Subgroups — C₉×#2283 #11767004

Order 1152.

852 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	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#383	C₂	1	2	yes	maximal, complemented
3	C₉×#382	C₂	1	2	yes	maximal, complemented
4	C₉×#381	C₂	1	2	yes	maximal
5	C₉×#380	C₂	1	2	yes	maximal
6	C₄×C₉×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
8	C₉×#376	C₂	1	2	yes	maximal, complemented
9	C₃×#2283	C₃	1	1	yes	maximal, characteristic
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄²×C₉	C₄	1	8	yes	complemented
16	C₂×C₄²×C₉	C₂²	1	4	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₉×#95	C₂²	1	4	yes	—
22	C₉×#95	C₂²	1	4	yes	—
23	C₉×#95	C₂²	1	4	yes	—
24	C₉×#95	C₂²	1	4	yes	—
25	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
26	C₃×#383	C₂×C₃	1	2	yes	—
27	C₃×#382	C₂×C₃	1	2	yes	—
28	C₃×#381	C₂×C₃	1	2	yes	—
29	C₃×#380	C₂×C₃	1	2	yes	—
30	C₃×C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
31	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
32	C₃×#376	C₂×C₃	1	2	yes	—
33	C₂⁴×C₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	4	no	—
35	C₂²×C₄×C₉	—	2	4	no	—
36	C₂²×C₄×C₉	—	2	4	no	—
37	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₉	C₂³	1	8	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×C₄⋊C₄	—	2	8	no	—
48	C₉×C₂²⋊C₄	—	2	8	no	—
49	C₄²×C₉	—	2	8	no	—
50	C₄²×C₉	C₂×C₄	1	4	yes	—
51	C₄²×C₉	C₂×C₄	1	4	yes	—
52	C₄²×C₉	C₂×C₄	1	4	yes	—
53	C₄²×C₉	C₂×C₄	1	4	yes	—
54	#2283	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
60	C₂×C₃×C₄²	C₃×C₄	1	8	yes	—
61	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
62	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	C₃×#95	C₂²×C₃	1	4	yes	—
67	C₃×#95	C₂²×C₃	1	4	yes	—
68	C₃×#95	C₂²×C₃	1	4	yes	—
69	C₃×#95	C₂²×C₃	1	4	yes	—
70	C₂³×C₉	—	2	2	no	—
71	C₂³×C₉	—	2	2	no	—
72	C₂³×C₉	C₂²×C₄	1	2	yes	—
73	C₂³×C₉	#41	1	4	yes	—
74	C₂³×C₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₉	—	2	8	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	8	no	—
81	C₂×C₄×C₉	—	2	8	no	—
82	C₂×C₄×C₉	—	2	8	no	—
83	C₂×C₄×C₉	—	2	8	no	—
84	C₂×C₄×C₉	—	2	8	no	—
85	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₉	#41	1	4	yes	—
92	C₂×C₄×C₉	#41	1	4	yes	—
93	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
94	#383	C₂×C₉	1	2	yes	complemented
95	#382	C₂×C₉	1	2	yes	complemented
96	#381	C₂×C₉	1	2	yes	—
97	#380	C₂×C₉	1	2	yes	—
98	C₄×C₄⋊C₄	C₂×C₉	1	2	yes	complemented
99	C₄×C₂²⋊C₄	C₂×C₉	1	2	yes	—
100	#376	C₂×C₉	1	2	yes	complemented
101	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	—	2	4	no	—
104	C₂²×C₃×C₄	—	2	4	no	—
105	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
106	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
107	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	2	8	no	—
116	C₃×C₂²⋊C₄	—	2	8	no	—
117	C₃×C₄²	—	2	8	no	—
118	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
119	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
120	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
121	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	8	no	—
125	C₂²×C₉	#117	1	4	yes	—
126	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₉	C₂×#41	1	2	yes	—
128	C₂²×C₉	C₂×#41	1	2	yes	—
129	C₂²×C₉	C₂×#41	1	2	yes	—
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	4	8	no	—
135	C₄×C₉	#117	1	4	yes	—
136	C₄×C₉	#117	1	4	yes	—
137	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
138	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
139	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
142	C₂×C₄²	C₄×C₉	1	8	yes	complemented
143	C₂×C₄²	C₂²×C₉	1	4	yes	—
144	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
148	#95	C₂²×C₉	1	4	yes	—
149	#95	C₂²×C₉	1	4	yes	—
150	#95	C₂²×C₉	1	4	yes	—
151	#95	C₂²×C₉	1	4	yes	—
152	C₂³×C₃	—	2	2	no	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
155	C₂³×C₃	C₃×#41	1	4	yes	—
156	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
168	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
169	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
170	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
171	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
173	C₂×C₃×C₄	C₃×#41	1	4	yes	—
174	C₂×C₃×C₄	C₃×#41	1	4	yes	—
175	C₂×C₉	—	2	4	no	—
176	C₂×C₉	C₂×#117	1	1	yes	characteristic
177	C₂×C₉	C₂×#117	1	1	yes	characteristic
178	C₂×C₉	C₂×#117	1	1	yes	characteristic
179	C₂×C₉	#529	1	4	yes	—
180	C₂⁴	C₂³×C₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
185	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
186	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
187	C₂²×C₄	C₂³×C₉	1	8	yes	—
188	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	8	no	—
195	C₂²⋊C₄	—	2	8	no	—
196	C₄²	—	2	8	no	—
197	C₄²	C₂×C₄×C₉	1	4	yes	—
198	C₄²	C₂×C₄×C₉	1	4	yes	—
199	C₄²	C₂×C₄×C₉	1	4	yes	—
200	C₄²	C₂×C₄×C₉	1	4	yes	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	C₃×#117	1	4	yes	—
205	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
206	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
207	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
208	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	4	8	no	—
214	C₃×C₄	C₃×#117	1	4	yes	—
215	C₃×C₄	C₃×#117	1	4	yes	—
216	C₉	#2283	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂²×C₄×C₉	1	2	yes	—
220	C₂³	C₉×#41	1	4	yes	—
221	C₂³	C₂⁴×C₉	1	1	yes	characteristic
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
233	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
234	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
235	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
238	C₂×C₄	C₉×#41	1	4	yes	—
239	C₂×C₄	C₉×#41	1	4	yes	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
242	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
243	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
244	C₂×C₃	C₃×#529	1	4	yes	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	C₉×#117	1	4	yes	—
249	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
250	C₂²	C₂×C₉×#41	1	2	yes	—
251	C₂²	C₂×C₉×#41	1	2	yes	—
252	C₂²	C₂×C₉×#41	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	8	no	—
258	C₄	C₉×#117	1	4	yes	—
259	C₄	C₉×#117	1	4	yes	—
260	C₃	C₃×#2283	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	C₂×C₉×#117	1	1	yes	characteristic
263	C₂	C₂×C₉×#117	1	1	yes	characteristic
264	C₂	C₂×C₉×#117	1	1	yes	characteristic
265	C₂	C₉×#529	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#383	C₂	semidirect
3. C₉×#382	C₂	semidirect
6. C₄×C₉×C₄⋊C₄	C₂	semidirect
8. C₉×#376	C₂	semidirect
15. C₂×C₄²×C₉	C₄	semidirect
54. #2283	C₉	direct (N × H)
94. #383	C₂×C₉	semidirect
95. #382	C₂×C₉	semidirect
98. C₄×C₄⋊C₄	C₂×C₉	semidirect
100. #376	C₂×C₉	semidirect
142. C₂×C₄²	C₄×C₉	semidirect
216. C₉	#2283	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₉	2
C₉×#383	2
C₉×#382	2
C₉×#381	2
C₉×#380	2
C₄×C₉×C₄⋊C₄	2
C₄×C₉×C₂²⋊C₄	2
C₉×#376	2
C₃×#2283	3

Subgroups — C9×#2283 #11767004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2283 #11767004