Subgroups — #19127607

Order 5312.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₈₃⋊#95	C₂	1	4	yes	maximal
4	C₂×C₄²×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₆₆	—	2	2	no	—
6	C₂×Dic₁₆₆	—	2	2	no	—
7	C₂×Dic₁₆₆	C₄	1	4	yes	complemented
8	C₂²×Dic₈₃	C₂²	1	2	yes	—
9	C₂²×Dic₈₃	C₂²	1	2	yes	—
10	C₄⋊Dic₈₃	—	2	2	no	—
11	C₂²×C₄×C₈₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₈₃	C₂²	1	1	yes	characteristic
13	C₄²×C₈₃	—	2	2	no	—
14	C₂³×C₈₃	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₈₃	—	2	2	no	—
16	C₂×C₄×C₈₃	—	2	2	no	—
17	C₂×C₄×C₈₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈₃	D₄	1	2	yes	—
19	C₂×C₄×C₈₃	D₄	1	2	yes	—
20	C₂×C₄×C₈₃	Q₈	1	4	yes	complemented
21	C₂×Dic₈₃	—	2	2	no	—
22	C₂×Dic₈₃	—	2	2	no	—
23	C₂×Dic₈₃	—	2	2	no	—
24	C₂×Dic₈₃	—	2	4	no	—
25	C₂×Dic₈₃	C₂×C₄	1	4	yes	—
26	Dic₁₆₆	—	2	8	no	—
27	C₂²×C₈₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₈₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₈₃	C₂×Q₈	1	2	yes	—
31	C₂²×C₈₃	#41	1	2	yes	—
32	C₄×C₈₃	—	2	4	no	—
33	C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
34	Dic₈₃	—	2	4	no	—
35	Dic₈₃	—	4	2	no	—
36	C₂×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈₃	C₄×Q₈	1	2	yes	—
38	C₂×C₈₃	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₈₃	#124	1	1	yes	characteristic
40	C₂×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₈₃	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	83	1	no	maximal, Sylow
43	C₂²×Q₈	—	83	1	no	—
44	C₂×C₄⋊C₄	—	83	1	no	—
45	C₂×C₄²	D₈₃	1	1	yes	characteristic
46	#95	—	83	4	no	—
47	C₂×Q₈	—	83	4	no	—
48	C₂×Q₈	—	166	2	no	—
49	C₂×Q₈	—	166	2	no	—
50	C₂²×C₄	—	83	2	no	—
51	C₂²×C₄	—	83	2	no	—
52	C₂²×C₄	C₂×D₈₃	1	2	yes	—
53	C₂²×C₄	C₂×D₈₃	1	1	yes	characteristic
54	C₄⋊C₄	—	166	2	no	—
55	C₄²	—	2	2	no	—
56	C₂³	C₂²×D₈₃	1	1	yes	characteristic
57	Q₈	—	166	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	83	4	no	—
61	C₂×C₄	—	166	2	no	—
62	C₂×C₄	—	166	4	no	—
63	C₂×C₄	—	166	2	no	—
64	C₂×C₄	—	166	2	no	—
65	C₂×C₄	Dic₁₆₆	1	4	yes	complemented
66	C₂×C₄	C₄×D₈₃	1	2	yes	—
67	C₂×C₄	D₃₃₂	1	2	yes	—
68	C₂×C₄	C₂²⋊D₈₃	1	2	yes	—
69	C₂²	C₂×C₄×D₈₃	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
71	C₂²	C₂×D₃₃₂	1	1	yes	characteristic
72	C₂²	C₂×Dic₁₆₆	1	2	yes	—
73	C₂²	#13050590	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	332	2	no	—
76	C₄	—	166	4	no	—
77	C₄	(C₂×D₈₃)⋊C₄	1	4	yes	—
78	C₂	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
79	C₂	C₄×Dic₁₆₆	1	2	yes	—
80	C₂	#14581771	1	2	yes	—
81	C₂	#14581780	1	1	yes	characteristic
82	C₂	#14581792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₄
Fitting	C₂×C₄²×C₈₃	C₂
socle	C₂³×C₈₃	C₂³
hypercenter	C₂×C₄²	D₈₃
hypocenter	C₈₃	#391
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×Dic₁₆₆	C₄	semidirect
20. C₂×C₄×C₈₃	Q₈	semidirect
41. C₈₃	#391	semidirect
65. C₂×C₄	Dic₁₆₆	semidirect