Subgroups — Q₈×C₂₃₉⋊C₁₆ #44486452

Order 30592.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃₉⋊C₁₆)	C₂	1	3	yes	maximal
2	C₄×C₂₃₉⋊C₁₆	C₂	1	3	yes	maximal
3	C₈×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃₉⋊C₁₆	C₂²	1	3	yes	—
5	C₂×C₂₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₂₃₉	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₃₉	C₄	1	3	yes	—
8	C₄×C₈×C₂₃₉	C₂²	1	3	yes	—
9	C₂₃₉⋊C₁₆	—	2	3	no	—
10	C₂₃₉⋊C₁₆	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₂₃₉	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₃₉	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₂₃₉	C₈	1	3	yes	—
15	C₄²×C₂₃₉	C₂×C₄	1	3	yes	—
16	Q₈×C₂₃₉	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₂₃₉	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₂₃₉	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₂₃₉	—	2	3	no	—
21	C₈×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₂₃₉	#41	1	1	yes	characteristic
23	C₂²×C₂₃₉	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₂₃₉	—	2	3	no	—
25	C₄×C₂₃₉	C₂×C₁₆	1	6	yes	—
26	C₄×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₂₃₉	#131	1	1	yes	characteristic
28	C₂×C₂₃₉	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₂₃₉	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₂₃₉	#504	1	1	yes	characteristic
31	C₂₃₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	239	1	no	maximal, Sylow
33	C₈×Q₈	D₂₃₉	1	1	yes	characteristic
34	C₄⋊C₁₆	—	239	3	no	—
35	C₄×C₁₆	—	239	3	no	—
36	C₄×Q₈	Dic₂₃₉	1	1	yes	characteristic
37	C₂×C₁₆	—	239	3	no	—
38	C₂×C₁₆	—	239	1	no	—
39	C₄⋊C₈	Dic₂₃₉	1	3	yes	—
40	C₄×C₈	C₂×D₂₃₉	1	3	yes	—
41	C₂×Q₈	C₂₃₉⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₂₃₉	1	3	yes	—
43	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
44	C₄⋊C₄	C₂₃₉⋊C₈	1	3	yes	—
45	C₄²	C₂×Dic₂₃₉	1	3	yes	—
46	C₁₆	—	478	3	no	—
47	C₁₆	—	239	2	no	—
48	Q₈	C₂₃₉⋊C₁₆	1	4	yes	direct product factor
49	C₂×C₄	C₂×C₂₃₉⋊C₈	1	3	yes	—
50	C₂×C₄	C₂×C₂₃₉⋊C₈	1	3	yes	—
51	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₂₃₉	1	1	yes	characteristic
54	C₈	#16838549	1	1	yes	characteristic
55	C₂²	C₂²×C₂₃₉⋊C₈	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×C₂₃₉⋊C₁₆	1	6	yes	—
58	C₄	Q₈×Dic₂₃₉	1	1	yes	characteristic
59	C₄	#21439967	1	1	yes	characteristic
60	C₂	Q₈×C₂₃₉⋊C₈	1	1	yes	characteristic
61	C₂	C₂²×C₂₃₉⋊C₁₆	1	1	yes	characteristic
62	C₂	#29505189	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃₉
commutator	C₂×C₂₃₉	C₂²×C₁₆
Fitting	C₈×Q₈×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂²×C₈
cosocle	C₂×C₈×C₂₃₉	C₂³
hypercenter	C₈×Q₈	D₂₃₉
hypocenter	C₂₃₉	Q₈×C₁₆
LCS	G ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₂₃₉⋊C₁₆	Q₈	direct (N × H)
16. Q₈×C₂₃₉	C₁₆	semidirect
31. C₂₃₉	Q₈×C₁₆	semidirect
48. Q₈	C₂₃₉⋊C₁₆	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⋊(C₂₃₉⋊C₁₆)	2
C₄×C₂₃₉⋊C₁₆	2
C₈×Q₈×C₂₃₉	2
Q₈×C₁₆	239

Subgroups — Q8×C239⋊C16 #44486452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₃₉⋊C₁₆ #44486452