Subgroups — #5065

Order 192.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic
2	#870	C₂	1	1	yes	maximal, characteristic
3	#808	C₂	1	2	yes	maximal
4	#807	C₂	1	2	yes	maximal
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#499	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊C₄	—	2	2	no	—
8	C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃	C₂²	1	2	yes	—
11	Dic₃⋊C₄	—	2	2	no	—
12	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#125	—	3	1	no	—
15	#125	D₃	1	1	yes	characteristic
16	#107	—	3	2	no	—
17	#106	—	3	2	no	—
18	C₄×C₈	—	3	1	no	—
19	C₂×C₃×C₄	—	2	2	no	—
20	C₂×C₃×C₄	D₄	1	2	yes	—
21	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×Dic₃	—	2	2	no	—
23	C₃⋊C₈	Q₈	1	2	yes	—
24	C₃⋊C₈	Q₈	1	2	yes	—
25	C₂×C₈	—	3	1	no	—
26	C₂×C₈	—	3	1	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	—	3	2	no	—
29	C₄⋊C₄	—	6	2	no	—
30	C₄⋊C₄	C₂×D₃	1	2	yes	—
31	C₄²	C₂×D₃	1	1	yes	characteristic
32	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
33	C₃×C₄	—	2	2	no	—
34	C₃×C₄	—	4	2	no	—
35	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
36	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
37	Dic₃	—	4	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	6	2	no	—
40	C₂×C₄	#67	1	2	yes	—
41	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
42	C₈	—	3	2	no	—
43	C₈	—	3	2	no	—
44	C₂×C₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₃	#135	1	2	yes	—
46	C₂²	C₂×#67	1	1	yes	characteristic
47	C₄	—	2	2	no	—
48	C₄	—	4	2	no	—
49	C₄	—	12	2	no	—
50	C₄	D₃×Q₈	1	1	yes	characteristic
51	C₄	D₃×Q₈	1	1	yes	characteristic
52	C₃	#499	1	1	yes	characteristic, complemented, Sylow
53	C₂	#944	1	1	yes	characteristic
54	C₂	#950	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#125	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	#125	D₃
hypocenter	C₃	#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
52. C₃	#499	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
#870	2
#808	2
#807	2
C₄×C₃⋊C₈	2
#499	3