Subgroups — #31771046

Order 17216.

82 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	#22575399	C₂	1	2	yes	maximal, complemented
3	#22575396	C₂	1	2	yes	maximal, complemented
4	#22575393	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864402	—	2	1	no	—
7	Dic₁₀₇₆	C₂²	1	1	yes	characteristic
8	#16864391	—	2	1	no	—
9	D₂₁₅₂	C₂²	1	1	yes	characteristic
10	#16864377	C₂²	1	2	yes	—
11	C₂×Q₈×C₂₆₉	—	2	1	no	—
12	Q₁₆×C₂₆₉	—	2	1	no	—
13	Q₁₆×C₂₆₉	C₂²	1	2	yes	—
14	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
15	Dic₅₃₈	—	2	1	no	—
16	D₁₀₇₆	—	2	1	no	—
17	C₂²⋊D₂₆₉	—	4	1	no	—
18	C₄×D₂₆₉	—	4	1	no	—
19	Q₈×C₂₆₉	—	2	2	no	—
20	Q₈×C₂₆₉	—	4	1	no	—
21	C₂×C₄×C₂₆₉	—	4	1	no	—
22	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
23	C₈×C₂₆₉	D₄	1	1	yes	characteristic
24	C₈×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂²×C₂₆₉	D₈	1	1	yes	characteristic
26	C₂×D₂₆₉	—	4	1	no	—
27	C₄×C₂₆₉	—	4	2	no	—
28	C₄×C₂₆₉	D₈	1	1	yes	characteristic
29	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
30	Dic₂₆₉	—	4	1	no	—
31	C₂×C₂₆₉	—	2	1	no	—
32	C₂×C₂₆₉	C₂×D₈	1	1	yes	characteristic
33	D₂₆₉	—	8	1	no	—
34	C₂₆₉	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	269	1	no	maximal, Sylow
36	#135	—	269	1	no	—
37	C₂×Q₁₆	D₂₆₉	1	1	yes	characteristic, complemented
38	Q₃₂	—	269	2	no	—
39	SD₃₂	—	269	2	no	—
40	M₃₂	—	269	1	no	—
41	#41	—	538	1	no	—
42	C₂×Q₈	—	2	1	no	—
43	Q₁₆	—	2	1	no	—
44	Q₁₆	—	269	1	no	—
45	Q₁₆	C₂×D₂₆₉	1	2	yes	—
46	SD₁₆	—	538	1	no	—
47	D₈	—	269	1	no	—
48	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
49	C₁₆	—	269	2	no	—
50	Q₈	—	2	2	no	—
51	Q₈	—	4	1	no	—
52	Q₈	—	538	1	no	—
53	D₄	—	538	1	no	—
54	D₄	—	1076	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	1076	1	no	—
57	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
58	C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
59	C₈	C₂²×D₂₆₉	1	1	yes	characteristic
60	C₂²	—	1076	1	no	—
61	C₂²	#16864387	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	1076	1	no	—
64	C₄	#16864387	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	2152	1	no	—
68	C₂	C₂×#16864387	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16864387
commutator, cosocle	C₈×C₂₆₉	C₂³
Frattini	C₈	C₂²×D₂₆₉
Fitting	C₂×Q₁₆×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂×D₈
prime radical	#22575475	C₂
hypercenter	C₂×Q₁₆	D₂₆₉
hypocenter	C₂₆₉	#510
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22575399	C₂	semidirect
3. #22575396	C₂	semidirect
4. #22575393	C₂	semidirect
5. C₂×Q₁₆×C₂₆₉	C₂	semidirect
34. C₂₆₉	#510	semidirect
37. C₂×Q₁₆	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
#22575399	2
#22575396	2
#22575393	2
C₂×Q₁₆×C₂₆₉	2
#510	269