Subgroups — #34001278

Order 18688.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625100	C₂	1	2	yes	maximal
2	C₄×#14556497	C₂	1	1	yes	maximal, characteristic
3	#17989426	—	2	2	no	—
4	C₂×#14556497	C₄	1	1	yes	characteristic
5	C₂×#14556497	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
7	#14556497	—	2	1	no	—
8	#14556497	D₄	1	1	yes	characteristic
9	#14556497	Q₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
13	C₂×C₈×C₇₃	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₇₃	C₁₆	1	1	yes	characteristic
16	C₁₆×C₇₃	—	2	1	no	—
17	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₇₃	—	2	1	no	—
20	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₇₃	C₃₂	1	2	yes	—
22	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₇₃	M₃₂	1	1	yes	characteristic
24	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
25	C₄×C₇₃	—	2	2	no	—
26	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
27	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
28	#7544	—	73	1	no	maximal, Sylow
29	C₂×C₇₃	—	2	1	no	—
30	C₂×C₇₃	C₄⋊C₃₂	1	1	yes	characteristic
31	M₁₂₈	—	73	2	no	—
32	C₄×C₃₂	—	73	1	no	—
33	C₇₃	#7544	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃₂	—	73	1	no	—
35	C₂×C₃₂	—	73	1	no	—
36	C₄×C₁₆	#64408	1	1	yes	characteristic
37	C₆₄	—	146	2	no	—
38	C₂×C₁₆	#10596833	1	1	yes	characteristic
39	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
40	C₄×C₈	#10596833	1	1	yes	characteristic
41	C₃₂	—	73	1	no	—
42	C₃₂	—	73	1	no	—
43	C₃₂	—	146	1	no	—
44	C₂×C₈	#11924031	1	1	yes	characteristic
45	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
46	C₄²	#11924031	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₄⋊#64408	1	1	yes	characteristic
49	C₁₆	#11924038	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
52	C₈	#14556641	1	2	yes	—
53	C₈	C₄⋊#10596833	1	1	yes	characteristic
54	C₈	#14556665	1	1	yes	characteristic
55	C₂²	#17989519	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊#11924031	1	1	yes	characteristic
58	C₄	C₂×#14556641	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₄⋊#14556641	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₄⋊C₃₂
prime radical	C₂²×C₇₃	M₆₄
cosocle	C₂×#14556497	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7544
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
33. C₇₃	#7544	semidirect

Maximal subgroups

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

subgroup	index
#22625100	2
C₄×#14556497	2
#7544	73