Subgroups — C₉×#1891 #11766612

Order 1152.

513 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#525	C₂	1	1	yes	maximal, characteristic
3	C₉×#413	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#407	C₂	1	2	yes	maximal, complemented
5	C₉×#344	C₂	1	2	yes	maximal, complemented
6	C₃×#1891	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₉×#131	—	2	4	no	—
10	C₂×C₉×M₁₆	—	2	2	no	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂×C₉×M₁₆	C₂²	1	2	yes	—
13	C₉×#123	C₄	1	2	yes	—
14	C₉×C₂²⋊Q₈	—	2	1	no	—
15	C₉×#121	—	2	1	no	—
16	C₉×#120	C₄	1	2	yes	—
17	C₄×D₄×C₉	—	2	2	no	—
18	C₉×#117	C₂²	1	1	yes	characteristic
19	C₉×#101	—	2	1	no	—
20	C₉×#100	—	2	1	no	—
21	C₉×C₂²⋊C₈	—	2	4	no	—
22	C₃×#568	C₂×C₃	1	1	yes	characteristic
23	C₃×#525	C₂×C₃	1	1	yes	characteristic
24	C₃×#413	C₂×C₃	1	1	yes	characteristic
25	C₃×#407	C₂×C₃	1	2	yes	—
26	C₃×#344	C₂×C₃	1	2	yes	—
27	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
28	C₉×#41	—	2	2	no	—
29	C₉×#41	—	2	2	no	—
30	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	D₄	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₉×M₁₆	—	2	4	no	—
42	C₉×M₁₆	—	2	2	no	—
43	C₉×M₁₆	—	2	2	no	—
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	—	2	2	no	—
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	D₄	1	4	yes	complemented
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₂²⋊C₄	—	2	2	no	—
52	C₉×C₂²⋊C₄	—	2	2	no	—
53	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₉	—	2	1	no	—
55	#1891	C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
58	C₃×#131	—	2	4	no	—
59	C₂×C₃×M₁₆	—	2	2	no	—
60	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
61	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
62	C₃×#123	C₃×C₄	1	2	yes	—
63	C₃×C₂²⋊Q₈	—	2	1	no	—
64	C₃×#121	—	2	1	no	—
65	C₃×#120	C₃×C₄	1	2	yes	—
66	C₃×C₄×D₄	—	2	2	no	—
67	C₃×#117	C₂²×C₃	1	1	yes	characteristic
68	C₃×#101	—	2	1	no	—
69	C₃×#100	—	2	1	no	—
70	C₃×C₂²⋊C₈	—	2	4	no	—
71	C₂³×C₉	—	2	1	no	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
75	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₉	—	2	2	no	—
77	D₄×C₉	—	2	4	no	—
78	D₄×C₉	—	2	2	no	—
79	D₄×C₉	—	4	4	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	4	1	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	—	4	1	no	—
88	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₉	#41	1	2	yes	—
93	C₈×C₉	—	2	4	no	—
94	C₈×C₉	—	4	2	no	—
95	#568	C₂×C₉	1	1	yes	characteristic, complemented
96	#525	C₂×C₉	1	1	yes	characteristic
97	#413	C₂×C₉	1	1	yes	characteristic, complemented
98	#407	C₂×C₉	1	2	yes	complemented
99	#344	C₂×C₉	1	2	yes	complemented
100	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
101	C₃×#41	—	2	2	no	—
102	C₃×#41	—	2	2	no	—
103	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
106	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	2	no	—
111	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₃×M₁₆	—	2	4	no	—
115	C₃×M₁₆	—	2	2	no	—
116	C₃×M₁₆	—	2	2	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	—	2	2	no	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₂²⋊C₄	—	2	2	no	—
125	C₃×C₂²⋊C₄	—	2	2	no	—
126	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
127	C₃×C₄²	—	2	1	no	—
128	C₂²×C₉	—	2	2	no	—
129	C₂²×C₉	—	2	4	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	4	1	no	—
132	C₂²×C₉	—	4	2	no	—
133	C₂²×C₉	—	4	1	no	—
134	C₂²×C₉	—	4	2	no	—
135	C₂²×C₉	—	4	1	no	—
136	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₉	C₄×D₄	1	2	yes	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	4	2	no	—
140	C₄×C₉	—	4	1	no	—
141	C₄×C₉	#120	1	1	yes	characteristic
142	C₄×C₉	#121	1	2	yes	—
143	C₄×C₉	#123	1	1	yes	characteristic
144	C₂×#41	C₂²×C₉	1	1	yes	characteristic
145	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
146	#131	—	2	4	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	C₂²×C₉	1	2	yes	—
149	C₂×M₁₆	C₂²×C₉	1	2	yes	—
150	#123	C₄×C₉	1	2	yes	—
151	C₂²⋊Q₈	—	2	1	no	—
152	#121	—	2	1	no	—
153	#120	C₄×C₉	1	2	yes	—
154	C₄×D₄	—	2	2	no	—
155	#117	C₂²×C₉	1	1	yes	characteristic
156	#101	—	2	1	no	—
157	#100	—	2	1	no	—
158	C₂²⋊C₈	—	2	4	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	4	2	no	—
162	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
163	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	4	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	4	1	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	4	1	no	—
176	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₃×#41	1	2	yes	—
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	—	4	2	no	—
183	C₂×C₉	—	2	1	no	—
184	C₂×C₉	—	2	2	no	—
185	C₂×C₉	—	4	2	no	—
186	C₂×C₉	—	4	2	no	—
187	C₂×C₉	#386	1	1	yes	characteristic
188	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
194	C₂×D₄	D₄×C₉	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
200	C₂²×C₄	D₄×C₉	1	2	yes	—
201	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	D₄×C₉	1	4	yes	complemented
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
215	C₄²	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	4	1	no	—
229	C₃×C₄	C₃×#120	1	1	yes	characteristic
230	C₃×C₄	C₃×#121	1	2	yes	—
231	C₃×C₄	C₃×#123	1	1	yes	characteristic
232	C₉	#1891	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
237	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
252	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
254	C₂×C₄	C₉×#41	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	4	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	C₃×#386	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
271	C₂²	C₄×D₄×C₉	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	C₉×#120	1	1	yes	characteristic
276	C₄	C₉×#121	1	2	yes	—
277	C₄	C₉×#123	1	1	yes	characteristic
278	C₃	C₃×#1891	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	4	2	no	—
282	C₂	—	4	2	no	—
283	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	#120
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₃×#525	C₂×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#568	C₂	semidirect
3. C₉×#413	C₂	semidirect
4. C₉×#407	C₂	semidirect
5. C₉×#344	C₂	semidirect
48. C₂×C₈×C₉	D₄	semidirect
55. #1891	C₉	direct (N × H)
95. #568	C₂×C₉	semidirect
97. #413	C₂×C₉	semidirect
98. #407	C₂×C₉	semidirect
99. #344	C₂×C₉	semidirect
209. C₂×C₈	D₄×C₉	semidirect
232. C₉	#1891	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×#568	2
C₉×#525	2
C₉×#413	2
C₉×#407	2
C₉×#344	2
C₃×#1891	3

Subgroups — C9×#1891 #11766612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1891 #11766612