Subgroups — #34001047

Order 18688.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624964	C₂	1	1	yes	maximal, characteristic
2	#22624964	C₂	1	1	yes	maximal, characteristic
3	#22608064	C₂	1	1	yes	maximal, characteristic
4	#17989450	—	2	1	no	—
5	#17989450	—	2	1	no	—
6	C₂×#14556497	C₄	1	1	yes	characteristic
7	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
9	C₄⋊#10596833	—	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	C₈	1	2	yes	complemented
14	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₃	—	2	1	no	—
16	C₂×#10596833	—	8	1	no	—
17	C₂×#10596833	—	8	1	no	—
18	C₂×#10596831	Q₁₆	1	1	yes	characteristic
19	C₄×Dic₇₃	D₈	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	#107	1	1	yes	characteristic
25	C₂×Dic₇₃	SD₃₂	1	1	yes	characteristic
26	C₂×Dic₇₃	SD₃₂	1	1	yes	characteristic
27	#10596833	—	16	1	no	—
28	#10596833	—	16	1	no	—
29	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₇₃	#108	1	1	yes	characteristic
31	#10596831	—	2	1	no	—
32	C₂²×C₇₃	#367	1	1	yes	characteristic
33	C₄×C₇₃	#335	1	1	yes	characteristic
34	C₄×C₇₃	#368	1	1	yes	characteristic
35	Dic₇₃	—	2	1	no	—
36	Dic₇₃	—	2	1	no	—
37	#7463	—	73	1	no	maximal, Sylow
38	C₂×C₇₃	#1372	1	1	yes	characteristic
39	C₂×C₇₃	#1426	1	1	yes	characteristic
40	C₂×C₇₃	#1426	1	1	yes	characteristic
41	#1397	—	73	1	no	—
42	#1372	—	73	1	no	—
43	#1372	—	73	1	no	—
44	C₇₃	#7463	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃₂	—	73	1	no	—
46	C₂×C₃₂	#64408	1	1	yes	characteristic
47	C₄×C₁₆	—	73	1	no	—
48	#335	—	146	1	no	—
49	#335	—	146	1	no	—
50	C₂×C₁₆	—	73	1	no	—
51	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
52	C₄⋊C₈	—	292	1	no	—
53	C₄⋊C₈	—	292	1	no	—
54	C₄×C₈	—	73	1	no	—
55	C₃₂	—	2	1	no	—
56	C₃₂	—	73	2	no	—
57	C₂×C₈	—	584	1	no	—
58	C₂×C₈	—	584	1	no	—
59	C₂×C₈	—	73	1	no	—
60	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
61	C₄²	—	73	1	no	—
62	C₁₆	—	146	1	no	—
63	C₁₆	#11924036	1	1	yes	characteristic
64	C₁₆	#11924037	1	1	yes	characteristic
65	C₂×C₄	—	73	1	no	—
66	C₂×C₄	—	73	1	no	—
67	C₂×C₄	#14556660	1	1	yes	characteristic
68	C₈	—	1168	1	no	—
69	C₈	—	1168	1	no	—
70	C₈	—	146	1	no	—
71	C₈	#14556658	1	1	yes	characteristic
72	C₈	#14556661	1	1	yes	characteristic
73	C₂²	#17989514	1	1	yes	characteristic
74	C₄	—	146	1	no	—
75	C₄	—	146	1	no	—
76	C₄	#17989451	1	1	yes	characteristic
77	C₄	#17989515	1	1	yes	characteristic
78	C₂	#22624965	1	1	yes	characteristic
79	C₂	#22625096	1	1	yes	characteristic
80	C₂	#22625097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17989514
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂×C₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	#367
cosocle	C₁₆×Dic₇₃	C₂²
hypercenter	C₂×C₃₂	#64408
hypocenter	C₇₃	#7463
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
13. #14556497	C₈	semidirect
44. C₇₃	#7463	semidirect

Maximal subgroups

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

subgroup	index
#22624964	2
#22608064	2
#7463	73