Subgroups — #19127993

Order 5312.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#14581725	C₂	1	2	yes	maximal
3	C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#14581715	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₂×C₈₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
9	Dic₈₃⋊C₄	—	2	1	no	—
10	C₄×Dic₈₃	—	2	1	no	—
11	C₂×Q₈×C₈₃	—	2	1	no	—
12	C₄⋊C₄×C₈₃	—	2	1	no	—
13	C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
14	C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	Q₈×C₈₃	—	4	2	no	—
16	C₂×C₄×C₈₃	—	2	2	no	—
17	C₂×C₄×C₈₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₈₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₈₃	—	2	1	no	—
21	C₂×Dic₈₃	—	2	1	no	—
22	Dic₁₆₆	—	2	1	no	—
23	Dic₁₆₆	Q₈	1	2	yes	complemented
24	C₈₃⋊C₈	—	2	2	no	—
25	C₂²×C₈₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₈₃	—	2	1	no	—
27	C₄×C₈₃	—	4	2	no	—
28	C₄×C₈₃	Q₁₆	1	2	yes	—
29	C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₈₃	#41	1	1	yes	characteristic
31	Dic₈₃	—	2	2	no	—
32	Dic₈₃	—	4	1	no	—
33	C₂×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₈₃	#477	1	1	yes	characteristic, complemented, Sylow
37	#477	—	83	1	no	maximal, Sylow
38	C₄⋊Q₈	D₈₃	1	1	yes	characteristic
39	C₄×Q₈	—	83	1	no	—
40	#107	—	83	2	no	—
41	C₄⋊C₈	—	83	1	no	—
42	#103	—	83	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	83	1	no	—
45	C₂×C₈	—	83	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	83	1	no	—
48	C₄⋊C₄	—	166	1	no	—
49	C₄⋊C₄	C₂×D₈₃	1	2	yes	—
50	C₄²	—	166	1	no	—
51	C₄²	C₂×D₈₃	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	83	2	no	—
54	Q₈	—	166	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	166	1	no	—
57	C₂×C₄	—	166	1	no	—
58	C₂×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₈₃	1	1	yes	characteristic
61	C₈	—	166	2	no	—
62	C₂²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	332	1	no	—
66	C₄	—	166	2	no	—
67	C₄	#13050580	1	2	yes	—
68	C₄	Q₈×D₈₃	1	1	yes	characteristic
69	C₄	#13050591	1	1	yes	characteristic
70	C₂	#14581768	1	1	yes	characteristic
71	C₂	C₂×#13050580	1	1	yes	characteristic
72	C₂	#14581818	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂³
Frattini	C₂×C₄	C₂²×D₈₃
Fitting	C₄⋊Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₈₃
hypocenter	C₈₃	#477
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. Dic₁₆₆	Q₈	semidirect
36. C₈₃	#477	semidirect