Subgroups — #31770413

Order 17216.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575475	C₂	1	1	yes	maximal, characteristic
2	#22575389	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16864402	—	2	1	no	—
5	Dic₁₀₇₆	C₄	1	1	yes	characteristic
6	#16864391	—	2	1	no	—
7	D₂₁₅₂	C₄	1	1	yes	characteristic
8	#16864386	—	2	1	no	—
9	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₂₆₉	—	2	1	no	—
11	Dic₅₃₈	—	2	1	no	—
12	D₁₀₇₆	—	2	1	no	—
13	C₂²⋊D₂₆₉	—	4	1	no	—
14	C₄×D₂₆₉	—	4	1	no	—
15	#14524130	—	4	1	no	—
16	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
17	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₆₉	D₄	1	1	yes	characteristic
19	C₂²×C₂₆₉	SD₁₆	1	1	yes	characteristic
20	C₂×D₂₆₉	—	4	1	no	—
21	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₂₆₉	D₈	1	1	yes	characteristic
23	Dic₂₆₉	—	4	1	no	—
24	C₂×C₂₆₉	—	2	1	no	—
25	C₂×C₂₆₉	#102	1	1	yes	characteristic
26	D₂₆₉	—	8	1	no	—
27	C₂₆₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	269	1	no	maximal, Sylow
29	#135	—	269	1	no	—
30	C₂×C₁₆	D₂₆₉	1	1	yes	characteristic, complemented
31	#108	—	269	1	no	—
32	#41	—	538	1	no	—
33	Q₁₆	—	269	1	no	—
34	SD₁₆	—	538	1	no	—
35	D₈	—	269	1	no	—
36	M₁₆	—	538	1	no	—
37	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	Q₈	—	538	1	no	—
40	D₄	—	538	1	no	—
41	D₄	—	1076	1	no	—
42	C₂×C₄	—	1076	1	no	—
43	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
44	C₈	—	1076	1	no	—
45	C₈	C₄×D₂₆₉	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
47	C₂²	—	1076	1	no	—
48	C₂²	#16864391	1	1	yes	characteristic
49	C₄	—	1076	1	no	—
50	C₄	#16864379	1	1	yes	characteristic
51	C₄	D₂₁₅₂	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	2152	1	no	—
54	C₂	#22575375	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₁₅₂
commutator	C₈×C₂₆₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₆₉
Fitting	C₂×C₁₆×C₂₆₉	C₂
socle	C₂×C₂₆₉	#102
prime radical	#22575475	C₂
cosocle	C₂×C₈×C₂₆₉	C₂²
hypercenter	C₂×C₁₆	D₂₆₉
hypocenter	C₂₆₉	#359
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22575389	C₂	semidirect
3. C₂×C₁₆×C₂₆₉	C₂	semidirect
27. C₂₆₉	#359	semidirect
30. 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
#22575475	2
#22575389	2
C₂×C₁₆×C₂₆₉	2
#359	269