Subgroups — #568×C₂₆₉ #31770305

Order 17216.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₆₉	C₂	1	1	yes	maximal, characteristic
2	#131×C₂₆₉	C₂	1	8	yes	maximal, complemented
3	C₂×M₁₆×C₂₆₉	C₂	1	6	yes	maximal, complemented
4	#41×C₂₆₉	C₄	1	4	yes	—
5	#41×C₂₆₉	C₂²	1	4	yes	—
6	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
7	C₂×D₄×C₂₆₉	C₄	1	3	yes	—
8	C₂²×C₄×C₂₆₉	C₂²	1	3	yes	—
9	M₁₆×C₂₆₉	C₂²	1	4	yes	—
10	M₁₆×C₂₆₉	C₂²	1	12	yes	complemented
11	C₂×C₈×C₂₆₉	C₂²	1	12	yes	complemented
12	C₂³×C₂₆₉	C₂×C₄	1	3	yes	—
13	Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
14	D₄×C₂₆₉	C₂×C₄	1	12	yes	—
15	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
16	C₂×C₄×C₂₆₉	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₂₆₉	C₂³	1	6	yes	—
18	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
19	C₈×C₂₆₉	C₂³	1	8	yes	—
20	C₂²×C₂₆₉	—	4	3	no	—
21	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₂₆₉	C₂²×C₄	1	6	yes	—
23	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
25	C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
26	C₂×C₂₆₉	—	2	1	no	—
27	C₂×C₂₆₉	—	2	6	no	—
28	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
29	C₂₆₉	#568	1	1	yes	characteristic, direct product factor, Sylow
30	#568	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×#41	C₂×C₂₆₉	1	1	yes	characteristic
32	#131	C₂×C₂₆₉	1	8	yes	complemented
33	C₂×M₁₆	C₂×C₂₆₉	1	6	yes	complemented
34	#41	C₄×C₂₆₉	1	4	yes	—
35	#41	C₂²×C₂₆₉	1	4	yes	—
36	C₂×Q₈	C₄×C₂₆₉	1	1	yes	characteristic
37	C₂×D₄	C₄×C₂₆₉	1	3	yes	—
38	C₂²×C₄	C₂²×C₂₆₉	1	3	yes	—
39	M₁₆	C₂²×C₂₆₉	1	4	yes	—
40	M₁₆	C₂²×C₂₆₉	1	12	yes	complemented
41	C₂×C₈	C₂²×C₂₆₉	1	12	yes	complemented
42	C₂³	C₂×C₄×C₂₆₉	1	3	yes	—
43	Q₈	C₂×C₄×C₂₆₉	1	4	yes	—
44	D₄	C₂×C₄×C₂₆₉	1	12	yes	—
45	C₂×C₄	C₂×C₄×C₂₆₉	1	3	yes	—
46	C₂×C₄	C₂×C₄×C₂₆₉	1	6	yes	—
47	C₂×C₄	C₂³×C₂₆₉	1	6	yes	—
48	C₂×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
49	C₈	C₂³×C₂₆₉	1	8	yes	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×C₄×C₂₆₉	1	1	yes	characteristic
52	C₂²	C₂²×C₄×C₂₆₉	1	6	yes	—
53	C₄	C₂²×C₄×C₂₆₉	1	1	yes	characteristic
54	C₄	C₂²×C₄×C₂₆₉	1	6	yes	—
55	C₄	C₂⁴×C₂₆₉	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×C₄×C₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₆₉	C₂⁴
commutator	C₂	C₂³×C₄×C₂₆₉
Frattini, cosocle	C₄	C₂⁴×C₂₆₉
socle	C₂×C₂₆₉	C₂³×C₄
prime radical	C₂×#41×C₂₆₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #131×C₂₆₉	C₂	semidirect
3. C₂×M₁₆×C₂₆₉	C₂	semidirect
10. M₁₆×C₂₆₉	C₂²	semidirect
11. C₂×C₈×C₂₆₉	C₂²	semidirect
29. C₂₆₉	#568	direct (N × H)
30. #568	C₂₆₉	direct (N × H)
32. #131	C₂×C₂₆₉	semidirect
33. C₂×M₁₆	C₂×C₂₆₉	semidirect
40. M₁₆	C₂²×C₂₆₉	semidirect
41. C₂×C₈	C₂²×C₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₂₆₉	2
#131×C₂₆₉	2
C₂×M₁₆×C₂₆₉	2
#568	269

Subgroups — #568×C269 #31770305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #568×C₂₆₉ #31770305