Subgroups — #5819

Order 192.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	#948	C₂	1	8	yes	maximal, complemented
3	C₂×#208	C₂	1	6	yes	maximal, complemented
4	#568	—	3	1	no	maximal, Sylow
5	C₃×#41	C₄	1	4	yes	—
6	C₃×#41	C₂²	1	4	yes	—
7	C₂×C₃×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₃×D₄	C₄	1	3	yes	—
9	C₂²×C₃×C₄	C₂²	1	3	yes	—
10	#208	C₂²	1	4	yes	—
11	#208	C₂²	1	12	yes	complemented
12	C₂×C₃⋊C₈	C₂²	1	12	yes	complemented
13	C₂×#41	D₃	1	1	yes	characteristic
14	#131	—	3	8	no	—
15	C₂×M₁₆	—	3	6	no	—
16	C₂³×C₃	C₂×C₄	1	3	yes	—
17	C₃×Q₈	C₂×C₄	1	4	yes	—
18	C₃×D₄	C₂×C₄	1	12	yes	—
19	C₂×C₃×C₄	C₂×C₄	1	3	yes	—
20	C₂×C₃×C₄	C₂×C₄	1	6	yes	—
21	C₂×C₃×C₄	C₂³	1	6	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₃⋊C₈	C₂³	1	8	yes	—
24	#41	Dic₃	1	4	yes	—
25	#41	C₂×D₃	1	4	yes	—
26	C₂×Q₈	Dic₃	1	1	yes	characteristic
27	C₂×D₄	Dic₃	1	3	yes	—
28	C₂²×C₄	C₂×D₃	1	3	yes	—
29	M₁₆	—	3	4	no	—
30	M₁₆	—	3	12	no	—
31	C₂×C₈	—	3	12	no	—
32	C₂²×C₃	—	4	3	no	—
33	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₃	C₂²×C₄	1	6	yes	—
35	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₃×C₄	C₂²×C₄	1	6	yes	—
37	C₃×C₄	C₂⁴	1	1	yes	characteristic
38	C₂³	C₂×Dic₃	1	3	yes	—
39	Q₈	C₂×Dic₃	1	4	yes	—
40	D₄	C₂×Dic₃	1	12	yes	—
41	C₂×C₄	C₂×Dic₃	1	3	yes	—
42	C₂×C₄	C₂×Dic₃	1	6	yes	—
43	C₂×C₄	C₂²×D₃	1	6	yes	—
44	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
45	C₈	—	3	8	no	—
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	—	2	6	no	—
48	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
49	C₂²	—	4	3	no	—
50	C₂²	C₂²×Dic₃	1	1	yes	characteristic
51	C₂²	C₂²×Dic₃	1	6	yes	—
52	C₄	C₂²×Dic₃	1	1	yes	characteristic
53	C₄	C₂²×Dic₃	1	6	yes	—
54	C₄	C₂³×D₃	1	1	yes	characteristic
55	C₃	#568	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₃
commutator, socle	C₂×C₃	C₂³×C₄
Fitting, prime radical	C₂×C₃×#41	C₂
cosocle	C₃×C₄	C₂⁴
hypercenter	C₂×#41	D₃
hypocenter	C₃	#568
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #948	C₂	semidirect
3. C₂×#208	C₂	semidirect
11. #208	C₂²	semidirect
12. C₂×C₃⋊C₈	C₂²	semidirect
55. C₃	#568	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#41	2
#948	2
C₂×#208	2
#568	3