Subgroups — C₂²×#16864411 #31771677

Order 17216.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14524137	C₂	1	2	yes	maximal, complemented
3	C₂³×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864411	C₂²	1	16	yes	direct product factor
5	C₂×#14524137	C₂²	1	12	yes	complemented
6	C₂²×Dic₂₆₉	C₄	1	6	yes	—
7	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
8	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₆₉	C₂²	1	6	yes	—
10	C₂⁴×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	C₂³	1	8	yes	complemented
12	C₂×Dic₂₆₉	C₂×C₄	1	12	yes	—
13	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	—
14	C₂×Dic₂₆₉	C₂×C₄	1	6	yes	—
15	C₂×Dic₂₆₉	C₂³	1	3	yes	—
16	C₂×Dic₂₆₉	C₂³	1	4	yes	—
17	C₂³×C₂₆₉	—	2	4	no	—
18	C₂³×C₂₆₉	C₂×C₄	1	6	yes	—
19	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₆₉	—	2	12	no	—
21	C₂²×C₂₆₉	M₁₆	1	4	yes	complemented
22	C₂²×C₂₆₉	C₂²×C₄	1	4	yes	—
23	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
24	Dic₂₆₉	C₂²×C₄	1	3	yes	—
25	Dic₂₆₉	C₂²×C₄	1	4	yes	—
26	Dic₂₆₉	C₂⁴	1	1	yes	characteristic
27	C₂×C₂₆₉	—	2	4	no	—
28	C₂×C₂₆₉	C₂×M₁₆	1	6	yes	complemented
29	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
30	C₂₆₉	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²×M₁₆	—	269	1	no	maximal, Sylow
32	C₂³×C₄	—	269	1	no	—
33	C₂×M₁₆	—	269	12	no	—
34	C₂²×C₈	—	269	2	no	—
35	C₂⁴	#11761640	1	1	yes	characteristic
36	C₂²×C₄	—	269	6	no	—
37	C₂²×C₄	—	269	6	no	—
38	C₂²×C₄	—	269	1	no	—
39	C₂²×C₄	—	269	1	no	—
40	M₁₆	—	269	16	no	—
41	C₂×C₈	—	269	12	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×#11761640	1	6	yes	—
44	C₂³	C₂×#11761640	1	1	yes	characteristic
45	C₂×C₄	—	269	3	no	—
46	C₂×C₄	—	269	4	no	—
47	C₂×C₄	—	269	12	no	—
48	C₂×C₄	—	269	6	no	—
49	C₂×C₄	—	269	3	no	—
50	C₈	—	269	8	no	—
51	C₂²	—	2	12	no	—
52	C₂²	#16864411	1	4	yes	direct product factor
53	C₂²	C₂²×#11761640	1	4	yes	—
54	C₂²	C₂²×#11761640	1	3	yes	—
55	C₄	—	269	1	no	—
56	C₄	—	269	3	no	—
57	C₄	—	269	4	no	—
58	C₂	—	2	4	no	—
59	C₂	C₂×#16864411	1	6	yes	direct product factor
60	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761640
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂	C₂³×#11761640
Fitting, prime radical	C₂⁴×C₂₆₉	C₄
socle	C₂³×C₂₆₉	C₂×C₄
cosocle	Dic₂₆₉	C₂⁴
hypercenter	C₂⁴	#11761640
hypocenter	C₂₆₉	C₂²×M₁₆
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#16864411	C₂	direct (N × H)
2. C₂²×#14524137	C₂	semidirect
4. #16864411	C₂²	direct (N × H)
5. C₂×#14524137	C₂²	semidirect
11. #14524137	C₂³	semidirect
21. C₂²×C₂₆₉	M₁₆	semidirect
28. C₂×C₂₆₉	C₂×M₁₆	semidirect
30. C₂₆₉	C₂²×M₁₆	semidirect
52. C₂²	#16864411	direct (N × H)
59. C₂	C₂×#16864411	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×#16864411	2
C₂²×#14524137	2
C₂³×Dic₂₆₉	2
C₂²×M₁₆	269

Subgroups — C22×#16864411 #31771677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16864411 #31771677