Subgroups — Q₆₄×D₁₂₇ #30641033

Order 16256.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₃₂×D₁₂₇	C₂	1	2	yes	maximal
2	Dic₂₀₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549716	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₁₂₇	C₂	1	1	yes	maximal, characteristic
5	Q₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₁₂₇	—	2	2	no	—
7	Dic₁₀₁₆	C₂²	1	2	yes	—
8	#16861155	—	2	2	no	—
9	C₁₆×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₁₂₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₂₇	C₂²	1	2	yes	—
12	C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
13	Q₈×D₁₂₇	—	4	2	no	—
14	Dic₅₀₈	—	2	2	no	—
15	#14523238	—	4	2	no	—
16	C₈×D₁₂₇	D₄	1	1	yes	characteristic
17	C₁₂₇⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₂₇	—	2	2	no	—
19	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
20	Dic₂₅₄	—	4	2	no	—
21	Dic₂₅₄	—	8	2	no	—
22	C₄×D₁₂₇	—	8	2	no	—
23	C₄×D₁₂₇	D₈	1	1	yes	characteristic
24	C₁₂₇⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₁₂₇	—	4	2	no	—
26	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₂₇	D₁₆	1	1	yes	characteristic
28	C₄×C₁₂₇	—	8	2	no	—
29	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
30	Dic₁₂₇	—	8	2	no	—
31	Dic₁₂₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
33	D₁₂₇	Q₆₄	1	2	yes	direct product factor
34	C₂×Q₆₄	—	127	1	no	maximal, Sylow
35	C₁₂₇	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₃₂	—	127	2	no	—
37	Q₆₄	—	127	2	no	—
38	Q₆₄	—	127	1	no	—
39	Q₆₄	D₁₂₇	1	1	yes	characteristic, direct product factor
40	C₂×C₃₂	—	127	1	no	—
41	C₂×Q₁₆	—	254	2	no	—
42	Q₃₂	—	127	2	no	—
43	Q₃₂	—	254	2	no	—
44	Q₃₂	C₂×D₁₂₇	1	2	yes	—
45	C₂×C₁₆	—	127	1	no	—
46	C₃₂	—	127	1	no	—
47	C₃₂	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂×Q₈	—	508	2	no	—
49	Q₁₆	—	2	2	no	—
50	Q₁₆	—	508	2	no	—
51	Q₁₆	—	254	2	no	—
52	C₂×C₈	—	127	1	no	—
53	C₁₆	—	127	1	no	—
54	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
55	Q₈	—	4	2	no	—
56	Q₈	—	1016	2	no	—
57	Q₈	—	508	2	no	—
58	C₂×C₄	—	127	1	no	—
59	C₂×C₄	—	1016	2	no	—
60	C₈	—	127	1	no	—
61	C₈	D₄×D₁₂₇	1	1	yes	characteristic
62	C₂²	—	127	1	no	—
63	C₄	—	8	2	no	—
64	C₄	—	127	1	no	—
65	C₄	—	1016	2	no	—
66	C₄	D₈×D₁₂₇	1	1	yes	characteristic
67	C₂	—	127	2	no	—
68	C₂	D₁₆×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₂₇
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	Q₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₁₆
prime radical	C₂×D₁₂₇	D₁₆
hypercenter	Q₆₄	D₁₂₇
hypocenter	C₁₂₇	C₂×Q₆₄
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
2. Dic₂₀₃₂	C₂	semidirect
3. #22549716	C₂	semidirect
5. Q₆₄×C₁₂₇	C₂	semidirect
33. D₁₂₇	Q₆₄	direct (N × H)
35. C₁₂₇	C₂×Q₆₄	semidirect
39. Q₆₄	D₁₂₇	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×D₁₂₇	2
Dic₂₀₃₂	2
#22549716	2
C₃₂×D₁₂₇	2
Q₆₄×C₁₂₇	2
C₂×Q₆₄	127

Subgroups — Q64×D127 #30641033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₆₄×D₁₂₇ #30641033