Subgroups — C₉×#4751 #13141691

Order 1728.

488 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#239	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#836	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#823	C₂	1	2	yes	maximal, complemented
6	C₃×#4751	C₃	1	1	yes	maximal, characteristic
7	C₉×#413	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₉×#239	—	2	2	no	—
10	C₉×#239	—	2	2	no	—
11	C₂×C₉×D₁₂	C₄	1	2	yes	—
12	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃×C₉	C₄	1	2	yes	—
14	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₉×M₁₆	C₂²	1	2	yes	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
18	D₃×C₈×C₉	—	2	2	no	—
19	C₂×C₃×#239	C₂×C₃	1	1	yes	characteristic
20	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
21	C₂×C₉×#41	—	3	1	no	—
22	C₃²×#101	C₂×C₃	1	1	yes	characteristic
23	C₃×#836	C₂×C₃	1	1	yes	characteristic
24	C₃×#823	C₂×C₃	1	2	yes	—
25	C₂×C₉×M₁₆	—	3	2	no	—
26	C₉×#101	—	2	1	no	—
27	C₉×#101	—	3	1	no	—
28	C₉×#101	D₃	1	1	yes	characteristic, complemented
29	C₉×#100	—	3	2	no	—
30	C₂²×D₃×C₉	C₂×C₄	1	2	yes	—
31	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
32	C₃×Q₈×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₉×D₁₂	—	2	4	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₄×D₃×C₉	—	2	2	no	—
39	C₄×D₃×C₉	—	2	2	no	—
40	C₄×D₃×C₉	D₄	1	2	yes	—
41	C₄×D₃×C₉	D₄	1	2	yes	—
42	C₃×C₈×C₉	—	2	2	no	—
43	C₉×C₃⋊C₈	—	2	2	no	—
44	C₃×#413	—	3	1	no	—
45	#4751	C₉	1	1	yes	characteristic, direct product factor
46	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
47	C₃×#239	—	2	2	no	—
48	C₃×#239	—	2	2	no	—
49	C₂×C₃×D₁₂	C₃×C₄	1	2	yes	—
50	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄×D₃	C₃×C₄	1	2	yes	—
52	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
53	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×#208	C₂²×C₃	1	2	yes	—
55	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
56	C₃×D₃×C₈	—	2	2	no	—
57	C₉×#41	—	6	2	no	—
58	C₉×#41	—	6	2	no	—
59	C₂×Q₈×C₉	—	2	1	no	—
60	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂×D₄×C₉	—	3	2	no	—
62	C₂×D₄×C₉	—	3	1	no	—
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₂²×C₄×C₉	—	3	2	no	—
65	C₉×M₁₆	—	2	2	no	—
66	C₉×M₁₆	—	3	2	no	—
67	C₉×M₁₆	—	6	2	no	—
68	C₉×M₁₆	C₂×D₃	1	2	yes	—
69	C₂×C₈×C₉	—	6	2	no	—
70	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×D₃×C₉	—	2	2	no	—
72	C₂×D₃×C₉	—	4	1	no	—
73	C₂×D₃×C₉	—	4	2	no	—
74	C₂×D₃×C₉	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄×C₉	—	2	2	no	—
76	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₉×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₉×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×#41	—	3	1	no	—
80	C₂×#239	C₂×C₉	1	1	yes	characteristic
81	C₂×C₃×M₁₆	—	3	2	no	—
82	D₃×M₁₆	C₂×C₉	1	2	yes	complemented
83	C₃×#101	—	2	1	no	—
84	C₃×#101	—	3	1	no	—
85	C₃×#101	C₂×C₉	1	1	yes	characteristic, complemented
86	C₃×#101	C₃×D₃	1	1	yes	characteristic
87	C₃×#100	—	3	2	no	—
88	#836	C₂×C₉	1	1	yes	characteristic, complemented
89	#823	C₂×C₉	1	2	yes	complemented
90	C₂²×C₃×D₃	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₃²×Q₈	—	2	2	no	—
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
96	C₃×D₁₂	—	2	4	no	—
97	C₃×D₁₂	—	2	2	no	—
98	C₃×C₄×D₃	—	2	2	no	—
99	C₃×C₄×D₃	—	2	2	no	—
100	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
101	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
102	C₂³×C₉	—	3	1	no	—
103	C₂³×C₉	—	3	2	no	—
104	C₃²×C₈	—	2	2	no	—
105	C₃×C₃⋊C₈	—	2	2	no	—
106	Q₈×C₉	—	2	2	no	—
107	Q₈×C₉	—	4	2	no	—
108	D₄×C₉	—	6	4	no	—
109	D₄×C₉	—	6	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	3	1	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	—	3	2	no	—
115	C₂×C₄×C₉	—	6	2	no	—
116	C₂×C₄×C₉	—	6	2	no	—
117	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
118	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
119	C₈×C₉	—	2	2	no	—
120	C₈×C₉	—	4	2	no	—
121	C₈×C₉	—	6	2	no	—
122	#413	—	3	1	no	Sylow
123	C₂×C₃×C₉	—	2	1	no	—
124	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	D₃×C₉	—	2	2	no	—
126	D₃×C₉	—	4	2	no	—
127	C₃×#41	—	6	2	no	—
128	C₃×#41	—	6	2	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
131	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂×C₃×D₄	—	3	2	no	—
133	C₂×C₃×D₄	—	3	1	no	—
134	C₂²×C₃×C₄	—	3	1	no	—
135	C₂²×C₃×C₄	—	3	2	no	—
136	#239	—	2	2	no	—
137	#239	—	2	2	no	—
138	C₂×D₁₂	C₄×C₉	1	2	yes	—
139	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₄×D₃	C₄×C₉	1	2	yes	—
141	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
142	C₃×M₁₆	—	2	2	no	—
143	C₃×M₁₆	—	3	2	no	—
144	C₃×M₁₆	—	6	2	no	—
145	C₃×M₁₆	C₂²×C₉	1	2	yes	—
146	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
147	C₂×C₃×C₈	—	6	2	no	—
148	#208	C₂²×C₉	1	2	yes	—
149	ΓL(1,𝔽₂₅)	—	2	2	no	—
150	D₃×C₈	—	2	2	no	—
151	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₃×D₃	—	2	2	no	—
153	C₂×C₃×D₃	—	4	1	no	—
154	C₂×C₃×D₃	—	4	2	no	—
155	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	2	yes	—
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
158	C₃×Dic₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₃×Dic₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	3	2	no	—
162	C₂²×C₉	—	6	2	no	—
163	C₂²×C₉	—	12	1	no	—
164	C₂²×C₉	—	12	2	no	—
165	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	3	1	no	—
169	C₄×C₉	—	3	1	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	D₃×D₄	1	2	yes	—
172	C₂×#41	—	3	1	no	—
173	C₂×M₁₆	—	3	2	no	—
174	#101	—	3	1	no	—
175	#101	D₃×C₉	1	1	yes	characteristic, complemented
176	#100	—	3	2	no	—
177	C₃×C₉	#413	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	3	1	no	—
179	C₂³×C₃	—	3	2	no	—
180	C₂²×D₃	C₂×C₄×C₉	1	2	yes	—
181	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
182	C₃×Q₈	—	2	2	no	—
183	C₃×Q₈	—	2	2	no	—
184	C₃×Q₈	—	4	2	no	—
185	C₃×D₄	—	6	4	no	—
186	C₃×D₄	—	6	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	2	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
195	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
196	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
197	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
198	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
199	D₁₂	—	2	4	no	—
200	D₁₂	—	2	2	no	—
201	C₄×D₃	—	2	2	no	—
202	C₄×D₃	—	2	2	no	—
203	C₄×D₃	D₄×C₉	1	2	yes	—
204	C₄×D₃	D₄×C₉	1	2	yes	—
205	C₃×C₈	—	2	2	no	—
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	4	2	no	—
208	C₃×C₈	—	6	2	no	—
209	C₃⋊C₈	—	2	2	no	—
210	C₂×C₃²	—	2	1	no	—
211	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₃×D₃	—	2	2	no	—
213	C₃×D₃	—	4	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	4	1	no	—
217	C₂×C₉	—	6	2	no	—
218	C₂×C₉	—	12	2	no	—
219	C₂×C₉	D₃×C₂²⋊C₄	1	1	yes	characteristic
220	#41	—	6	2	no	—
221	#41	—	6	2	no	—
222	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
223	C₂×D₄	—	3	2	no	—
224	C₂×D₄	—	3	1	no	—
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	3	2	no	—
227	M₁₆	—	3	2	no	—
228	M₁₆	—	6	2	no	—
229	M₁₆	C₂×D₃×C₉	1	2	yes	—
230	C₂×C₈	—	6	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	3	2	no	—
233	C₂²×C₃	—	6	2	no	—
234	C₂²×C₃	—	12	1	no	—
235	C₂²×C₃	—	12	2	no	—
236	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
238	C₂×D₃	—	2	2	no	—
239	C₂×D₃	—	4	1	no	—
240	C₂×D₃	—	4	2	no	—
241	C₂×D₃	C₉×C₂²⋊C₄	1	2	yes	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	3	1	no	—
246	C₃×C₄	—	3	1	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
249	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
250	Dic₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
251	Dic₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
252	C₃²	C₃×#413	1	1	yes	characteristic
253	C₉	—	2	1	no	—
254	C₉	#4751	1	1	yes	characteristic, direct product factor
255	C₂³	—	3	1	no	—
256	C₂³	—	3	2	no	—
257	Q₈	—	2	2	no	—
258	D₄	—	6	4	no	—
259	D₄	—	6	2	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	6	2	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
266	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	6	2	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	—	6	2	no	—
274	C₂×C₃	—	12	2	no	—
275	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
277	D₃	—	2	2	no	—
278	D₃	—	4	2	no	—
279	C₂²	—	3	2	no	—
280	C₂²	—	6	2	no	—
281	C₂²	—	12	1	no	—
282	C₂²	—	12	2	no	—
283	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	3	1	no	—
286	C₄	—	3	1	no	—
287	C₄	D₃×D₄×C₉	1	2	yes	—
288	C₃	—	2	1	no	—
289	C₃	C₉×#413	1	1	yes	characteristic, complemented
290	C₃	C₃×#4751	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	6	2	no	—
293	C₂	—	12	2	no	—
294	C₂	D₃×C₉×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#101	C₂
socle	C₂×C₃²	C₂×C₃×C₂²⋊C₄
prime radical	C₂×C₃×#239	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#101	D₃
hypocenter	C₃	C₉×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#101

Complements

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

N	H ≅ G/N	type
2. D₃×C₉×M₁₆	C₂	semidirect
3. C₃×C₉×#101	C₂	semidirect
4. C₉×#836	C₂	semidirect
5. C₉×#823	C₂	semidirect
28. C₉×#101	D₃	semidirect
45. #4751	C₉	direct (N × H)
82. D₃×M₁₆	C₂×C₉	semidirect
85. C₃×#101	C₂×C₉	semidirect
88. #836	C₂×C₉	semidirect
89. #823	C₂×C₉	semidirect
175. #101	D₃×C₉	semidirect
177. C₃×C₉	#413	semidirect
254. C₉	#4751	direct (N × H)
289. C₃	C₉×#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#239	2
D₃×C₉×M₁₆	2
C₃×C₉×#101	2
C₉×#836	2
C₉×#823	2
C₃×#4751	3
C₉×#413	3

Subgroups — C9×#4751 #13141691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4751 #13141691