Subgroups — C₂²×#16864385 #31771364

Order 17216.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16864385	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₆₉	C₄	1	1	yes	characteristic
6	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₆₉	C₄	1	6	yes	—
8	C₂×C₄×D₂₆₉	C₂²	1	6	yes	—
9	#16864385	C₂²	1	16	yes	direct product factor
10	C₂×#14524130	C₂²	1	6	yes	complemented
11	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₆₉	C₂²	1	6	yes	complemented
13	C₂²×D₂₆₉	—	2	4	no	—
14	C₂²×D₂₆₉	C₂×C₄	1	6	yes	—
15	C₂×Dic₂₆₉	C₂×C₄	1	6	yes	—
16	C₄×D₂₆₉	C₂×C₄	1	12	yes	—
17	C₄×D₂₆₉	C₂³	1	4	yes	—
18	#14524130	C₂³	1	4	yes	complemented
19	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
21	C₂×C₄×C₂₆₉	C₂³	1	3	yes	—
22	C₈×C₂₆₉	C₂³	1	4	yes	complemented
23	C₂²×C₂₆₉	M₁₆	1	4	yes	complemented
24	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
25	C₂×D₂₆₉	—	2	12	no	—
26	C₂×D₂₆₉	C₂²×C₄	1	4	yes	—
27	C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
28	C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
29	Dic₂₆₉	C₂²×C₄	1	4	yes	—
30	C₂×C₂₆₉	C₂×M₁₆	1	6	yes	complemented
31	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
32	D₂₆₉	—	2	4	no	—
33	C₂₆₉	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂²×M₁₆	—	269	1	no	maximal, Sylow
35	C₂³×C₄	—	269	1	no	—
36	C₂×M₁₆	—	269	12	no	—
37	C₂²×C₈	—	269	1	no	—
38	C₂²×C₈	D₂₆₉	1	1	yes	characteristic, complemented
39	C₂⁴	—	269	1	no	—
40	C₂²×C₄	—	269	1	no	—
41	C₂²×C₄	—	269	6	no	—
42	C₂²×C₄	—	269	6	no	—
43	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
44	M₁₆	—	269	16	no	—
45	C₂×C₈	—	269	6	no	—
46	C₂×C₈	C₂×D₂₆₉	1	6	yes	complemented
47	C₂³	—	269	6	no	—
48	C₂³	—	538	4	no	—
49	C₂³	C₄×D₂₆₉	1	1	yes	characteristic
50	C₂×C₄	—	269	6	no	—
51	C₂×C₄	—	269	12	no	—
52	C₂×C₄	—	269	4	no	—
53	C₂×C₄	C₄×D₂₆₉	1	3	yes	—
54	C₂×C₄	C₂²×D₂₆₉	1	3	yes	—
55	C₈	—	269	4	no	—
56	C₈	C₂²×D₂₆₉	1	4	yes	complemented
57	C₂²	—	269	4	no	—
58	C₂²	—	538	12	no	—
59	C₂²	#16864385	1	4	yes	direct product factor
60	C₂²	C₂×C₄×D₂₆₉	1	3	yes	—
61	C₄	—	269	4	no	—
62	C₄	C₂×C₄×D₂₆₉	1	3	yes	—
63	C₄	C₂³×D₂₆₉	1	1	yes	characteristic
64	C₂	—	538	4	no	—
65	C₂	C₂×#16864385	1	6	yes	direct product factor
66	C₂	C₂²×C₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₉
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₄	C₂³×D₂₆₉
Fitting	C₂²×C₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂×C₄
prime radical	C₂³×D₂₆₉	C₄
cosocle	C₄×C₂₆₉	C₂⁴
hypercenter	C₂²×C₈	D₂₆₉
hypocenter	C₂₆₉	C₂²×M₁₆
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16864385	C₂	direct (N × H)
3. C₂²×#14524130	C₂	semidirect
4. C₂²×C₈×C₂₆₉	C₂	semidirect
9. #16864385	C₂²	direct (N × H)
10. C₂×#14524130	C₂²	semidirect
12. C₂×C₈×C₂₆₉	C₂²	semidirect
18. #14524130	C₂³	semidirect
22. C₈×C₂₆₉	C₂³	semidirect
23. C₂²×C₂₆₉	M₁₆	semidirect
30. C₂×C₂₆₉	C₂×M₁₆	semidirect
33. C₂₆₉	C₂²×M₁₆	semidirect
38. C₂²×C₈	D₂₆₉	semidirect
46. C₂×C₈	C₂×D₂₆₉	semidirect
56. C₈	C₂²×D₂₆₉	semidirect
59. C₂²	#16864385	direct (N × H)
65. C₂	C₂×#16864385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₆₉	2
C₂×#16864385	2
C₂²×#14524130	2
C₂²×C₈×C₂₆₉	2
C₂²×M₁₆	269

Subgroups — C22×#16864385 #31771364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16864385 #31771364