Subgroups — #1004

Order 96.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁴
Fitting	C₂×C₃×D₄	C₂
hypercenter	C₂×D₄	D₃
hypocenter	C₃	H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄	C₂	semidirect
2. C₂×#67	C₂	semidirect
3. #237	C₂	semidirect
4. D₃×D₄	C₂	semidirect
5. #235	C₂	semidirect
7. C₂³×C₃	C₂²	semidirect
8. C₂²×D₃	C₂²	semidirect
9. C₃×D₄	C₂²	semidirect
10. C₂×C₃×C₄	C₂²	semidirect
11. #67	C₂²	semidirect
12. #67	C₂²	semidirect
13. C₂×Dic₃	C₂²	semidirect
14. D₁₂	C₂²	semidirect
15. C₄×D₃	C₂²	semidirect
16. Dic₆	C₂²	semidirect
21. C₂×D₄	D₃	semidirect
30. C₂³	C₂×D₃	semidirect
35. D₄	C₂×D₃	semidirect
38. C₂×C₄	C₂×D₃	semidirect
50. C₃	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₄	2
C₂×#67	2
#237	2
D₃×D₄	2
#235	2
H(5,𝔽₂)	3