Subgroups — #5968

Order 192.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#1009	C₂	1	10	yes	maximal, complemented
3	D₃×#41	C₂	1	10	yes	maximal, complemented
4	#1007	C₂	1	5	yes	maximal, complemented
5	C₂×#239	C₂	1	5	yes	maximal, complemented
6	#585	—	3	1	no	maximal, Sylow
7	C₃×#41	C₂²	1	10	yes	complemented
8	C₂×C₃×Q₈	C₂²	1	5	yes	complemented
9	#239	C₂²	1	10	yes	complemented
10	#239	C₂²	1	20	yes	complemented
11	D₃×Q₈	C₂²	1	10	yes	complemented
12	#237	C₂²	1	10	yes	complemented
13	D₃×D₄	C₂²	1	30	yes	complemented
14	#235	C₂²	1	30	yes	complemented
15	C₂×D₁₂	C₂²	1	15	yes	complemented
16	C₂×C₄×D₃	C₂²	1	15	yes	complemented
17	2₋¹⁺⁴	—	3	5	no	—
18	2₋¹⁺⁴	D₃	1	1	yes	characteristic, complemented
19	H(5,𝔽₂)	—	3	10	no	—
20	C₂×#41	—	3	10	no	—
21	C₂×#41	—	3	5	no	—
22	C₂²×D₃	C₂³	1	15	yes	—
23	C₃×Q₈	C₂³	1	10	yes	—
24	C₃×D₄	C₂³	1	10	yes	—
25	C₂×C₃×C₄	C₂³	1	15	yes	—
26	#67	C₂³	1	20	yes	—
27	C₂×Dic₃	C₂³	1	5	yes	—
28	D₁₂	C₂³	1	30	yes	—
29	C₄×D₃	C₂³	1	10	yes	—
30	C₄×D₃	C₂³	1	30	yes	—
31	Dic₆	C₂³	1	10	yes	—
32	#41	—	3	10	no	—
33	#41	—	3	20	no	—
34	#41	—	3	10	no	—
35	#41	—	3	30	no	—
36	#41	C₂×D₃	1	10	yes	complemented
37	C₂×Q₈	—	3	10	no	—
38	C₂×Q₈	C₂×D₃	1	5	yes	complemented
39	C₂×D₄	—	3	30	no	—
40	C₂×D₄	—	3	15	no	—
41	C₂²×C₄	—	3	15	no	—
42	C₂²×C₃	C₂⁴	1	5	yes	—
43	C₂×D₃	—	4	15	no	—
44	C₂×D₃	C₂⁴	1	10	yes	—
45	C₃×C₄	C₂⁴	1	10	yes	—
46	Dic₃	C₂⁴	1	5	yes	—
47	Dic₃	C₂⁴	1	1	yes	characteristic
48	C₂³	—	3	15	no	—
49	Q₈	—	3	10	no	—
50	Q₈	C₂²×D₃	1	10	yes	—
51	D₄	—	3	20	no	—
52	D₄	—	3	30	no	—
53	D₄	C₂²×D₃	1	10	yes	—
54	C₂×C₄	—	3	5	no	—
55	C₂×C₄	—	3	10	no	—
56	C₂×C₄	—	3	30	no	—
57	C₂×C₄	C₂²×D₃	1	15	yes	—
58	C₂×C₃	—	2	5	no	—
59	C₂×C₃	C₂⁵	1	1	yes	characteristic
60	D₃	—	2	10	no	—
61	C₂²	—	3	10	no	—
62	C₂²	—	12	15	no	—
63	C₂²	C₂³×D₃	1	5	yes	—
64	C₄	—	3	1	no	—
65	C₄	—	3	5	no	—
66	C₄	C₂³×D₃	1	10	yes	—
67	C₃	#585	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	5	no	—
69	C₂	—	6	10	no	—
70	C₂	C₂⁴×D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁵
Fitting	C₃×2₋¹⁺⁴	C₂
hypercenter	2₋¹⁺⁴	D₃
hypocenter	C₃	#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃×2₋¹⁺⁴	C₂	semidirect
2. #1009	C₂	semidirect
3. D₃×#41	C₂	semidirect
4. #1007	C₂	semidirect
5. C₂×#239	C₂	semidirect
7. C₃×#41	C₂²	semidirect
8. C₂×C₃×Q₈	C₂²	semidirect
9. #239	C₂²	semidirect
10. #239	C₂²	semidirect
11. D₃×Q₈	C₂²	semidirect
12. #237	C₂²	semidirect
13. D₃×D₄	C₂²	semidirect
14. #235	C₂²	semidirect
15. C₂×D₁₂	C₂²	semidirect
16. C₂×C₄×D₃	C₂²	semidirect
18. 2₋¹⁺⁴	D₃	semidirect
36. #41	C₂×D₃	semidirect
38. C₂×Q₈	C₂×D₃	semidirect
67. C₃	#585	semidirect

Maximal subgroups

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

subgroup	index
C₃×2₋¹⁺⁴	2
#1009	2
D₃×#41	2
#1007	2
C₂×#239	2
#585	3