Subgroups — C₄×Dic₂₁₆₄ #46830271

Order 34624.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₀₈₂	C₂	1	2	yes	maximal
3	#31772562	C₂	1	1	yes	maximal, characteristic
4	#31772553	C₂	1	2	yes	maximal
5	C₄×C₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₈₂	C₂²	1	2	yes	—
7	Dic₂₁₆₄	C₄	1	4	yes	direct product factor
8	C₄⋊Dic₅₄₁	C₂²	1	2	yes	—
9	#22575815	—	2	2	no	—
10	C₄×Dic₅₄₁	—	2	2	no	—
11	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
13	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
14	Dic₁₀₈₂	—	2	2	no	—
15	Dic₁₀₈₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₅₄₁	—	2	2	no	—
17	C₂×Dic₅₄₁	—	2	2	no	—
18	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
21	C₈×C₅₄₁	—	2	1	no	—
22	C₈×C₅₄₁	C₂×C₄	1	2	yes	—
23	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₅₄₁	—	2	1	no	—
25	C₄×C₅₄₁	Q₁₆	1	2	yes	complemented
26	C₄×C₅₄₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₅₄₁	#41	1	1	yes	characteristic
28	Dic₅₄₁	—	2	4	no	—
29	Dic₅₄₁	—	4	2	no	—
30	C₂×C₅₄₁	C₄×D₄	1	1	yes	characteristic
31	C₂×C₅₄₁	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₅₄₁	#135	1	1	yes	characteristic
33	C₅₄₁	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	541	1	no	maximal, Sylow
35	C₂×Q₁₆	—	541	1	no	—
36	C₄×Q₈	—	541	2	no	—
37	#107	—	541	1	no	—
38	#103	—	541	2	no	—
39	C₄×C₈	D₅₄₁	1	1	yes	characteristic
40	C₂×Q₈	—	541	2	no	—
41	Q₁₆	—	541	4	no	—
42	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
43	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
44	C₄⋊C₄	—	541	2	no	—
45	C₄⋊C₄	—	1082	2	no	—
46	C₄²	—	1082	2	no	—
47	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
48	Q₈	—	541	4	no	—
49	Q₈	—	1082	2	no	—
50	C₂×C₄	—	1082	2	no	—
51	C₂×C₄	—	1082	2	no	—
52	C₂×C₄	D₂₁₆₄	1	1	yes	characteristic
53	C₂×C₄	D₂₁₆₄	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₄×D₅₄₁	1	2	yes	—
57	C₂²	C₂×D₂₁₆₄	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	1082	4	no	—
60	C₄	—	2164	2	no	—
61	C₄	Dic₂₁₆₄	1	2	yes	direct product factor
62	C₄	C₂×C₄×D₅₄₁	1	1	yes	characteristic
63	C₄	#22575836	1	1	yes	characteristic
64	C₂	C₄×D₂₁₆₄	1	1	yes	characteristic
65	C₂	C₂×Dic₂₁₆₄	1	1	yes	characteristic
66	C₂	#31772650	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₁₆₄
commutator	C₄×C₅₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	C₄×C₈×C₅₄₁	C₂
socle, prime radical	C₂²×C₅₄₁	C₂×D₄
cosocle	C₂×C₄×C₅₄₁	C₂³
hypercenter	C₄×C₈	D₅₄₁
hypocenter	C₅₄₁	C₄×Q₁₆
LCS	G ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
7. Dic₂₁₆₄	C₄	direct (N × H)
25. C₄×C₅₄₁	Q₁₆	semidirect
33. C₅₄₁	C₄×Q₁₆	semidirect
61. C₄	Dic₂₁₆₄	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×Dic₂₁₆₄	2
C₄×Dic₁₀₈₂	2
#31772562	2
#31772553	2
C₄×C₈×C₅₄₁	2
C₄×Q₁₆	541

Subgroups — C4×Dic2164 #46830271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₂₁₆₄ #46830271