Subgroups — #19132786

Order 5504.

78 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	#14582729	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₄₄	—	2	1	no	—
5	D₆₈₈	—	2	1	no	—
6	D₆₈₈	C₄	1	2	yes	—
7	#13063261	—	2	1	no	—
8	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₄₃	—	2	1	no	—
10	C₂×D₁₇₂	—	4	1	no	—
11	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
12	#10621877	—	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	D₈	1	1	yes	characteristic
19	D₁₇₂	—	4	2	no	—
20	D₁₇₂	—	8	1	no	—
21	C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₄₃	SD₁₆	1	1	yes	characteristic
23	C₄₃⋊C₈	—	8	1	no	—
24	C₂²×C₄₃	SD₃₂	1	1	yes	characteristic
25	C₂×D₄₃	—	8	2	no	—
26	C₂×D₄₃	—	16	2	no	—
27	C₄×C₄₃	#102	1	1	yes	characteristic
28	C₄×C₄₃	D₁₆	1	1	yes	characteristic
29	#1419	—	43	1	no	maximal, Sylow
30	C₂×C₄₃	—	2	1	no	—
31	C₂×C₄₃	#357	1	1	yes	characteristic
32	D₄₃	—	16	2	no	—
33	C₂×D₁₆	—	43	1	no	—
34	M₆₄	D₄₃	1	1	yes	characteristic, complemented
35	#368	—	43	1	no	—
36	C₄₃	#1419	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₈	—	86	1	no	—
38	D₁₆	—	43	2	no	—
39	D₁₆	—	86	1	no	—
40	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
41	#108	—	86	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	172	1	no	—
44	D₈	—	86	2	no	—
45	D₈	—	172	1	no	—
46	M₁₆	—	172	1	no	—
47	C₂×C₈	D₁₇₂	1	1	yes	characteristic
48	C₁₆	C₄×D₄₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
50	C₂³	—	344	1	no	—
51	D₄	—	344	1	no	—
52	D₄	—	172	2	no	—
53	C₂×C₄	D₃₄₄	1	1	yes	characteristic
54	C₈	—	344	1	no	—
55	C₈	#10621873	1	1	yes	characteristic
56	C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
57	C₂²	—	344	2	no	—
58	C₂²	—	688	2	no	—
59	C₂²	#13063270	1	1	yes	characteristic
60	C₄	#13063247	1	1	yes	characteristic
61	C₄	D₆₈₈	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	688	2	no	—
64	C₂	#14582701	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14582701
commutator	C₁₆×C₄₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₄₃
Fitting	C₄₃×M₆₄	C₂
socle	C₂×C₄₃	#357
hypercenter	M₆₄	D₄₃
hypocenter	C₄₃	#1419
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #14582729	C₂	semidirect
3. C₄₃×M₆₄	C₂	semidirect
34. M₆₄	D₄₃	semidirect
36. C₄₃	#1419	semidirect