Subgroups — C₉×#3197 #11767918

Order 1152.

627 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#525	C₂	1	2	yes	maximal, complemented
2	C₉×#485	C₂	1	2	yes	maximal, complemented
3	C₉×#482	C₂	1	2	yes	maximal, complemented
4	C₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#469	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#468	C₂	1	2	yes	maximal, complemented
7	C₉×#452	C₂	1	2	yes	maximal, complemented
8	C₉×#449	C₂	1	2	yes	maximal, complemented
9	C₉×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3197	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	2	yes	—
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₉×SD₁₆	C₂²	1	2	yes	complemented
15	C₂×C₉×D₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₉×#123	—	2	2	no	—
19	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₉×#121	C₂²	1	2	yes	—
22	C₉×#121	C₂²	1	2	yes	—
23	C₉×#120	—	2	2	no	—
24	C₄×D₄×C₉	—	2	2	no	—
25	C₄×D₄×C₉	—	2	2	no	—
26	C₉×#117	C₂²	1	2	yes	—
27	C₉×#107	C₂²	1	2	yes	complemented
28	C₉×#106	C₂²	1	1	yes	characteristic, complemented
29	C₉×#106	C₂²	1	1	yes	characteristic, complemented
30	C₉×#103	C₂²	1	2	yes	complemented
31	C₉×#103	C₂²	1	2	yes	complemented
32	C₉×#102	C₂²	1	2	yes	complemented
33	C₉×#102	C₂²	1	2	yes	complemented
34	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₃×#525	C₂×C₃	1	2	yes	—
37	C₃×#485	C₂×C₃	1	2	yes	—
38	C₃×#482	C₂×C₃	1	2	yes	—
39	C₃×#470	C₂×C₃	1	1	yes	characteristic
40	C₃×#469	C₂×C₃	1	1	yes	characteristic
41	C₃×#468	C₂×C₃	1	2	yes	—
42	C₃×#452	C₂×C₃	1	2	yes	—
43	C₃×#449	C₂×C₃	1	2	yes	—
44	C₃×#407	C₂×C₃	1	1	yes	characteristic
45	C₂⁴×C₉	D₄	1	1	yes	characteristic
46	C₉×#41	—	4	2	no	—
47	C₉×#41	—	4	2	no	—
48	C₂×Q₈×C₉	C₂³	1	2	yes	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	—	2	2	no	—
51	C₂×D₄×C₉	—	2	2	no	—
52	C₂×D₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	—	4	1	no	—
55	C₂²×C₄×C₉	—	4	1	no	—
56	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₉	D₄	1	2	yes	—
58	C₂²×C₄×C₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
60	C₉×Q₁₆	—	4	1	no	—
61	C₉×SD₁₆	—	4	2	no	—
62	C₉×D₈	—	4	1	no	—
63	C₉×M₁₆	—	4	1	no	—
64	C₉×M₁₆	—	4	1	no	—
65	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₉	C₂³	1	2	yes	—
67	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
68	C₉×C₄⋊C₄	—	2	2	no	—
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	C₂³	1	2	yes	—
71	C₉×C₄⋊C₄	C₂³	1	2	yes	—
72	C₉×C₂²⋊C₄	—	2	2	no	—
73	C₉×C₂²⋊C₄	—	2	2	no	—
74	C₉×C₂²⋊C₄	—	2	2	no	—
75	C₉×C₂²⋊C₄	—	2	2	no	—
76	C₉×C₂²⋊C₄	—	2	2	no	—
77	C₄²×C₉	—	2	2	no	—
78	#3197	C₉	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
80	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
83	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
86	C₃×#123	—	2	2	no	—
87	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
88	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#121	C₂²×C₃	1	2	yes	—
90	C₃×#121	C₂²×C₃	1	2	yes	—
91	C₃×#120	—	2	2	no	—
92	C₃×C₄×D₄	—	2	2	no	—
93	C₃×C₄×D₄	—	2	2	no	—
94	C₃×#117	C₂²×C₃	1	2	yes	—
95	C₃×#107	C₂²×C₃	1	2	yes	—
96	C₃×#106	C₂²×C₃	1	1	yes	characteristic
97	C₃×#106	C₂²×C₃	1	1	yes	characteristic
98	C₃×#103	C₂²×C₃	1	2	yes	—
99	C₃×#103	C₂²×C₃	1	2	yes	—
100	C₃×#102	C₂²×C₃	1	2	yes	—
101	C₃×#102	C₂²×C₃	1	2	yes	—
102	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
103	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
104	C₂³×C₉	—	2	2	no	—
105	C₂³×C₉	—	4	1	no	—
106	C₂³×C₉	—	4	2	no	—
107	C₂³×C₉	C₂×D₄	1	2	yes	—
108	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₉	—	2	2	no	—
110	Q₈×C₉	—	2	2	no	—
111	D₄×C₉	—	2	2	no	—
112	D₄×C₉	—	2	2	no	—
113	D₄×C₉	—	4	4	no	—
114	D₄×C₉	—	4	4	no	—
115	D₄×C₉	—	4	4	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	4	1	no	—
124	C₂×C₄×C₉	—	4	2	no	—
125	C₂×C₄×C₉	—	4	4	no	—
126	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
129	C₈×C₉	—	2	1	no	—
130	C₈×C₉	—	2	2	no	—
131	C₈×C₉	—	2	1	no	—
132	#525	C₂×C₉	1	2	yes	complemented
133	#485	C₂×C₉	1	2	yes	complemented
134	#482	C₂×C₉	1	2	yes	complemented
135	#470	C₂×C₉	1	1	yes	characteristic, complemented
136	#469	C₂×C₉	1	1	yes	characteristic, complemented
137	#468	C₂×C₉	1	2	yes	complemented
138	#452	C₂×C₉	1	2	yes	complemented
139	#449	C₂×C₉	1	2	yes	complemented
140	#407	C₂×C₉	1	1	yes	characteristic, complemented
141	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
142	C₃×#41	—	4	2	no	—
143	C₃×#41	—	4	2	no	—
144	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
149	C₂²×C₃×C₄	—	2	2	no	—
150	C₂²×C₃×C₄	—	4	1	no	—
151	C₂²×C₃×C₄	—	4	1	no	—
152	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
154	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
155	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
156	C₃×Q₁₆	—	4	1	no	—
157	C₃×SD₁₆	—	4	2	no	—
158	C₃×D₈	—	4	1	no	—
159	C₃×M₁₆	—	4	1	no	—
160	C₃×M₁₆	—	4	1	no	—
161	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
162	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
163	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
167	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
168	C₃×C₂²⋊C₄	—	2	2	no	—
169	C₃×C₂²⋊C₄	—	2	2	no	—
170	C₃×C₂²⋊C₄	—	2	2	no	—
171	C₃×C₂²⋊C₄	—	2	2	no	—
172	C₃×C₂²⋊C₄	—	2	2	no	—
173	C₃×C₄²	—	2	2	no	—
174	C₂²×C₉	—	2	2	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	2	2	no	—
178	C₂²×C₉	—	2	2	no	—
179	C₂²×C₉	—	4	4	no	—
180	C₂²×C₉	—	4	2	no	—
181	C₂²×C₉	—	4	2	no	—
182	C₂²×C₉	—	4	2	no	—
183	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	4	2	no	—
187	C₄×C₉	—	4	2	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
190	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×#41	C₂²×C₉	1	2	yes	—
192	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
193	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic, complemented
194	C₂×SD₁₆	C₂²×C₉	1	2	yes	complemented
195	C₂×D₈	C₂²×C₉	1	1	yes	characteristic, complemented
196	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
197	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
198	#123	—	2	2	no	—
199	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
200	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
201	#121	C₂²×C₉	1	2	yes	—
202	#121	C₂²×C₉	1	2	yes	—
203	#120	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	#117	C₂²×C₉	1	2	yes	—
207	#107	C₂²×C₉	1	2	yes	complemented
208	#106	C₂²×C₉	1	1	yes	characteristic, complemented
209	#106	C₂²×C₉	1	1	yes	characteristic, complemented
210	#103	C₂²×C₉	1	2	yes	complemented
211	#103	C₂²×C₉	1	2	yes	complemented
212	#102	C₂²×C₉	1	2	yes	complemented
213	#102	C₂²×C₉	1	2	yes	complemented
214	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
215	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
216	C₂³×C₃	—	2	2	no	—
217	C₂³×C₃	—	4	1	no	—
218	C₂³×C₃	—	4	2	no	—
219	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
221	C₃×Q₈	—	2	2	no	—
222	C₃×Q₈	—	2	2	no	—
223	C₃×D₄	—	2	2	no	—
224	C₃×D₄	—	2	2	no	—
225	C₃×D₄	—	4	4	no	—
226	C₃×D₄	—	4	4	no	—
227	C₃×D₄	—	4	4	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	—	4	1	no	—
236	C₂×C₃×C₄	—	4	2	no	—
237	C₂×C₃×C₄	—	4	4	no	—
238	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	2	2	no	—
243	C₃×C₈	—	2	1	no	—
244	C₂×C₉	—	4	2	no	—
245	C₂×C₉	—	4	1	no	—
246	C₂×C₉	—	8	2	no	—
247	C₂×C₉	#534	1	1	yes	characteristic
248	C₂×C₉	#575	1	2	yes	—
249	C₂⁴	D₄×C₉	1	1	yes	characteristic
250	#41	—	4	2	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	C₂³×C₉	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	C₂³×C₉	1	2	yes	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
261	C₂²×C₄	D₄×C₉	1	2	yes	—
262	C₂²×C₄	C₂³×C₉	1	2	yes	—
263	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
264	Q₁₆	—	4	1	no	—
265	SD₁₆	—	4	2	no	—
266	D₈	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×C₉	1	2	yes	—
271	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₂³×C₉	1	2	yes	—
275	C₄⋊C₄	C₂³×C₉	1	2	yes	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₂²×C₃	—	2	2	no	—
283	C₂²×C₃	—	2	2	no	—
284	C₂²×C₃	—	2	2	no	—
285	C₂²×C₃	—	2	2	no	—
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	4	4	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	—	4	2	no	—
290	C₂²×C₃	—	4	2	no	—
291	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	—	4	2	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
299	C₉	#3197	1	1	yes	characteristic, direct product factor, Sylow
300	C₂³	—	2	2	no	—
301	C₂³	—	4	1	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂×D₄×C₉	1	2	yes	—
304	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
323	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
324	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₂×C₃	—	4	1	no	—
329	C₂×C₃	—	4	2	no	—
330	C₂×C₃	—	8	2	no	—
331	C₂×C₃	C₃×#534	1	1	yes	characteristic
332	C₂×C₃	C₃×#575	1	2	yes	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
349	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
350	C₃	C₃×#3197	1	1	yes	characteristic
351	C₂	—	4	2	no	—
352	C₂	—	4	1	no	—
353	C₂	—	8	2	no	—
354	C₂	C₉×#534	1	1	yes	characteristic
355	C₂	C₉×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3197	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#525	C₂	semidirect
2. C₉×#485	C₂	semidirect
3. C₉×#482	C₂	semidirect
4. C₉×#470	C₂	semidirect
5. C₉×#469	C₂	semidirect
6. C₉×#468	C₂	semidirect
7. C₉×#452	C₂	semidirect
8. C₉×#449	C₂	semidirect
9. C₉×#407	C₂	semidirect
13. C₂×C₉×Q₁₆	C₂²	semidirect
14. C₂×C₉×SD₁₆	C₂²	semidirect
15. C₂×C₉×D₈	C₂²	semidirect
27. C₉×#107	C₂²	semidirect
28. C₉×#106	C₂²	semidirect
29. C₉×#106	C₂²	semidirect
30. C₉×#103	C₂²	semidirect
31. C₉×#103	C₂²	semidirect
32. C₉×#102	C₂²	semidirect
33. C₉×#102	C₂²	semidirect
78. #3197	C₉	direct (N × H)
132. #525	C₂×C₉	semidirect
133. #485	C₂×C₉	semidirect
134. #482	C₂×C₉	semidirect
135. #470	C₂×C₉	semidirect
136. #469	C₂×C₉	semidirect
137. #468	C₂×C₉	semidirect
138. #452	C₂×C₉	semidirect
139. #449	C₂×C₉	semidirect
140. #407	C₂×C₉	semidirect
193. C₂×Q₁₆	C₂²×C₉	semidirect
194. C₂×SD₁₆	C₂²×C₉	semidirect
195. C₂×D₈	C₂²×C₉	semidirect
207. #107	C₂²×C₉	semidirect
208. #106	C₂²×C₉	semidirect
209. #106	C₂²×C₉	semidirect
210. #103	C₂²×C₉	semidirect
211. #103	C₂²×C₉	semidirect
212. #102	C₂²×C₉	semidirect
213. #102	C₂²×C₉	semidirect
299. C₉	#3197	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#525	2
C₉×#485	2
C₉×#482	2
C₉×#470	2
C₉×#469	2
C₉×#468	2
C₉×#452	2
C₉×#449	2
C₉×#407	2
C₃×#3197	3

Subgroups — C9×#3197 #11767918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3197 #11767918