Subgroups — #32973667

Order 18688.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608005	C₂	1	2	yes	maximal
2	C₇₃×#1397	C₂	1	1	yes	maximal, characteristic
3	#17988340	—	2	2	no	—
4	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
7	C₄⋊#10596831	—	4	2	no	—
8	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₇₃	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
11	C₃₂×C₇₃	—	2	1	no	—
12	C₃₂×C₇₃	C₈	1	2	yes	complemented
13	C₂×#10596831	—	8	2	no	—
14	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
16	C₄²×C₇₃	D₈	1	1	yes	characteristic
17	C₁₆×C₇₃	—	2	1	no	—
18	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
21	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
22	C₈×C₇₃	—	2	1	no	—
23	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₃	#108	1	1	yes	characteristic
25	#10596831	—	16	2	no	—
26	C₂²×C₇₃	#367	1	1	yes	characteristic
27	C₄×C₇₃	—	2	2	no	—
28	C₄×C₇₃	#335	1	1	yes	characteristic
29	C₄×C₇₃	#368	1	1	yes	characteristic
30	#7463	—	73	1	no	maximal, Sylow
31	C₂×C₇₃	#1372	1	1	yes	characteristic
32	C₂×C₇₃	#1426	1	2	yes	—
33	#1397	D₇₃	1	1	yes	characteristic
34	#1372	—	73	2	no	—
35	C₇₃	#7463	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃₂	Dic₇₃	1	1	yes	characteristic
37	C₂×C₃₂	Dic₇₃	1	1	yes	characteristic
38	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
39	#335	—	146	2	no	—
40	C₂×C₁₆	Dic₁₄₆	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
42	C₄⋊C₈	—	292	2	no	—
43	C₄×C₈	D₂₉₂	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#10596831	1	2	yes	complemented
46	C₂×C₈	—	584	2	no	—
47	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
48	C₂×C₈	Dic₂₉₂	1	1	yes	characteristic
49	C₄²	D₅₈₄	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×#10596831	1	1	yes	characteristic
52	C₁₆	#11924012	1	1	yes	characteristic
53	C₂×C₄	#14556525	1	1	yes	characteristic
54	C₂×C₄	#14556536	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	1168	2	no	—
57	C₈	C₄⋊#10596831	1	1	yes	characteristic
58	C₈	#14556527	1	1	yes	characteristic
59	C₂²	#17988415	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#17988340	1	1	yes	characteristic
62	C₄	#17988417	1	1	yes	characteristic
63	C₂	#22608005	1	1	yes	characteristic
64	C₂	#22608137	1	2	yes	—

	subgroup	G/H
center	C₂²	#17988415
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₃
Fitting	C₇₃×#1397	C₂
socle, prime radical	C₂²×C₇₃	#367
cosocle	C₄×C₁₆×C₇₃	C₂²
hypercenter	#1397	D₇₃
hypocenter	C₇₃	#7463
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₇₃	C₈	semidirect
35. C₇₃	#7463	semidirect
45. C₃₂	#10596831	semidirect

Maximal subgroups

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

subgroup	index
#22608005	2
C₇₃×#1397	2
#7463	73