Subgroups — Q₈×D₁₆₄ #14581003

Order 2624.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₄₁	C₂	1	2	yes	maximal, complemented
2	#13050164	C₂	1	3	yes	maximal, complemented
3	#13050144	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₆₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
7	Q₈×D₄₁	—	2	8	no	—
8	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₁	C₂²	1	6	yes	—
10	C₂×Dic₈₂	C₂²	1	6	yes	—
11	C₄⋊C₄×C₄₁	C₂²	1	3	yes	—
12	C₄²×C₄₁	C₂²	1	3	yes	—
13	#10620281	C₂²	1	6	yes	—
14	C₄⋊Dic₄₁	C₂²	1	3	yes	—
15	C₄⋊Dic₄₁	C₂²	1	6	yes	—
16	C₂²×D₄₁	C₂³	1	2	yes	—
17	Q₈×C₄₁	D₄	1	4	yes	complemented
18	C₂×C₄×C₄₁	C₂³	1	3	yes	—
19	C₂×C₄×C₄₁	C₂³	1	3	yes	—
20	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₁	C₂³	1	6	yes	—
22	D₁₆₄	Q₈	1	4	yes	direct product factor
23	C₄×D₄₁	—	2	12	no	—
24	Dic₈₂	—	2	12	no	—
25	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₁	—	2	2	no	—
27	C₂×D₄₁	—	2	2	no	—
28	C₂×D₄₁	C₂×Q₈	1	4	yes	complemented
29	C₄×C₄₁	—	2	3	no	—
30	C₄×C₄₁	C₂×D₄	1	6	yes	—
31	C₄×C₄₁	C₂×Q₈	1	2	yes	complemented
32	Dic₄₁	—	2	6	no	—
33	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₄₁	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₁	—	2	4	no	—
37	D₄×Q₈	—	41	1	no	maximal, Sylow
38	C₄₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²×Q₈	—	41	2	no	—
40	C₄⋊Q₈	—	41	3	no	—
41	C₂²⋊Q₈	—	41	6	no	—
42	C₄×Q₈	D₄₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	41	3	no	—
44	C₂×Q₈	—	41	6	no	—
45	C₂×Q₈	—	82	8	no	—
46	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
47	C₂×D₄	—	41	1	no	—
48	C₂²×C₄	—	41	6	no	—
49	C₄⋊C₄	—	41	3	no	—
50	C₄⋊C₄	—	41	6	no	—
51	C₄⋊C₄	C₂×D₄₁	1	3	yes	—
52	C₂²⋊C₄	—	41	6	no	—
53	C₄²	C₂×D₄₁	1	3	yes	—
54	C₂³	—	41	2	no	—
55	Q₈	—	82	12	no	—
56	Q₈	D₁₆₄	1	4	yes	direct product factor
57	D₄	—	41	4	no	—
58	C₂×C₄	—	41	6	no	—
59	C₂×C₄	—	82	12	no	—
60	C₂×C₄	C₂²×D₄₁	1	3	yes	—
61	C₂×C₄	C₂²×D₄₁	1	3	yes	—
62	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
63	C₂²	—	41	4	no	—
64	C₂²	—	82	2	no	—
65	C₂²	—	82	2	no	—
66	C₂²	C₂³×D₄₁	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	82	6	no	—
69	C₄	C₂×D₁₆₄	1	6	yes	—
70	C₄	Q₈×D₄₁	1	2	yes	complemented
71	C₂	—	82	4	no	—
72	C₂	C₂²×D₁₆₄	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₄₁	1	1	yes	characteristic
74	C₂	#13050211	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₄×Q₈×C₄₁	C₂
prime radical	C₂×D₁₆₄	C₂²
hypercenter	C₄×Q₈	D₄₁
hypocenter	C₄₁	D₄×Q₈
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₄₁	C₂	semidirect
2. #13050164	C₂	semidirect
3. #13050144	C₂	semidirect
5. C₄×Q₈×C₄₁	C₂	semidirect
17. Q₈×C₄₁	D₄	semidirect
22. D₁₆₄	Q₈	direct (N × H)
28. C₂×D₄₁	C₂×Q₈	semidirect
31. C₄×C₄₁	C₂×Q₈	semidirect
38. C₄₁	D₄×Q₈	semidirect
42. C₄×Q₈	D₄₁	semidirect
56. Q₈	D₁₆₄	direct (N × H)
70. C₄	Q₈×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₄₁	2
#13050164	2
#13050144	2
C₄×D₁₆₄	2
C₄×Q₈×C₄₁	2
D₄×Q₈	41

Subgroups — Q8×D164 #14581003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₆₄ #14581003