Subgroups — C₉×#1893 #11766614

Order 1152.

648 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#121	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
4	C₉×#380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#340	C₂	1	2	yes	maximal, complemented
6	C₃×#1893	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	—	2	1	no	—
8	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	—	2	2	no	—
11	C₂²×C₈×C₉	—	2	4	no	—
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₉×#121	—	2	2	no	—
14	C₉×#121	C₄	1	4	yes	complemented
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
18	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
19	C₉×#102	—	2	4	no	—
20	C₉×#95	—	2	1	no	—
21	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#121	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#102	C₂×C₃	1	2	yes	—
24	C₃×#380	C₂×C₃	1	1	yes	characteristic
25	C₃×#340	C₂×C₃	1	2	yes	—
26	C₂⁴×C₉	—	2	1	no	—
27	C₂⁴×C₉	D₄	1	1	yes	characteristic
28	C₂×D₄×C₉	—	2	4	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	4	2	no	—
33	C₂×D₄×C₉	—	4	2	no	—
34	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₉	—	2	2	no	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	D₄	1	2	yes	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₉	—	2	8	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	—	2	2	no	—
46	C₂×C₈×C₉	D₄	1	4	yes	complemented
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₉×C₂²⋊C₄	—	2	4	no	—
52	C₉×C₂²⋊C₄	—	4	2	no	—
53	#1893	C₉	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×D₄	—	2	1	no	—
55	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃×C₈	—	2	2	no	—
58	C₂²×C₃×C₈	—	2	4	no	—
59	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
60	C₃×#121	—	2	2	no	—
61	C₃×#121	C₃×C₄	1	4	yes	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
65	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
66	C₃×#102	—	2	4	no	—
67	C₃×#95	—	2	1	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	—	2	2	no	—
71	C₂³×C₉	—	4	2	no	—
72	C₂³×C₉	—	4	1	no	—
73	C₂³×C₉	—	4	1	no	—
74	C₂³×C₉	—	4	2	no	—
75	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
76	C₂³×C₉	D₈	1	2	yes	—
77	C₂³×C₉	SD₁₆	1	2	yes	—
78	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
79	D₄×C₉	—	2	4	no	—
80	D₄×C₉	—	4	8	no	—
81	D₄×C₉	—	4	1	no	—
82	D₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	1	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	—	4	2	no	—
92	C₂×C₄×C₉	—	4	2	no	—
93	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	#41	1	2	yes	—
98	C₈×C₉	—	2	4	no	—
99	C₂³×C₈	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂×#121	C₂×C₉	1	1	yes	characteristic
101	C₂×#102	C₂×C₉	1	2	yes	complemented
102	#380	C₂×C₉	1	1	yes	characteristic, complemented
103	#340	C₂×C₉	1	2	yes	complemented
104	C₂⁴×C₃	—	2	1	no	—
105	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
106	C₂×C₃×D₄	—	2	4	no	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	—	2	2	no	—
110	C₂×C₃×D₄	—	4	2	no	—
111	C₂×C₃×D₄	—	4	2	no	—
112	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₃×C₈	—	2	8	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
125	C₃×C₄⋊C₄	—	2	1	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
129	C₃×C₂²⋊C₄	—	2	4	no	—
130	C₃×C₂²⋊C₄	—	4	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	2	4	no	—
135	C₂²×C₉	—	4	2	no	—
136	C₂²×C₉	—	4	2	no	—
137	C₂²×C₉	—	4	2	no	—
138	C₂²×C₉	—	8	2	no	—
139	C₂²×C₉	—	8	2	no	—
140	C₂²×C₉	#102	1	4	yes	complemented
141	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₉	#121	1	2	yes	—
143	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
145	C₂²×C₉	#135	1	2	yes	—
146	C₄×C₉	—	2	2	no	—
147	C₄×C₉	—	4	2	no	—
148	C₄×C₉	—	8	2	no	—
149	C₄×C₉	C₄×D₄	1	2	yes	—
150	C₄×C₉	#120	1	1	yes	characteristic
151	C₄×C₉	#123	1	1	yes	characteristic
152	C₂²×D₄	—	2	1	no	—
153	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂²×C₉	1	2	yes	—
158	#121	—	2	2	no	—
159	#121	C₄×C₉	1	4	yes	complemented
160	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	—	2	1	no	—
164	#102	—	2	4	no	—
165	#95	—	2	1	no	—
166	C₂³×C₃	—	2	2	no	—
167	C₂³×C₃	—	2	2	no	—
168	C₂³×C₃	—	2	2	no	—
169	C₂³×C₃	—	4	2	no	—
170	C₂³×C₃	—	4	1	no	—
171	C₂³×C₃	—	4	1	no	—
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
174	C₂³×C₃	C₃×D₈	1	2	yes	—
175	C₂³×C₃	C₃×SD₁₆	1	2	yes	—
176	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
177	C₃×D₄	—	2	4	no	—
178	C₃×D₄	—	4	8	no	—
179	C₃×D₄	—	4	1	no	—
180	C₃×D₄	—	4	1	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	4	1	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	2	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
192	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₃×#41	1	2	yes	—
196	C₃×C₈	—	2	4	no	—
197	C₂×C₉	—	2	4	no	—
198	C₂×C₉	—	8	2	no	—
199	C₂×C₉	#386	1	1	yes	characteristic
200	C₂×C₉	C₂×#102	1	1	yes	characteristic
201	C₂×C₉	#416	1	1	yes	characteristic
202	C₂×C₉	#465	1	2	yes	—
203	C₂×C₉	#466	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂⁴	D₄×C₉	1	1	yes	characteristic
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
218	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
219	C₂²×C₄	D₄×C₉	1	2	yes	—
220	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	D₄×C₉	1	4	yes	complemented
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	4	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	—	4	2	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	8	2	no	—
239	C₂²×C₃	—	8	2	no	—
240	C₂²×C₃	C₃×#102	1	4	yes	—
241	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₃	C₃×#121	1	2	yes	—
243	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
244	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
245	C₂²×C₃	C₃×#135	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	8	2	no	—
249	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
250	C₃×C₄	C₃×#120	1	1	yes	characteristic
251	C₃×C₄	C₃×#123	1	1	yes	characteristic
252	C₉	#1893	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
261	C₂³	C₉×D₈	1	2	yes	—
262	C₂³	C₉×SD₁₆	1	2	yes	—
263	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
264	D₄	—	2	4	no	—
265	D₄	—	4	8	no	—
266	D₄	—	4	1	no	—
267	D₄	—	4	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
279	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
281	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
282	C₂×C₄	C₉×#41	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₂×C₃	—	2	4	no	—
285	C₂×C₃	—	8	2	no	—
286	C₂×C₃	C₃×#386	1	1	yes	characteristic
287	C₂×C₃	C₂×C₃×#102	1	1	yes	characteristic
288	C₂×C₃	C₃×#416	1	1	yes	characteristic
289	C₂×C₃	C₃×#465	1	2	yes	—
290	C₂×C₃	C₃×#466	1	2	yes	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₉×#102	1	4	yes	complemented
301	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
302	C₂²	C₉×#121	1	2	yes	—
303	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
304	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
305	C₂²	C₉×#135	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₄×D₄×C₉	1	2	yes	—
310	C₄	C₉×#120	1	1	yes	characteristic
311	C₄	C₉×#123	1	1	yes	characteristic
312	C₃	C₃×#1893	1	1	yes	characteristic
313	C₂	—	2	4	no	—
314	C₂	—	8	2	no	—
315	C₂	C₉×#386	1	1	yes	characteristic
316	C₂	C₂×C₉×#102	1	1	yes	characteristic
317	C₂	C₉×#416	1	1	yes	characteristic
318	C₂	C₉×#465	1	2	yes	—
319	C₂	C₉×#466	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#121	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×C₉	C₂	semidirect
3. C₂×C₉×#102	C₂	semidirect
4. C₉×#380	C₂	semidirect
5. C₉×#340	C₂	semidirect
14. C₉×#121	C₄	semidirect
46. C₂×C₈×C₉	D₄	semidirect
53. #1893	C₉	direct (N × H)
99. C₂³×C₈	C₂×C₉	semidirect
101. C₂×#102	C₂×C₉	semidirect
102. #380	C₂×C₉	semidirect
103. #340	C₂×C₉	semidirect
140. C₂²×C₉	#102	semidirect
159. #121	C₄×C₉	semidirect
224. C₂×C₈	D₄×C₉	semidirect
252. C₉	#1893	direct (N × H)
300. C₂²	C₉×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×C₉×#121	2
C₂×C₉×#102	2
C₉×#380	2
C₉×#340	2
C₃×#1893	3

Subgroups — C9×#1893 #11766614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1893 #11766614