Subgroups — C₂×C₃×C₉×#120 #13154545

Order 1728.

3310 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄×C₉	C₂	1	3	yes	maximal, complemented
3	C₃×C₉×#120	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂×C₃²×#120	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#120	C₃	1	3	yes	maximal, direct product factor
7	C₂⁴×C₃×C₉	—	2	12	no	—
8	C₂⁴×C₃×C₉	C₂²	1	3	yes	—
9	C₂⁴×C₃×C₉	C₂²	1	4	yes	complemented
10	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₃×D₄×C₉	—	2	12	no	—
12	C₂×C₃×D₄×C₉	C₂²	1	12	yes	complemented
13	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	complemented
14	C₃×C₉×C₂²⋊C₄	C₂²	1	12	yes	complemented
15	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃²×D₄	C₂×C₃	1	3	yes	—
17	C₂⁵×C₉	C₂×C₃	1	3	yes	complemented
18	C₃²×#120	C₂×C₃	1	8	yes	—
19	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
20	C₂²×D₄×C₉	C₂×C₃	1	9	yes	complemented
21	C₉×#120	C₂×C₃	1	24	yes	direct product factor
22	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	9	yes	complemented
23	C₂³×C₃×C₉	—	2	32	no	—
24	C₂³×C₃×C₉	—	2	12	no	—
25	C₂³×C₃×C₉	—	2	24	no	—
26	C₂³×C₃×C₉	—	2	6	no	—
27	C₂³×C₃×C₉	D₄	1	12	yes	complemented
28	C₂³×C₃×C₉	C₂³	1	6	yes	—
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂³×C₃×C₉	C₂³	1	2	yes	complemented
31	C₃×D₄×C₉	—	2	24	no	—
32	C₂×C₃×C₄×C₉	—	2	6	no	—
33	C₂×C₃×C₄×C₉	C₂³	1	6	yes	complemented
34	C₂×C₃×#120	C₉	1	3	yes	direct product factor
35	C₂×C₃×#120	C₃²	1	1	yes	characteristic
36	C₂⁴×C₃²	—	2	12	no	—
37	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
38	C₂⁴×C₃²	C₂²×C₃	1	4	yes	—
39	C₂⁴×C₃²	C₂²×C₃	1	3	yes	—
40	C₂×C₃²×D₄	—	2	12	no	—
41	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
43	C₂⁴×C₉	—	2	36	no	—
44	C₂⁴×C₉	C₂²×C₃	1	9	yes	—
45	C₂⁴×C₉	C₂²×C₃	1	12	yes	complemented
46	C₂⁴×C₉	C₂²×C₃	1	3	yes	complemented
47	C₃²×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
48	C₂×D₄×C₉	—	2	36	no	—
49	C₂×D₄×C₉	C₂²×C₃	1	36	yes	complemented
50	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	complemented
51	C₉×C₂²⋊C₄	C₂²×C₃	1	36	yes	complemented
52	C₂²×C₃×C₉	—	2	32	no	—
53	C₂²×C₃×C₉	—	2	24	no	—
54	C₂²×C₃×C₉	—	2	12	no	—
55	C₂²×C₃×C₉	—	2	6	no	—
56	C₂²×C₃×C₉	—	4	4	no	—
57	C₂²×C₃×C₉	C₂×D₄	1	6	yes	—
58	C₂²×C₃×C₉	C₂×D₄	1	12	yes	complemented
59	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	6	no	—
61	C₂⁵×C₃	C₂×C₉	1	3	yes	complemented
62	C₂⁵×C₃	C₂×C₃²	1	1	yes	characteristic
63	C₂²×C₃×D₄	C₂×C₉	1	9	yes	complemented
64	C₂²×C₃×D₄	C₂×C₃²	1	3	yes	—
65	C₃×#120	C₂×C₉	1	24	yes	direct product factor
66	C₃×#120	C₂×C₃²	1	8	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	9	yes	complemented
68	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
69	C₂³×C₃²	—	2	32	no	—
70	C₂³×C₃²	—	2	12	no	—
71	C₂³×C₃²	—	2	24	no	—
72	C₂³×C₃²	—	2	6	no	—
73	C₂³×C₃²	C₃×D₄	1	12	yes	—
74	C₂³×C₃²	C₂³×C₃	1	6	yes	—
75	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
76	C₂³×C₃²	C₂³×C₃	1	2	yes	—
77	C₃²×D₄	—	2	24	no	—
78	C₂×C₃²×C₄	—	2	6	no	—
79	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
80	C₂³×C₉	—	2	96	no	—
81	C₂³×C₉	—	2	36	no	—
82	C₂³×C₉	—	2	72	no	—
83	C₂³×C₉	—	2	18	no	—
84	C₂³×C₉	C₃×D₄	1	36	yes	complemented
85	C₂³×C₉	C₂³×C₃	1	18	yes	—
86	C₂³×C₉	C₂³×C₃	1	3	yes	—
87	C₂³×C₉	C₂³×C₃	1	6	yes	complemented
88	D₄×C₉	—	2	72	no	—
89	C₂×C₄×C₉	—	2	18	no	—
90	C₂×C₄×C₉	C₂³×C₃	1	18	yes	complemented
91	C₂×#120	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×C₃×C₉	—	2	12	no	—
93	C₂×C₃×C₉	—	4	2	no	—
94	C₂×C₃×C₉	#120	1	4	yes	direct product factor
95	C₂×C₃×C₉	C₂²×D₄	1	3	yes	—
96	C₂⁴×C₃	—	2	36	no	—
97	C₂⁴×C₃	—	2	12	no	—
98	C₂⁴×C₃	C₂²×C₉	1	9	yes	—
99	C₂⁴×C₃	C₂²×C₉	1	12	yes	complemented
100	C₂⁴×C₃	C₂²×C₉	1	3	yes	complemented
101	C₂⁴×C₃	C₂²×C₃²	1	3	yes	—
102	C₂⁴×C₃	C₂²×C₃²	1	4	yes	—
103	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	36	no	—
105	C₂×C₃×D₄	—	2	12	no	—
106	C₂×C₃×D₄	C₂²×C₉	1	36	yes	complemented
107	C₂×C₃×D₄	C₂²×C₃²	1	12	yes	—
108	C₂²×C₃×C₄	C₂²×C₉	1	9	yes	complemented
109	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
110	C₃×C₂²⋊C₄	C₂²×C₉	1	36	yes	complemented
111	C₃×C₂²⋊C₄	C₂²×C₃²	1	12	yes	—
112	C₂²×C₃²	—	2	32	no	—
113	C₂²×C₃²	—	2	24	no	—
114	C₂²×C₃²	—	2	12	no	—
115	C₂²×C₃²	—	2	6	no	—
116	C₂²×C₃²	—	4	4	no	—
117	C₂²×C₃²	C₂×C₃×D₄	1	6	yes	—
118	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
119	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
120	C₃²×C₄	—	2	6	no	—
121	C₂²×C₉	—	2	96	no	—
122	C₂²×C₉	—	2	72	no	—
123	C₂²×C₉	—	2	36	no	—
124	C₂²×C₉	—	2	18	no	—
125	C₂²×C₉	—	4	12	no	—
126	C₂²×C₉	C₂×C₃×D₄	1	18	yes	—
127	C₂²×C₉	C₂×C₃×D₄	1	36	yes	complemented
128	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
129	C₄×C₉	—	2	18	no	—
130	C₂⁵	C₂×C₃×C₉	1	1	yes	characteristic, complemented
131	C₂²×D₄	C₂×C₃×C₉	1	3	yes	complemented
132	#120	C₂×C₃×C₉	1	8	yes	direct product factor
133	C₂×C₂²⋊C₄	C₂×C₃×C₉	1	3	yes	complemented
134	C₃×C₉	C₂×#120	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₃	—	2	32	no	—
136	C₂³×C₃	—	2	96	no	—
137	C₂³×C₃	—	2	36	no	—
138	C₂³×C₃	—	2	72	no	—
139	C₂³×C₃	—	2	12	no	—
140	C₂³×C₃	—	2	24	no	—
141	C₂³×C₃	—	2	18	no	—
142	C₂³×C₃	—	2	6	no	—
143	C₂³×C₃	D₄×C₉	1	36	yes	complemented
144	C₂³×C₃	C₂³×C₉	1	18	yes	—
145	C₂³×C₃	C₂³×C₉	1	3	yes	—
146	C₂³×C₃	C₂³×C₉	1	6	yes	complemented
147	C₂³×C₃	C₃²×D₄	1	12	yes	—
148	C₂³×C₃	C₂³×C₃²	1	6	yes	—
149	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
150	C₂³×C₃	C₂³×C₃²	1	2	yes	—
151	C₃×D₄	—	2	72	no	—
152	C₃×D₄	—	2	24	no	—
153	C₂×C₃×C₄	—	2	18	no	—
154	C₂×C₃×C₄	—	2	6	no	—
155	C₂×C₃×C₄	C₂³×C₉	1	18	yes	complemented
156	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
157	C₂×C₃²	—	2	12	no	—
158	C₂×C₃²	—	4	2	no	—
159	C₂×C₃²	C₃×#120	1	4	yes	—
160	C₂×C₃²	C₂²×C₃×D₄	1	3	yes	—
161	C₂×C₉	—	2	36	no	—
162	C₂×C₉	—	4	6	no	—
163	C₂×C₉	C₃×#120	1	12	yes	direct product factor
164	C₂×C₉	C₂²×C₃×D₄	1	9	yes	—
165	C₂⁴	—	2	12	no	—
166	C₂⁴	C₂²×C₃×C₉	1	3	yes	—
167	C₂⁴	C₂²×C₃×C₉	1	4	yes	complemented
168	C₂⁴	C₂²×C₃×C₉	1	1	yes	characteristic, complemented
169	C₂×D₄	—	2	12	no	—
170	C₂×D₄	C₂²×C₃×C₉	1	12	yes	complemented
171	C₂²×C₄	C₂²×C₃×C₉	1	3	yes	complemented
172	C₂²⋊C₄	C₂²×C₃×C₉	1	12	yes	complemented
173	C₂²×C₃	—	2	32	no	—
174	C₂²×C₃	—	2	96	no	—
175	C₂²×C₃	—	2	72	no	—
176	C₂²×C₃	—	2	12	no	—
177	C₂²×C₃	—	2	24	no	—
178	C₂²×C₃	—	2	36	no	—
179	C₂²×C₃	—	2	6	no	—
180	C₂²×C₃	—	2	18	no	—
181	C₂²×C₃	—	4	12	no	—
182	C₂²×C₃	—	4	4	no	—
183	C₂²×C₃	C₂×D₄×C₉	1	18	yes	—
184	C₂²×C₃	C₂×D₄×C₉	1	36	yes	complemented
185	C₂²×C₃	C₂⁴×C₉	1	3	yes	—
186	C₂²×C₃	C₂×C₃²×D₄	1	6	yes	—
187	C₂²×C₃	C₂×C₃²×D₄	1	12	yes	—
188	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	2	18	no	—
191	C₃²	C₂×C₃×#120	1	1	yes	characteristic
192	C₉	C₂×C₃×#120	1	3	yes	direct product factor
193	C₂³	—	2	32	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	24	no	—
196	C₂³	—	2	6	no	—
197	C₂³	C₃×D₄×C₉	1	12	yes	complemented
198	C₂³	C₂³×C₃×C₉	1	6	yes	—
199	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
200	C₂³	C₂³×C₃×C₉	1	2	yes	complemented
201	D₄	—	2	24	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	C₂³×C₃×C₉	1	6	yes	complemented
204	C₂×C₃	—	2	36	no	—
205	C₂×C₃	—	2	12	no	—
206	C₂×C₃	—	4	6	no	—
207	C₂×C₃	—	4	2	no	—
208	C₂×C₃	C₉×#120	1	12	yes	direct product factor
209	C₂×C₃	C₂²×D₄×C₉	1	9	yes	—
210	C₂×C₃	C₃²×#120	1	4	yes	—
211	C₂×C₃	C₂²×C₃²×D₄	1	3	yes	—
212	C₂²	—	2	32	no	—
213	C₂²	—	2	24	no	—
214	C₂²	—	2	12	no	—
215	C₂²	—	2	6	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂×C₃×D₄×C₉	1	6	yes	—
218	C₂²	C₂×C₃×D₄×C₉	1	12	yes	complemented
219	C₂²	C₂⁴×C₃×C₉	1	1	yes	characteristic
220	C₄	—	2	6	no	—
221	C₃	C₂×C₉×#120	1	3	yes	direct product factor
222	C₃	C₂×C₃²×#120	1	1	yes	characteristic
223	C₂	—	2	12	no	—
224	C₂	—	4	2	no	—
225	C₂	C₃×C₉×#120	1	4	yes	direct product factor
226	C₂	C₂²×C₃×D₄×C₉	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂²	C₂⁴×C₃×C₉
Frattini, cosocle	C₂²×C₃	C₂⁴×C₃²
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂×C₃²×#120	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₃×C₉	C₂	semidirect
2. C₂²×C₃×D₄×C₉	C₂	semidirect
3. C₃×C₉×#120	C₂	direct (N × H)
4. C₂×C₃×C₉×C₂²⋊C₄	C₂	semidirect
6. C₂×C₉×#120	C₃	direct (N × H)
9. C₂⁴×C₃×C₉	C₂²	semidirect
10. C₂⁴×C₃×C₉	C₂²	semidirect
12. C₂×C₃×D₄×C₉	C₂²	semidirect
13. C₂²×C₃×C₄×C₉	C₂²	semidirect
14. C₃×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₂⁵×C₉	C₂×C₃	semidirect
20. C₂²×D₄×C₉	C₂×C₃	semidirect
21. C₉×#120	C₂×C₃	direct (N × H)
22. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
27. C₂³×C₃×C₉	D₄	semidirect
30. C₂³×C₃×C₉	C₂³	semidirect
33. C₂×C₃×C₄×C₉	C₂³	semidirect
34. C₂×C₃×#120	C₉	direct (N × H)
45. C₂⁴×C₉	C₂²×C₃	semidirect
46. C₂⁴×C₉	C₂²×C₃	semidirect
49. C₂×D₄×C₉	C₂²×C₃	semidirect
50. C₂²×C₄×C₉	C₂²×C₃	semidirect
51. C₉×C₂²⋊C₄	C₂²×C₃	semidirect
58. C₂²×C₃×C₉	C₂×D₄	semidirect
61. C₂⁵×C₃	C₂×C₉	semidirect
63. C₂²×C₃×D₄	C₂×C₉	semidirect
65. C₃×#120	C₂×C₉	direct (N × H)
67. C₂×C₃×C₂²⋊C₄	C₂×C₉	semidirect
84. C₂³×C₉	C₃×D₄	semidirect
87. C₂³×C₉	C₂³×C₃	semidirect
90. C₂×C₄×C₉	C₂³×C₃	semidirect
91. C₂×#120	C₃×C₉	direct (N × H)
94. C₂×C₃×C₉	#120	direct (N × H)
99. C₂⁴×C₃	C₂²×C₉	semidirect
100. C₂⁴×C₃	C₂²×C₉	semidirect
106. C₂×C₃×D₄	C₂²×C₉	semidirect
108. C₂²×C₃×C₄	C₂²×C₉	semidirect
110. C₃×C₂²⋊C₄	C₂²×C₉	semidirect
127. C₂²×C₉	C₂×C₃×D₄	semidirect
130. C₂⁵	C₂×C₃×C₉	semidirect
131. C₂²×D₄	C₂×C₃×C₉	semidirect
132. #120	C₂×C₃×C₉	direct (N × H)
133. C₂×C₂²⋊C₄	C₂×C₃×C₉	semidirect
134. C₃×C₉	C₂×#120	direct (N × H)
143. C₂³×C₃	D₄×C₉	semidirect
146. C₂³×C₃	C₂³×C₉	semidirect
155. C₂×C₃×C₄	C₂³×C₉	semidirect
163. C₂×C₉	C₃×#120	direct (N × H)
167. C₂⁴	C₂²×C₃×C₉	semidirect
168. C₂⁴	C₂²×C₃×C₉	semidirect
170. C₂×D₄	C₂²×C₃×C₉	semidirect
171. C₂²×C₄	C₂²×C₃×C₉	semidirect
172. C₂²⋊C₄	C₂²×C₃×C₉	semidirect
184. C₂²×C₃	C₂×D₄×C₉	semidirect
192. C₉	C₂×C₃×#120	direct (N × H)
197. C₂³	C₃×D₄×C₉	semidirect
200. C₂³	C₂³×C₃×C₉	semidirect
203. C₂×C₄	C₂³×C₃×C₉	semidirect
208. C₂×C₃	C₉×#120	direct (N × H)
218. C₂²	C₂×C₃×D₄×C₉	semidirect
221. C₃	C₂×C₉×#120	direct (N × H)
225. C₂	C₃×C₉×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×C₉	2
C₂²×C₃×D₄×C₉	2
C₃×C₉×#120	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₂×C₃²×#120	3
C₂×C₉×#120	3

Subgroups — C2×C3×C9×#120 #13154545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₉×#120 #13154545