Subgroups — #33000808

Order 18688.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988363	C₂	1	3	yes	maximal
2	C₂×C₄×#11924003	C₂	1	3	yes	maximal
3	C₇₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	C₂²	1	1	yes	characteristic
5	C₂²×#11924003	C₂²	1	3	yes	—
6	#17988363	C₄	1	12	yes	complemented
7	C₄×#11924003	C₄	1	12	yes	complemented
8	C₂×#97×C₇₃	C₄	1	3	yes	—
9	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
10	#376×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924003	C₂×C₄	1	6	yes	—
12	C₂×#11924003	C₂×C₄	1	6	yes	—
13	C₂×#11924003	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
17	#97×C₇₃	C₈	1	12	yes	—
18	C₄×C₈×C₇₃	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₇₃	C₂×C₄	1	6	yes	—
20	#95×C₇₃	C₈	1	4	yes	—
21	#11924003	C₄²	1	4	yes	—
22	#11924003	C₄²	1	12	yes	—
23	C₂²×C₄×C₇₃	C₂×C₈	1	6	yes	—
24	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₄²	1	6	yes	—
26	C₂×C₈×C₇₃	C₂×C₈	1	12	yes	—
27	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
28	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
29	C₄²×C₇₃	C₄²	1	6	yes	—
30	C₄²×C₇₃	C₂×C₈	1	6	yes	—
31	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	12	no	—
33	C₂×C₄×C₇₃	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
36	C₈×C₇₃	C₄×C₈	1	12	yes	—
37	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
38	C₈×C₇₃	C₂×C₄²	1	3	yes	—
39	C₂²×C₇₃	C₂×C₄×C₈	1	3	yes	—
40	C₂²×C₇₃	#504	1	4	yes	—
41	C₄×C₇₃	—	2	12	no	—
42	C₄×C₇₃	C₄³	1	1	yes	characteristic
43	C₄×C₇₃	C₂×C₄×C₈	1	3	yes	—
44	#12539	—	73	1	no	maximal, Sylow
45	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
46	C₂×C₇₃	#2108	1	6	yes	—
47	C₂×#346	—	73	3	no	—
48	C₂×C₄×C₁₆	—	73	3	no	—
49	#1726	D₇₃	1	1	yes	characteristic
50	C₇₃	#12539	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	—	73	1	no	—
52	C₂²×C₁₆	—	73	3	no	—
53	C₂×#97	Dic₇₃	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
55	#376	Dic₇₃	1	1	yes	characteristic
56	#346	—	73	12	no	—
57	C₄×C₁₆	—	73	12	no	—
58	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
59	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₇₃	1	3	yes	—
61	C₂×C₁₆	—	73	6	no	—
62	C₂×C₁₆	—	73	6	no	—
63	C₂×C₁₆	—	73	12	no	—
64	#97	#10596831	1	12	yes	—
65	C₄×C₈	C₄×D₇₃	1	6	yes	—
66	C₄×C₈	C₄×D₇₃	1	6	yes	—
67	#95	#10596831	1	4	yes	—
68	C₂²×C₄	C₂×#10596831	1	6	yes	—
69	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
70	C₂×C₈	C₄×Dic₇₃	1	6	yes	—
71	C₂×C₈	C₂×#10596831	1	12	yes	—
72	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
73	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
74	C₄²	C₄×Dic₇₃	1	6	yes	—
75	C₄²	C₂×#10596831	1	6	yes	—
76	C₁₆	—	73	4	no	—
77	C₁₆	—	73	12	no	—
78	C₂³	C₂²×#10596831	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×#10596831	1	12	yes	—
81	C₂×C₄	C₂×C₄×Dic₇₃	1	3	yes	—
82	C₂×C₄	C₂²×#10596831	1	3	yes	—
83	C₈	C₄×#10596831	1	12	yes	—
84	C₈	C₄²×D₇₃	1	1	yes	characteristic
85	C₈	C₄²×D₇₃	1	3	yes	—
86	C₂²	C₂×C₄×#10596831	1	3	yes	—
87	C₂²	#17989004	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄²×Dic₇₃	1	1	yes	characteristic
90	C₄	C₂×C₄×#10596831	1	3	yes	—
91	C₂	C₄²×#10596831	1	1	yes	characteristic
92	C₂	#22611528	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂×C₇₃	C₄²×C₈
Fitting	C₇₃×#1726	C₂
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1726	D₇₃
hypocenter	C₇₃	#12539
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #17988363	C₄	semidirect
7. C₄×#11924003	C₄	semidirect
50. C₇₃	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988363	2
C₂×C₄×#11924003	2
C₇₃×#1726	2
#12539	73