Subgroups — #5243

Order 192.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#935	C₂	1	3	yes	maximal, complemented
3	#834	C₂	1	3	yes	maximal, complemented
4	#458	—	3	1	no	maximal, Sylow
5	C₃×#41	—	2	3	no	—
6	C₂×C₃×D₄	—	2	3	no	—
7	C₂×C₃×D₄	C₂²	1	3	yes	—
8	C₂²×Dic₃	C₂²	1	1	yes	characteristic, complemented
9	#217	—	2	3	no	—
10	#217	C₂²	1	3	yes	complemented
11	Dic₃⋊C₄	—	2	3	no	—
12	H(5,𝔽₂)	D₃	1	1	yes	characteristic
13	#123	—	3	3	no	—
14	#99	—	3	3	no	—
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	D₄	1	3	yes	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₃×Q₈	—	2	1	no	—
19	C₃×D₄	—	2	3	no	—
20	C₃×D₄	—	2	6	no	—
21	C₂×C₃×C₄	—	2	3	no	—
22	C₂×C₃×C₄	D₄	1	3	yes	—
23	C₂×Dic₃	—	2	3	no	—
24	C₂×Dic₃	—	2	3	no	—
25	#41	—	2	3	no	—
26	C₂×D₄	—	2	3	no	—
27	C₂×D₄	C₂×D₃	1	3	yes	—
28	C₂²×C₄	—	3	1	no	—
29	C₄⋊C₄	—	6	3	no	—
30	C₂²⋊C₄	—	3	3	no	—
31	C₂²⋊C₄	—	6	3	no	—
32	C₂²×C₃	—	2	3	no	—
33	C₂²×C₃	—	4	1	no	—
34	C₂²×C₃	—	4	3	no	—
35	C₂²×C₃	—	8	1	no	—
36	C₂²×C₃	C₂×D₄	1	3	yes	—
37	C₃×C₄	—	2	3	no	—
38	Dic₃	—	4	3	no	—
39	Dic₃	—	4	1	no	—
40	C₂³	—	2	1	no	—
41	C₂³	#67	1	3	yes	—
42	C₂³	C₂²×D₃	1	1	yes	characteristic
43	Q₈	—	2	1	no	—
44	D₄	—	2	3	no	—
45	D₄	—	2	6	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	6	3	no	—
48	C₂×C₄	—	6	3	no	—
49	C₂×C₄	#67	1	3	yes	—
50	C₂×C₃	—	2	3	no	—
51	C₂×C₃	—	4	3	no	—
52	C₂×C₃	#120	1	1	yes	characteristic
53	C₂²	—	2	3	no	—
54	C₂²	—	4	1	no	—
55	C₂²	—	4	3	no	—
56	C₂²	—	8	1	no	—
57	C₂²	C₂×#67	1	3	yes	—
58	C₄	—	2	3	no	—
59	C₄	—	12	3	no	—
60	C₄	—	12	1	no	—
61	C₃	#458	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#953
commutator, cosocle	C₂³×C₃	C₂³
Frattini	C₂³	C₂²×D₃
Fitting, prime radical	C₃×H(5,𝔽₂)	C₂
socle	C₂×C₃	#120
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	#458
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #935	C₂	semidirect
3. #834	C₂	semidirect
8. C₂²×Dic₃	C₂²	semidirect
10. #217	C₂²	semidirect
61. C₃	#458	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₂)	2
#935	2
#834	2
#458	3