Subgroups — #34001208

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	#22608065	C₂	1	1	yes	maximal, characteristic
3	#17989426	C₄	1	4	yes	complemented
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	C₈	1	2	yes	—
8	#14556497	C₂×C₄	1	2	yes	—
9	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
11	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₇₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₇₃	C₁₆	1	1	yes	characteristic
15	C₁₆×C₇₃	C₁₆	1	2	yes	—
16	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₇₃	—	2	1	no	—
19	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
20	C₈×C₇₃	—	2	1	no	—
21	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
22	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
23	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
24	C₄×C₇₃	—	4	1	no	—
25	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₇₃	M₆₄	1	1	yes	characteristic
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	—	73	1	no	—
35	C₄×C₁₆	#64408	1	1	yes	characteristic
36	C₆₄	—	73	4	no	—
37	C₂×C₁₆	#10596833	1	1	yes	characteristic
38	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
39	C₄×C₈	#10596833	1	1	yes	characteristic
40	C₃₂	—	73	2	no	—
41	C₃₂	—	73	2	no	—
42	C₂×C₈	#11924031	1	1	yes	characteristic
43	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
44	C₄²	#11924031	1	1	yes	characteristic
45	C₁₆	#11924031	1	2	yes	—
46	C₁₆	C₄×#64408	1	1	yes	characteristic
47	C₁₆	#11924036	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×#10596833	1	1	yes	characteristic
52	C₈	C₂×#11924031	1	1	yes	characteristic
53	C₂²	#17989519	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	C₄×#11924031	1	1	yes	characteristic
56	C₄	#17989519	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#22625038	1	1	yes	characteristic

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

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
#22608065	2
#7511	73