Subgroups — C₂⁵×#87604 #27158733

Order 12416.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#87604	C₂	1	62	yes	maximal, direct product factor
2	C₂⁵×D₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×#87604	C₂²	1	620	yes	direct product factor
4	C₂⁴×D₉₇	C₄	1	31	yes	complemented
5	C₂⁴×D₉₇	C₂²	1	31	yes	—
6	C₂⁵×C₉₇	C₄	1	1	yes	characteristic, complemented
7	C₂²×#87604	C₂³	1	1240	yes	direct product factor
8	C₂³×D₉₇	C₂×C₄	1	465	yes	complemented
9	C₂³×D₉₇	C₂³	1	155	yes	—
10	C₂⁴×C₉₇	C₂×C₄	1	31	yes	complemented
11	C₂³×C₉₇	C₂²×C₄	1	155	yes	complemented
12	C₂²×D₉₇	C₂²×C₄	1	1085	yes	complemented
13	C₂²×D₉₇	C₂⁴	1	155	yes	—
14	C₂×#87604	C₂⁴	1	496	yes	direct product factor
15	C₂²×C₉₇	C₂³×C₄	1	155	yes	complemented
16	C₂×D₉₇	C₂³×C₄	1	465	yes	complemented
17	C₂×D₉₇	C₂⁵	1	31	yes	—
18	#87604	C₂⁵	1	32	yes	direct product factor
19	C₂×C₉₇	C₂⁴×C₄	1	31	yes	complemented
20	D₉₇	C₂⁴×C₄	1	31	yes	complemented
21	D₉₇	C₂⁶	1	1	yes	characteristic
22	C₂⁵×C₄	—	97	1	no	maximal, Sylow
23	C₉₇	C₂⁵×C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂⁶	—	97	1	no	—
25	C₂⁴×C₄	—	97	62	no	—
26	C₂⁵	—	97	31	no	—
27	C₂⁵	—	97	31	no	—
28	C₂⁵	#87604	1	1	yes	characteristic, direct product factor
29	C₂³×C₄	—	97	620	no	—
30	C₂⁴	—	97	465	no	—
31	C₂⁴	—	97	155	no	—
32	C₂⁴	C₂×#87604	1	31	yes	direct product factor
33	C₂²×C₄	—	97	1240	no	—
34	C₂³	—	97	155	no	—
35	C₂³	—	97	1085	no	—
36	C₂³	C₂²×#87604	1	155	yes	direct product factor
37	C₂×C₄	—	97	496	no	—
38	C₂²	—	97	31	no	—
39	C₂²	—	97	465	no	—
40	C₂²	C₂³×#87604	1	155	yes	direct product factor
41	C₄	—	97	32	no	—
42	C₂	—	97	1	no	—
43	C₂	—	97	31	no	—
44	C₂	C₂⁴×#87604	1	31	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁵	#87604
commutator, hypocenter	C₉₇	C₂⁵×C₄
Frattini	C₁	G
Fitting, socle	C₂⁵×C₉₇	C₄
prime radical	C₂⁵×D₉₇	C₂
cosocle	D₉₇	C₂⁶
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#87604	C₂	direct (N × H)
3. C₂³×#87604	C₂²	direct (N × H)
4. C₂⁴×D₉₇	C₄	semidirect
6. C₂⁵×C₉₇	C₄	semidirect
7. C₂²×#87604	C₂³	direct (N × H)
8. C₂³×D₉₇	C₂×C₄	semidirect
10. C₂⁴×C₉₇	C₂×C₄	semidirect
11. C₂³×C₉₇	C₂²×C₄	semidirect
12. C₂²×D₉₇	C₂²×C₄	semidirect
14. C₂×#87604	C₂⁴	direct (N × H)
15. C₂²×C₉₇	C₂³×C₄	semidirect
16. C₂×D₉₇	C₂³×C₄	semidirect
18. #87604	C₂⁵	direct (N × H)
19. C₂×C₉₇	C₂⁴×C₄	semidirect
20. D₉₇	C₂⁴×C₄	semidirect
23. C₉₇	C₂⁵×C₄	semidirect
28. C₂⁵	#87604	direct (N × H)
32. C₂⁴	C₂×#87604	direct (N × H)
36. C₂³	C₂²×#87604	direct (N × H)
40. C₂²	C₂³×#87604	direct (N × H)
44. C₂	C₂⁴×#87604	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×#87604	2
C₂⁵×D₉₇	2
C₂⁵×C₄	97

Subgroups — C25×#87604 #27158733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×#87604 #27158733