
Frattini	C₁	G
Fitting, socle	C₂⁵×C₁₃₇	C₄
prime radical	C₂⁵×D₁₃₇	C₂
cosocle	D₁₃₇	C₂⁶
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#10587739	C₂	direct (N × H)
3. C₂³×#10587739	C₂²	direct (N × H)
4. C₂⁴×D₁₃₇	C₄	semidirect
6. C₂⁵×C₁₃₇	C₄	semidirect
7. C₂²×#10587739	C₂³	direct (N × H)
8. C₂³×D₁₃₇	C₂×C₄	semidirect
10. C₂⁴×C₁₃₇	C₂×C₄	semidirect
11. C₂×#10587739	C₂⁴	direct (N × H)
12. C₂²×D₁₃₇	C₂²×C₄	semidirect
14. C₂³×C₁₃₇	C₂²×C₄	semidirect
15. C₂²×C₁₃₇	C₂³×C₄	semidirect
16. C₂×D₁₃₇	C₂³×C₄	semidirect
18. #10587739	C₂⁵	direct (N × H)
19. C₂×C₁₃₇	C₂⁴×C₄	semidirect
20. D₁₃₇	C₂⁴×C₄	semidirect
22. C₁₃₇	C₂⁵×C₄	semidirect
28. C₂⁵	#10587739	direct (N × H)
32. C₂⁴	C₂×#10587739	direct (N × H)
36. C₂³	C₂²×#10587739	direct (N × H)
40. C₂²	C₂³×#10587739	direct (N × H)
44. C₂	C₂⁴×#10587739	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×#10587739	2
C₂⁵×D₁₃₇	2
C₂⁵×C₄	137

Subgroups — C25×#10587739 #31797786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×#10587739 #31797786