Subgroups — C₂²×#16864390 #31771378

Order 17216.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16864390	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₃₈	—	2	3	no	—
6	C₂×Dic₅₃₈	C₂²	1	6	yes	complemented
7	C₂²×Dic₂₆₉	—	2	1	no	—
8	#16864390	C₂²	1	16	yes	direct product factor
9	C₂×#14524130	C₂²	1	6	yes	complemented
10	C₂⁴×C₂₆₉	—	2	1	no	—
11	C₂×D₄×C₂₆₉	—	2	3	no	—
12	C₂×D₄×C₂₆₉	C₂²	1	6	yes	—
13	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
14	Dic₅₃₈	—	2	6	no	—
15	Dic₅₃₈	C₂³	1	4	yes	complemented
16	C₂×Dic₂₆₉	—	2	6	no	—
17	#14524130	C₂³	1	4	yes	complemented
18	C₂³×C₂₆₉	—	2	6	no	—
19	C₂³×C₂₆₉	—	4	4	no	—
20	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
21	D₄×C₂₆₉	—	2	6	no	—
22	D₄×C₂₆₉	C₂³	1	4	yes	—
23	C₂×C₄×C₂₆₉	D₄	1	3	yes	—
24	C₂×C₄×C₂₆₉	C₂³	1	3	yes	—
25	C₂²×C₂₆₉	—	2	4	no	—
26	C₂²×C₂₆₉	—	4	12	no	—
27	C₂²×C₂₆₉	SD₁₆	1	4	yes	complemented
28	C₂²×C₂₆₉	C₂×D₄	1	3	yes	—
29	C₄×C₂₆₉	C₂×D₄	1	3	yes	—
30	C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
31	Dic₂₆₉	—	2	4	no	—
32	C₂×C₂₆₉	—	4	4	no	—
33	C₂×C₂₆₉	C₂×SD₁₆	1	6	yes	complemented
34	C₂×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
35	C₂₆₉	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂²×SD₁₆	—	269	1	no	maximal, Sylow
37	C₂²×Q₈	—	269	1	no	—
38	C₂²×D₄	D₂₆₉	1	1	yes	characteristic
39	C₂×SD₁₆	—	269	12	no	—
40	C₂²×C₈	—	269	1	no	—
41	C₂⁴	—	2	1	no	—
42	C₂×Q₈	—	269	6	no	—
43	C₂×Q₈	—	538	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	C₂×D₂₆₉	1	6	yes	—
46	C₂²×C₄	—	538	1	no	—
47	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
48	SD₁₆	—	269	16	no	—
49	C₂×C₈	—	269	6	no	—
50	C₂³	—	2	6	no	—
51	C₂³	—	4	4	no	—
52	C₂³	C₂²⋊D₂₆₉	1	1	yes	characteristic
53	Q₈	—	269	4	no	—
54	Q₈	—	538	6	no	—
55	D₄	—	2	6	no	—
56	D₄	C₂²×D₂₆₉	1	4	yes	—
57	C₂×C₄	—	538	6	no	—
58	C₂×C₄	C₂²⋊D₂₆₉	1	3	yes	—
59	C₂×C₄	C₂²×D₂₆₉	1	3	yes	—
60	C₈	—	269	4	no	—
61	C₂²	—	2	4	no	—
62	C₂²	—	4	12	no	—
63	C₂²	#16864390	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₂₆₉	1	3	yes	—
65	C₄	—	538	4	no	—
66	C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
67	C₄	C₂³×D₂₆₉	1	1	yes	characteristic
68	C₂	—	4	4	no	—
69	C₂	C₂×#16864390	1	6	yes	direct product factor
70	C₂	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₆₉
commutator, cosocle	C₄×C₂₆₉	C₂⁴
Frattini	C₄	C₂³×D₂₆₉
Fitting, prime radical	C₂²×D₄×C₂₆₉	C₂
socle	C₂³×C₂₆₉	D₄
hypercenter	C₂²×D₄	D₂₆₉
hypocenter	C₂₆₉	C₂²×SD₁₆
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₃₈	C₂	semidirect
2. C₂×#16864390	C₂	direct (N × H)
3. C₂²×#14524130	C₂	semidirect
6. C₂×Dic₅₃₈	C₂²	semidirect
8. #16864390	C₂²	direct (N × H)
9. C₂×#14524130	C₂²	semidirect
15. Dic₅₃₈	C₂³	semidirect
17. #14524130	C₂³	semidirect
27. C₂²×C₂₆₉	SD₁₆	semidirect
33. C₂×C₂₆₉	C₂×SD₁₆	semidirect
35. C₂₆₉	C₂²×SD₁₆	semidirect
63. C₂²	#16864390	direct (N × H)
69. C₂	C₂×#16864390	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×Dic₅₃₈	2
C₂×#16864390	2
C₂²×#14524130	2
C₂²×D₄×C₂₆₉	2
C₂²×SD₁₆	269

Subgroups — C22×#16864390 #31771378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16864390 #31771378