Subgroups — C₄×Dic₁₆₄ #14580432

Order 2624.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₈₂	C₂	1	2	yes	maximal
3	#13050098	C₂	1	1	yes	maximal, characteristic
4	#13050089	C₂	1	2	yes	maximal
5	C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₂	C₂²	1	2	yes	—
7	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₁	C₂²	1	2	yes	—
11	#10620279	—	2	2	no	—
12	C₄×Dic₄₁	—	2	2	no	—
13	Dic₁₆₄	C₄	1	4	yes	direct product factor
14	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
15	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
17	C₂×Dic₄₁	—	2	2	no	—
18	C₂×Dic₄₁	—	2	2	no	—
19	Dic₈₂	—	2	2	no	—
20	Dic₈₂	C₂×C₄	1	4	yes	—
21	C₈×C₄₁	—	2	1	no	—
22	C₈×C₄₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₄₁	—	2	1	no	—
25	C₄×C₄₁	Q₁₆	1	2	yes	complemented
26	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₄₁	#41	1	1	yes	characteristic
28	Dic₄₁	—	2	4	no	—
29	Dic₄₁	—	4	2	no	—
30	C₂×C₄₁	C₄×D₄	1	1	yes	characteristic
31	C₂×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₄₁	#135	1	1	yes	characteristic
33	C₄×Q₁₆	—	41	1	no	maximal, Sylow
34	C₄₁	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
35	C₂×Q₁₆	—	41	1	no	—
36	C₄×Q₈	—	41	2	no	—
37	#107	—	41	1	no	—
38	#103	—	41	2	no	—
39	C₄×C₈	D₄₁	1	1	yes	characteristic
40	C₂×Q₈	—	41	2	no	—
41	Q₁₆	—	41	4	no	—
42	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
43	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
44	C₄⋊C₄	—	41	2	no	—
45	C₄⋊C₄	—	82	2	no	—
46	C₄²	—	82	2	no	—
47	C₄²	C₂×D₄₁	1	1	yes	characteristic
48	Q₈	—	41	4	no	—
49	Q₈	—	82	2	no	—
50	C₂×C₄	—	82	2	no	—
51	C₂×C₄	—	82	2	no	—
52	C₂×C₄	D₁₆₄	1	1	yes	characteristic
53	C₂×C₄	D₁₆₄	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₄×D₄₁	1	2	yes	—
57	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	82	4	no	—
60	C₄	—	164	2	no	—
61	C₄	Dic₁₆₄	1	2	yes	direct product factor
62	C₄	C₂×C₄×D₄₁	1	1	yes	characteristic
63	C₄	#10620306	1	1	yes	characteristic
64	C₂	C₄×D₁₆₄	1	1	yes	characteristic
65	C₂	C₂×Dic₁₆₄	1	1	yes	characteristic
66	C₂	#13050186	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₆₄
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄×C₈×C₄₁	C₂
socle, prime radical	C₂²×C₄₁	C₂×D₄
cosocle	C₂×C₄×C₄₁	C₂³
hypercenter	C₄×C₈	D₄₁
hypocenter	C₄₁	C₄×Q₁₆
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
13. Dic₁₆₄	C₄	direct (N × H)
25. C₄×C₄₁	Q₁₆	semidirect
34. C₄₁	C₄×Q₁₆	semidirect
61. C₄	Dic₁₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₁₆₄	2
C₄×Dic₈₂	2
#13050098	2
#13050089	2
C₄×C₈×C₄₁	2
C₄×Q₁₆	41

Subgroups — C4×Dic164 #14580432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₁₆₄ #14580432