Subgroups — Q₈×#14524137 #31771653

Order 17216.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14524137	C₂	1	3	yes	maximal
2	C₄×#14524137	C₂	1	3	yes	maximal
3	Q₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524137	C₂²	1	3	yes	—
5	C₂×#14524137	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₂₆₉	C₄	1	3	yes	—
7	C₄×Dic₂₆₉	C₂²	1	3	yes	—
8	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
9	#14524137	—	2	3	no	—
10	#14524137	Q₈	1	2	yes	direct product factor
11	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	—
12	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
13	Q₈×C₂₆₉	C₈	1	4	yes	complemented
14	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
15	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₂₆₉	C₂×C₈	1	6	yes	—
17	Dic₂₆₉	—	2	3	no	—
18	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
19	Dic₂₆₉	#41	1	1	yes	characteristic
20	C₂×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₂₆₉	#131	1	1	yes	characteristic
23	C₂₆₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	269	1	no	maximal, Sylow
25	C₄×Q₈	—	269	1	no	—
26	C₄⋊C₈	—	269	3	no	—
27	C₄×C₈	—	269	3	no	—
28	C₂×Q₈	#11761640	1	1	yes	characteristic
29	C₂×C₈	—	269	1	no	—
30	C₂×C₈	—	269	3	no	—
31	C₄⋊C₄	—	269	3	no	—
32	C₄²	—	269	3	no	—
33	Q₈	#14524137	1	4	yes	direct product factor
34	C₂×C₄	—	269	1	no	—
35	C₂×C₄	—	269	3	no	—
36	C₂×C₄	C₂×#11761640	1	3	yes	—
37	C₈	—	269	2	no	—
38	C₈	—	538	3	no	—
39	C₂²	C₂²×#11761640	1	1	yes	characteristic
40	C₄	—	269	1	no	—
41	C₄	—	269	1	no	—
42	C₄	—	538	3	no	—
43	C₄	C₂×#14524137	1	6	yes	—
44	C₂	Q₈×#11761640	1	1	yes	characteristic
45	C₂	C₂²×#14524137	1	1	yes	characteristic
46	C₂	#22575541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
Fitting	C₂×Q₈×C₂₆₉	C₄
socle, prime radical	C₂²×C₂₆₉	C₂²×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂×Q₈	#11761640
hypocenter	C₂₆₉	C₈×Q₈
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
10. #14524137	Q₈	direct (N × H)
13. Q₈×C₂₆₉	C₈	semidirect
23. C₂₆₉	C₈×Q₈	semidirect
33. Q₈	#14524137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊#14524137	2
C₄×#14524137	2
Q₈×Dic₂₆₉	2
C₈×Q₈	269

Subgroups — Q8×#14524137 #31771653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14524137 #31771653