Subgroups — C₉×#4967 #13142583

Order 1728.

434 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#930	C₂	1	1	yes	maximal, characteristic
3	C₉×#830	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#809	C₂	1	2	yes	maximal, complemented
5	C₉×#807	C₂	1	2	yes	maximal, complemented
6	C₃×#4967	C₃	1	1	yes	maximal, characteristic
7	C₉×#480	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	—	2	1	no	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×#67	—	2	1	no	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₄⋊C₄	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₃×#930	C₂×C₃	1	1	yes	characteristic
19	C₃×#830	C₂×C₃	1	1	yes	characteristic
20	C₃×#809	C₂×C₃	1	2	yes	—
21	C₃×#807	C₂×C₃	1	2	yes	—
22	C₉×#121	—	3	1	no	—
23	C₂×C₉×C₄⋊C₄	—	2	1	no	—
24	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
25	C₉×#107	—	3	2	no	—
26	C₉×#102	—	3	2	no	—
27	C₉×C₂²⋊C₈	—	3	1	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂²×D₃×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	4	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₉×#67	—	4	2	no	—
36	C₂×C₉×Dic₃	—	2	1	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₉×C₃⋊C₈	—	2	2	no	—
39	C₃×#480	—	3	1	no	—
40	#4967	C₉	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×#67	—	2	1	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
45	C₃²×C₄⋊C₄	—	2	1	no	—
46	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
48	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
50	C₂×D₄×C₉	—	3	1	no	—
51	C₂×D₄×C₉	—	6	1	no	—
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	—	2	1	no	—
55	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂×C₈×C₉	—	3	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	3	1	no	—
61	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
62	C₉×C₂²⋊C₄	—	6	1	no	—
63	C₂²×C₃×C₉	—	2	1	no	—
64	C₂²×C₃×C₉	—	2	1	no	—
65	C₂²×C₃×C₉	D₈	1	2	yes	—
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×D₃×C₉	—	4	1	no	—
68	C₂×D₃×C₉	—	4	1	no	—
69	C₂×D₃×C₉	—	4	1	no	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	—	4	2	no	—
72	C₃×C₄×C₉	#41	1	2	yes	—
73	C₉×Dic₃	—	4	1	no	—
74	C₃×#121	—	3	1	no	—
75	C₂×C₃×C₄⋊C₄	—	2	1	no	—
76	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
77	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
78	#930	C₂×C₉	1	1	yes	characteristic
79	C₃×#107	—	3	2	no	—
80	C₃×#102	—	3	2	no	—
81	C₃×C₂²⋊C₈	—	3	1	no	—
82	#830	C₂×C₉	1	1	yes	characteristic, complemented
83	#809	C₂×C₉	1	2	yes	complemented
84	#807	C₂×C₉	1	2	yes	complemented
85	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
86	C₂²×C₃×D₃	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	4	2	no	—
90	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₃×#67	—	4	2	no	—
93	C₂×C₃×Dic₃	—	2	1	no	—
94	C₃×D₁₂	—	2	2	no	—
95	C₂³×C₉	—	2	1	no	—
96	C₂³×C₉	—	6	1	no	—
97	C₂³×C₉	#67	1	1	yes	characteristic
98	C₃×C₃⋊C₈	—	2	2	no	—
99	D₄×C₉	—	6	2	no	—
100	D₄×C₉	—	12	2	no	—
101	C₂×C₄×C₉	—	2	4	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	4	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	1	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	4	4	no	—
108	C₂×C₄×C₉	—	4	2	no	—
109	C₂×C₄×C₉	—	6	1	no	—
110	C₂×C₄×C₉	#67	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
112	C₈×C₉	—	6	2	no	—
113	#480	—	3	1	no	Sylow
114	C₂×C₃×C₉	—	2	2	no	—
115	C₂×C₃×C₉	#123	1	1	yes	characteristic
116	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₃×C₉	#137	1	1	yes	characteristic
118	D₃×C₉	—	8	1	no	—
119	C₂×C₃×D₄	—	3	1	no	—
120	C₂×C₃×D₄	—	6	1	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂×#67	—	2	1	no	—
128	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
129	C₂×C₃×C₈	—	3	2	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	1	no	—
132	C₃×C₄⋊C₄	—	2	1	no	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	3	1	no	—
135	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
136	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
137	C₃×C₂²⋊C₄	—	6	1	no	—
138	(C₂×D₃)⋊C₄	—	2	1	no	—
139	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	C₃×D₈	1	2	yes	—
144	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×D₃	—	4	1	no	—
146	C₂×C₃×D₃	—	4	1	no	—
147	C₂×C₃×D₃	—	4	1	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	—	4	2	no	—
150	C₃²×C₄	C₃×#41	1	2	yes	—
151	C₃×Dic₃	—	4	1	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	2	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	12	1	no	—
159	C₂²×C₉	—	12	1	no	—
160	C₂²×C₉	—	12	1	no	—
161	C₂²×C₉	#213	1	2	yes	—
162	C₂²×C₉	C₂×#67	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	4	4	no	—
167	C₄×C₉	—	4	2	no	—
168	C₄×C₉	—	12	1	no	—
169	C₄×C₉	#235	1	2	yes	—
170	#121	—	3	1	no	—
171	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic, complemented
172	#107	—	3	2	no	—
173	#102	—	3	2	no	—
174	C₂²⋊C₈	—	3	1	no	—
175	C₃×C₉	#480	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₃	—	2	1	no	—
177	C₂³×C₃	—	6	1	no	—
178	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
179	C₂³×C₃	C₃×#67	1	1	yes	characteristic
180	C₂²×D₃	—	2	1	no	—
181	C₃×D₄	—	6	2	no	—
182	C₃×D₄	—	12	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	4	4	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	6	1	no	—
195	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
197	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
199	#67	—	4	2	no	—
200	C₂×Dic₃	—	2	1	no	—
201	D₁₂	—	2	2	no	—
202	C₃×C₈	—	6	2	no	—
203	C₃⋊C₈	—	2	2	no	—
204	C₂×C₃²	—	2	2	no	—
205	C₂×C₃²	C₃×#123	1	1	yes	characteristic
206	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
207	C₂×C₃²	C₃×#137	1	1	yes	characteristic
208	C₃×D₃	—	8	1	no	—
209	C₂×C₉	—	2	4	no	—
210	C₂×C₉	—	2	2	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	24	1	no	—
215	C₂×C₉	#929	1	1	yes	characteristic
216	C₂×C₉	C₂×#213	1	1	yes	characteristic
217	C₂×C₉	#942	1	1	yes	characteristic
218	C₂×D₄	—	3	1	no	—
219	C₂×D₄	—	6	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
222	C₂×C₈	—	3	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
226	C₂²⋊C₄	—	6	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	12	1	no	—
236	C₂²×C₃	—	12	1	no	—
237	C₂²×C₃	—	12	1	no	—
238	C₂²×C₃	C₉×D₈	1	2	yes	—
239	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
240	C₂²×C₃	C₃×#213	1	2	yes	—
241	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
242	C₂×D₃	—	4	1	no	—
243	C₂×D₃	—	4	1	no	—
244	C₂×D₃	—	4	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	4	4	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	—	12	1	no	—
253	C₃×C₄	C₉×#41	1	2	yes	—
254	C₃×C₄	C₃×#235	1	2	yes	—
255	Dic₃	—	4	1	no	—
256	C₃²	C₃×#480	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	#4967	1	1	yes	characteristic, direct product factor
259	C₂³	—	6	1	no	—
260	C₂³	C₉×#67	1	1	yes	characteristic
261	D₄	—	6	2	no	—
262	D₄	—	12	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	C₉×#67	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
269	C₈	—	6	2	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	24	1	no	—
277	C₂×C₃	C₉×#123	1	1	yes	characteristic
278	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
279	C₂×C₃	C₉×#137	1	1	yes	characteristic
280	C₂×C₃	C₃×#929	1	1	yes	characteristic
281	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
282	C₂×C₃	C₃×#942	1	1	yes	characteristic
283	D₃	—	8	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	12	1	no	—
287	C₂²	—	12	1	no	—
288	C₂²	—	12	1	no	—
289	C₂²	C₉×#213	1	2	yes	—
290	C₂²	C₂×C₉×#67	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	—	12	1	no	—
294	C₄	C₉×#235	1	2	yes	—
295	C₃	—	2	1	no	—
296	C₃	C₉×#480	1	1	yes	characteristic, complemented
297	C₃	C₃×#4967	1	1	yes	characteristic
298	C₂	—	2	2	no	—
299	C₂	—	24	1	no	—
300	C₂	C₉×#929	1	1	yes	characteristic
301	C₂	C₂×C₉×#213	1	1	yes	characteristic
302	C₂	C₉×#942	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#930	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#480
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₉×#830	C₂	semidirect
4. C₉×#809	C₂	semidirect
5. C₉×#807	C₂	semidirect
24. C₂×C₉×C₄⋊C₄	D₃	semidirect
40. #4967	C₉	direct (N × H)
76. C₂×C₃×C₄⋊C₄	C₂×C₉	semidirect
82. #830	C₂×C₉	semidirect
83. #809	C₂×C₉	semidirect
84. #807	C₂×C₉	semidirect
171. C₂×C₄⋊C₄	D₃×C₉	semidirect
175. C₃×C₉	#480	semidirect
258. C₉	#4967	direct (N × H)
296. C₃	C₉×#480	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₉×#930	2
C₉×#830	2
C₉×#809	2
C₉×#807	2
C₃×#4967	3
C₉×#480	3

Subgroups — C9×#4967 #13142583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4967 #13142583