Subgroups — C₂²×H(5,𝔽₂)×C₂₆₉ #46808698

Order 34432.

6560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×C₂₆₉	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#41×C₂₆₉	C₂	1	6	yes	maximal, complemented
3	C₂³×D₄×C₂₆₉	C₂	1	9	yes	maximal, complemented
4	C₂⁵×C₂₆₉	C₂²	1	6	yes	complemented
5	H(5,𝔽₂)×C₂₆₉	C₂²	1	256	yes	direct product factor
6	C₂×#41×C₂₆₉	C₂²	1	144	yes	complemented
7	C₂²×Q₈×C₂₆₉	C₂²	1	2	yes	complemented
8	C₂²×D₄×C₂₆₉	C₂²	1	18	yes	complemented
9	C₂²×D₄×C₂₆₉	C₂²	1	216	yes	complemented
10	C₂³×C₄×C₂₆₉	C₂²	1	9	yes	complemented
11	C₂⁴×C₂₆₉	—	4	24	no	—
12	C₂⁴×C₂₆₉	C₂³	1	9	yes	—
13	C₂⁴×C₂₆₉	C₂³	1	72	yes	complemented
14	#41×C₂₆₉	C₂³	1	384	yes	complemented
15	C₂×Q₈×C₂₆₉	C₂³	1	24	yes	complemented
16	C₂×D₄×C₂₆₉	C₂³	1	216	yes	complemented
17	C₂×D₄×C₂₆₉	C₂³	1	576	yes	complemented
18	C₂²×C₄×C₂₆₉	C₂³	1	6	yes	—
19	C₂²×C₄×C₂₆₉	C₂³	1	108	yes	complemented
20	C₂³×C₂₆₉	—	2	36	no	—
21	C₂³×C₂₆₉	—	4	144	no	—
22	C₂³×C₂₆₉	C₂⁴	1	54	yes	—
23	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
24	C₂³×C₂₆₉	C₂⁴	1	96	yes	complemented
25	Q₈×C₂₆₉	C₂⁴	1	32	yes	complemented
26	D₄×C₂₆₉	C₂⁴	1	288	yes	complemented
27	C₂×C₄×C₂₆₉	C₂⁴	1	36	yes	—
28	C₂×C₄×C₂₆₉	C₂⁴	1	144	yes	complemented
29	C₂²×C₂₆₉	—	2	108	no	—
30	C₂²×C₂₆₉	—	4	96	no	—
31	C₂²×C₂₆₉	H(5,𝔽₂)	1	4	yes	direct product factor
32	C₂²×C₂₆₉	C₂⁵	1	36	yes	—
33	C₂²×C₂₆₉	C₂⁵	1	3	yes	—
34	C₄×C₂₆₉	C₂⁵	1	24	yes	—
35	C₂×C₂₆₉	—	2	36	no	—
36	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	6	yes	direct product factor
37	C₂×C₂₆₉	C₂⁶	1	1	yes	characteristic
38	C₂₆₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×H(5,𝔽₂)	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×H(5,𝔽₂)	C₂×C₂₆₉	1	48	yes	direct product factor
41	C₂²×#41	C₂×C₂₆₉	1	6	yes	complemented
42	C₂³×D₄	C₂×C₂₆₉	1	9	yes	complemented
43	C₂⁵	C₂²×C₂₆₉	1	6	yes	complemented
44	H(5,𝔽₂)	C₂²×C₂₆₉	1	256	yes	direct product factor
45	C₂×#41	C₂²×C₂₆₉	1	144	yes	complemented
46	C₂²×Q₈	C₂²×C₂₆₉	1	2	yes	complemented
47	C₂²×D₄	C₂²×C₂₆₉	1	18	yes	complemented
48	C₂²×D₄	C₂²×C₂₆₉	1	216	yes	complemented
49	C₂³×C₄	C₂²×C₂₆₉	1	9	yes	complemented
50	C₂⁴	—	4	24	no	—
51	C₂⁴	C₂³×C₂₆₉	1	9	yes	—
52	C₂⁴	C₂³×C₂₆₉	1	72	yes	complemented
53	#41	C₂³×C₂₆₉	1	384	yes	complemented
54	C₂×Q₈	C₂³×C₂₆₉	1	24	yes	complemented
55	C₂×D₄	C₂³×C₂₆₉	1	216	yes	complemented
56	C₂×D₄	C₂³×C₂₆₉	1	576	yes	complemented
57	C₂²×C₄	C₂³×C₂₆₉	1	6	yes	—
58	C₂²×C₄	C₂³×C₂₆₉	1	108	yes	complemented
59	C₂³	—	2	36	no	—
60	C₂³	—	4	144	no	—
61	C₂³	C₂⁴×C₂₆₉	1	54	yes	—
62	C₂³	C₂⁴×C₂₆₉	1	1	yes	characteristic
63	C₂³	C₂⁴×C₂₆₉	1	96	yes	complemented
64	Q₈	C₂⁴×C₂₆₉	1	32	yes	complemented
65	D₄	C₂⁴×C₂₆₉	1	288	yes	complemented
66	C₂×C₄	C₂⁴×C₂₆₉	1	36	yes	—
67	C₂×C₄	C₂⁴×C₂₆₉	1	144	yes	complemented
68	C₂²	—	2	108	no	—
69	C₂²	—	4	96	no	—
70	C₂²	H(5,𝔽₂)×C₂₆₉	1	4	yes	direct product factor
71	C₂²	C₂⁵×C₂₆₉	1	36	yes	—
72	C₂²	C₂⁵×C₂₆₉	1	3	yes	—
73	C₄	C₂⁵×C₂₆₉	1	24	yes	—
74	C₂	—	2	36	no	—
75	C₂	C₂×H(5,𝔽₂)×C₂₆₉	1	6	yes	direct product factor
76	C₂	C₂⁶×C₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₆₉	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₆₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)×C₂₆₉	C₂	direct (N × H)
2. C₂²×#41×C₂₆₉	C₂	semidirect
3. C₂³×D₄×C₂₆₉	C₂	semidirect
4. C₂⁵×C₂₆₉	C₂²	semidirect
5. H(5,𝔽₂)×C₂₆₉	C₂²	direct (N × H)
6. C₂×#41×C₂₆₉	C₂²	semidirect
7. C₂²×Q₈×C₂₆₉	C₂²	semidirect
8. C₂²×D₄×C₂₆₉	C₂²	semidirect
9. C₂²×D₄×C₂₆₉	C₂²	semidirect
10. C₂³×C₄×C₂₆₉	C₂²	semidirect
13. C₂⁴×C₂₆₉	C₂³	semidirect
14. #41×C₂₆₉	C₂³	semidirect
15. C₂×Q₈×C₂₆₉	C₂³	semidirect
16. C₂×D₄×C₂₆₉	C₂³	semidirect
17. C₂×D₄×C₂₆₉	C₂³	semidirect
19. C₂²×C₄×C₂₆₉	C₂³	semidirect
24. C₂³×C₂₆₉	C₂⁴	semidirect
25. Q₈×C₂₆₉	C₂⁴	semidirect
26. D₄×C₂₆₉	C₂⁴	semidirect
28. C₂×C₄×C₂₆₉	C₂⁴	semidirect
31. C₂²×C₂₆₉	H(5,𝔽₂)	direct (N × H)
36. C₂×C₂₆₉	C₂×H(5,𝔽₂)	direct (N × H)
38. C₂₆₉	C₂²×H(5,𝔽₂)	direct (N × H)
39. C₂²×H(5,𝔽₂)	C₂₆₉	direct (N × H)
40. C₂×H(5,𝔽₂)	C₂×C₂₆₉	direct (N × H)
41. C₂²×#41	C₂×C₂₆₉	semidirect
42. C₂³×D₄	C₂×C₂₆₉	semidirect
43. C₂⁵	C₂²×C₂₆₉	semidirect
44. H(5,𝔽₂)	C₂²×C₂₆₉	direct (N × H)
45. C₂×#41	C₂²×C₂₆₉	semidirect
46. C₂²×Q₈	C₂²×C₂₆₉	semidirect
47. C₂²×D₄	C₂²×C₂₆₉	semidirect
48. C₂²×D₄	C₂²×C₂₆₉	semidirect
49. C₂³×C₄	C₂²×C₂₆₉	semidirect
52. C₂⁴	C₂³×C₂₆₉	semidirect
53. #41	C₂³×C₂₆₉	semidirect
54. C₂×Q₈	C₂³×C₂₆₉	semidirect
55. C₂×D₄	C₂³×C₂₆₉	semidirect
56. C₂×D₄	C₂³×C₂₆₉	semidirect
58. C₂²×C₄	C₂³×C₂₆₉	semidirect
63. C₂³	C₂⁴×C₂₆₉	semidirect
64. Q₈	C₂⁴×C₂₆₉	semidirect
65. D₄	C₂⁴×C₂₆₉	semidirect
67. C₂×C₄	C₂⁴×C₂₆₉	semidirect
70. C₂²	H(5,𝔽₂)×C₂₆₉	direct (N × H)
75. C₂	C₂×H(5,𝔽₂)×C₂₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)×C₂₆₉	2
C₂²×#41×C₂₆₉	2
C₂³×D₄×C₂₆₉	2
C₂²×H(5,𝔽₂)	269

Subgroups — C22×H(5,𝔽2)×C269 #46808698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×H(5,𝔽₂)×C₂₆₉ #46808698