Subgroups — #5966

Order 192.

810 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, complemented
2	#1010	C₂	1	6	yes	maximal, complemented
3	D₃×#41	C₂	1	6	yes	maximal, complemented
4	#1004	C₂	1	9	yes	maximal, complemented
5	C₂×#237	C₂	1	9	yes	maximal, complemented
6	#585	—	3	1	no	maximal, Sylow
7	C₃×#41	C₂²	1	6	yes	complemented
8	C₂×C₃×D₄	C₂²	1	9	yes	complemented
9	C₂×#67	C₂²	1	18	yes	complemented
10	C₂²×Dic₃	C₂²	1	6	yes	complemented
11	#239	C₂²	1	2	yes	complemented
12	D₃×Q₈	C₂²	1	6	yes	complemented
13	#237	C₂²	1	18	yes	complemented
14	#237	C₂²	1	36	yes	complemented
15	D₃×D₄	C₂²	1	18	yes	complemented
16	#235	C₂²	1	18	yes	complemented
17	C₂×C₄×D₃	C₂²	1	9	yes	complemented
18	C₂×Dic₆	C₂²	1	9	yes	complemented
19	2₋¹⁺⁴	—	3	6	no	—
20	H(5,𝔽₂)	—	3	9	no	—
21	H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
22	C₂×#41	—	3	6	no	—
23	C₂×#41	—	3	9	no	—
24	C₂³×C₃	C₂³	1	6	yes	—
25	C₂²×D₃	C₂³	1	9	yes	—
26	C₃×Q₈	C₂³	1	2	yes	—
27	C₃×D₄	C₂³	1	18	yes	—
28	C₂×C₃×C₄	C₂³	1	9	yes	—
29	#67	C₂³	1	36	yes	—
30	C₂×Dic₃	C₂³	1	18	yes	—
31	C₂×Dic₃	C₂³	1	9	yes	—
32	D₁₂	C₂³	1	6	yes	—
33	C₄×D₃	C₂³	1	6	yes	—
34	C₄×D₃	C₂³	1	18	yes	—
35	Dic₆	C₂³	1	18	yes	—
36	#41	—	3	2	no	—
37	#41	—	3	18	no	—
38	#41	—	3	36	no	—
39	#41	—	3	18	no	—
40	#41	C₂×D₃	1	6	yes	complemented
41	C₂×Q₈	—	3	6	no	—
42	C₂×Q₈	—	3	9	no	—
43	C₂×D₄	—	3	18	no	—
44	C₂×D₄	—	3	18	no	—
45	C₂×D₄	C₂×D₃	1	9	yes	complemented
46	C₂²×C₄	—	3	6	no	—
47	C₂²×C₄	—	3	9	no	—
48	C₂²×C₃	—	4	6	no	—
49	C₂²×C₃	C₂⁴	1	9	yes	—
50	C₂×D₃	—	4	9	no	—
51	C₂×D₃	C₂⁴	1	6	yes	—
52	C₃×C₄	C₂⁴	1	6	yes	—
53	Dic₃	C₂⁴	1	9	yes	—
54	Dic₃	C₂⁴	1	1	yes	characteristic
55	C₂³	—	3	9	no	—
56	C₂³	C₂²×D₃	1	6	yes	—
57	Q₈	—	3	18	no	—
58	Q₈	C₂²×D₃	1	2	yes	—
59	D₄	—	3	36	no	—
60	D₄	—	3	6	no	—
61	D₄	C₂²×D₃	1	18	yes	—
62	C₂×C₄	—	3	9	no	—
63	C₂×C₄	—	3	18	no	—
64	C₂×C₄	—	3	6	no	—
65	C₂×C₄	—	3	18	no	—
66	C₂×C₄	C₂²×D₃	1	9	yes	—
67	C₂×C₃	—	2	9	no	—
68	C₂×C₃	C₂⁵	1	1	yes	characteristic
69	D₃	—	2	6	no	—
70	C₂²	—	3	6	no	—
71	C₂²	—	4	6	no	—
72	C₂²	—	12	9	no	—
73	C₂²	C₂³×D₃	1	9	yes	—
74	C₄	—	3	1	no	—
75	C₄	—	3	9	no	—
76	C₄	C₂³×D₃	1	6	yes	—
77	C₃	#585	1	1	yes	characteristic, complemented, Sylow
78	C₂	—	2	9	no	—
79	C₂	—	6	6	no	—
80	C₂	C₂⁴×D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁵
Fitting	C₃×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	semidirect
2. #1010	C₂	semidirect
3. D₃×#41	C₂	semidirect
4. #1004	C₂	semidirect
5. C₂×#237	C₂	semidirect
7. C₃×#41	C₂²	semidirect
8. C₂×C₃×D₄	C₂²	semidirect
9. C₂×#67	C₂²	semidirect
10. C₂²×Dic₃	C₂²	semidirect
11. #239	C₂²	semidirect
12. D₃×Q₈	C₂²	semidirect
13. #237	C₂²	semidirect
14. #237	C₂²	semidirect
15. D₃×D₄	C₂²	semidirect
16. #235	C₂²	semidirect
17. C₂×C₄×D₃	C₂²	semidirect
18. C₂×Dic₆	C₂²	semidirect
21. H(5,𝔽₂)	D₃	semidirect
40. #41	C₂×D₃	semidirect
45. C₂×D₄	C₂×D₃	semidirect
77. 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×H(5,𝔽₂)	2
#1010	2
D₃×#41	2
#1004	2
C₂×#237	2
#585	3