Subgroups — #33002386

Order 18688.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608009	C₂	1	6	yes	maximal
2	C₇₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924003	C₂²	1	4	yes	—
4	C₂×#97×C₇₃	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
6	#376×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924003	—	2	12	no	—
8	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
11	#97×C₇₃	—	2	6	no	—
12	C₄×C₈×C₇₃	—	2	6	no	—
13	#95×C₇₃	C₈	1	4	yes	—
14	#11924003	—	4	4	no	—
15	C₂²×C₄×C₇₃	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₈×C₇₃	—	2	3	no	—
18	C₂×C₈×C₇₃	—	2	3	no	—
19	C₂×C₈×C₇₃	M₁₆	1	6	yes	—
20	C₂×C₈×C₇₃	#41	1	6	yes	—
21	C₄²×C₇₃	—	2	3	no	—
22	C₄²×C₇₃	—	2	3	no	—
23	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₃	—	2	6	no	—
25	C₂×C₄×C₇₃	—	2	12	no	—
26	C₂×C₄×C₇₃	#117	1	3	yes	—
27	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
28	C₈×C₇₃	—	2	6	no	—
29	C₈×C₇₃	#126	1	4	yes	—
30	C₂²×C₇₃	#431	1	3	yes	—
31	C₂²×C₇₃	#504	1	4	yes	—
32	C₄×C₇₃	—	4	6	no	—
33	C₄×C₇₃	#383	1	1	yes	characteristic
34	C₄×C₇₃	#432	1	3	yes	—
35	#12895	—	73	1	no	maximal, Sylow
36	C₂×C₇₃	#1839	1	1	yes	characteristic
37	C₂×C₇₃	#2164	1	6	yes	—
38	#1726	D₇₃	1	1	yes	characteristic
39	#1374	—	73	6	no	—
40	C₇₃	#12895	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₁₆	—	73	4	no	—
42	C₂×#97	Dic₇₃	1	3	yes	—
43	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
44	#376	Dic₇₃	1	1	yes	characteristic
45	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
46	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₇₃	1	3	yes	—
48	C₂×C₁₆	—	146	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	#10596831	1	4	yes	—
52	C₂²×C₄	C₂×#10596831	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11924012	1	6	yes	—
57	C₂×C₈	#11924028	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	292	4	no	—
61	C₂³	C₂²×#10596831	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	#14556550	1	3	yes	—
65	C₂×C₄	C₂×#11924012	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#14556585	1	4	yes	—
68	C₂²	#17988640	1	3	yes	—
69	C₂²	#17989004	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#17988455	1	1	yes	characteristic
72	C₄	#17988644	1	3	yes	—
73	C₂	#22610074	1	1	yes	characteristic
74	C₂	#22611738	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	C₇₃×#1726	C₂
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1726	D₇₃
hypocenter	C₇₃	#12895
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
40. C₇₃	#12895	semidirect

Maximal subgroups

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

subgroup	index
#22608009	2
C₇₃×#1726	2
#12895	73