Subgroups — C₉×#135 #63537

Order 288.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#41	C₂	1	2	yes	maximal, complemented
2	C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₉×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#135	C₃	1	1	yes	maximal, characteristic
7	Q₈×C₉	C₂²	1	2	yes	—
8	D₄×C₉	—	2	2	no	—
9	D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₄×C₉	—	2	2	no	—
11	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₈×C₉	C₂²	1	1	yes	characteristic
13	C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×#41	C₂×C₃	1	2	yes	—
15	C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
16	C₃×SD₁₆	C₂×C₃	1	2	yes	—
17	C₃×D₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₉	—	2	2	no	—
20	C₂²×C₉	D₄	1	1	yes	characteristic
21	C₄×C₉	—	2	2	no	—
22	C₄×C₉	D₄	1	1	yes	characteristic
23	C₄×C₉	C₂³	1	1	yes	characteristic
24	#135	C₉	1	1	yes	characteristic, direct product factor, Sylow
25	C₃×Q₈	C₂²×C₃	1	2	yes	—
26	C₃×D₄	—	2	2	no	—
27	C₃×D₄	C₂²×C₃	1	2	yes	—
28	C₂×C₃×C₄	—	2	2	no	—
29	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₃×C₈	C₂²×C₃	1	1	yes	characteristic
31	C₃×C₈	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₉	—	2	1	no	—
33	C₂×C₉	—	4	2	no	—
34	C₂×C₉	C₂×D₄	1	1	yes	characteristic
35	#41	C₂×C₉	1	2	yes	complemented
36	Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
37	SD₁₆	C₂×C₉	1	2	yes	complemented
38	D₈	C₂×C₉	1	1	yes	characteristic, complemented
39	C₂×C₈	C₂×C₉	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	2	2	no	—
41	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
42	C₃×C₄	—	2	2	no	—
43	C₃×C₄	C₃×D₄	1	1	yes	characteristic
44	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
45	C₉	#135	1	1	yes	characteristic, direct product factor, Sylow
46	Q₈	C₂²×C₉	1	2	yes	—
47	D₄	—	2	2	no	—
48	D₄	C₂²×C₉	1	2	yes	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
51	C₈	C₂²×C₉	1	1	yes	characteristic
52	C₈	C₂²×C₉	1	1	yes	characteristic
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	4	2	no	—
55	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
56	C₂²	—	2	2	no	—
57	C₂²	D₄×C₉	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	D₄×C₉	1	1	yes	characteristic
60	C₄	C₂³×C₉	1	1	yes	characteristic
61	C₃	C₃×#135	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	2	no	—
64	C₂	C₂×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	D₄
commutator	C₄	C₂³×C₉
Frattini, cosocle	C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₂×C₃×D₄
prime radical	C₃×#135	C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#41	C₂	semidirect
2. C₉×Q₁₆	C₂	semidirect
3. C₉×SD₁₆	C₂	semidirect
4. C₉×D₈	C₂	semidirect
5. C₂×C₈×C₉	C₂	semidirect
24. #135	C₉	direct (N × H)
35. #41	C₂×C₉	semidirect
36. Q₁₆	C₂×C₉	semidirect
37. SD₁₆	C₂×C₉	semidirect
38. D₈	C₂×C₉	semidirect
39. C₂×C₈	C₂×C₉	semidirect
45. C₉	#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#41	2
C₉×Q₁₆	2
C₉×SD₁₆	2
C₉×D₈	2
C₂×C₈×C₉	2
C₃×#135	3

Subgroups — C9×#135 #63537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#135 #63537