Subgroups — #34001041

Order 18688.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624962	C₂	1	2	yes	maximal
2	C₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17989450	—	2	2	no	—
4	C₂×#14556497	C₄	1	1	yes	characteristic
5	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
6	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
7	C₄⋊#10596833	—	4	2	no	—
8	C₂×#11924003	Q₈	1	1	yes	characteristic
9	C₈×Dic₇₃	D₄	1	1	yes	characteristic
10	#14556497	—	2	1	no	—
11	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₇₃	C₈	1	2	yes	complemented
13	C₂×#10596833	—	8	2	no	—
14	C₂×#10596831	Q₁₆	1	1	yes	characteristic
15	C₄×Dic₇₃	D₈	1	1	yes	characteristic
16	#11924003	—	2	1	no	—
17	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
21	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
22	C₂×Dic₇₃	Q₃₂	1	1	yes	characteristic
23	#10596833	—	16	2	no	—
24	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₇₃	#108	1	1	yes	characteristic
26	#10596831	—	2	1	no	—
27	C₂²×C₇₃	#366	1	1	yes	characteristic
28	C₄×C₇₃	#335	1	1	yes	characteristic
29	C₄×C₇₃	#368	1	1	yes	characteristic
30	Dic₇₃	—	2	1	no	—
31	Dic₇₃	SD₆₄	1	2	yes	—
32	#7461	—	73	1	no	maximal, Sylow
33	C₂×C₇₃	#1371	1	1	yes	characteristic
34	C₂×C₇₃	#1424	1	1	yes	characteristic
35	C₂×C₇₃	#1425	1	1	yes	characteristic
36	C₄×C₃₂	—	73	1	no	—
37	#1371	—	73	2	no	—
38	C₇₃	#7461	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃₂	—	73	1	no	—
40	C₂×C₃₂	#64408	1	1	yes	characteristic
41	C₄×C₁₆	—	73	1	no	—
42	#335	—	146	2	no	—
43	C₂×C₁₆	—	73	1	no	—
44	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
45	C₄⋊C₈	—	292	2	no	—
46	C₄×C₈	—	73	1	no	—
47	C₃₂	—	146	1	no	—
48	C₃₂	#10596833	1	2	yes	complemented
49	C₂×C₈	—	584	2	no	—
50	C₂×C₈	—	73	1	no	—
51	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
52	C₄²	—	73	1	no	—
53	C₁₆	—	146	1	no	—
54	C₁₆	C₂×#10596833	1	1	yes	characteristic
55	C₁₆	#11924038	1	1	yes	characteristic
56	C₂×C₄	—	73	1	no	—
57	C₂×C₄	—	73	1	no	—
58	C₂×C₄	#14556660	1	1	yes	characteristic
59	C₈	—	1168	2	no	—
60	C₈	—	146	1	no	—
61	C₈	C₄⋊#10596833	1	1	yes	characteristic
62	C₈	#14556662	1	1	yes	characteristic
63	C₂²	#17989513	1	1	yes	characteristic
64	C₄	—	73	2	no	—
65	C₄	—	146	1	no	—
66	C₄	#17989450	1	1	yes	characteristic
67	C₄	#17989515	1	1	yes	characteristic
68	C₂	#22624962	1	1	yes	characteristic
69	C₂	#22625093	1	1	yes	characteristic
70	C₂	#22625095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17989513
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂×C₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	#366
cosocle	C₁₆×Dic₇₃	C₂²
hypercenter	C₂×C₃₂	#64408
hypocenter	C₇₃	#7461
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₇₃	C₈	semidirect
38. C₇₃	#7461	semidirect
48. C₃₂	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22624962	2
C₃₂×Dic₇₃	2
#7461	73