Subgroups — C₆₄×#64408 #34001200

Order 18688.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625099	C₂	1	1	yes	maximal, characteristic
2	C₃₂×#64408	C₂	1	1	yes	maximal, characteristic
3	C₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989518	C₄	1	1	yes	characteristic
5	C₁₆×#64408	C₄	1	1	yes	characteristic
6	#17989426	C₄	1	2	yes	complemented
7	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
8	#17988306	C₄	1	1	yes	characteristic, complemented
9	C₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
10	#14556664	C₈	1	1	yes	characteristic
11	C₈×#64408	C₈	1	1	yes	characteristic
12	#14556641	C₈	1	2	yes	—
13	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
14	#14556497	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
16	#11924036	C₁₆	1	1	yes	characteristic
17	C₄×#64408	C₁₆	1	1	yes	characteristic
18	#11924031	C₁₆	1	2	yes	—
19	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
20	#11924003	C₂×C₈	1	1	yes	characteristic
21	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
22	C₂×#64408	C₃₂	1	2	yes	—
23	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
24	#10596833	C₃₂	1	2	yes	—
25	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
26	#10596831	C₂×C₁₆	1	1	yes	characteristic
27	C₂×D₇₃	C₂×C₃₂	1	1	yes	characteristic
28	#64408	C₆₄	1	4	yes	direct product factor
29	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
30	Dic₇₃	C₂×C₃₂	1	1	yes	characteristic
31	C₄×C₆₄	—	73	1	no	maximal, Sylow
32	C₂×C₇₃	C₄×C₃₂	1	1	yes	characteristic
33	D₇₃	C₂×C₆₄	1	2	yes	—
34	C₂×C₆₄	—	73	1	no	—
35	C₂×C₆₄	—	73	1	no	—
36	C₄×C₃₂	—	73	1	no	—
37	C₇₃	C₄×C₆₄	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃₂	—	73	1	no	—
39	C₂×C₃₂	—	73	1	no	—
40	C₄×C₁₆	—	73	1	no	—
41	C₆₄	—	73	2	no	—
42	C₆₄	—	73	1	no	—
43	C₆₄	#64408	1	1	yes	characteristic, direct product factor
44	C₂×C₁₆	—	73	1	no	—
45	C₂×C₁₆	—	73	1	no	—
46	C₄×C₈	—	73	1	no	—
47	C₃₂	—	73	2	no	—
48	C₃₂	—	73	1	no	—
49	C₃₂	C₂×#64408	1	1	yes	characteristic
50	C₂×C₈	—	73	1	no	—
51	C₂×C₈	—	73	1	no	—
52	C₄²	—	73	1	no	—
53	C₁₆	—	73	2	no	—
54	C₁₆	—	73	1	no	—
55	C₁₆	C₄×#64408	1	1	yes	characteristic
56	C₂×C₄	—	73	2	no	—
57	C₂×C₄	—	73	1	no	—
58	C₈	—	73	2	no	—
59	C₈	—	73	1	no	—
60	C₈	C₈×#64408	1	1	yes	characteristic
61	C₂²	—	73	1	no	—
62	C₄	—	73	4	no	—
63	C₄	—	73	1	no	—
64	C₄	C₁₆×#64408	1	1	yes	characteristic
65	C₂	—	73	2	no	—
66	C₂	C₃₂×#64408	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₆₄	#64408
commutator, hypocenter	C₇₃	C₄×C₆₄
Frattini	C₃₂	C₂×#64408
Fitting	C₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₄×C₃₂
prime radical	C₂×D₇₃	C₂×C₃₂
cosocle	C₃₂×D₇₃	C₂²
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
6. #17989426	C₄	semidirect
8. #17988306	C₄	semidirect
9. C₆₄×C₇₃	C₄	semidirect
28. #64408	C₆₄	direct (N × H)
37. C₇₃	C₄×C₆₄	semidirect
43. C₆₄	#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
#22625099	2
C₃₂×#64408	2
C₆₄×D₇₃	2
C₄×C₆₄	73

Subgroups — C64×#64408 #34001200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₄×#64408 #34001200