Subgroups — C₉×#1922 #11766643

Order 1152.

279 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#427	C₂	1	1	yes	maximal, characteristic
2	C₉×#424	C₂	1	1	yes	maximal, characteristic
3	C₉×#405	C₂	1	1	yes	maximal, characteristic
4	C₉×#341	C₂	1	1	yes	maximal, characteristic
5	C₉×#339	C₂	1	1	yes	maximal, characteristic
6	C₉×#339	C₂	1	1	yes	maximal, characteristic
7	C₉×#336	C₂	1	1	yes	maximal, characteristic
8	C₃×#1922	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×#117	C₂²	1	1	yes	characteristic
14	C₉×#117	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#107	C₄	1	1	yes	characteristic, complemented
17	C₉×#107	C₄	1	1	yes	characteristic, complemented
18	C₉×#106	C₄	1	1	yes	characteristic, complemented
19	C₉×#106	C₄	1	1	yes	characteristic, complemented
20	C₉×C₄⋊C₈	—	2	1	no	—
21	C₉×C₄⋊C₈	—	2	1	no	—
22	C₉×#97	—	2	1	no	—
23	C₄×C₈×C₉	—	2	1	no	—
24	C₃×#427	C₂×C₃	1	1	yes	characteristic
25	C₃×#424	C₂×C₃	1	1	yes	characteristic
26	C₃×#405	C₂×C₃	1	1	yes	characteristic
27	C₃×#341	C₂×C₃	1	1	yes	characteristic
28	C₃×#339	C₂×C₃	1	1	yes	characteristic
29	C₃×#339	C₂×C₃	1	1	yes	characteristic
30	C₃×#336	C₂×C₃	1	1	yes	characteristic
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×M₁₆	—	2	1	no	—
34	C₉×M₁₆	—	2	1	no	—
35	C₉×M₁₆	—	2	1	no	—
36	C₉×M₁₆	Q₈	1	2	yes	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	—	2	1	no	—
39	C₂×C₈×C₉	—	2	1	no	—
40	C₂×C₈×C₉	—	2	1	no	—
41	C₂×C₈×C₉	—	2	1	no	—
42	C₂×C₈×C₉	—	2	1	no	—
43	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×C₉	D₄	1	1	yes	characteristic
46	C₂×C₈×C₉	D₄	1	1	yes	characteristic
47	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₉×C₂²⋊C₄	—	2	1	no	—
52	C₉×C₂²⋊C₄	—	2	1	no	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	—	2	1	no	—
57	#1922	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×#117	C₂²×C₃	1	1	yes	characteristic
63	C₃×#117	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
65	C₃×#107	C₃×C₄	1	1	yes	characteristic
66	C₃×#107	C₃×C₄	1	1	yes	characteristic
67	C₃×#106	C₃×C₄	1	1	yes	characteristic
68	C₃×#106	C₃×C₄	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	2	1	no	—
70	C₃×C₄⋊C₈	—	2	1	no	—
71	C₃×#97	—	2	1	no	—
72	C₃×C₄×C₈	—	2	1	no	—
73	C₂³×C₉	#41	1	1	yes	characteristic
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	2	1	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₉	#41	1	1	yes	characteristic
85	C₂×C₄×C₉	#41	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	1	yes	characteristic
87	C₈×C₉	—	2	1	no	—
88	C₈×C₉	—	2	1	no	—
89	C₈×C₉	—	2	2	no	—
90	C₈×C₉	—	2	1	no	—
91	C₈×C₉	—	2	1	no	—
92	C₈×C₉	—	4	1	no	—
93	#427	C₂×C₉	1	1	yes	characteristic
94	#424	C₂×C₉	1	1	yes	characteristic
95	#405	C₂×C₉	1	1	yes	characteristic
96	#341	C₂×C₉	1	1	yes	characteristic
97	#339	C₂×C₉	1	1	yes	characteristic
98	#339	C₂×C₉	1	1	yes	characteristic
99	#336	C₂×C₉	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×M₁₆	—	2	1	no	—
103	C₃×M₁₆	—	2	1	no	—
104	C₃×M₁₆	—	2	1	no	—
105	C₃×M₁₆	C₃×Q₈	1	2	yes	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
116	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
117	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₂²⋊C₄	—	2	1	no	—
122	C₃×C₄²	—	2	1	no	—
123	C₃×C₄²	—	2	1	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
129	C₂²×C₉	#125	1	1	yes	characteristic
130	C₂²×C₉	#126	1	1	yes	characteristic
131	C₄×C₉	—	4	2	no	—
132	C₄×C₉	—	4	1	no	—
133	C₄×C₉	—	4	1	no	—
134	C₄×C₉	—	4	1	no	—
135	C₄×C₉	—	4	1	no	—
136	C₄×C₉	#117	1	1	yes	characteristic
137	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
138	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₉	#123	1	1	yes	characteristic
140	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
141	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
144	#117	C₂²×C₉	1	1	yes	characteristic
145	#117	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
147	#107	C₄×C₉	1	1	yes	characteristic, complemented
148	#107	C₄×C₉	1	1	yes	characteristic, complemented
149	#106	C₄×C₉	1	1	yes	characteristic, complemented
150	#106	C₄×C₉	1	1	yes	characteristic, complemented
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	1	no	—
153	#97	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₂³×C₃	C₃×#41	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	—	2	1	no	—
174	C₃×C₈	—	4	1	no	—
175	C₂×C₉	—	2	1	no	—
176	C₂×C₉	—	2	1	no	—
177	C₂×C₉	#387	1	1	yes	characteristic
178	C₂×C₉	#443	1	1	yes	characteristic
179	C₂×C₉	#444	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
182	M₁₆	—	2	1	no	—
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	1	no	—
185	M₁₆	Q₈×C₉	1	2	yes	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
194	C₂×C₈	D₄×C₉	1	1	yes	characteristic
195	C₂×C₈	D₄×C₉	1	1	yes	characteristic
196	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
197	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
198	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
199	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₃	C₃×#125	1	1	yes	characteristic
210	C₂²×C₃	C₃×#126	1	1	yes	characteristic
211	C₃×C₄	—	4	2	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	—	4	1	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	C₃×#117	1	1	yes	characteristic
217	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
218	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
219	C₃×C₄	C₃×#123	1	1	yes	characteristic
220	C₉	#1922	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³	C₉×#41	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
232	C₂×C₄	C₉×#41	1	1	yes	characteristic
233	C₂×C₄	C₉×#41	1	1	yes	characteristic
234	C₂×C₄	C₉×#41	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	C₃×#387	1	1	yes	characteristic
244	C₂×C₃	C₃×#443	1	1	yes	characteristic
245	C₂×C₃	C₃×#444	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
249	C₂²	C₉×#125	1	1	yes	characteristic
250	C₂²	C₉×#126	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	C₉×#117	1	1	yes	characteristic
257	C₄	C₄×Q₈×C₉	1	1	yes	characteristic
258	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
259	C₄	C₉×#123	1	1	yes	characteristic
260	C₃	C₃×#1922	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	C₉×#387	1	1	yes	characteristic
264	C₂	C₉×#443	1	1	yes	characteristic
265	C₂	C₉×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×C₄×D₄
prime radical	C₂³×C₃	C₃×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₉×#107	C₄	semidirect
17. C₉×#107	C₄	semidirect
18. C₉×#106	C₄	semidirect
19. C₉×#106	C₄	semidirect
57. #1922	C₉	direct (N × H)
147. #107	C₄×C₉	semidirect
148. #107	C₄×C₉	semidirect
149. #106	C₄×C₉	semidirect
150. #106	C₄×C₉	semidirect
220. C₉	#1922	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×#427	2
C₉×#424	2
C₉×#405	2
C₉×#341	2
C₉×#339	2
C₉×#336	2
C₃×#1922	3

Subgroups — C9×#1922 #11766643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1922 #11766643