Subgroups — #34001281

Order 18688.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625092	C₂	1	2	yes	maximal
2	C₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#17989513	—	2	2	no	—
4	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
5	#17988306	C₄	1	1	yes	characteristic, complemented
6	C₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
7	#14556660	—	4	2	no	—
8	C₁₆×D₇₃	D₄	1	1	yes	characteristic
9	#14556497	Q₈	1	1	yes	characteristic
10	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
11	C₄⋊#64408	—	8	2	no	—
12	C₈×D₇₃	D₈	1	1	yes	characteristic
13	#11924003	Q₁₆	1	1	yes	characteristic
14	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×#64408	—	16	2	no	—
16	C₄×D₇₃	D₁₆	1	1	yes	characteristic
17	C₈×C₇₃	#107	1	1	yes	characteristic
18	#10596831	Q₃₂	1	1	yes	characteristic
19	C₂×D₇₃	D₃₂	1	1	yes	characteristic
20	#64408	—	32	2	no	—
21	C₄×C₇₃	#366	1	1	yes	characteristic
22	Dic₇₃	Q₆₄	1	1	yes	characteristic
23	#7546	—	73	1	no	maximal, Sylow
24	C₂×C₇₃	#1423	1	1	yes	characteristic
25	D₇₃	SD₁₂₈	1	2	yes	—
26	C₂×C₆₄	—	73	1	no	—
27	#1423	—	73	2	no	—
28	C₇₃	#7546	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃₂	—	73	1	no	—
30	#366	—	146	2	no	—
31	C₆₄	—	73	1	no	—
32	C₆₄	#64408	1	1	yes	characteristic, complemented
33	C₂×C₁₆	—	73	1	no	—
34	#107	—	292	2	no	—
35	C₃₂	—	73	1	no	—
36	C₃₂	C₂×#64408	1	1	yes	characteristic
37	C₂×C₈	—	73	1	no	—
38	C₄⋊C₄	—	584	2	no	—
39	C₁₆	—	73	1	no	—
40	C₁₆	C₄⋊#64408	1	1	yes	characteristic
41	C₂×C₄	—	73	1	no	—
42	C₂×C₄	—	1168	2	no	—
43	C₈	—	73	1	no	—
44	C₈	#14556660	1	1	yes	characteristic
45	C₂²	—	73	1	no	—
46	C₄	—	2336	2	no	—
47	C₄	—	73	1	no	—
48	C₄	#17989513	1	1	yes	characteristic
49	C₂	—	73	2	no	—
50	C₂	#22625092	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22625092
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₃₂	C₂×#64408
Fitting	C₆₄×C₇₃	C₄
socle	C₂×C₇₃	#1423
prime radical	C₂×D₇₃	D₃₂
cosocle	C₃₂×D₇₃	C₂²
hypercenter	C₆₄	#64408
hypocenter	C₇₃	#7546
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
5. #17988306	C₄	semidirect
6. C₆₄×C₇₃	C₄	semidirect
28. C₇₃	#7546	semidirect
32. C₆₄	#64408	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625092	2
C₆₄×D₇₃	2
#7546	73