Subgroups — #31771021

Order 17216.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575459	C₂	1	1	yes	maximal, characteristic
2	#22575393	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575392	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₂₆₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864402	C₄	1	1	yes	characteristic
7	#16864386	C₄	1	1	yes	characteristic
8	#16864385	C₄	1	2	yes	—
9	C₈×D₂₆₉	C₂²	1	2	yes	—
10	#16864377	C₂²	1	2	yes	—
11	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
14	Dic₅₃₈	C₈	1	1	yes	characteristic
15	D₁₀₇₆	C₈	1	1	yes	characteristic
16	C₂²⋊D₂₆₉	C₈	1	2	yes	—
17	C₄×D₂₆₉	C₂×C₄	1	2	yes	—
18	#14524130	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₈×C₂₆₉	C₂³	1	1	yes	characteristic
22	C₂²×C₂₆₉	C₂×C₈	1	1	yes	characteristic
23	C₂×D₂₆₉	C₂×C₈	1	2	yes	—
24	C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
25	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
26	Dic₂₆₉	C₂×C₈	1	2	yes	—
27	C₂×C₂₆₉	—	2	1	no	—
28	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
29	D₂₆₉	—	2	2	no	—
30	C₂₆₉	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	269	1	no	maximal, Sylow
32	#131	—	269	1	no	—
33	M₃₂	—	269	1	no	—
34	M₃₂	—	269	2	no	—
35	C₂×C₁₆	—	269	2	no	—
36	C₂×C₁₆	D₂₆₉	1	1	yes	characteristic, complemented
37	#41	—	269	1	no	—
38	M₁₆	—	269	1	no	—
39	M₁₆	—	269	2	no	—
40	C₂×C₈	—	269	2	no	—
41	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
42	C₁₆	—	269	2	no	—
43	C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
44	C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
45	Q₈	—	269	1	no	—
46	D₄	—	269	2	no	—
47	D₄	—	269	1	no	—
48	C₂×C₄	—	269	2	no	—
49	C₂×C₄	C₄×D₂₆₉	1	1	yes	characteristic
50	C₈	—	269	2	no	—
51	C₈	C₄×D₂₆₉	1	1	yes	characteristic
52	C₈	C₂²×D₂₆₉	1	1	yes	characteristic
53	C₂²	—	269	2	no	—
54	C₂²	C₈×D₂₆₉	1	1	yes	characteristic
55	C₄	—	269	2	no	—
56	C₄	C₈×D₂₆₉	1	1	yes	characteristic
57	C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	538	2	no	—
60	C₂	C₂×C₈×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₂₆₉
commutator, socle	C₂×C₂₆₉	C₂²×C₈
Frattini	C₈	C₂²×D₂₆₉
Fitting	C₂×C₁₆×C₂₆₉	C₂
prime radical	#16864402	C₄
cosocle	C₈×C₂₆₉	C₂³
hypercenter	C₂×C₁₆	D₂₆₉
hypocenter	C₂₆₉	#504
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22575393	C₂	semidirect
3. #22575392	C₂	semidirect
4. C₁₆×D₂₆₉	C₂	semidirect
5. C₂×C₁₆×C₂₆₉	C₂	semidirect
30. C₂₆₉	#504	semidirect
36. C₂×C₁₆	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575459	2
#22575393	2
#22575392	2
C₁₆×D₂₆₉	2
C₂×C₁₆×C₂₆₉	2
#504	269