Subgroups — #34001029

Order 18688.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625090	C₂	1	2	yes	maximal
2	#22607847	C₂	1	1	yes	maximal, characteristic
3	#17989519	—	2	2	no	—
4	C₄×#11924003	C₂²	1	1	yes	characteristic
5	C₈×#10596831	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
7	#14556641	—	4	2	no	—
8	C₂×#11924003	D₄	1	1	yes	characteristic
9	C₂×#11924003	Q₈	1	1	yes	characteristic
10	C₄×#10596831	—	2	1	no	—
11	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
13	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×#10596831	—	2	2	no	—
15	#11924003	—	2	1	no	—
16	#11924003	SD₁₆	1	2	yes	—
17	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
20	C₁₆×C₇₃	—	2	1	no	—
21	C₁₆×C₇₃	C₁₆	1	2	yes	—
22	C₂×C₄×C₇₃	—	2	1	no	—
23	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₃	#106	1	1	yes	characteristic
25	C₈×C₇₃	#108	1	1	yes	characteristic
26	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
27	C₈×C₇₃	M₃₂	1	1	yes	characteristic
28	#10596831	—	4	2	no	—
29	C₂²×C₇₃	#364	1	1	yes	characteristic
30	C₄×C₇₃	—	2	2	no	—
31	C₄×C₇₃	#334	1	1	yes	characteristic
32	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
33	#7456	—	73	1	no	maximal, Sylow
34	C₂×C₇₃	—	2	1	no	—
35	C₂×C₇₃	#1367	1	1	yes	characteristic
36	#1422	—	73	2	no	—
37	#1311	—	73	1	no	—
38	C₇₃	#7456	1	1	yes	characteristic, complemented, Sylow
39	M₆₄	—	146	2	no	—
40	C₄×C₁₆	—	73	1	no	—
41	C₄×C₁₆	#64408	1	1	yes	characteristic
42	C₈²	—	73	1	no	—
43	C₂×C₁₆	—	73	1	no	—
44	C₂×C₁₆	—	73	1	no	—
45	C₂×C₁₆	#10596833	1	1	yes	characteristic
46	C₂×C₁₆	#10596833	1	1	yes	characteristic
47	C₄×C₈	—	146	1	no	—
48	C₄×C₈	C₂×#64408	1	1	yes	characteristic
49	C₃₂	—	292	2	no	—
50	C₂×C₈	—	146	2	no	—
51	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
52	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
53	C₄²	#11924038	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	73	2	no	—
56	C₁₆	—	146	1	no	—
57	C₁₆	#11924031	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
60	C₈	—	292	2	no	—
61	C₈	#14556659	1	1	yes	characteristic
62	C₈	#14556661	1	1	yes	characteristic
63	C₈	C₂×#11924031	1	1	yes	characteristic
64	C₈	#14556665	1	1	yes	characteristic
65	C₂²	#17989509	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#17989448	1	1	yes	characteristic
68	C₄	C₄⋊#11924031	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#22624953	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556659
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂×C₇₃	#1367
prime radical	C₂²×C₇₃	#364
cosocle	C₄×#11924003	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7456
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
38. C₇₃	#7456	semidirect

Maximal subgroups

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

subgroup	index
#22625090	2
#22607847	2
#7456	73