Subgroups — C₉×#1902 #11766623

Order 1152.

456 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#137	C₂	1	1	yes	maximal, characteristic
2	C₉×#568	C₂	1	1	yes	maximal, characteristic
3	C₉×#421	C₂	1	1	yes	maximal, characteristic
4	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1902	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#137	—	2	2	no	—
12	C₉×#137	—	2	2	no	—
13	C₂×C₉×Q₁₆	C₄	1	2	yes	complemented
14	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
15	C₉×#131	—	2	2	no	—
16	C₉×#131	—	2	2	no	—
17	C₂×C₉×M₁₆	—	2	2	no	—
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
22	C₉×#117	C₂²	1	1	yes	characteristic
23	C₉×#104	—	2	2	no	—
24	C₉×#103	—	2	2	no	—
25	C₉×#101	—	2	2	no	—
26	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
27	C₃×#568	C₂×C₃	1	1	yes	characteristic
28	C₃×#421	C₂×C₃	1	1	yes	characteristic
29	C₃×#419	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#101	C₂×C₃	1	1	yes	characteristic
31	C₃×#344	C₂×C₃	1	1	yes	characteristic
32	C₃×#338	C₂×C₃	1	1	yes	characteristic
33	C₉×#41	—	2	1	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	D₄	1	2	yes	—
36	C₉×#41	D₄	1	2	yes	—
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	—	2	1	no	—
40	C₂×Q₈×C₉	—	2	1	no	—
41	C₂×Q₈×C₉	—	2	1	no	—
42	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
43	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×Q₁₆	—	2	4	no	—
50	C₉×SD₁₆	—	2	4	no	—
51	C₉×M₁₆	—	2	4	no	—
52	C₉×M₁₆	—	2	1	no	—
53	C₉×M₁₆	—	2	1	no	—
54	C₉×M₁₆	—	2	1	no	—
55	C₉×M₁₆	—	2	1	no	—
56	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	—	2	1	no	—
58	C₉×M₁₆	—	2	1	no	—
59	C₉×M₁₆	D₄	1	2	yes	—
60	C₂×C₈×C₉	—	2	4	no	—
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
64	C₂×C₈×C₉	D₄	1	2	yes	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	C₉×C₂²⋊C₄	—	2	1	no	—
67	C₄²×C₉	—	2	1	no	—
68	#1902	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₃×#137	—	2	2	no	—
72	C₃×#137	—	2	2	no	—
73	C₂×C₃×Q₁₆	C₃×C₄	1	2	yes	—
74	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
75	C₃×#131	—	2	2	no	—
76	C₃×#131	—	2	2	no	—
77	C₂×C₃×M₁₆	—	2	2	no	—
78	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₃×#117	C₂²×C₃	1	1	yes	characteristic
83	C₃×#104	—	2	2	no	—
84	C₃×#103	—	2	2	no	—
85	C₃×#101	—	2	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	#41	1	1	yes	characteristic
88	Q₈×C₉	—	2	1	no	—
89	Q₈×C₉	—	2	2	no	—
90	Q₈×C₉	—	2	2	no	—
91	Q₈×C₉	—	4	1	no	—
92	Q₈×C₉	—	4	1	no	—
93	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
94	D₄×C₉	—	2	2	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	—	2	1	no	—
97	D₄×C₉	C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	1	no	—
104	C₂×C₄×C₉	—	4	1	no	—
105	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	#41	1	1	yes	characteristic
111	C₈×C₉	—	2	2	no	—
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	4	1	no	—
115	C₂×#137	C₂×C₉	1	1	yes	characteristic
116	#568	C₂×C₉	1	1	yes	characteristic
117	#421	C₂×C₉	1	1	yes	characteristic
118	#419	C₂×C₉	1	1	yes	characteristic, complemented
119	C₂×#101	C₂×C₉	1	1	yes	characteristic, complemented
120	#344	C₂×C₉	1	1	yes	characteristic, complemented
121	#338	C₂×C₉	1	1	yes	characteristic, complemented
122	C₃×#41	—	2	1	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	C₃×D₄	1	2	yes	—
125	C₃×#41	C₃×D₄	1	2	yes	—
126	C₂×C₃×Q₈	—	2	1	no	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
132	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×Q₁₆	—	2	4	no	—
139	C₃×SD₁₆	—	2	4	no	—
140	C₃×M₁₆	—	2	4	no	—
141	C₃×M₁₆	—	2	1	no	—
142	C₃×M₁₆	—	2	1	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	—	2	1	no	—
145	C₃×M₁₆	—	2	1	no	—
146	C₃×M₁₆	—	2	1	no	—
147	C₃×M₁₆	—	2	1	no	—
148	C₃×M₁₆	C₃×D₄	1	2	yes	—
149	C₂×C₃×C₈	—	2	4	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
153	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₂²⋊C₄	—	2	1	no	—
156	C₃×C₄²	—	2	1	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	—	8	1	no	—
160	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₉	#121	1	1	yes	characteristic
162	C₂²×C₉	#123	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	8	1	no	—
166	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₉	C₄×D₄	1	1	yes	characteristic
168	C₄×C₉	#120	1	1	yes	characteristic
169	C₄×C₉	#121	1	1	yes	characteristic
170	C₂×#41	C₂²×C₉	1	1	yes	characteristic
171	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
172	#137	—	2	2	no	—
173	#137	—	2	2	no	—
174	C₂×Q₁₆	C₄×C₉	1	2	yes	complemented
175	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
176	#131	—	2	2	no	—
177	#131	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
180	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
181	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
182	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
183	#117	C₂²×C₉	1	1	yes	characteristic
184	#104	—	2	2	no	—
185	#103	—	2	2	no	—
186	#101	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	C₃×#41	1	1	yes	characteristic
189	C₃×Q₈	—	2	1	no	—
190	C₃×Q₈	—	2	2	no	—
191	C₃×Q₈	—	2	2	no	—
192	C₃×Q₈	—	4	1	no	—
193	C₃×Q₈	—	4	1	no	—
194	C₃×Q₈	C₃×C₂²⋊C₄	1	2	yes	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₃×C₂²⋊C₄	1	2	yes	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
212	C₃×C₈	—	2	2	no	—
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	4	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	4	2	no	—
220	C₂×C₉	#386	1	1	yes	characteristic
221	#41	—	2	1	no	—
222	#41	—	2	1	no	—
223	#41	D₄×C₉	1	2	yes	—
224	#41	D₄×C₉	1	2	yes	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
231	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
237	Q₁₆	—	2	4	no	—
238	SD₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	1	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	1	no	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	D₄×C₉	1	2	yes	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
252	C₂×C₈	D₄×C₉	1	2	yes	—
253	C₄⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	8	1	no	—
259	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
260	C₂²×C₃	C₃×#121	1	1	yes	characteristic
261	C₂²×C₃	C₃×#123	1	1	yes	characteristic
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	—	8	1	no	—
265	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
267	C₃×C₄	C₃×#120	1	1	yes	characteristic
268	C₃×C₄	C₃×#121	1	1	yes	characteristic
269	C₉	#1902	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³	—	2	1	no	—
271	C₂³	C₉×#41	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	2	2	no	—
275	Q₈	—	4	1	no	—
276	Q₈	—	4	1	no	—
277	Q₈	C₉×C₂²⋊C₄	1	2	yes	—
278	D₄	—	2	2	no	—
279	D₄	—	2	2	no	—
280	D₄	—	2	1	no	—
281	D₄	C₉×C₂²⋊C₄	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
294	C₂×C₄	C₉×#41	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	4	2	no	—
303	C₂×C₃	C₃×#386	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	8	1	no	—
307	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
308	C₂²	C₉×#121	1	1	yes	characteristic
309	C₂²	C₉×#123	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
314	C₄	C₄×D₄×C₉	1	1	yes	characteristic
315	C₄	C₉×#120	1	1	yes	characteristic
316	C₄	C₉×#121	1	1	yes	characteristic
317	C₃	C₃×#1902	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	4	2	no	—
322	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#419	C₂	semidirect
5. C₂×C₉×#101	C₂	semidirect
6. C₉×#344	C₂	semidirect
7. C₉×#338	C₂	semidirect
13. C₂×C₉×Q₁₆	C₄	semidirect
14. C₂×C₉×SD₁₆	C₄	semidirect
68. #1902	C₉	direct (N × H)
118. #419	C₂×C₉	semidirect
119. C₂×#101	C₂×C₉	semidirect
120. #344	C₂×C₉	semidirect
121. #338	C₂×C₉	semidirect
174. C₂×Q₁₆	C₄×C₉	semidirect
175. C₂×SD₁₆	C₄×C₉	semidirect
269. C₉	#1902	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#137	2
C₉×#568	2
C₉×#421	2
C₉×#419	2
C₂×C₉×#101	2
C₉×#344	2
C₉×#338	2
C₃×#1902	3

Subgroups — C9×#1902 #11766623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1902 #11766623