Subgroups — C₉×#137 #63539

Order 288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×Q₁₆	C₂	1	2	yes	maximal, complemented
4	C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#137	C₃	1	1	yes	maximal, characteristic
7	Q₈×C₉	—	2	1	no	—
8	Q₈×C₉	C₂²	1	2	yes	—
9	Q₈×C₉	C₂²	1	1	yes	characteristic
10	D₄×C₉	—	2	1	no	—
11	D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₉	—	2	1	no	—
13	C₂×C₄×C₉	—	2	1	no	—
14	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₈×C₉	C₂²	1	2	yes	—
16	C₃×#41	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₃×Q₁₆	C₂×C₃	1	2	yes	—
19	C₃×SD₁₆	C₂×C₃	1	2	yes	—
20	C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₉	—	2	1	no	—
22	C₂²×C₉	D₄	1	1	yes	characteristic
23	C₄×C₉	—	2	1	no	—
24	C₄×C₉	—	2	2	no	—
25	C₄×C₉	D₄	1	1	yes	characteristic
26	C₄×C₉	C₂³	1	1	yes	characteristic
27	#137	C₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×Q₈	—	2	1	no	—
29	C₃×Q₈	C₂²×C₃	1	2	yes	—
30	C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
31	C₃×D₄	—	2	1	no	—
32	C₃×D₄	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₃×C₈	C₂²×C₃	1	2	yes	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	—	4	1	no	—
39	C₂×C₉	C₂×D₄	1	1	yes	characteristic
40	#41	C₂×C₉	1	1	yes	characteristic
41	C₂×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
42	Q₁₆	C₂×C₉	1	2	yes	complemented
43	SD₁₆	C₂×C₉	1	2	yes	complemented
44	M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
45	C₂²×C₃	—	2	1	no	—
46	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	C₃×D₄	1	1	yes	characteristic
50	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
51	C₉	#137	1	1	yes	characteristic, direct product factor, Sylow
52	Q₈	—	2	1	no	—
53	Q₈	C₂²×C₉	1	2	yes	—
54	Q₈	C₂²×C₉	1	1	yes	characteristic
55	D₄	—	2	1	no	—
56	D₄	C₂²×C₉	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
60	C₈	C₂²×C₉	1	2	yes	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
64	C₂²	—	2	1	no	—
65	C₂²	D₄×C₉	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2	2	no	—
68	C₄	D₄×C₉	1	1	yes	characteristic
69	C₄	C₂³×C₉	1	1	yes	characteristic
70	C₃	C₃×#137	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	4	1	no	—
73	C₂	C₂×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂×D₄
commutator	C₄	C₂³×C₉
Frattini, cosocle	C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₂×C₃×D₄
prime radical	C₃×#41	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×C₉	C₂	semidirect
3. C₉×Q₁₆	C₂	semidirect
4. C₉×SD₁₆	C₂	semidirect
5. C₉×M₁₆	C₂	semidirect
27. #137	C₉	direct (N × H)
41. C₂×Q₈	C₂×C₉	semidirect
42. Q₁₆	C₂×C₉	semidirect
43. SD₁₆	C₂×C₉	semidirect
44. M₁₆	C₂×C₉	semidirect
51. C₉	#137	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
C₂×Q₈×C₉	2
C₉×Q₁₆	2
C₉×SD₁₆	2
C₉×M₁₆	2
C₃×#137	3

Subgroups — C9×#137 #63539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#137 #63539