Subgroups — C₂⁴×D₃₈₈ #27156529

Order 12416.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₉₇	C₂	1	2	yes	maximal, complemented
2	C₂³×D₃₈₈	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₉₇	—	2	32	no	—
5	C₂⁴×D₉₇	C₂²	1	60	yes	complemented
6	C₂²×D₃₈₈	C₂²	1	560	yes	direct product factor
7	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉₇	C₂²	1	30	yes	complemented
9	C₂³×D₉₇	—	2	480	no	—
10	C₂³×D₉₇	C₂³	1	280	yes	complemented
11	C₂×D₃₈₈	C₂³	1	960	yes	direct product factor
12	C₂⁴×C₉₇	D₄	1	16	yes	complemented
13	C₂⁴×C₉₇	C₂³	1	15	yes	—
14	C₂²×C₄×C₉₇	C₂³	1	140	yes	complemented
15	C₂³×C₉₇	C₂×D₄	1	120	yes	complemented
16	C₂³×C₉₇	C₂⁴	1	35	yes	—
17	C₂²×D₉₇	—	2	1120	no	—
18	C₂²×D₉₇	C₂⁴	1	240	yes	complemented
19	C₂×C₄×C₉₇	C₂⁴	1	120	yes	complemented
20	D₃₈₈	C₂⁴	1	256	yes	direct product factor
21	C₂²×C₉₇	C₂²×D₄	1	140	yes	complemented
22	C₂²×C₉₇	C₂⁵	1	15	yes	—
23	C₂×D₉₇	—	2	480	no	—
24	C₂×D₉₇	C₂⁵	1	32	yes	complemented
25	C₄×C₉₇	C₂⁵	1	16	yes	complemented
26	C₂×C₉₇	C₂³×D₄	1	30	yes	complemented
27	C₂×C₉₇	C₂⁶	1	1	yes	characteristic
28	D₉₇	—	2	32	no	—
29	C₂⁴×D₄	—	97	1	no	maximal, Sylow
30	C₉₇	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
31	C₂⁶	—	97	2	no	—
32	C₂³×D₄	—	97	60	no	—
33	C₂⁴×C₄	D₉₇	1	1	yes	characteristic, complemented
34	C₂⁵	—	97	60	no	—
35	C₂⁵	—	194	32	no	—
36	C₂⁵	C₂×D₉₇	1	1	yes	characteristic
37	C₂²×D₄	—	97	560	no	—
38	C₂³×C₄	C₂×D₉₇	1	30	yes	complemented
39	C₂⁴	—	97	280	no	—
40	C₂⁴	—	194	480	no	—
41	C₂⁴	D₃₈₈	1	16	yes	direct product factor
42	C₂⁴	C₂²×D₉₇	1	15	yes	—
43	C₂×D₄	—	97	960	no	—
44	C₂²×C₄	C₂²×D₉₇	1	140	yes	complemented
45	C₂³	—	97	240	no	—
46	C₂³	—	194	1120	no	—
47	C₂³	C₂×D₃₈₈	1	120	yes	direct product factor
48	C₂³	C₂³×D₉₇	1	35	yes	—
49	D₄	—	97	256	no	—
50	C₂×C₄	C₂³×D₉₇	1	120	yes	complemented
51	C₂²	—	97	32	no	—
52	C₂²	—	194	480	no	—
53	C₂²	C₂²×D₃₈₈	1	140	yes	direct product factor
54	C₂²	C₂⁴×D₉₇	1	15	yes	—
55	C₄	C₂⁴×D₉₇	1	16	yes	complemented
56	C₂	—	194	32	no	—
57	C₂	C₂³×D₃₈₈	1	30	yes	direct product factor
58	C₂	C₂⁵×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₉₇
commutator, cosocle	C₂×C₉₇	C₂⁶
Frattini	C₂	C₂⁵×D₉₇
Fitting	C₂⁴×C₄×C₉₇	C₂
socle	C₂⁵×C₉₇	C₂²
hypercenter	C₂⁴×C₄	D₉₇
hypocenter	C₉₇	C₂⁴×D₄
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₉₇	C₂	semidirect
2. C₂³×D₃₈₈	C₂	direct (N × H)
3. C₂⁴×C₄×C₉₇	C₂	semidirect
5. C₂⁴×D₉₇	C₂²	semidirect
6. C₂²×D₃₈₈	C₂²	direct (N × H)
8. C₂³×C₄×C₉₇	C₂²	semidirect
10. C₂³×D₉₇	C₂³	semidirect
11. C₂×D₃₈₈	C₂³	direct (N × H)
12. C₂⁴×C₉₇	D₄	semidirect
14. C₂²×C₄×C₉₇	C₂³	semidirect
15. C₂³×C₉₇	C₂×D₄	semidirect
18. C₂²×D₉₇	C₂⁴	semidirect
19. C₂×C₄×C₉₇	C₂⁴	semidirect
20. D₃₈₈	C₂⁴	direct (N × H)
21. C₂²×C₉₇	C₂²×D₄	semidirect
24. C₂×D₉₇	C₂⁵	semidirect
25. C₄×C₉₇	C₂⁵	semidirect
26. C₂×C₉₇	C₂³×D₄	semidirect
30. C₉₇	C₂⁴×D₄	semidirect
33. C₂⁴×C₄	D₉₇	semidirect
38. C₂³×C₄	C₂×D₉₇	semidirect
41. C₂⁴	D₃₈₈	direct (N × H)
44. C₂²×C₄	C₂²×D₉₇	semidirect
47. C₂³	C₂×D₃₈₈	direct (N × H)
50. C₂×C₄	C₂³×D₉₇	semidirect
53. C₂²	C₂²×D₃₈₈	direct (N × H)
55. C₄	C₂⁴×D₉₇	semidirect
57. C₂	C₂³×D₃₈₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₉₇	2
C₂³×D₃₈₈	2
C₂⁴×C₄×C₉₇	2
C₂⁴×D₄	97

Subgroups — C24×D388 #27156529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₃₈₈ #27156529