Subgroups — #34001269

Order 18688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625099	C₂	1	1	yes	maximal, characteristic
2	#22625092	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989513	—	2	1	no	—
5	#17989426	—	2	1	no	—
6	Q₃₂×D₇₃	—	2	1	no	—
7	Dic₁₁₆₈	C₄	1	1	yes	characteristic, complemented
8	#17988427	—	2	1	no	—
9	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
10	Q₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
11	#14556660	—	4	1	no	—
12	Q₁₆×D₇₃	—	4	1	no	—
13	Dic₅₈₄	—	2	1	no	—
14	#14556537	—	4	1	no	—
15	C₁₆×D₇₃	D₄	1	1	yes	characteristic
16	#14556497	D₄	1	1	yes	characteristic
17	Q₃₂×C₇₃	—	2	1	no	—
18	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₄⋊#64408	—	8	1	no	—
20	Q₈×D₇₃	—	8	1	no	—
21	Dic₂₉₂	—	4	1	no	—
22	#11924018	—	8	1	no	—
23	C₈×D₇₃	D₈	1	1	yes	characteristic
24	#11924003	SD₁₆	1	1	yes	characteristic
25	Q₁₆×C₇₃	—	4	1	no	—
26	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×#64408	—	16	1	no	—
28	Q₈×C₇₃	—	8	1	no	—
29	C₄×D₇₃	—	16	1	no	—
30	C₄×D₇₃	D₁₆	1	1	yes	characteristic
31	Dic₁₄₆	—	8	1	no	—
32	Dic₁₄₆	—	16	1	no	—
33	C₈×C₇₃	#102	1	1	yes	characteristic
34	#10596831	SD₃₂	1	1	yes	characteristic
35	C₂×D₇₃	D₃₂	1	1	yes	characteristic
36	#64408	—	32	1	no	—
37	C₄×C₇₃	—	16	1	no	—
38	C₄×C₇₃	#357	1	1	yes	characteristic
39	Dic₇₃	—	16	1	no	—
40	Dic₇₃	SD₆₄	1	1	yes	characteristic
41	#7538	—	73	1	no	maximal, Sylow
42	C₂×C₇₃	#1415	1	1	yes	characteristic
43	D₇₃	SD₁₂₈	1	1	yes	characteristic
44	D₇₃	Q₁₂₈	1	1	yes	characteristic
45	C₂×Q₆₄	—	73	1	no	—
46	C₂×C₆₄	—	73	1	no	—
47	#1423	—	73	1	no	—
48	C₇₃	#7538	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₃₂	—	146	1	no	—
50	Q₆₄	—	73	1	no	—
51	Q₆₄	—	146	1	no	—
52	Q₆₄	#64408	1	1	yes	characteristic, complemented
53	C₂×C₃₂	—	73	1	no	—
54	#366	—	146	1	no	—
55	C₆₄	—	146	1	no	—
56	C₂×Q₁₆	—	292	1	no	—
57	Q₃₂	—	2	1	no	—
58	Q₃₂	—	292	1	no	—
59	Q₃₂	—	146	1	no	—
60	C₂×C₁₆	—	73	1	no	—
61	#107	—	292	1	no	—
62	C₃₂	—	73	1	no	—
63	C₃₂	C₂×#64408	1	1	yes	characteristic
64	C₂×Q₈	—	584	1	no	—
65	Q₁₆	—	4	1	no	—
66	Q₁₆	—	292	1	no	—
67	Q₁₆	—	584	1	no	—
68	C₂×C₈	—	73	1	no	—
69	C₄⋊C₄	—	584	1	no	—
70	C₁₆	—	73	1	no	—
71	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	Q₈	—	584	1	no	—
74	Q₈	—	1168	1	no	—
75	C₂×C₄	—	73	1	no	—
76	C₂×C₄	—	1168	1	no	—
77	C₂×C₄	—	1168	1	no	—
78	C₈	—	73	1	no	—
79	C₈	#14556653	1	1	yes	characteristic
80	C₂²	—	73	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	2336	1	no	—
83	C₄	—	73	1	no	—
84	C₄	—	1168	1	no	—
85	C₄	#17989499	1	1	yes	characteristic
86	C₂	—	73	1	no	—
87	C₂	—	73	1	no	—
88	C₂	#22625080	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22625080
commutator	C₃₂×C₇₃	C₂×C₄
Frattini	C₃₂	C₂×#64408
Fitting	Q₆₄×C₇₃	C₄
socle	C₂×C₇₃	#1415
prime radical	C₂×D₇₃	D₃₂
cosocle	C₃₂×D₇₃	C₂²
hypercenter	Q₆₄	#64408
hypocenter	C₇₃	#7538
LCS	G ▷ C₃₂×C₇₃ ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
7. Dic₁₁₆₈	C₄	semidirect
10. Q₆₄×C₇₃	C₄	semidirect
48. C₇₃	#7538	semidirect
52. Q₆₄	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22625099	2
#22625092	2
Q₆₄×D₇₃	2
#7538	73