Subgroups — #19127444

Order 5312.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14581789	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₃₂	—	2	1	no	—
5	C₂×D₃₃₂	—	2	1	no	—
6	C₂×D₃₃₂	C₄	1	2	yes	—
7	C₂×C₈₃⋊C₈	—	2	1	no	—
8	C₂×C₈×C₈₃	—	2	1	no	—
9	C₄²×C₈₃	C₂²	1	1	yes	characteristic
10	C₂²×D₈₃	—	2	2	no	—
11	C₂×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₈₃	D₄	1	1	yes	characteristic
13	C₂×C₄×C₈₃	D₄	1	1	yes	characteristic
14	D₃₃₂	—	4	2	no	—
15	D₃₃₂	—	4	1	no	—
16	D₃₃₂	—	4	1	no	—
17	D₃₃₂	—	4	1	no	—
18	D₃₃₂	—	4	1	no	—
19	C₈×C₈₃	—	4	1	no	—
20	C₈₃⋊C₈	—	4	1	no	—
21	C₂²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₈₃	—	4	2	no	—
23	C₂×D₈₃	—	4	2	no	—
24	C₂×D₈₃	—	4	2	no	—
25	C₄×C₈₃	—	2	1	no	—
26	C₄×C₈₃	D₈	1	1	yes	characteristic
27	C₄×C₈₃	D₈	1	1	yes	characteristic
28	C₄×C₈₃	SD₁₆	1	1	yes	characteristic
29	C₄×C₈₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₈₃	#100	1	1	yes	characteristic
31	C₂×C₈₃	#102	1	1	yes	characteristic
32	C₂×C₈₃	#102	1	1	yes	characteristic
33	D₈₃	—	8	2	no	—
34	C₈₃	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	83	1	no	maximal, Sylow
36	#127	—	83	1	no	—
37	C₄⋊C₈	—	83	1	no	—
38	C₄⋊C₈	D₈₃	1	1	yes	characteristic, complemented
39	C₂×D₄	—	83	2	no	—
40	C₂×D₄	—	166	1	no	—
41	C₂×D₄	—	166	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	166	1	no	—
44	C₄²	C₂×D₈₃	1	1	yes	characteristic
45	C₂³	—	166	2	no	—
46	D₄	—	332	2	no	—
47	D₄	—	332	1	no	—
48	D₄	—	332	1	no	—
49	D₄	—	332	1	no	—
50	D₄	—	332	1	no	—
51	C₂×C₄	C₄×D₈₃	1	1	yes	characteristic
52	C₂×C₄	D₃₃₂	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	332	1	no	—
56	C₂²	—	332	2	no	—
57	C₂²	—	332	2	no	—
58	C₂²	—	332	2	no	—
59	C₂²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#13050575	1	1	yes	characteristic
62	C₄	D₆₆₄	1	1	yes	characteristic
63	C₄	#13050577	1	1	yes	characteristic
64	C₄	#13050579	1	1	yes	characteristic
65	C₂	—	664	2	no	—
66	C₂	#14581708	1	1	yes	characteristic
67	C₂	#14581712	1	1	yes	characteristic
68	C₂	#14581714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂×C₄
Frattini	C₄²	C₂×D₈₃
Fitting	C₄⋊C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²⋊C₄
hypercenter	C₄⋊C₈	D₈₃
hypocenter	C₈₃	#331
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₈₃⋊C₈)	C₂	semidirect
3. C₄⋊C₈×C₈₃	C₂	semidirect
34. C₈₃	#331	semidirect
38. C₄⋊C₈	D₈₃	semidirect