Subgroups — C₉×#121 #63523

Order 288.

141 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₉	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#121	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₉	C₂²	1	2	yes	—
8	C₂³×C₉	C₂²	1	1	yes	characteristic
9	D₄×C₉	—	2	4	no	—
10	D₄×C₉	—	2	2	no	—
11	C₂×C₄×C₉	—	2	2	no	—
12	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₉	C₂²	1	2	yes	—
14	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₄	C₂×C₃	1	2	yes	—
16	C₂×C₃×D₄	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
20	C₂²×C₉	—	2	1	no	—
21	C₂²×C₉	—	2	1	no	—
22	C₂²×C₉	—	2	2	no	—
23	C₂²×C₉	—	2	2	no	—
24	C₂²×C₉	—	2	2	no	—
25	C₂²×C₉	D₄	1	2	yes	complemented
26	C₂²×C₉	C₂³	1	1	yes	characteristic
27	C₄×C₉	—	2	1	no	—
28	C₄×C₉	—	2	2	no	—
29	C₄×C₉	D₄	1	2	yes	complemented
30	#121	C₉	1	1	yes	characteristic, direct product factor, Sylow
31	C₂³×C₃	C₂²×C₃	1	2	yes	—
32	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
33	C₃×D₄	—	2	4	no	—
34	C₃×D₄	—	2	2	no	—
35	C₂×C₃×C₄	—	2	2	no	—
36	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂²×C₃	1	2	yes	—
38	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₉	—	2	2	no	—
40	C₂×C₉	—	4	2	no	—
41	C₂×C₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₉	#41	1	1	yes	characteristic
44	C₂×D₄	C₂×C₉	1	2	yes	complemented
45	C₂×D₄	C₂×C₉	1	1	yes	characteristic, complemented
46	C₂²×C₄	C₂×C₉	1	1	yes	characteristic, complemented
47	C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	C₂×C₉	1	2	yes	complemented
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	—	2	1	no	—
51	C₂²×C₃	—	2	2	no	—
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	—	2	2	no	—
54	C₂²×C₃	C₃×D₄	1	2	yes	—
55	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	—	2	2	no	—
58	C₃×C₄	C₃×D₄	1	2	yes	—
59	C₉	#121	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³	C₂²×C₉	1	2	yes	—
61	C₂³	C₂²×C₉	1	1	yes	characteristic
62	D₄	—	2	4	no	—
63	D₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₉	1	2	yes	—
67	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	—	4	2	no	—
70	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
71	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
72	C₂×C₃	C₃×#41	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	D₄×C₉	1	2	yes	complemented
79	C₂²	C₂³×C₉	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	D₄×C₉	1	2	yes	complemented
83	C₃	C₃×#121	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	—	4	2	no	—
86	C₂	C₂×D₄×C₉	1	1	yes	characteristic
87	C₂	C₂×D₄×C₉	1	1	yes	characteristic
88	C₂	C₉×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³
commutator	C₂²	C₂³×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃
prime radical	C₃×#121	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₉	C₂	semidirect
2. C₂×D₄×C₉	C₂	semidirect
3. C₂²×C₄×C₉	C₂	semidirect
4. C₉×C₄⋊C₄	C₂	semidirect
5. C₉×C₂²⋊C₄	C₂	semidirect
25. C₂²×C₉	D₄	semidirect
29. C₄×C₉	D₄	semidirect
30. #121	C₉	direct (N × H)
44. C₂×D₄	C₂×C₉	semidirect
45. C₂×D₄	C₂×C₉	semidirect
46. C₂²×C₄	C₂×C₉	semidirect
47. C₄⋊C₄	C₂×C₉	semidirect
48. C₂²⋊C₄	C₂×C₉	semidirect
59. C₉	#121	direct (N × H)
78. C₂²	D₄×C₉	semidirect
82. C₄	D₄×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₉	2
C₂²×C₄×C₉	2
C₉×C₄⋊C₄	2
C₉×C₂²⋊C₄	2
C₃×#121	3

Subgroups — C9×#121 #63523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#121 #63523