Subgroups — C₉×#1999 #11766720

Order 1152.

744 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	C₂²×C₉×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#121	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
4	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1999	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	—	2	2	no	—
8	C₂²×D₄×C₉	C₂²	1	2	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×D₈	—	2	4	no	—
11	C₂×C₉×D₈	—	2	2	no	—
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×#121	—	2	4	no	—
15	C₉×#121	—	2	4	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
18	C₉×#102	—	2	4	no	—
19	C₉×C₂²⋊C₈	—	2	2	no	—
20	C₂²×C₃×D₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#121	C₂×C₃	1	2	yes	—
22	C₂×C₃×#102	C₂×C₃	1	2	yes	—
23	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#340	C₂×C₃	1	1	yes	characteristic
25	C₂⁴×C₉	—	2	2	no	—
26	C₂⁴×C₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₉	—	2	8	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	2	2	no	—
33	C₂×D₄×C₉	—	4	4	no	—
34	C₂×D₄×C₉	—	4	4	no	—
35	C₂×D₄×C₉	D₄	1	4	yes	complemented
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₉×D₈	—	2	8	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₂×C₈×C₉	D₄	1	2	yes	complemented
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	D₄	1	4	yes	complemented
50	C₉×C₂²⋊C₄	—	2	8	no	—
51	#1999	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×D₄	—	2	2	no	—
53	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×D₈	—	2	4	no	—
56	C₂×C₃×D₈	—	2	2	no	—
57	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×#121	—	2	4	no	—
60	C₃×#121	—	2	4	no	—
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
63	C₃×#102	—	2	4	no	—
64	C₃×C₂²⋊C₈	—	2	2	no	—
65	C₂³×C₉	—	2	2	no	—
66	C₂³×C₉	—	2	1	no	—
67	C₂³×C₉	—	2	1	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	4	no	—
70	C₂³×C₉	—	4	2	no	—
71	C₂³×C₉	—	4	2	no	—
72	C₂³×C₉	—	4	2	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	—	4	2	no	—
75	C₂³×C₉	—	4	2	no	—
76	C₂³×C₉	D₈	1	2	yes	complemented
77	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
78	D₄×C₉	—	2	4	no	—
79	D₄×C₉	—	2	4	no	—
80	D₄×C₉	—	4	16	no	—
81	D₄×C₉	—	4	2	no	—
82	D₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	—	4	2	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	2	no	—
90	C₂×C₄×C₉	—	4	1	no	—
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₉	#41	1	1	yes	characteristic
95	C₈×C₉	—	2	2	no	—
96	C₈×C₉	—	4	1	no	—
97	C₂²×D₈	C₂×C₉	1	1	yes	characteristic, complemented
98	C₂×#121	C₂×C₉	1	2	yes	complemented
99	C₂×#102	C₂×C₉	1	2	yes	complemented
100	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
101	#340	C₂×C₉	1	1	yes	characteristic, complemented
102	C₂⁴×C₃	—	2	2	no	—
103	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	8	no	—
105	C₂×C₃×D₄	—	2	2	no	—
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	—	2	2	no	—
110	C₂×C₃×D₄	—	4	4	no	—
111	C₂×C₃×D₄	—	4	4	no	—
112	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₃×D₈	—	2	8	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
127	C₃×C₂²⋊C₄	—	2	8	no	—
128	C₂²×C₉	—	2	4	no	—
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	4	1	no	—
131	C₂²×C₉	—	4	2	no	—
132	C₂²×C₉	—	4	1	no	—
133	C₂²×C₉	—	4	4	no	—
134	C₂²×C₉	—	4	4	no	—
135	C₂²×C₉	—	4	4	no	—
136	C₂²×C₉	—	8	2	no	—
137	C₂²×C₉	—	8	2	no	—
138	C₂²×C₉	—	8	2	no	—
139	C₂²×C₉	—	8	2	no	—
140	C₂²×C₉	#120	1	2	yes	—
141	C₂²×C₉	#121	1	1	yes	characteristic
142	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
143	C₂²×C₉	#135	1	1	yes	characteristic
144	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
145	C₄×C₉	—	4	1	no	—
146	C₄×C₉	—	4	4	no	—
147	C₄×C₉	#120	1	1	yes	characteristic
148	C₄×C₉	#121	1	2	yes	—
149	C₄×C₉	#127	1	1	yes	characteristic
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	C₂²×C₉	1	2	yes	—
152	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
157	#121	—	2	4	no	—
158	#121	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	#102	—	2	4	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₂³×C₃	—	2	2	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	—	2	2	no	—
167	C₂³×C₃	—	2	4	no	—
168	C₂³×C₃	—	4	2	no	—
169	C₂³×C₃	—	4	2	no	—
170	C₂³×C₃	—	4	2	no	—
171	C₂³×C₃	—	4	2	no	—
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	—	4	2	no	—
174	C₂³×C₃	C₃×D₈	1	2	yes	—
175	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
176	C₃×D₄	—	2	4	no	—
177	C₃×D₄	—	2	4	no	—
178	C₃×D₄	—	4	16	no	—
179	C₃×D₄	—	4	2	no	—
180	C₃×D₄	—	4	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	1	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
192	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	—	4	1	no	—
195	C₂×C₉	—	4	2	no	—
196	C₂×C₉	—	8	4	no	—
197	C₂×C₉	#392	1	1	yes	characteristic
198	C₂×C₉	#447	1	2	yes	—
199	C₂×C₉	#449	1	2	yes	—
200	C₂×C₉	#466	1	1	yes	characteristic
201	C₂×C₉	#469	1	1	yes	characteristic
202	C₂⁴	—	2	2	no	—
203	C₂⁴	D₄×C₉	1	1	yes	characteristic
204	C₂×D₄	—	2	8	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	D₄×C₉	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
220	D₈	—	2	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	D₄×C₉	1	2	yes	complemented
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	D₄×C₉	1	4	yes	complemented
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²×C₃	—	2	4	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	4	1	no	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	—	4	1	no	—
233	C₂²×C₃	—	4	4	no	—
234	C₂²×C₃	—	4	4	no	—
235	C₂²×C₃	—	4	4	no	—
236	C₂²×C₃	—	8	2	no	—
237	C₂²×C₃	—	8	2	no	—
238	C₂²×C₃	—	8	2	no	—
239	C₂²×C₃	—	8	2	no	—
240	C₂²×C₃	C₃×#120	1	2	yes	—
241	C₂²×C₃	C₃×#121	1	1	yes	characteristic
242	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
243	C₂²×C₃	C₃×#135	1	1	yes	characteristic
244	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
245	C₃×C₄	—	4	1	no	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	C₃×#120	1	1	yes	characteristic
248	C₃×C₄	C₃×#121	1	2	yes	—
249	C₃×C₄	C₃×#127	1	1	yes	characteristic
250	C₉	#1999	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	2	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	C₉×D₈	1	2	yes	complemented
263	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	4	16	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
278	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
280	C₂×C₄	C₉×#41	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₂×C₃	—	4	2	no	—
284	C₂×C₃	—	8	4	no	—
285	C₂×C₃	C₃×#392	1	1	yes	characteristic
286	C₂×C₃	C₃×#447	1	2	yes	—
287	C₂×C₃	C₃×#449	1	2	yes	—
288	C₂×C₃	C₃×#466	1	1	yes	characteristic
289	C₂×C₃	C₃×#469	1	1	yes	characteristic
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	C₉×#120	1	2	yes	—
303	C₂²	C₉×#121	1	1	yes	characteristic
304	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
305	C₂²	C₉×#135	1	1	yes	characteristic
306	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
307	C₄	—	4	1	no	—
308	C₄	—	4	4	no	—
309	C₄	C₉×#120	1	1	yes	characteristic
310	C₄	C₉×#121	1	2	yes	—
311	C₄	C₉×#127	1	1	yes	characteristic
312	C₃	C₃×#1999	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	—	8	4	no	—
315	C₂	C₉×#392	1	1	yes	characteristic
316	C₂	C₉×#447	1	2	yes	—
317	C₂	C₉×#449	1	2	yes	—
318	C₂	C₉×#466	1	1	yes	characteristic
319	C₂	C₉×#469	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₃×#1999	C₃
LCS	G ▷ C₂²×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. C₂²×C₉×D₈	C₂	semidirect
2. C₂×C₉×#121	C₂	semidirect
3. C₂×C₉×#102	C₂	semidirect
4. C₂×C₉×C₂²⋊C₈	C₂	semidirect
5. C₉×#340	C₂	semidirect
35. C₂×D₄×C₉	D₄	semidirect
47. C₂×C₈×C₉	D₄	semidirect
49. C₉×C₄⋊C₄	D₄	semidirect
51. #1999	C₉	direct (N × H)
76. C₂³×C₉	D₈	semidirect
97. C₂²×D₈	C₂×C₉	semidirect
98. C₂×#121	C₂×C₉	semidirect
99. C₂×#102	C₂×C₉	semidirect
100. C₂×C₂²⋊C₈	C₂×C₉	semidirect
101. #340	C₂×C₉	semidirect
212. C₂×D₄	D₄×C₉	semidirect
224. C₂×C₈	D₄×C₉	semidirect
226. C₄⋊C₄	D₄×C₉	semidirect
250. C₉	#1999	direct (N × H)
262. C₂³	C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₈	2
C₂×C₉×#121	2
C₂×C₉×#102	2
C₂×C₉×C₂²⋊C₈	2
C₉×#340	2
C₃×#1999	3

Subgroups — C9×#1999 #11766720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1999 #11766720