Subgroups — Q₆₄×D₁₀₉ #28365335

Order 13952.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₃₂×D₁₀₉	C₂	1	2	yes	maximal
2	Dic₁₇₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294866	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	Q₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₁₀₉	—	2	2	no	—
7	Dic₈₇₂	C₂²	1	2	yes	—
8	#16812645	—	2	2	no	—
9	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812605	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₀₉	C₂²	1	2	yes	—
12	C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
13	Q₈×D₁₀₉	—	4	2	no	—
14	Dic₄₃₆	—	2	2	no	—
15	#13182019	—	4	2	no	—
16	C₈×D₁₀₉	D₄	1	1	yes	characteristic
17	#13182004	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₀₉	—	2	2	no	—
19	C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
20	Q₈×C₁₀₉	—	4	2	no	—
21	C₄×D₁₀₉	—	8	2	no	—
22	C₄×D₁₀₉	D₈	1	1	yes	characteristic
23	Dic₂₁₈	—	4	2	no	—
24	Dic₂₁₈	—	8	2	no	—
25	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
26	#11724784	D₈	1	1	yes	characteristic
27	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
28	C₄×C₁₀₉	—	8	2	no	—
29	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
30	Dic₁₀₉	—	8	2	no	—
31	Dic₁₀₉	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
33	D₁₀₉	Q₆₄	1	2	yes	direct product factor
34	C₂×Q₆₄	—	109	1	no	maximal, Sylow
35	C₁₀₉	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₃₂	—	109	2	no	—
37	Q₆₄	—	109	2	no	—
38	Q₆₄	—	109	1	no	—
39	Q₆₄	D₁₀₉	1	1	yes	characteristic, direct product factor
40	C₂×C₃₂	—	109	1	no	—
41	C₂×Q₁₆	—	218	2	no	—
42	Q₃₂	—	109	2	no	—
43	Q₃₂	—	218	2	no	—
44	Q₃₂	C₂×D₁₀₉	1	2	yes	—
45	C₂×C₁₆	—	109	1	no	—
46	C₃₂	—	109	1	no	—
47	C₃₂	C₂×D₁₀₉	1	1	yes	characteristic
48	C₂×Q₈	—	436	2	no	—
49	Q₁₆	—	2	2	no	—
50	Q₁₆	—	436	2	no	—
51	Q₁₆	—	218	2	no	—
52	C₂×C₈	—	109	1	no	—
53	C₁₆	—	109	1	no	—
54	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
55	Q₈	—	4	2	no	—
56	Q₈	—	872	2	no	—
57	Q₈	—	436	2	no	—
58	C₂×C₄	—	872	2	no	—
59	C₂×C₄	—	109	1	no	—
60	C₈	—	109	1	no	—
61	C₈	D₄×D₁₀₉	1	1	yes	characteristic
62	C₂²	—	109	1	no	—
63	C₄	—	8	2	no	—
64	C₄	—	872	2	no	—
65	C₄	—	109	1	no	—
66	C₄	D₈×D₁₀₉	1	1	yes	characteristic
67	C₂	—	109	2	no	—
68	C₂	D₁₆×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₀₉
commutator, cosocle	C₁₆×C₁₀₉	C₂³
Frattini	C₁₆	C₂²×D₁₀₉
Fitting	Q₆₄×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₁₆
prime radical	C₂×D₁₀₉	D₁₆
hypercenter	Q₆₄	D₁₀₉
hypocenter	C₁₀₉	C₂×Q₆₄
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
2. Dic₁₇₄₄	C₂	semidirect
3. #20294866	C₂	semidirect
5. Q₆₄×C₁₀₉	C₂	semidirect
33. D₁₀₉	Q₆₄	direct (N × H)
35. C₁₀₉	C₂×Q₆₄	semidirect
39. Q₆₄	D₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₃₂×D₁₀₉	2
Dic₁₇₄₄	2
#20294866	2
C₃₂×D₁₀₉	2
Q₆₄×C₁₀₉	2
C₂×Q₆₄	109

Subgroups — Q64×D109 #28365335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₆₄×D₁₀₉ #28365335