Subgroups — C₂⁴×C₂²⋊D₉₇ #27156528

Order 12416.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂²⋊D₉₇	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁶×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₉₇	—	2	16	no	—
6	C₂⁴×D₉₇	C₂²	1	30	yes	complemented
7	C₂²×C₂²⋊D₉₇	C₂²	1	560	yes	direct product factor
8	C₂³×Dic₉₇	C₂²	1	30	yes	complemented
9	C₂⁵×C₉₇	—	2	16	no	—
10	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
11	C₂⁵×C₉₇	C₂²	1	30	yes	complemented
12	C₂³×D₉₇	—	2	240	no	—
13	C₂³×D₉₇	C₂³	1	140	yes	complemented
14	C₂×C₂²⋊D₉₇	C₂³	1	960	yes	direct product factor
15	C₂²×Dic₉₇	C₂³	1	140	yes	complemented
16	C₂⁴×C₉₇	—	2	240	no	—
17	C₂⁴×C₉₇	D₄	1	16	yes	complemented
18	C₂⁴×C₉₇	C₂³	1	15	yes	—
19	C₂⁴×C₉₇	C₂³	1	140	yes	complemented
20	C₂³×C₉₇	—	2	560	no	—
21	C₂³×C₉₇	C₂×D₄	1	120	yes	complemented
22	C₂³×C₉₇	C₂⁴	1	35	yes	—
23	C₂³×C₉₇	C₂⁴	1	120	yes	complemented
24	C₂²×D₉₇	—	2	560	no	—
25	C₂²×D₉₇	C₂⁴	1	120	yes	complemented
26	C₂²⋊D₉₇	C₂⁴	1	256	yes	direct product factor
27	C₂×Dic₉₇	C₂⁴	1	120	yes	complemented
28	C₂²×C₉₇	—	2	240	no	—
29	C₂²×C₉₇	C₂²×D₄	1	140	yes	complemented
30	C₂²×C₉₇	C₂⁵	1	15	yes	—
31	C₂²×C₉₇	C₂⁵	1	16	yes	complemented
32	C₂×D₉₇	—	2	240	no	—
33	C₂×D₉₇	C₂⁵	1	16	yes	complemented
34	Dic₉₇	C₂⁵	1	16	yes	complemented
35	C₂×C₉₇	—	2	16	no	—
36	C₂×C₉₇	C₂³×D₄	1	30	yes	complemented
37	C₂×C₉₇	C₂⁶	1	1	yes	characteristic
38	D₉₇	—	2	16	no	—
39	C₂⁴×D₄	—	97	1	no	maximal, Sylow
40	C₉₇	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
41	C₂⁶	—	97	1	no	—
42	C₂⁶	D₉₇	1	1	yes	characteristic, complemented
43	C₂³×D₄	—	97	60	no	—
44	C₂⁴×C₄	—	97	1	no	—
45	C₂⁵	—	2	16	no	—
46	C₂⁵	—	97	30	no	—
47	C₂⁵	—	194	16	no	—
48	C₂⁵	C₂×D₉₇	1	1	yes	characteristic
49	C₂⁵	C₂×D₉₇	1	30	yes	complemented
50	C₂²×D₄	—	97	560	no	—
51	C₂³×C₄	—	97	30	no	—
52	C₂⁴	—	2	240	no	—
53	C₂⁴	—	97	140	no	—
54	C₂⁴	—	194	240	no	—
55	C₂⁴	C₂²⋊D₉₇	1	16	yes	direct product factor
56	C₂⁴	C₂²×D₉₇	1	15	yes	—
57	C₂⁴	C₂²×D₉₇	1	140	yes	complemented
58	C₂×D₄	—	97	960	no	—
59	C₂²×C₄	—	97	140	no	—
60	C₂³	—	2	560	no	—
61	C₂³	—	97	120	no	—
62	C₂³	—	194	560	no	—
63	C₂³	C₂×C₂²⋊D₉₇	1	120	yes	direct product factor
64	C₂³	C₂³×D₉₇	1	35	yes	—
65	C₂³	C₂³×D₉₇	1	120	yes	complemented
66	D₄	—	97	256	no	—
67	C₂×C₄	—	97	120	no	—
68	C₂²	—	2	240	no	—
69	C₂²	—	97	16	no	—
70	C₂²	—	194	240	no	—
71	C₂²	C₂²×C₂²⋊D₉₇	1	140	yes	direct product factor
72	C₂²	C₂⁴×D₉₇	1	15	yes	—
73	C₂²	C₂⁴×D₉₇	1	16	yes	complemented
74	C₄	—	97	16	no	—
75	C₂	—	2	16	no	—
76	C₂	—	194	16	no	—
77	C₂	C₂³×C₂²⋊D₉₇	1	30	yes	direct product factor
78	C₂	C₂⁵×D₉₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₉₇	C₂	semidirect
2. C₂³×C₂²⋊D₉₇	C₂	direct (N × H)
3. C₂⁴×Dic₉₇	C₂	semidirect
4. C₂⁶×C₉₇	C₂	semidirect
6. C₂⁴×D₉₇	C₂²	semidirect
7. C₂²×C₂²⋊D₉₇	C₂²	direct (N × H)
8. C₂³×Dic₉₇	C₂²	semidirect
11. C₂⁵×C₉₇	C₂²	semidirect
13. C₂³×D₉₇	C₂³	semidirect
14. C₂×C₂²⋊D₉₇	C₂³	direct (N × H)
15. C₂²×Dic₉₇	C₂³	semidirect
17. C₂⁴×C₉₇	D₄	semidirect
19. C₂⁴×C₉₇	C₂³	semidirect
21. C₂³×C₉₇	C₂×D₄	semidirect
23. C₂³×C₉₇	C₂⁴	semidirect
25. C₂²×D₉₇	C₂⁴	semidirect
26. C₂²⋊D₉₇	C₂⁴	direct (N × H)
27. C₂×Dic₉₇	C₂⁴	semidirect
29. C₂²×C₉₇	C₂²×D₄	semidirect
31. C₂²×C₉₇	C₂⁵	semidirect
33. C₂×D₉₇	C₂⁵	semidirect
34. Dic₉₇	C₂⁵	semidirect
36. C₂×C₉₇	C₂³×D₄	semidirect
40. C₉₇	C₂⁴×D₄	semidirect
42. C₂⁶	D₉₇	semidirect
49. C₂⁵	C₂×D₉₇	semidirect
55. C₂⁴	C₂²⋊D₉₇	direct (N × H)
57. C₂⁴	C₂²×D₉₇	semidirect
63. C₂³	C₂×C₂²⋊D₉₇	direct (N × H)
65. C₂³	C₂³×D₉₇	semidirect
71. C₂²	C₂²×C₂²⋊D₉₇	direct (N × H)
73. C₂²	C₂⁴×D₉₇	semidirect
77. C₂	C₂³×C₂²⋊D₉₇	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×C₂²⋊D₉₇	2
C₂⁴×Dic₉₇	2
C₂⁶×C₉₇	2
C₂⁴×D₄	97

Subgroups — C24×C22⋊D97 #27156528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂²⋊D₉₇ #27156528