Subgroups — #10682871

Order 768.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1373	C₂	1	1	yes	maximal, characteristic
2	#67508	C₂	1	1	yes	maximal, characteristic
3	#67502	C₂	1	1	yes	maximal, characteristic
4	#7464	—	3	1	no	maximal, Sylow
5	C₃×#364	—	2	1	no	—
6	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₃₂	C₄	1	1	yes	characteristic
8	C₂×C₃⋊C₃₂	C₄	1	1	yes	characteristic
9	#4463	—	2	1	no	—
10	#1397	—	3	1	no	—
11	#1373	—	3	1	no	—
12	#1373	D₃	1	1	yes	characteristic
13	C₃×M₃₂	—	4	1	no	—
14	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₁₆	Q₈	1	1	yes	characteristic
16	C₃×C₄×C₈	D₄	1	1	yes	characteristic
17	#812	—	4	1	no	—
18	C₃⋊C₃₂	—	2	1	no	—
19	C₃⋊C₃₂	C₈	1	2	yes	—
20	C₂×C₃₂	—	3	1	no	—
21	C₂×C₃₂	—	3	1	no	—
22	#364	—	2	1	no	—
23	#364	—	6	1	no	—
24	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₃×C₈	D₈	1	1	yes	characteristic
27	C₃×C₄²	Q₁₆	1	1	yes	characteristic
28	C₃×C₁₆	—	2	1	no	—
29	C₃×C₁₆	—	8	1	no	—
30	C₃×C₁₆	C₂×C₈	1	1	yes	characteristic
31	C₃×C₁₆	M₁₆	1	1	yes	characteristic
32	C₃⋊C₁₆	—	8	1	no	—
33	M₃₂	—	4	1	no	—
34	M₃₂	—	12	1	no	—
35	C₂×C₁₆	Dic₆	1	1	yes	characteristic
36	C₂×C₁₆	C₄×D₃	1	1	yes	characteristic
37	C₄×C₈	#67	1	1	yes	characteristic
38	C₃₂	—	3	2	no	—
39	C₃₂	—	6	1	no	—
40	C₂×C₃×C₄	—	2	1	no	—
41	C₂×C₃×C₄	#107	1	1	yes	characteristic
42	C₃×C₈	C₄⋊C₈	1	1	yes	characteristic
43	C₃×C₈	#108	1	1	yes	characteristic
44	C₃×C₈	SD₃₂	1	1	yes	characteristic
45	C₃×C₈	SD₃₂	1	1	yes	characteristic
46	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
47	C₂×C₈	#213	1	1	yes	characteristic
48	C₄²	#216	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	8	1	no	—
51	C₁₆	—	24	1	no	—
52	C₁₆	D₃×C₈	1	1	yes	characteristic
53	C₁₆	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
54	C₂²×C₃	#368	1	1	yes	characteristic
55	C₃×C₄	—	4	1	no	—
56	C₃×C₄	#335	1	1	yes	characteristic
57	C₃×C₄	#367	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	#807	1	1	yes	characteristic
60	C₈	Dic₃⋊C₈	1	1	yes	characteristic
61	C₈	#822	1	1	yes	characteristic
62	C₈	#827	1	1	yes	characteristic
63	C₈	#828	1	1	yes	characteristic
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	#1372	1	1	yes	characteristic
66	C₂²	#4493	1	1	yes	characteristic
67	C₄	—	4	1	no	—
68	C₄	#4479	1	1	yes	characteristic
69	C₄	#4490	1	1	yes	characteristic
70	C₃	#7464	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	1	no	—
72	C₂	#67470	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4490
commutator	C₃×C₁₆	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₃
Fitting	C₃×#1373	C₂
socle	C₂×C₃	#1372
prime radical	C₂²×C₃	#368
cosocle	C₃×C₄×C₁₆	C₂²
hypercenter	#1373	D₃
hypocenter	C₃	#7464
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
70. C₃	#7464	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1373	2
#67508	2
#67502	2
#7464	3