Subgroups — #34001209

Order 18688.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625100	C₂	1	2	yes	maximal
2	#22608064	C₂	1	1	yes	maximal, characteristic
3	#17989426	C₄	1	4	yes	complemented
4	C₂×#14556497	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924003	C₈	1	1	yes	characteristic
8	C₈×Dic₇₃	C₈	1	1	yes	characteristic
9	#14556497	C₂×C₄	1	2	yes	—
10	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
11	C₃₂×C₇₃	C₈	1	2	yes	—
12	C₂×#10596831	C₁₆	1	1	yes	characteristic
13	C₄×Dic₇₃	C₁₆	1	1	yes	characteristic
14	#11924003	C₁₆	1	2	yes	—
15	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
19	C₂×Dic₇₃	—	2	1	no	—
20	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
21	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
22	#10596831	—	2	1	no	—
23	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
24	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
25	C₄×C₇₃	M₆₄	1	1	yes	characteristic
26	Dic₇₃	—	4	1	no	—
27	#7511	—	73	1	no	maximal, Sylow
28	C₂×C₇₃	—	2	1	no	—
29	C₂×C₇₃	#1397	1	1	yes	characteristic
30	M₁₂₈	—	73	2	no	—
31	#1397	—	73	1	no	—
32	C₇₃	#7511	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃₂	—	73	1	no	—
34	C₂×C₃₂	#64408	1	1	yes	characteristic
35	C₄×C₁₆	—	73	1	no	—
36	C₆₄	—	73	4	no	—
37	C₂×C₁₆	—	73	1	no	—
38	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
39	C₄×C₈	—	73	1	no	—
40	C₃₂	—	73	2	no	—
41	C₃₂	#10596833	1	2	yes	—
42	C₂×C₈	—	73	1	no	—
43	C₂×C₈	#11924036	1	1	yes	characteristic
44	C₄²	—	73	1	no	—
45	C₁₆	—	73	2	no	—
46	C₁₆	C₄×#64408	1	1	yes	characteristic
47	C₁₆	C₂×#10596833	1	1	yes	characteristic
48	C₂×C₄	—	146	1	no	—
49	C₂×C₄	#14556664	1	1	yes	characteristic
50	C₈	—	146	1	no	—
51	C₈	C₄×#10596833	1	1	yes	characteristic
52	C₈	#14556664	1	1	yes	characteristic
53	C₂²	#17989520	1	1	yes	characteristic
54	C₄	—	292	1	no	—
55	C₄	#17989473	1	1	yes	characteristic
56	C₄	#17989520	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#22625039	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂×C₃₂×C₇₃	C₄
socle	C₂×C₇₃	#1397
prime radical	C₂²×C₇₃	M₆₄
cosocle	C₂×#14556497	C₂²
hypercenter	C₂×C₃₂	#64408
hypocenter	C₇₃	#7511
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
3. #17989426	C₄	semidirect
32. C₇₃	#7511	semidirect

Maximal subgroups

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

subgroup	index
#22625100	2
#22608064	2
#7511	73