Subgroups — #19132772

Order 5504.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₄₄	—	2	1	no	—
5	D₆₈₈	—	2	1	no	—
6	D₆₈₈	C₄	1	2	yes	complemented
7	C₄₃⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
9	#107×C₄₃	—	2	1	no	—
10	C₂×D₁₇₂	—	4	1	no	—
11	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
12	C₄⋊C₄×C₄₃	—	4	1	no	—
13	D₃₄₄	—	2	2	no	—
14	D₃₄₄	—	4	1	no	—
15	C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×D₄₃	—	8	1	no	—
18	C₂×C₄×C₄₃	—	8	1	no	—
19	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
20	D₁₇₂	—	4	2	no	—
21	D₁₇₂	—	8	1	no	—
22	C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₄₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₄₃	D₁₆	1	1	yes	characteristic
25	C₂×D₄₃	—	8	2	no	—
26	C₂×D₄₃	—	16	1	no	—
27	C₂×D₄₃	—	16	1	no	—
28	C₄×C₄₃	—	16	1	no	—
29	C₄×C₄₃	#102	1	1	yes	characteristic
30	C₄×C₄₃	SD₃₂	1	1	yes	characteristic
31	#1415	—	43	1	no	maximal, Sylow
32	C₂×C₄₃	#357	1	1	yes	characteristic
33	C₂×C₄₃	D₃₂	1	1	yes	characteristic
34	C₂×C₄₃	SD₆₄	1	1	yes	characteristic
35	D₄₃	—	16	2	no	—
36	C₂×D₁₆	—	43	1	no	—
37	C₂×C₃₂	—	43	1	no	—
38	#366	D₄₃	1	1	yes	characteristic, complemented
39	C₄₃	#1415	1	1	yes	characteristic, complemented, Sylow
40	C₂×D₈	—	86	1	no	—
41	D₁₆	—	43	2	no	—
42	D₁₆	—	86	1	no	—
43	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	86	1	no	—
46	C₂×D₄	—	172	1	no	—
47	D₈	—	86	2	no	—
48	D₈	—	172	1	no	—
49	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₄₃	1	1	yes	characteristic
52	C₁₆	D₁₇₂	1	1	yes	characteristic
53	C₂³	—	344	1	no	—
54	D₄	—	344	1	no	—
55	D₄	—	172	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#10621882	1	1	yes	characteristic
58	C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
59	C₈	#10621884	1	1	yes	characteristic
60	C₂²	—	344	2	no	—
61	C₂²	—	688	1	no	—
62	C₂²	—	688	1	no	—
63	C₂²	#13063266	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#13063245	1	1	yes	characteristic
66	C₄	#13063268	1	1	yes	characteristic
67	C₂	—	688	2	no	—
68	C₂	#14582699	1	1	yes	characteristic
69	C₂	#14582734	1	1	yes	characteristic
70	C₂	#14582736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13063266
commutator	C₁₆×C₄₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₄₃
Fitting	C₄₃×#366	C₂
socle	C₂²×C₄₃	D₁₆
hypercenter	#366	D₄₃
hypocenter	C₄₃	#1415
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃⋊C₃₂	C₂	semidirect
3. C₄₃×#366	C₂	semidirect
6. D₆₈₈	C₄	semidirect
38. #366	D₄₃	semidirect
39. C₄₃	#1415	semidirect