Subgroups — #10683086

Order 768.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1423	C₂	1	1	yes	maximal, characteristic
2	#67591	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic
4	#7545	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
6	C₃×#366	—	2	1	no	—
7	#4504	—	2	1	no	—
8	C₃⋊C₆₄	C₄	1	2	yes	complemented
9	C₂×C₆₄	—	3	1	no	—
10	#1423	—	3	1	no	—
11	#1423	D₃	1	1	yes	characteristic
12	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
13	C₃×#107	—	4	1	no	—
14	#818	—	4	1	no	—
15	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₃×C₃₂	Q₈	1	1	yes	characteristic
17	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
18	#366	—	2	1	no	—
19	#366	—	6	1	no	—
20	C₆₄	—	3	2	no	—
21	C₂×C₃×C₈	D₈	1	1	yes	characteristic
22	C₃×C₄⋊C₄	—	8	1	no	—
23	C₄⋊Dic₃	—	8	1	no	—
24	C₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
25	C₃×C₁₆	Q₁₆	1	1	yes	characteristic
26	C₂×C₁₆	#67	1	1	yes	characteristic
27	#107	—	4	1	no	—
28	#107	—	12	1	no	—
29	C₃₂	Dic₆	1	1	yes	characteristic
30	C₃₂	C₄×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	—	16	1	no	—
32	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
33	C₂×Dic₃	—	16	1	no	—
34	C₃×C₈	#107	1	1	yes	characteristic
35	C₃×C₈	Q₃₂	1	1	yes	characteristic
36	C₂×C₈	#213	1	1	yes	characteristic
37	C₄⋊C₄	—	8	1	no	—
38	C₄⋊C₄	—	24	1	no	—
39	C₁₆	Dic₃⋊C₄	1	1	yes	characteristic
40	C₁₆	#216	1	1	yes	characteristic
41	C₂²×C₃	D₃₂	1	1	yes	characteristic
42	C₃×C₄	—	32	1	no	—
43	C₃×C₄	#366	1	1	yes	characteristic
44	C₃×C₄	Q₆₄	1	1	yes	characteristic
45	Dic₃	—	32	1	no	—
46	C₂×C₄	—	16	1	no	—
47	C₂×C₄	—	48	1	no	—
48	C₂×C₄	#826	1	1	yes	characteristic
49	C₈	#807	1	1	yes	characteristic
50	C₈	#829	1	1	yes	characteristic
51	C₂×C₃	#1423	1	1	yes	characteristic
52	C₂×C₃	D₆₄	1	1	yes	characteristic
53	C₂×C₃	Q₁₂₈	1	1	yes	characteristic
54	C₂²	#4519	1	1	yes	characteristic
55	C₄	—	32	1	no	—
56	C₄	—	96	1	no	—
57	C₄	#4489	1	1	yes	characteristic
58	C₄	#4522	1	1	yes	characteristic
59	C₃	#7545	1	1	yes	characteristic, complemented, Sylow
60	C₂	#67575	1	1	yes	characteristic
61	C₂	#67607	1	1	yes	characteristic
62	C₂	#67610	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4519
commutator	C₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₃
Fitting	C₃×#1423	C₂
socle, prime radical	C₂²×C₃	D₃₂
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	#1423	D₃
hypocenter	C₃	#7545
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
8. C₃⋊C₆₄	C₄	semidirect
59. C₃	#7545	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1423	2
#67591	2
C₂×C₃⋊C₆₄	2
#7545	3