Subgroups — #32973320

Order 18688.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608067	C₂	1	1	yes	maximal, characteristic
2	#22608067	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1313	C₂	1	1	yes	maximal, characteristic
4	#17988421	C₄	1	2	yes	—
5	#17988421	C₄	1	2	yes	—
6	#346×C₇₃	C₄	1	1	yes	characteristic
7	#346×C₇₃	C₂²	1	1	yes	characteristic
8	#322×C₇₃	C₄	1	1	yes	characteristic
9	#14556497	—	2	2	no	—
10	#14556497	C₈	1	4	yes	complemented
11	C₂×C₁₆×C₇₃	—	2	1	no	—
12	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₇₃	—	2	1	no	—
15	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₇₃	—	4	1	no	—
17	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
18	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
20	C₁₆×C₇₃	—	4	1	no	—
21	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
22	C₁₆×C₇₃	M₁₆	1	2	yes	—
23	C₂×C₄×C₇₃	—	2	1	no	—
24	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
25	C₈×C₇₃	—	2	1	no	—
26	C₈×C₇₃	—	8	1	no	—
27	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
28	C₈×C₇₃	#97	1	1	yes	characteristic
29	C₂²×C₇₃	#347	1	1	yes	characteristic
30	C₄×C₇₃	—	4	1	no	—
31	C₄×C₇₃	#322	1	1	yes	characteristic
32	C₄×C₇₃	#347	1	1	yes	characteristic
33	#7333	—	73	1	no	maximal, Sylow
34	C₂×C₇₃	—	2	1	no	—
35	C₂×C₇₃	#1313	1	1	yes	characteristic
36	#1398	—	73	1	no	—
37	#1398	—	73	1	no	—
38	#1313	D₇₃	1	1	yes	characteristic
39	C₇₃	#7333	1	1	yes	characteristic, complemented, Sylow
40	M₆₄	—	73	2	no	—
41	M₆₄	—	73	2	no	—
42	#346	Dic₇₃	1	1	yes	characteristic
43	#346	C₂×D₇₃	1	1	yes	characteristic
44	#322	Dic₇₃	1	1	yes	characteristic
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
47	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
50	C₃₂	—	73	4	no	—
51	C₃₂	—	146	2	no	—
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
54	C₂×C₈	#11924012	1	1	yes	characteristic
55	C₄²	#11924012	1	1	yes	characteristic
56	C₁₆	—	4	1	no	—
57	C₁₆	C₈×D₇₃	1	2	yes	—
58	C₁₆	#11924011	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#14556502	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	C₈×Dic₇₃	1	1	yes	characteristic
64	C₈	#14556501	1	1	yes	characteristic
65	C₂²	#17988365	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#17988309	1	1	yes	characteristic
68	C₄	#17988365	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#22607852	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17988309
commutator	C₈×C₇₃	C₄×C₈
Frattini	#346	C₂×D₇₃
Fitting	C₇₃×#1313	C₂
socle	C₂×C₇₃	#1313
prime radical	C₂²×C₇₃	#347
cosocle	#346×C₇₃	C₂²
hypercenter	#1313	D₇₃
hypocenter	C₇₃	#7333
LCS	G ▷ C₈×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #346 ◁ #1313

Complements

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

N	H ≅ G/N	type
10. #14556497	C₈	semidirect
39. C₇₃	#7333	semidirect

Maximal subgroups

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

subgroup	index
#22608067	2
C₇₃×#1313	2
#7333	73