Subgroups — #32973871

Order 18688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608142	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608133	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#2262	C₂	1	1	yes	maximal, characteristic
4	#17988413	—	2	1	no	—
5	#17988306	—	2	1	no	—
6	#508×C₇₃	—	2	1	no	—
7	Q₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
8	SD₆₄×C₇₃	—	2	1	no	—
9	D₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
11	#14556525	—	4	1	no	—
12	#135×C₇₃	—	4	1	no	—
13	Q₃₂×C₇₃	—	2	1	no	—
14	SD₃₂×C₇₃	—	4	1	no	—
15	D₁₆×C₇₃	—	2	1	no	—
16	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
17	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₃	D₄	1	1	yes	characteristic
19	C₄⋊Dic₇₃	—	8	1	no	—
20	#41×C₇₃	—	8	1	no	—
21	Q₁₆×C₇₃	—	4	1	no	—
22	SD₁₆×C₇₃	—	8	1	no	—
23	D₈×C₇₃	—	4	1	no	—
24	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
25	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
27	Q₈×C₇₃	—	8	1	no	—
28	D₄×C₇₃	—	8	1	no	—
29	D₄×C₇₃	—	16	1	no	—
30	C₂×C₄×C₇₃	—	16	1	no	—
31	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
32	C₂×Dic₇₃	—	16	1	no	—
33	C₈×C₇₃	#102	1	1	yes	characteristic
34	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
35	C₂²×C₇₃	—	16	1	no	—
36	C₂²×C₇₃	D₃₂	1	1	yes	characteristic
37	C₄×C₇₃	—	16	1	no	—
38	C₄×C₇₃	#357	1	1	yes	characteristic
39	C₄×C₇₃	SD₆₄	1	1	yes	characteristic
40	Dic₇₃	—	32	1	no	—
41	#7540	—	73	1	no	maximal, Sylow
42	C₂×C₇₃	—	2	1	no	—
43	C₂×C₇₃	—	32	1	no	—
44	C₂×C₇₃	#1415	1	1	yes	characteristic
45	#2262	D₇₃	1	1	yes	characteristic
46	M₁₂₈	—	73	1	no	—
47	#1424	—	73	1	no	—
48	C₇₃	#7540	1	1	yes	characteristic, complemented, Sylow
49	#508	—	2	1	no	—
50	Q₆₄	Dic₇₃	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	Dic₇₃	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
54	#366	—	146	1	no	—
55	C₆₄	—	146	1	no	—
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
61	#107	—	292	1	no	—
62	C₃₂	C₂×Dic₇₃	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₇₃	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	#11924013	1	1	yes	characteristic
69	C₄⋊C₄	—	584	1	no	—
70	C₁₆	C₂²⋊Dic₇₃	1	1	yes	characteristic
71	C₁₆	#11924016	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	1168	1	no	—
77	C₂×C₄	#14556532	1	1	yes	characteristic
78	C₈	#14556512	1	1	yes	characteristic
79	C₈	#14556535	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	#17988422	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	2336	1	no	—
84	C₄	#17988388	1	1	yes	characteristic
85	C₄	#17988425	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#22608109	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22608109
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₃
Fitting, prime radical	C₇₃×#2262	C₂
socle	C₂×C₇₃	#1415
cosocle	C₂×C₃₂×C₇₃	C₂²
hypercenter	#2262	D₇₃
hypocenter	C₇₃	#7540
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #22608142	C₂	semidirect
2. #22608133	C₂	semidirect
7. Q₆₄×C₇₃	C₄	semidirect
9. D₃₂×C₇₃	C₄	semidirect
48. C₇₃	#7540	semidirect
50. Q₆₄	Dic₇₃	semidirect
52. D₃₂	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22608142	2
#22608133	2
C₇₃×#2262	2
#7540	73