Subgroups — #34001032

Order 18688.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624958	C₂	1	1	yes	maximal, characteristic
2	#22624956	C₂	1	1	yes	maximal, characteristic
3	#22608066	C₂	1	1	yes	maximal, characteristic
4	#17989509	—	2	1	no	—
5	#17989448	—	2	1	no	—
6	#17988421	C₄	1	1	yes	characteristic
7	#17988362	C₂²	1	1	yes	characteristic
8	M₆₄×C₇₃	C₄	1	1	yes	characteristic
9	#14556665	—	4	1	no	—
10	C₄⋊#10596833	—	4	1	no	—
11	C₂×#11924003	Q₈	1	1	yes	characteristic
12	C₈×Dic₇₃	D₄	1	1	yes	characteristic
13	#14556497	—	2	1	no	—
14	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₃	C₈	1	2	yes	complemented
16	C₂×#10596833	—	8	1	no	—
17	#11924031	—	8	1	no	—
18	C₂×#10596831	SD₁₆	1	1	yes	characteristic
19	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
20	#11924003	—	2	1	no	—
21	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
22	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
25	C₂×Dic₇₃	—	2	1	no	—
26	#10596833	—	16	1	no	—
27	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₇₃	#108	1	1	yes	characteristic
29	#10596831	—	2	1	no	—
30	C₂²×C₇₃	#365	1	1	yes	characteristic
31	C₄×C₇₃	#334	1	1	yes	characteristic
32	C₄×C₇₃	#365	1	1	yes	characteristic
33	Dic₇₃	—	4	1	no	—
34	#7457	—	73	1	no	maximal, Sylow
35	C₂×C₇₃	—	2	1	no	—
36	C₂×C₇₃	#1368	1	1	yes	characteristic
37	#1398	—	73	1	no	—
38	#1369	—	73	1	no	—
39	#1368	—	73	1	no	—
40	C₇₃	#7457	1	1	yes	characteristic, complemented, Sylow
41	M₆₄	—	73	1	no	—
42	M₆₄	#64408	1	1	yes	characteristic
43	#364	—	146	1	no	—
44	#346	—	73	1	no	—
45	#334	—	146	1	no	—
46	M₃₂	—	292	1	no	—
47	C₂×C₁₆	—	73	1	no	—
48	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
49	C₄⋊C₈	—	292	1	no	—
50	C₄×C₈	—	73	1	no	—
51	C₃₂	—	146	1	no	—
52	C₃₂	#10596833	1	2	yes	complemented
53	C₂×C₈	—	584	1	no	—
54	C₂×C₈	—	73	1	no	—
55	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
56	C₄²	—	73	1	no	—
57	C₁₆	—	584	1	no	—
58	C₁₆	—	146	1	no	—
59	C₁₆	C₂×#10596833	1	1	yes	characteristic
60	C₁₆	#11924038	1	1	yes	characteristic
61	C₂×C₄	—	146	1	no	—
62	C₂×C₄	#14556659	1	1	yes	characteristic
63	C₈	—	1168	1	no	—
64	C₈	—	146	1	no	—
65	C₈	C₄⋊#10596833	1	1	yes	characteristic
66	C₈	#14556661	1	1	yes	characteristic
67	C₂²	#17989512	1	1	yes	characteristic
68	C₄	—	292	1	no	—
69	C₄	#17989448	1	1	yes	characteristic
70	C₄	#17989512	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#22624956	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22624956
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#64408
Fitting	M₆₄×C₇₃	C₄
socle	C₂×C₇₃	#1368
prime radical	C₂²×C₇₃	#365
cosocle	#17988362	C₂²
hypercenter	M₆₄	#64408
hypocenter	C₇₃	#7457
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₇₃	C₈	semidirect
40. C₇₃	#7457	semidirect
52. C₃₂	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22624958	2
#22624956	2
#22608066	2
#7457	73