Subgroups — C₉×#5143 #13143253

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	C₉×#951	C₂	1	2	yes	maximal
3	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#824	C₂	1	2	yes	maximal, complemented
5	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5143	C₃	1	1	yes	maximal, characteristic
7	C₉×#473	—	3	1	no	maximal
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₃×C₉×M₁₆	—	2	1	no	—
11	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	—	2	1	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×#216	—	2	2	no	—
15	C₉×#214	—	2	2	no	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₉×Dic₁₂	—	2	1	no	—
18	C₉×Dic₁₂	—	2	1	no	—
19	C₃²×#131	C₂×C₃	1	1	yes	characteristic
20	C₃×#951	C₂×C₃	1	2	yes	—
21	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₃×#824	C₂×C₃	1	2	yes	—
23	C₃×#819	C₂×C₃	1	1	yes	characteristic
24	C₉×#137	—	3	2	no	—
25	C₂×C₉×Q₁₆	—	3	1	no	—
26	C₉×#131	—	2	1	no	—
27	C₉×#131	D₃	1	1	yes	characteristic
28	C₉×#108	—	3	1	no	—
29	C₉×#101	—	3	2	no	—
30	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
31	C₃×D₄×C₉	—	2	1	no	—
32	C₃×D₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₉×Dic₆	—	2	2	no	—
38	C₃×C₈×C₉	—	2	1	no	—
39	C₃×C₈×C₉	D₄	1	1	yes	characteristic
40	C₃×C₈×C₉	D₄	1	1	yes	characteristic
41	C₉×C₃⋊C₈	—	2	2	no	—
42	C₃×#473	—	3	1	no	—
43	#5143	C₉	1	1	yes	characteristic, direct product factor
44	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
46	C₃²×M₁₆	—	2	1	no	—
47	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃²×C₈	—	2	1	no	—
49	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₃×#216	—	2	2	no	—
51	C₃×#214	—	2	2	no	—
52	C₃×#208	C₂²×C₃	1	2	yes	—
53	C₃×Dic₁₂	—	2	1	no	—
54	C₃×Dic₁₂	—	2	1	no	—
55	C₉×#41	—	2	1	no	—
56	C₉×#41	C₂×D₃	1	1	yes	characteristic
57	C₂×Q₈×C₉	—	3	2	no	—
58	C₉×Q₁₆	—	6	2	no	—
59	C₉×Q₁₆	—	6	1	no	—
60	C₉×Q₁₆	—	6	1	no	—
61	C₉×SD₁₆	—	6	2	no	—
62	C₉×M₁₆	—	2	2	no	—
63	C₉×M₁₆	—	2	1	no	—
64	C₉×M₁₆	—	2	1	no	—
65	C₉×M₁₆	—	3	2	no	—
66	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	2	2	no	—
68	C₂×C₈×C₉	—	2	1	no	—
69	C₂×C₈×C₉	—	2	1	no	—
70	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₉	—	2	1	no	—
72	C₂²×C₃×C₉	#41	1	1	yes	characteristic
73	C₃×C₄×C₉	—	2	1	no	—
74	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₉×Dic₃	—	4	2	no	—
77	C₃×#137	—	3	2	no	—
78	C₂×C₃×Q₁₆	—	3	1	no	—
79	C₃×#131	—	2	1	no	—
80	C₃×#131	C₂×C₉	1	1	yes	characteristic
81	C₃×#131	C₃×D₃	1	1	yes	characteristic
82	#951	C₂×C₉	1	2	yes	—
83	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×#108	—	3	1	no	—
85	C₃×#101	—	3	2	no	—
86	#824	C₂×C₉	1	2	yes	complemented
87	#819	C₂×C₉	1	1	yes	characteristic, complemented
88	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
89	C₃²×D₄	—	2	1	no	—
90	C₃²×D₄	C₃×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₃×Dic₃	—	2	2	no	—
94	C₃×Dic₆	—	2	2	no	—
95	C₃×Dic₆	—	2	2	no	—
96	C₃²×C₈	—	2	1	no	—
97	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
98	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
99	C₃×C₃⋊C₈	—	2	2	no	—
100	Q₈×C₉	—	2	1	no	—
101	Q₈×C₉	—	6	2	no	—
102	Q₈×C₉	—	6	2	no	—
103	Q₈×C₉	D₁₂	1	1	yes	characteristic
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	1	no	—
106	D₄×C₉	—	2	1	no	—
107	D₄×C₉	D₁₂	1	1	yes	characteristic
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	6	2	no	—
112	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	2	1	no	—
115	C₈×C₉	—	2	1	no	—
116	C₈×C₉	—	2	1	no	—
117	C₈×C₉	—	6	2	no	—
118	C₈×C₉	#67	1	1	yes	characteristic
119	C₈×C₉	#67	1	1	yes	characteristic
120	#473	—	3	1	no	Sylow
121	C₂×C₃×C₉	—	2	1	no	—
122	C₂×C₃×C₉	—	4	1	no	—
123	C₂×C₃×C₉	#121	1	1	yes	characteristic
124	C₃×#41	—	2	1	no	—
125	C₃×#41	C₂²×C₉	1	1	yes	characteristic
126	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂×C₃×Q₈	—	3	2	no	—
128	C₂×Dic₆	C₂²×C₉	1	2	yes	—
129	C₃×Q₁₆	—	6	2	no	—
130	C₃×Q₁₆	—	6	1	no	—
131	C₃×Q₁₆	—	6	1	no	—
132	C₃×SD₁₆	—	6	2	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	1	no	—
135	C₃×M₁₆	—	2	1	no	—
136	C₃×M₁₆	—	2	1	no	—
137	C₃×M₁₆	—	3	2	no	—
138	C₃×M₁₆	C₂²×C₉	1	1	yes	characteristic
139	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
146	#216	—	2	2	no	—
147	#214	—	2	2	no	—
148	#208	C₂²×C₉	1	2	yes	—
149	Dic₁₂	—	2	1	no	—
150	Dic₁₂	—	2	1	no	—
151	C₂²×C₃²	—	2	1	no	—
152	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
153	C₃²×C₄	—	2	1	no	—
154	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
155	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
156	C₃×Dic₃	—	4	2	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	1	no	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	#235	1	1	yes	characteristic
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	12	2	no	—
166	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
167	C₄×C₉	C₂×#67	1	1	yes	characteristic
168	#137	—	3	2	no	—
169	C₂×Q₁₆	—	3	1	no	—
170	#131	D₃×C₉	1	1	yes	characteristic
171	#108	—	3	1	no	—
172	#101	—	3	2	no	—
173	C₃×C₉	#473	1	1	yes	characteristic, complemented, Sylow
174	C₃×Q₈	—	2	1	no	—
175	C₃×Q₈	—	6	2	no	—
176	C₃×Q₈	—	6	2	no	—
177	C₃×Q₈	D₄×C₉	1	1	yes	characteristic
178	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	—	2	1	no	—
181	C₃×D₄	—	2	1	no	—
182	C₃×D₄	—	2	1	no	—
183	C₃×D₄	D₄×C₉	1	1	yes	characteristic
184	C₃×D₄	C₃×D₁₂	1	1	yes	characteristic
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
192	C₂×Dic₃	—	2	2	no	—
193	Dic₆	—	2	2	no	—
194	Dic₆	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃×C₈	D₄×C₉	1	1	yes	characteristic
202	C₃×C₈	D₄×C₉	1	1	yes	characteristic
203	C₃×C₈	C₃×#67	1	1	yes	characteristic
204	C₃×C₈	C₃×#67	1	1	yes	characteristic
205	C₃⋊C₈	—	2	2	no	—
206	C₂×C₃²	—	2	1	no	—
207	C₂×C₃²	—	4	1	no	—
208	C₂×C₃²	C₃×#121	1	1	yes	characteristic
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	4	1	no	—
212	C₂×C₉	—	4	2	no	—
213	C₂×C₉	—	4	1	no	—
214	C₂×C₉	#930	1	1	yes	characteristic
215	#41	C₂×D₃×C₉	1	1	yes	characteristic
216	C₂×Q₈	—	3	2	no	—
217	Q₁₆	—	6	2	no	—
218	Q₁₆	—	6	1	no	—
219	Q₁₆	—	6	1	no	—
220	SD₁₆	—	6	2	no	—
221	M₁₆	—	2	1	no	—
222	M₁₆	—	3	2	no	—
223	M₁₆	C₂×D₃×C₉	1	1	yes	characteristic
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	C₉×#41	1	1	yes	characteristic
231	C₂²×C₃	C₃×#235	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	12	2	no	—
238	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
239	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
240	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
242	Dic₃	—	4	2	no	—
243	C₃²	C₃×#473	1	1	yes	characteristic
244	C₉	—	2	1	no	—
245	C₉	#5143	1	1	yes	characteristic, direct product factor
246	Q₈	—	6	2	no	—
247	Q₈	—	6	2	no	—
248	Q₈	C₉×D₁₂	1	1	yes	characteristic
249	D₄	—	2	1	no	—
250	D₄	C₉×D₁₂	1	1	yes	characteristic
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	6	2	no	—
256	C₈	C₉×#67	1	1	yes	characteristic
257	C₈	C₉×#67	1	1	yes	characteristic
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	4	1	no	—
262	C₂×C₃	—	4	2	no	—
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	C₉×#121	1	1	yes	characteristic
266	C₂×C₃	C₃×#930	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	C₉×#235	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	12	2	no	—
271	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
272	C₄	C₂×C₉×#67	1	1	yes	characteristic
273	C₃	—	2	1	no	—
274	C₃	C₉×#473	1	1	yes	characteristic, complemented
275	C₃	C₃×#5143	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	4	1	no	—
278	C₂	C₉×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#930
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	C₃×#121
prime radical	C₃²×#41	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#131	D₃
hypocenter	C₃	C₉×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×Dic₁₂	C₂	semidirect
4. C₉×#824	C₂	semidirect
5. C₉×#819	C₂	semidirect
43. #5143	C₉	direct (N × H)
83. C₂×Dic₁₂	C₂×C₉	semidirect
86. #824	C₂×C₉	semidirect
87. #819	C₂×C₉	semidirect
173. C₃×C₉	#473	semidirect
245. C₉	#5143	direct (N × H)
274. C₃	C₉×#473	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#131	2
C₉×#951	2
C₂×C₉×Dic₁₂	2
C₉×#824	2
C₉×#819	2
C₃×#5143	3
C₉×#473	3

Subgroups — C9×#5143 #13143253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5143 #13143253