Subgroups — C₉×#5753 #13153408

Order 1728.

806 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#1010	C₂	1	2	yes	maximal, complemented
2	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#909	C₂	1	6	yes	maximal, complemented
4	C₂×C₉×Dic₁₂	C₂	1	3	yes	maximal, complemented
5	C₉×#904	C₂	1	3	yes	maximal, complemented
6	C₃×#5753	C₃	1	1	yes	maximal, characteristic
7	C₉×#578	—	3	1	no	maximal
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	D₃×Q₈×C₉	—	2	2	no	—
10	C₉×#237	—	2	6	no	—
11	C₉×#235	C₂²	1	6	yes	—
12	C₂×C₉×Dic₆	C₂²	1	6	yes	—
13	C₃×C₉×M₁₆	C₂²	1	3	yes	—
14	C₂×C₃×C₈×C₉	C₂²	1	3	yes	—
15	C₉×Dic₁₂	C₂²	1	3	yes	—
16	C₉×Dic₁₂	C₂²	1	6	yes	—
17	C₉×D₂₄	C₂²	1	1	yes	characteristic
18	C₉×#204	C₂²	1	6	yes	—
19	C₃×#1010	C₂×C₃	1	2	yes	—
20	C₃²×#131	C₂×C₃	1	1	yes	characteristic
21	C₃×#909	C₂×C₃	1	6	yes	—
22	C₂×C₃×Dic₁₂	C₂×C₃	1	3	yes	—
23	C₃×#904	C₂×C₃	1	3	yes	—
24	C₉×2₋¹⁺⁴	—	3	2	no	—
25	C₉×#137	—	3	6	no	—
26	C₉×#135	—	3	3	no	—
27	C₂×C₉×Q₁₆	—	3	3	no	—
28	C₉×#131	—	2	1	no	—
29	C₉×#131	D₃	1	1	yes	characteristic, complemented
30	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
31	C₃×D₄×C₉	D₄	1	3	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₉×#67	—	2	6	no	—
34	C₂×C₉×Dic₃	—	2	6	no	—
35	C₉×D₁₂	C₂³	1	2	yes	—
36	C₄×D₃×C₉	—	2	6	no	—
37	C₉×Dic₆	—	2	6	no	—
38	C₉×Dic₆	C₂³	1	6	yes	—
39	C₃×C₈×C₉	C₂³	1	3	yes	—
40	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
41	C₃×#578	—	3	1	no	—
42	#5753	C₉	1	1	yes	characteristic, direct product factor
43	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
44	C₃×D₃×Q₈	—	2	2	no	—
45	C₃×#237	—	2	6	no	—
46	C₃×#235	C₂²×C₃	1	6	yes	—
47	C₂×C₃×Dic₆	C₂²×C₃	1	6	yes	—
48	C₃²×M₁₆	C₂²×C₃	1	3	yes	—
49	C₂×C₃²×C₈	C₂²×C₃	1	3	yes	—
50	C₃×Dic₁₂	C₂²×C₃	1	3	yes	—
51	C₃×Dic₁₂	C₂²×C₃	1	6	yes	—
52	C₃×D₂₄	C₂²×C₃	1	1	yes	characteristic
53	C₃×#204	C₂²×C₃	1	6	yes	—
54	C₉×#41	—	2	1	no	—
55	C₉×#41	—	3	6	no	—
56	C₉×#41	—	6	6	no	—
57	C₉×#41	C₂×D₃	1	1	yes	characteristic
58	C₂×Q₈×C₉	—	3	6	no	—
59	C₂×Q₈×C₉	—	6	2	no	—
60	C₉×Q₁₆	—	3	3	no	—
61	C₉×Q₁₆	—	3	6	no	—
62	C₉×SD₁₆	—	3	6	no	—
63	C₉×D₈	—	3	1	no	—
64	C₉×M₁₆	—	2	3	no	—
65	C₉×M₁₆	C₂×D₃	1	3	yes	—
66	C₂×C₈×C₉	—	2	3	no	—
67	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
68	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
69	C₂×D₃×C₉	—	2	2	no	—
70	C₃×C₄×C₉	C₂×D₄	1	3	yes	—
71	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
72	C₉×Dic₃	—	2	6	no	—
73	C₃×2₋¹⁺⁴	—	3	2	no	—
74	#1010	C₂×C₉	1	2	yes	complemented
75	C₃×#137	—	3	6	no	—
76	C₃×#135	—	3	3	no	—
77	C₂×C₃×Q₁₆	—	3	3	no	—
78	C₃×#131	—	2	1	no	—
79	C₃×#131	C₂×C₉	1	1	yes	characteristic, complemented
80	C₃×#131	C₃×D₃	1	1	yes	characteristic
81	#909	C₂×C₉	1	6	yes	complemented
82	C₂×Dic₁₂	C₂×C₉	1	3	yes	complemented
83	#904	C₂×C₉	1	3	yes	complemented
84	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
85	C₃²×D₄	C₃×D₄	1	3	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
87	C₃×#67	—	2	6	no	—
88	C₂×C₃×Dic₃	—	2	6	no	—
89	C₃×D₁₂	C₂³×C₃	1	2	yes	—
90	C₃×C₄×D₃	—	2	6	no	—
91	C₃×Dic₆	—	2	6	no	—
92	C₃×Dic₆	C₂³×C₃	1	6	yes	—
93	C₃²×C₈	C₂³×C₃	1	3	yes	—
94	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
95	Q₈×C₉	—	2	1	no	—
96	Q₈×C₉	—	3	6	no	—
97	Q₈×C₉	—	6	6	no	—
98	Q₈×C₉	D₁₂	1	1	yes	characteristic
99	D₄×C₉	—	2	3	no	—
100	D₄×C₉	—	3	2	no	—
101	D₄×C₉	—	6	6	no	—
102	D₄×C₉	D₁₂	1	3	yes	—
103	C₂×C₄×C₉	—	2	3	no	—
104	C₂×C₄×C₉	—	6	6	no	—
105	C₂×C₄×C₉	—	6	6	no	—
106	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
107	C₈×C₉	—	2	1	no	—
108	C₈×C₉	—	2	3	no	—
109	C₈×C₉	C₂²×D₃	1	1	yes	characteristic
110	C₈×C₉	C₂²×D₃	1	3	yes	—
111	#578	—	3	1	no	Sylow
112	C₂×C₃×C₉	—	2	3	no	—
113	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
114	D₃×C₉	—	4	2	no	—
115	C₃×#41	—	2	1	no	—
116	C₃×#41	—	3	6	no	—
117	C₃×#41	—	6	6	no	—
118	C₃×#41	C₂²×C₉	1	1	yes	characteristic
119	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
120	C₂×C₃×Q₈	—	3	6	no	—
121	C₂×C₃×Q₈	—	6	2	no	—
122	D₃×Q₈	—	2	2	no	—
123	#237	—	2	6	no	—
124	#235	C₂²×C₉	1	6	yes	—
125	C₂×Dic₆	C₂²×C₉	1	6	yes	—
126	C₃×Q₁₆	—	3	3	no	—
127	C₃×Q₁₆	—	3	6	no	—
128	C₃×SD₁₆	—	3	6	no	—
129	C₃×D₈	—	3	1	no	—
130	C₃×M₁₆	—	2	3	no	—
131	C₃×M₁₆	C₂²×C₉	1	3	yes	—
132	C₃×M₁₆	C₂×C₃×D₃	1	3	yes	—
133	C₂×C₃×C₈	—	2	3	no	—
134	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
135	C₂×C₃×C₈	C₂×C₃×D₃	1	3	yes	—
136	Dic₁₂	C₂²×C₉	1	3	yes	—
137	Dic₁₂	C₂²×C₉	1	6	yes	—
138	D₂₄	C₂²×C₉	1	1	yes	characteristic
139	#204	C₂²×C₉	1	6	yes	—
140	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
141	C₂×C₃×D₃	—	2	2	no	—
142	C₃²×C₄	C₂×C₃×D₄	1	3	yes	—
143	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
144	C₃×Dic₃	—	2	6	no	—
145	C₂²×C₉	—	2	3	no	—
146	C₂²×C₉	—	6	2	no	—
147	C₂²×C₉	C₂×D₁₂	1	3	yes	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	3	no	—
150	C₄×C₉	—	6	6	no	—
151	C₄×C₉	C₂×D₁₂	1	3	yes	—
152	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
153	2₋¹⁺⁴	—	3	2	no	—
154	#137	—	3	6	no	—
155	#135	—	3	3	no	—
156	C₂×Q₁₆	—	3	3	no	—
157	#131	D₃×C₉	1	1	yes	characteristic, complemented
158	C₃×C₉	#578	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	2	1	no	—
160	C₃×Q₈	—	3	6	no	—
161	C₃×Q₈	—	6	6	no	—
162	C₃×Q₈	D₄×C₉	1	1	yes	characteristic
163	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
164	C₃×D₄	—	2	3	no	—
165	C₃×D₄	—	3	2	no	—
166	C₃×D₄	—	6	6	no	—
167	C₃×D₄	D₄×C₉	1	3	yes	—
168	C₃×D₄	C₃×D₁₂	1	3	yes	—
169	C₂×C₃×C₄	—	2	3	no	—
170	C₂×C₃×C₄	—	6	6	no	—
171	C₂×C₃×C₄	—	6	6	no	—
172	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
173	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
174	#67	—	2	6	no	—
175	C₂×Dic₃	—	2	6	no	—
176	D₁₂	C₂³×C₉	1	2	yes	—
177	C₄×D₃	—	2	6	no	—
178	Dic₆	—	2	6	no	—
179	Dic₆	C₂³×C₉	1	6	yes	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	2	3	no	—
182	C₃×C₈	C₂³×C₉	1	3	yes	—
183	C₃×C₈	C₂³×C₉	1	1	yes	characteristic
184	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₃×C₈	C₂²×C₃×D₃	1	3	yes	—
186	C₂×C₃²	—	2	3	no	—
187	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₃×D₃	—	4	2	no	—
189	C₂×C₉	—	2	3	no	—
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	4	3	no	—
192	C₂×C₉	—	12	2	no	—
193	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
194	#41	—	3	6	no	—
195	#41	—	6	6	no	—
196	#41	C₂×D₃×C₉	1	1	yes	characteristic
197	C₂×Q₈	—	3	6	no	—
198	C₂×Q₈	—	6	2	no	—
199	Q₁₆	—	3	3	no	—
200	Q₁₆	—	3	6	no	—
201	SD₁₆	—	3	6	no	—
202	D₈	—	3	1	no	—
203	M₁₆	C₂×D₃×C₉	1	3	yes	—
204	C₂×C₈	C₂×D₃×C₉	1	3	yes	—
205	C₂²×C₃	—	2	3	no	—
206	C₂²×C₃	—	6	2	no	—
207	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
208	C₂²×C₃	C₂×C₃×D₁₂	1	3	yes	—
209	C₂×D₃	—	2	2	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	3	no	—
212	C₃×C₄	—	6	6	no	—
213	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
214	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
215	C₃×C₄	C₂×C₃×D₁₂	1	3	yes	—
216	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
217	Dic₃	—	2	6	no	—
218	C₃²	C₃×#578	1	1	yes	characteristic
219	C₉	—	2	1	no	—
220	C₉	#5753	1	1	yes	characteristic, direct product factor
221	Q₈	—	3	6	no	—
222	Q₈	—	6	6	no	—
223	Q₈	C₉×D₁₂	1	1	yes	characteristic
224	D₄	—	3	2	no	—
225	D₄	—	6	6	no	—
226	D₄	C₉×D₁₂	1	3	yes	—
227	C₂×C₄	—	6	6	no	—
228	C₂×C₄	—	6	6	no	—
229	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
230	C₈	C₂²×D₃×C₉	1	1	yes	characteristic
231	C₈	C₂²×D₃×C₉	1	3	yes	—
232	C₂×C₃	—	2	3	no	—
233	C₂×C₃	—	2	3	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	4	3	no	—
236	C₂×C₃	—	12	2	no	—
237	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
238	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
239	D₃	—	4	2	no	—
240	C₂²	—	6	2	no	—
241	C₂²	C₂×C₉×D₁₂	1	3	yes	—
242	C₄	—	6	6	no	—
243	C₄	C₂×C₉×D₁₂	1	3	yes	—
244	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
245	C₃	—	2	1	no	—
246	C₃	C₉×#578	1	1	yes	characteristic, complemented
247	C₃	C₃×#5753	1	1	yes	characteristic
248	C₂	—	2	3	no	—
249	C₂	—	12	2	no	—
250	C₂	C₂²×C₉×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×D₁₂
commutator	C₃×C₄	C₂⁴×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	C₂²×C₃×D₄
prime radical	C₃×#5753	C₃
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₉×#131	D₃
hypocenter	C₃	C₉×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₉×#41 ◁ C₉×#131

Complements

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

N	H ≅ G/N	type
1. C₉×#1010	C₂	semidirect
2. C₃×C₉×#131	C₂	semidirect
3. C₉×#909	C₂	semidirect
4. C₂×C₉×Dic₁₂	C₂	semidirect
5. C₉×#904	C₂	semidirect
29. C₉×#131	D₃	semidirect
42. #5753	C₉	direct (N × H)
74. #1010	C₂×C₉	semidirect
79. C₃×#131	C₂×C₉	semidirect
81. #909	C₂×C₉	semidirect
82. C₂×Dic₁₂	C₂×C₉	semidirect
83. #904	C₂×C₉	semidirect
157. #131	D₃×C₉	semidirect
158. C₃×C₉	#578	semidirect
220. C₉	#5753	direct (N × H)
246. C₃	C₉×#578	semidirect

Maximal subgroups

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

subgroup	index
C₉×#1010	2
C₃×C₉×#131	2
C₉×#909	2
C₂×C₉×Dic₁₂	2
C₉×#904	2
C₃×#5753	3
C₉×#578	3

Subgroups — C9×#5753 #13153408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5753 #13153408