Subgroups — C₃×C₉×#121 #11721898

Order 864.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₄×C₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃²×#121	C₃	1	1	yes	maximal, characteristic
7	C₉×#121	C₃	1	3	yes	maximal, direct product factor
8	C₂³×C₃×C₉	C₂²	1	2	yes	—
9	C₂³×C₃×C₉	C₂²	1	1	yes	characteristic
10	C₃×D₄×C₉	—	2	4	no	—
11	C₃×D₄×C₉	—	2	2	no	—
12	C₂×C₃×C₄×C₉	—	2	2	no	—
13	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₉	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₃²×D₄	C₂×C₃	1	2	yes	—
17	C₂×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
21	C₂×D₄×C₉	C₂×C₃	1	6	yes	complemented
22	C₂×D₄×C₉	C₂×C₃	1	3	yes	complemented
23	C₂²×C₄×C₉	C₂×C₃	1	3	yes	complemented
24	C₉×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
25	C₉×C₂²⋊C₄	C₂×C₃	1	6	yes	complemented
26	C₂²×C₃×C₉	—	2	1	no	—
27	C₂²×C₃×C₉	—	2	1	no	—
28	C₂²×C₃×C₉	—	2	2	no	—
29	C₂²×C₃×C₉	—	2	2	no	—
30	C₂²×C₃×C₉	—	2	2	no	—
31	C₂²×C₃×C₉	D₄	1	2	yes	complemented
32	C₂²×C₃×C₉	C₂³	1	1	yes	characteristic
33	C₃×C₄×C₉	—	2	1	no	—
34	C₃×C₄×C₉	—	2	2	no	—
35	C₃×C₄×C₉	D₄	1	2	yes	complemented
36	C₃×#121	C₉	1	3	yes	direct product factor
37	C₃×#121	C₃²	1	1	yes	characteristic
38	C₂³×C₃²	C₂²×C₃	1	2	yes	—
39	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
40	C₃²×D₄	—	2	4	no	—
41	C₃²×D₄	—	2	2	no	—
42	C₂×C₃²×C₄	—	2	2	no	—
43	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₉	C₂²×C₃	1	6	yes	—
47	C₂³×C₉	C₂²×C₃	1	3	yes	—
48	D₄×C₉	—	2	12	no	—
49	D₄×C₉	—	2	6	no	—
50	C₂×C₄×C₉	—	2	6	no	—
51	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
52	C₂×C₄×C₉	C₂²×C₃	1	6	yes	—
53	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
54	C₂×C₃×C₉	—	2	2	no	—
55	C₂×C₃×C₉	—	4	2	no	—
56	C₂×C₃×C₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₉	#41	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂×C₉	1	6	yes	complemented
60	C₂×C₃×D₄	C₂×C₉	1	3	yes	complemented
61	C₂×C₃×D₄	C₂×C₃²	1	2	yes	—
62	C₂×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×C₉	1	3	yes	complemented
64	C₂²×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
65	C₃×C₄⋊C₄	C₂×C₉	1	3	yes	complemented
66	C₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
67	C₃×C₂²⋊C₄	C₂×C₉	1	6	yes	complemented
68	C₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
69	C₂²×C₃²	—	2	1	no	—
70	C₂²×C₃²	—	2	1	no	—
71	C₂²×C₃²	—	2	2	no	—
72	C₂²×C₃²	—	2	2	no	—
73	C₂²×C₃²	—	2	2	no	—
74	C₂²×C₃²	C₃×D₄	1	2	yes	—
75	C₂²×C₃²	C₂³×C₃	1	1	yes	characteristic
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	—	2	2	no	—
78	C₃²×C₄	C₃×D₄	1	2	yes	—
79	C₂²×C₉	—	2	3	no	—
80	C₂²×C₉	—	2	3	no	—
81	C₂²×C₉	—	2	6	no	—
82	C₂²×C₉	—	2	6	no	—
83	C₂²×C₉	—	2	6	no	—
84	C₂²×C₉	C₃×D₄	1	6	yes	complemented
85	C₂²×C₉	C₂³×C₃	1	3	yes	—
86	C₄×C₉	—	2	3	no	—
87	C₄×C₉	—	2	6	no	—
88	C₄×C₉	C₃×D₄	1	6	yes	complemented
89	#121	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
90	C₃×C₉	#121	1	1	yes	characteristic, direct product factor, Sylow
91	C₂³×C₃	C₂²×C₉	1	6	yes	—
92	C₂³×C₃	C₂²×C₉	1	3	yes	—
93	C₂³×C₃	C₂²×C₃²	1	2	yes	—
94	C₂³×C₃	C₂²×C₃²	1	1	yes	characteristic
95	C₃×D₄	—	2	12	no	—
96	C₃×D₄	—	2	4	no	—
97	C₃×D₄	—	2	6	no	—
98	C₃×D₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	6	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	C₂²×C₉	1	3	yes	—
102	C₂×C₃×C₄	C₂²×C₉	1	6	yes	—
103	C₂×C₃×C₄	C₂²×C₉	1	3	yes	—
104	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²×C₃²	1	2	yes	—
106	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
107	C₂×C₃²	—	2	2	no	—
108	C₂×C₃²	—	4	2	no	—
109	C₂×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
110	C₂×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
111	C₂×C₃²	C₃×#41	1	1	yes	characteristic
112	C₂×C₉	—	2	6	no	—
113	C₂×C₉	—	4	6	no	—
114	C₂×C₉	C₂×C₃×D₄	1	3	yes	—
115	C₂×C₉	C₂×C₃×D₄	1	3	yes	—
116	C₂×C₉	C₃×#41	1	3	yes	—
117	C₂×D₄	C₂×C₃×C₉	1	2	yes	complemented
118	C₂×D₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
119	C₂²×C₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
120	C₄⋊C₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
121	C₂²⋊C₄	C₂×C₃×C₉	1	2	yes	complemented
122	C₂²×C₃	—	2	3	no	—
123	C₂²×C₃	—	2	3	no	—
124	C₂²×C₃	—	2	1	no	—
125	C₂²×C₃	—	2	1	no	—
126	C₂²×C₃	—	2	6	no	—
127	C₂²×C₃	—	2	6	no	—
128	C₂²×C₃	—	2	6	no	—
129	C₂²×C₃	—	2	2	no	—
130	C₂²×C₃	—	2	2	no	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	D₄×C₉	1	6	yes	complemented
133	C₂²×C₃	C₂³×C₉	1	3	yes	—
134	C₂²×C₃	C₃²×D₄	1	2	yes	—
135	C₂²×C₃	C₂³×C₃²	1	1	yes	characteristic
136	C₃×C₄	—	2	3	no	—
137	C₃×C₄	—	2	6	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	D₄×C₉	1	6	yes	complemented
141	C₃×C₄	C₃²×D₄	1	2	yes	—
142	C₃²	C₃×#121	1	1	yes	characteristic
143	C₉	C₃×#121	1	3	yes	direct product factor
144	C₂³	C₂²×C₃×C₉	1	2	yes	—
145	C₂³	C₂²×C₃×C₉	1	1	yes	characteristic
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₃×C₉	1	2	yes	—
151	C₂×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
152	C₂×C₃	—	2	6	no	—
153	C₂×C₃	—	2	2	no	—
154	C₂×C₃	—	4	6	no	—
155	C₂×C₃	—	4	2	no	—
156	C₂×C₃	C₂×D₄×C₉	1	3	yes	—
157	C₂×C₃	C₂×D₄×C₉	1	3	yes	—
158	C₂×C₃	C₉×#41	1	3	yes	—
159	C₂×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
160	C₂×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
161	C₂×C₃	C₃²×#41	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	C₃×D₄×C₉	1	2	yes	complemented
168	C₂²	C₂³×C₃×C₉	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	C₃×D₄×C₉	1	2	yes	complemented
172	C₃	C₉×#121	1	3	yes	direct product factor
173	C₃	C₃²×#121	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	4	2	no	—
176	C₂	C₂×C₃×D₄×C₉	1	1	yes	characteristic
177	C₂	C₂×C₃×D₄×C₉	1	1	yes	characteristic
178	C₂	C₃×C₉×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	C₂³
commutator	C₂²	C₂³×C₃×C₉
Frattini, cosocle	C₂²×C₃	C₂³×C₃²
socle	C₂²×C₃²	C₂³×C₃
prime radical	C₃²×#121	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄×C₉	C₂	semidirect
2. C₂×C₃×D₄×C₉	C₂	semidirect
3. C₂²×C₃×C₄×C₉	C₂	semidirect
4. C₃×C₉×C₄⋊C₄	C₂	semidirect
5. C₃×C₉×C₂²⋊C₄	C₂	semidirect
7. C₉×#121	C₃	direct (N × H)
21. C₂×D₄×C₉	C₂×C₃	semidirect
22. C₂×D₄×C₉	C₂×C₃	semidirect
23. C₂²×C₄×C₉	C₂×C₃	semidirect
24. C₉×C₄⋊C₄	C₂×C₃	semidirect
25. C₉×C₂²⋊C₄	C₂×C₃	semidirect
31. C₂²×C₃×C₉	D₄	semidirect
35. C₃×C₄×C₉	D₄	semidirect
36. C₃×#121	C₉	direct (N × H)
59. C₂×C₃×D₄	C₂×C₉	semidirect
60. C₂×C₃×D₄	C₂×C₉	semidirect
63. C₂²×C₃×C₄	C₂×C₉	semidirect
65. C₃×C₄⋊C₄	C₂×C₉	semidirect
67. C₃×C₂²⋊C₄	C₂×C₉	semidirect
84. C₂²×C₉	C₃×D₄	semidirect
88. C₄×C₉	C₃×D₄	semidirect
89. #121	C₃×C₉	direct (N × H)
90. C₃×C₉	#121	direct (N × H)
117. C₂×D₄	C₂×C₃×C₉	semidirect
118. C₂×D₄	C₂×C₃×C₉	semidirect
119. C₂²×C₄	C₂×C₃×C₉	semidirect
120. C₄⋊C₄	C₂×C₃×C₉	semidirect
121. C₂²⋊C₄	C₂×C₃×C₉	semidirect
132. C₂²×C₃	D₄×C₉	semidirect
140. C₃×C₄	D₄×C₉	semidirect
143. C₉	C₃×#121	direct (N × H)
167. C₂²	C₃×D₄×C₉	semidirect
171. C₄	C₃×D₄×C₉	semidirect
172. C₃	C₉×#121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₄×C₉	2
C₂²×C₃×C₄×C₉	2
C₃×C₉×C₄⋊C₄	2
C₃×C₉×C₂²⋊C₄	2
C₃²×#121	3
C₉×#121	3

Subgroups — C3×C9×#121 #11721898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#121 #11721898