Subgroups — C₉×#5518 #13152479

Order 1728.

652 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	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₉×#924	C₂	1	2	yes	maximal
3	C₉×#889	C₂	1	4	yes	maximal, complemented
4	C₉×#879	C₂	1	4	yes	maximal
5	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×Dic₆	C₂	1	2	yes	maximal, complemented
8	C₃×#5518	C₃	1	1	yes	maximal, characteristic
9	C₉×#541	—	3	1	no	maximal
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×Dic₆	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#217	C₂²	1	4	yes	—
17	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
19	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₄×C₉×Dic₃	C₂²	1	4	yes	—
23	C₃²×#117	C₂×C₃	1	1	yes	characteristic
24	C₃×#924	C₂×C₃	1	2	yes	—
25	C₃×#889	C₂×C₃	1	4	yes	—
26	C₃×#879	C₂×C₃	1	4	yes	—
27	C₃×#870	C₂×C₃	1	1	yes	characteristic
28	C₃×#869	C₂×C₃	1	1	yes	characteristic
29	C₃×C₄×Dic₆	C₂×C₃	1	2	yes	—
30	C₉×C₄⋊Q₈	—	3	1	no	—
31	C₉×#126	—	3	4	no	—
32	C₉×#125	—	3	4	no	—
33	C₉×#125	—	3	1	no	—
34	C₉×C₂²⋊Q₈	—	3	2	no	—
35	C₄×Q₈×C₉	—	3	2	no	—
36	C₉×#117	—	2	1	no	—
37	C₉×#117	D₃	1	1	yes	characteristic
38	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	—	2	2	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₉×Dic₃	C₂³	1	4	yes	—
45	C₂×C₉×Dic₃	C₂³	1	4	yes	—
46	C₉×Dic₆	—	2	4	no	—
47	C₃×#541	—	3	1	no	—
48	#5518	C₉	1	1	yes	characteristic, direct product factor
49	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
51	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₃²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
53	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×#217	C₂²×C₃	1	4	yes	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
57	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
58	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
59	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
60	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
61	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
62	C₂×Q₈×C₉	—	3	2	no	—
63	C₂²×C₄×C₉	—	2	1	no	—
64	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₉×C₄⋊C₄	—	2	2	no	—
66	C₉×C₄⋊C₄	—	3	2	no	—
67	C₉×C₄⋊C₄	—	3	4	no	—
68	C₉×C₄⋊C₄	—	3	4	no	—
69	C₉×C₄⋊C₄	—	3	4	no	—
70	C₉×C₄⋊C₄	—	3	4	no	—
71	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
72	C₉×C₂²⋊C₄	—	2	2	no	—
73	C₉×C₂²⋊C₄	—	3	4	no	—
74	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	—	2	1	no	—
77	C₄²×C₉	—	3	4	no	—
78	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
79	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₉	—	2	1	no	—
81	C₂²×C₃×C₉	—	2	2	no	—
82	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
83	C₃×C₄×C₉	—	2	1	no	—
84	C₃×C₄×C₉	—	2	2	no	—
85	C₃×C₄×C₉	—	2	2	no	—
86	C₃×C₄×C₉	#41	1	2	yes	—
87	C₉×Dic₃	—	2	4	no	—
88	C₉×Dic₃	—	2	4	no	—
89	C₃×C₄⋊Q₈	—	3	1	no	—
90	C₃×#126	—	3	4	no	—
91	C₃×#125	—	3	4	no	—
92	C₃×#125	—	3	1	no	—
93	C₃×C₂²⋊Q₈	—	3	2	no	—
94	C₃×C₄×Q₈	—	3	2	no	—
95	C₃×#117	—	2	1	no	—
96	C₃×#117	C₂×C₉	1	1	yes	characteristic
97	C₃×#117	C₃×D₃	1	1	yes	characteristic
98	#924	C₂×C₉	1	2	yes	—
99	#889	C₂×C₉	1	4	yes	complemented
100	#879	C₂×C₉	1	4	yes	—
101	#870	C₂×C₉	1	1	yes	characteristic, complemented
102	#869	C₂×C₉	1	1	yes	characteristic, complemented
103	C₄×Dic₆	C₂×C₉	1	2	yes	complemented
104	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
107	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
111	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
112	C₃×Dic₆	—	2	4	no	—
113	C₂³×C₉	—	2	1	no	—
114	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
115	Q₈×C₉	—	6	4	no	—
116	C₂×C₄×C₉	—	2	4	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	3	4	no	—
123	C₂×C₄×C₉	—	3	4	no	—
124	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
125	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
126	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
128	#541	—	3	1	no	Sylow
129	C₂×C₃×C₉	—	4	1	no	—
130	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
131	C₂×C₃×C₉	2₋¹⁺⁴	1	2	yes	—
132	C₂×C₃×Q₈	—	3	2	no	—
133	C₂²×C₃×C₄	—	2	1	no	—
134	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
136	C₂×Dic₆	C₂²×C₉	1	2	yes	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	3	4	no	—
139	C₃×C₄⋊C₄	—	3	4	no	—
140	C₃×C₄⋊C₄	—	3	4	no	—
141	C₃×C₄⋊C₄	—	3	2	no	—
142	C₃×C₄⋊C₄	—	3	4	no	—
143	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
145	C₃×C₂²⋊C₄	—	2	2	no	—
146	C₃×C₂²⋊C₄	—	3	4	no	—
147	C₃×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
148	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
149	C₃×C₄²	—	2	1	no	—
150	C₃×C₄²	—	2	1	no	—
151	C₃×C₄²	—	3	4	no	—
152	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
154	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
155	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
156	#217	C₂²×C₉	1	4	yes	—
157	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
158	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
159	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
160	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
161	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
162	C₄×Dic₃	C₂²×C₉	1	4	yes	—
163	C₂²×C₃²	—	2	1	no	—
164	C₂²×C₃²	—	2	2	no	—
165	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
166	C₃²×C₄	—	2	1	no	—
167	C₃²×C₄	—	2	2	no	—
168	C₃²×C₄	—	2	2	no	—
169	C₃²×C₄	C₃×#41	1	2	yes	—
170	C₃×Dic₃	—	2	4	no	—
171	C₃×Dic₃	—	2	4	no	—
172	C₂²×C₉	—	2	2	no	—
173	C₂²×C₉	—	2	1	no	—
174	C₂²×C₉	—	2	2	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
178	C₄×C₉	—	2	2	no	—
179	C₄×C₉	—	2	1	no	—
180	C₄×C₉	—	2	2	no	—
181	C₄×C₉	—	2	2	no	—
182	C₄×C₉	—	2	2	no	—
183	C₄×C₉	—	4	2	no	—
184	C₄×C₉	—	4	2	no	—
185	C₄×C₉	—	6	4	no	—
186	C₄×C₉	—	6	4	no	—
187	C₄×C₉	#235	1	2	yes	—
188	C₄⋊Q₈	—	3	1	no	—
189	#126	—	3	4	no	—
190	#125	—	3	4	no	—
191	#125	—	3	1	no	—
192	C₂²⋊Q₈	—	3	2	no	—
193	C₄×Q₈	—	3	2	no	—
194	#117	D₃×C₉	1	1	yes	characteristic
195	C₃×C₉	#541	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	2	1	no	—
197	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
198	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₃×Q₈	—	6	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	3	4	no	—
208	C₂×C₃×C₄	—	3	4	no	—
209	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
210	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
212	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
214	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
215	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
217	C₂×Dic₃	C₂³×C₉	1	4	yes	—
218	C₂×Dic₃	C₂³×C₉	1	4	yes	—
219	Dic₆	—	2	4	no	—
220	C₂×C₃²	—	4	1	no	—
221	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
222	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	2	no	—
225	C₂×C₉	—	4	2	no	—
226	C₂×C₉	—	4	1	no	—
227	C₂×C₉	C₂×#235	1	1	yes	characteristic
228	C₂×C₉	#1010	1	2	yes	—
229	C₂×Q₈	—	3	2	no	—
230	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
231	C₄⋊C₄	—	3	2	no	—
232	C₄⋊C₄	—	3	4	no	—
233	C₄⋊C₄	—	3	4	no	—
234	C₄⋊C₄	—	3	4	no	—
235	C₄⋊C₄	—	3	4	no	—
236	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
237	C₂²⋊C₄	—	3	4	no	—
238	C₂²⋊C₄	C₂×D₃×C₉	1	2	yes	—
239	C₄²	—	3	4	no	—
240	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
241	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	4	2	no	—
249	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
250	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	6	4	no	—
262	C₃×C₄	—	6	4	no	—
263	C₃×C₄	C₉×#41	1	2	yes	—
264	C₃×C₄	C₃×#235	1	2	yes	—
265	Dic₃	—	2	4	no	—
266	Dic₃	—	2	4	no	—
267	C₃²	C₃×#541	1	1	yes	characteristic
268	C₉	—	2	1	no	—
269	C₉	#5518	1	1	yes	characteristic, direct product factor
270	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
271	Q₈	—	6	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	3	4	no	—
274	C₂×C₄	—	3	4	no	—
275	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
276	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
277	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	—	4	1	no	—
283	C₂×C₃	—	4	1	no	—
284	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
285	C₂×C₃	C₉×2₋¹⁺⁴	1	2	yes	—
286	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
287	C₂×C₃	C₃×#1010	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	6	4	no	—
295	C₄	—	6	4	no	—
296	C₄	C₉×#235	1	2	yes	—
297	C₃	—	2	1	no	—
298	C₃	C₉×#541	1	1	yes	characteristic, complemented
299	C₃	C₃×#5518	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	C₂×C₉×#235	1	1	yes	characteristic
302	C₂	C₉×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂³×C₃²	C₂³×C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117

Complements

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

N	H ≅ G/N	type
3. C₉×#889	C₂	semidirect
5. C₉×#870	C₂	semidirect
6. C₉×#869	C₂	semidirect
7. C₄×C₉×Dic₆	C₂	semidirect
48. #5518	C₉	direct (N × H)
99. #889	C₂×C₉	semidirect
101. #870	C₂×C₉	semidirect
102. #869	C₂×C₉	semidirect
103. C₄×Dic₆	C₂×C₉	semidirect
195. C₃×C₉	#541	semidirect
269. C₉	#5518	direct (N × H)
298. C₃	C₉×#541	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₉×#924	2
C₉×#889	2
C₉×#879	2
C₉×#870	2
C₉×#869	2
C₄×C₉×Dic₆	2
C₃×#5518	3
C₉×#541	3

Subgroups — C9×#5518 #13152479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5518 #13152479