Subgroups — C₉×#1797 #11766518

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₉×#336	C₂	1	2	yes	maximal, complemented
5	C₃×#1797	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	4	yes	—
10	C₂×C₄²×C₉	—	2	4	no	—
11	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊C₈	—	2	4	no	—
16	C₉×C₂²⋊C₈	—	2	4	no	—
17	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₈	C₂×C₃	1	2	yes	—
19	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	2	yes	—
20	C₃×#336	C₂×C₃	1	2	yes	—
21	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₉	—	2	2	no	—
23	C₂²×C₄×C₉	—	2	2	no	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	—	2	2	no	—
29	C₂²×C₄×C₉	C₈	1	4	yes	complemented
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	2	yes	—
36	C₂²×C₄×C₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₉	—	2	4	no	—
39	C₂×C₈×C₉	—	2	4	no	—
40	C₂×C₈×C₉	—	2	4	no	—
41	C₄²×C₉	—	2	2	no	—
42	C₄²×C₉	—	2	2	no	—
43	C₄²×C₉	—	2	2	no	—
44	C₄²×C₉	D₄	1	4	yes	—
45	#1797	C₉	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
47	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
48	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
50	C₂×C₃×C₄²	—	2	4	no	—
51	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	4	no	—
56	C₃×C₂²⋊C₈	—	2	4	no	—
57	C₂³×C₉	—	2	1	no	—
58	C₂³×C₉	—	2	1	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₄⋊C₄	1	2	yes	—
62	C₂³×C₉	C₂×C₈	1	2	yes	—
63	C₂³×C₉	M₁₆	1	2	yes	—
64	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	—	2	8	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	2	8	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
81	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
82	C₂×C₄×C₉	M₁₆	1	2	yes	—
83	C₂×C₄×C₉	M₁₆	1	2	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₉	#41	1	2	yes	—
88	C₈×C₉	—	4	4	no	—
89	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
90	C₂×C₄⋊C₈	C₂×C₉	1	2	yes	complemented
91	C₂×C₂²⋊C₈	C₂×C₉	1	2	yes	—
92	#336	C₂×C₉	1	2	yes	complemented
93	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	2	2	no	—
101	C₂²×C₃×C₄	C₃×C₈	1	4	yes	—
102	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
108	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₃×C₄²	—	2	2	no	—
116	C₃×C₄²	C₃×D₄	1	4	yes	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	2	no	—
119	C₂²×C₉	—	2	2	no	—
120	C₂²×C₉	—	2	2	no	—
121	C₂²×C₉	—	2	2	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	C₄⋊C₈	1	4	yes	complemented
124	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₉	#117	1	1	yes	characteristic
127	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
129	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	C₂²⋊C₈	1	4	yes	complemented
136	C₄×C₉	#121	1	2	yes	—
137	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
138	C₂³×C₄	C₄×C₉	1	1	yes	characteristic
139	C₂³×C₄	C₄×C₉	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₉	1	4	yes	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₄×C₉	1	1	yes	characteristic
144	C₂×C₄²	C₄×C₉	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
147	C₄⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	—	2	1	no	—
153	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
154	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
155	C₂³×C₃	C₃×M₁₆	1	2	yes	—
156	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
170	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
174	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
175	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
179	C₂×C₃×C₄	C₃×#41	1	2	yes	—
180	C₃×C₈	—	4	4	no	—
181	C₂×C₉	—	2	4	no	—
182	C₂×C₉	#380	1	1	yes	characteristic
183	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₂×C₉	#407	1	1	yes	characteristic
185	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
186	C₂×C₉	#423	1	1	yes	characteristic
187	C₂×C₉	#431	1	1	yes	characteristic
188	C₂×C₉	#432	1	1	yes	characteristic
189	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₈×C₉	1	4	yes	complemented
198	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
203	C₂²×C₄	D₄×C₉	1	2	yes	—
204	C₂²×C₄	Q₈×C₉	1	2	yes	—
205	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	D₄×C₉	1	4	yes	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	C₃×C₄⋊C₈	1	4	yes	—
220	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
222	C₂²×C₃	C₃×#117	1	1	yes	characteristic
223	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
225	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	C₃×C₂²⋊C₈	1	4	yes	—
232	C₃×C₄	C₃×#121	1	2	yes	—
233	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
234	C₉	#1797	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₉×C₄⋊C₄	1	2	yes	—
240	C₂³	C₂×C₈×C₉	1	2	yes	—
241	C₂³	C₉×M₁₆	1	2	yes	—
242	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
258	C₂×C₄	C₂×C₈×C₉	1	2	yes	—
259	C₂×C₄	C₂×C₈×C₉	1	2	yes	—
260	C₂×C₄	C₉×M₁₆	1	2	yes	—
261	C₂×C₄	C₉×M₁₆	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
264	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
265	C₂×C₄	C₉×#41	1	2	yes	—
266	C₈	—	4	4	no	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	C₃×#380	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
270	C₂×C₃	C₃×#407	1	1	yes	characteristic
271	C₂×C₃	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
272	C₂×C₃	C₃×#423	1	1	yes	characteristic
273	C₂×C₃	C₃×#431	1	1	yes	characteristic
274	C₂×C₃	C₃×#432	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	C₉×C₄⋊C₈	1	4	yes	complemented
282	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
284	C₂²	C₉×#117	1	1	yes	characteristic
285	C₂²	C₂²×C₈×C₉	1	1	yes	characteristic
286	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
287	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
288	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	C₉×C₂²⋊C₈	1	4	yes	complemented
294	C₄	C₉×#121	1	2	yes	—
295	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
296	C₃	C₃×#1797	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₉×#380	1	1	yes	characteristic
299	C₂	C₂×C₉×C₂²⋊C₈	1	1	yes	characteristic
300	C₂	C₉×#407	1	1	yes	characteristic
301	C₂	C₂×C₉×C₄⋊C₈	1	1	yes	characteristic
302	C₂	C₉×#423	1	1	yes	characteristic
303	C₂	C₉×#431	1	1	yes	characteristic
304	C₂	C₉×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₈×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
4. C₉×#336	C₂	semidirect
29. C₂²×C₄×C₉	C₈	semidirect
45. #1797	C₉	direct (N × H)
90. C₂×C₄⋊C₈	C₂×C₉	semidirect
92. #336	C₂×C₉	semidirect
123. C₂²×C₉	C₄⋊C₈	semidirect
135. C₄×C₉	C₂²⋊C₈	semidirect
197. C₂²×C₄	C₈×C₉	semidirect
234. C₉	#1797	direct (N × H)
281. C₂²	C₉×C₄⋊C₈	semidirect
293. C₄	C₉×C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₉	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×C₂²⋊C₈	2
C₉×#336	2
C₃×#1797	3

Subgroups — C9×#1797 #11766518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1797 #11766518