Subgroups — #31771001

Order 17216.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575453	C₂	1	1	yes	maximal, characteristic
2	#22575384	C₂	1	4	yes	maximal
3	C₄×#14524130	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₃₈	—	2	1	no	—
6	C₂×#14524130	C₂²	1	2	yes	—
7	C₄⋊Dic₂₆₉	—	2	1	no	—
8	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₂₆₉	—	2	1	no	—
10	C₄⋊C₄×C₂₆₉	—	2	1	no	—
11	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
12	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
13	Dic₅₃₈	—	4	2	no	—
14	C₂×Dic₂₆₉	—	2	2	no	—
15	#14524130	Q₈	1	4	yes	complemented
16	Q₈×C₂₆₉	—	4	2	no	—
17	C₂×C₄×C₂₆₉	—	2	2	no	—
18	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
21	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₆₉	—	4	2	no	—
23	C₄×C₂₆₉	SD₁₆	1	2	yes	—
24	C₄×C₂₆₉	SD₁₆	1	2	yes	—
25	C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
26	Dic₂₆₉	—	4	2	no	—
27	C₂×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
30	C₂₆₉	#498	1	1	yes	characteristic, complemented, Sylow
31	#498	—	269	1	no	maximal, Sylow
32	C₄⋊Q₈	—	269	1	no	—
33	C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic
34	#106	—	269	4	no	—
35	C₄×C₈	—	269	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	538	1	no	—
38	C₂×C₈	—	269	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	269	2	no	—
41	C₄⋊C₄	—	538	1	no	—
42	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
43	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	1076	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	538	2	no	—
48	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
51	C₈	—	269	4	no	—
52	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	1076	2	no	—
55	C₄	#16864389	1	2	yes	—
56	C₄	#16864390	1	2	yes	—
57	C₄	Q₈×D₂₆₉	1	2	yes	—
58	C₂	#22575451	1	1	yes	characteristic
59	C₂	C₂×#16864389	1	1	yes	characteristic
60	C₂	C₂×#16864390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₄⋊Q₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	#498
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
15. #14524130	Q₈	semidirect
30. C₂₆₉	#498	semidirect

Maximal subgroups

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

subgroup	index
#22575453	2
#22575384	2
C₄×#14524130	2
C₄⋊Q₈×C₂₆₉	2
#498	269