Subgroups — C₂²×#239 #5959

Order 192.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#239	C₂	1	24	yes	maximal, direct product factor
3	C₂²×D₁₂	C₂	1	3	yes	maximal, complemented
4	C₂²×C₄×D₃	C₂	1	3	yes	maximal, complemented
5	C₂²×#41	—	3	1	no	maximal, Sylow
6	C₂³×D₃	C₂²	1	3	yes	—
7	C₂×C₃×Q₈	C₂²	1	12	yes	complemented
8	C₂²×C₃×C₄	C₂²	1	3	yes	—
9	C₂²×Dic₃	C₂²	1	1	yes	characteristic
10	#239	C₂²	1	64	yes	direct product factor
11	C₂×D₁₂	C₂²	1	36	yes	complemented
12	C₂×C₄×D₃	C₂²	1	36	yes	complemented
13	C₂×#41	—	3	24	no	—
14	C₂²×Q₈	D₃	1	1	yes	characteristic, complemented
15	C₂²×D₄	—	3	3	no	—
16	C₂³×C₄	—	3	3	no	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₂²×D₃	—	2	12	no	—
19	C₂²×D₃	C₂³	1	18	yes	—
20	C₃×Q₈	C₂³	1	16	yes	complemented
21	C₂×C₃×C₄	C₂³	1	18	yes	—
22	C₂×Dic₃	C₂³	1	6	yes	—
23	D₁₂	C₂³	1	48	yes	complemented
24	C₄×D₃	C₂³	1	48	yes	complemented
25	C₂⁴	—	3	3	no	—
26	#41	—	3	64	no	—
27	C₂×Q₈	C₂×D₃	1	12	yes	complemented
28	C₂×D₄	—	3	36	no	—
29	C₂²×C₄	—	3	1	no	—
30	C₂²×C₄	—	3	36	no	—
31	C₂²×C₄	C₂×D₃	1	3	yes	—
32	C₂²×C₃	#41	1	4	yes	complemented
33	C₂²×C₃	C₂⁴	1	3	yes	—
34	C₂×D₃	—	2	36	no	—
35	C₂×D₃	C₂⁴	1	12	yes	—
36	C₃×C₄	C₂⁴	1	12	yes	—
37	Dic₃	C₂⁴	1	4	yes	—
38	C₂³	—	3	18	no	—
39	C₂³	—	6	12	no	—
40	C₂³	C₂²×D₃	1	1	yes	characteristic
41	Q₈	C₂²×D₃	1	16	yes	complemented
42	D₄	—	3	48	no	—
43	C₂×C₄	—	3	6	no	—
44	C₂×C₄	—	3	48	no	—
45	C₂×C₄	C₂²×D₃	1	18	yes	—
46	C₂×C₃	C₂×#41	1	6	yes	complemented
47	C₂×C₃	C₂⁵	1	1	yes	characteristic
48	D₃	—	2	12	no	—
49	C₂²	—	3	12	no	—
50	C₂²	—	6	36	no	—
51	C₂²	#239	1	4	yes	direct product factor
52	C₂²	C₂³×D₃	1	3	yes	—
53	C₄	—	3	4	no	—
54	C₄	C₂³×D₃	1	12	yes	—
55	C₃	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	6	12	no	—
57	C₂	C₂×#239	1	6	yes	direct product factor
58	C₂	C₂⁴×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁵
Frattini	C₂	C₂⁴×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
2. C₂×#239	C₂	direct (N × H)
3. C₂²×D₁₂	C₂	semidirect
4. C₂²×C₄×D₃	C₂	semidirect
7. C₂×C₃×Q₈	C₂²	semidirect
10. #239	C₂²	direct (N × H)
11. C₂×D₁₂	C₂²	semidirect
12. C₂×C₄×D₃	C₂²	semidirect
14. C₂²×Q₈	D₃	semidirect
20. C₃×Q₈	C₂³	semidirect
23. D₁₂	C₂³	semidirect
24. C₄×D₃	C₂³	semidirect
27. C₂×Q₈	C₂×D₃	semidirect
32. C₂²×C₃	#41	semidirect
41. Q₈	C₂²×D₃	semidirect
46. C₂×C₃	C₂×#41	semidirect
51. C₂²	#239	direct (N × H)
55. C₃	C₂²×#41	semidirect
57. C₂	C₂×#239	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂×#239	2
C₂²×D₁₂	2
C₂²×C₄×D₃	2
C₂²×#41	3

Subgroups — C22×#239 #5959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#239 #5959