Subgroups — C₉×#5688 #13153163

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#896	C₂	1	4	yes	maximal
3	C₉×#892	C₂	1	2	yes	maximal
4	C₉×#888	C₂	1	2	yes	maximal, complemented
5	C₉×#887	C₂	1	4	yes	maximal, complemented
6	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic
7	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5688	C₃	1	1	yes	maximal, characteristic
9	C₉×#533	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	4	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
17	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
19	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₄×C₉×Dic₃	C₂²	1	2	yes	—
21	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
22	C₄×C₉×Dic₃	C₂²	1	4	yes	—
23	C₃²×#125	C₂×C₃	1	1	yes	characteristic
24	C₃×#896	C₂×C₃	1	4	yes	—
25	C₃×#892	C₂×C₃	1	2	yes	—
26	C₃×#888	C₂×C₃	1	2	yes	—
27	C₃×#887	C₂×C₃	1	4	yes	—
28	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
29	C₃×#870	C₂×C₃	1	1	yes	characteristic
30	C₉×C₄⋊Q₈	—	3	2	no	—
31	C₉×#125	—	2	1	no	—
32	C₉×#125	—	3	1	no	—
33	C₉×#125	D₃	1	1	yes	characteristic, complemented
34	C₉×C₂²⋊Q₈	—	3	4	no	—
35	C₄×Q₈×C₉	—	3	4	no	—
36	C₉×#117	—	3	2	no	—
37	C₂×C₄²×C₉	—	3	1	no	—
38	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	C₂³	1	4	yes	—
43	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
44	C₂×C₉×Dic₃	C₂³	1	2	yes	—
45	C₄×D₃×C₉	—	2	4	no	—
46	C₄×D₃×C₉	Q₈	1	4	yes	complemented
47	C₉×Dic₆	—	2	8	no	—
48	C₃×#533	—	3	1	no	—
49	#5688	C₉	1	1	yes	characteristic, direct product factor
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
53	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₃×(C₂×D₃)⋊C₄	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×C₄×Dic₃	C₂²×C₃	1	4	yes	—
61	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
62	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
63	C₂×Q₈×C₉	—	3	4	no	—
64	C₂²×C₄×C₉	—	3	1	no	—
65	C₂²×C₄×C₉	—	3	2	no	—
66	C₉×C₄⋊C₄	—	2	2	no	—
67	C₉×C₄⋊C₄	—	2	4	no	—
68	C₉×C₄⋊C₄	—	3	2	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×C₄⋊C₄	C₂×D₃	1	4	yes	—
73	C₉×C₂²⋊C₄	—	3	4	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	—	3	2	no	—
76	C₄²×C₉	—	3	1	no	—
77	C₄²×C₉	—	3	4	no	—
78	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
80	C₂×D₃×C₉	—	2	2	no	—
81	C₂×D₃×C₉	C₂×Q₈	1	2	yes	—
82	C₃×C₄×C₉	—	2	4	no	—
83	C₃×C₄×C₉	—	2	2	no	—
84	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
85	C₉×Dic₃	—	2	4	no	—
86	C₉×Dic₃	C₂×Q₈	1	2	yes	—
87	C₉×Dic₃	#41	1	4	yes	—
88	C₃×C₄⋊Q₈	—	3	2	no	—
89	C₃×#125	—	2	1	no	—
90	C₃×#125	—	3	1	no	—
91	C₃×#125	C₂×C₉	1	1	yes	characteristic, complemented
92	C₃×#125	C₃×D₃	1	1	yes	characteristic
93	C₃×C₂²⋊Q₈	—	3	4	no	—
94	C₃×C₄×Q₈	—	3	4	no	—
95	C₃×#117	—	3	2	no	—
96	C₂×C₃×C₄²	—	3	1	no	—
97	#896	C₂×C₉	1	4	yes	—
98	#892	C₂×C₉	1	2	yes	—
99	#888	C₂×C₉	1	2	yes	complemented
100	#887	C₂×C₉	1	4	yes	complemented
101	C₄²×D₃	C₂×C₉	1	1	yes	characteristic
102	#870	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
105	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
106	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
108	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
110	C₃×C₄×D₃	—	2	4	no	—
111	C₃×C₄×D₃	C₃×Q₈	1	4	yes	—
112	C₃×Dic₆	—	2	8	no	—
113	C₂³×C₉	—	3	1	no	—
114	Q₈×C₉	—	6	8	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	2	4	no	—
118	C₂×C₄×C₉	—	3	1	no	—
119	C₂×C₄×C₉	—	3	2	no	—
120	C₂×C₄×C₉	—	3	4	no	—
121	C₂×C₄×C₉	—	3	4	no	—
122	C₂×C₄×C₉	—	6	4	no	—
123	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
124	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
125	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
126	#533	—	3	1	no	Sylow
127	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₉	C₂×#41	1	2	yes	—
129	D₃×C₉	—	2	2	no	—
130	C₂×C₃×Q₈	—	3	4	no	—
131	C₂²×C₃×C₄	—	3	1	no	—
132	C₂²×C₃×C₄	—	3	2	no	—
133	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
134	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
135	C₂×Dic₆	C₂²×C₉	1	4	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₄⋊C₄	—	3	4	no	—
139	C₃×C₄⋊C₄	—	3	4	no	—
140	C₃×C₄⋊C₄	—	3	2	no	—
141	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
142	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
143	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
145	C₃×C₂²⋊C₄	—	3	4	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	3	2	no	—
148	C₃×C₄²	—	3	1	no	—
149	C₃×C₄²	—	3	4	no	—
150	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
151	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
152	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
153	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
154	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
155	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
156	C₄×Dic₃	C₂²×C₉	1	2	yes	—
157	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
158	C₄×Dic₃	C₂²×C₉	1	4	yes	—
159	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	2	no	—
161	C₂×C₃×D₃	C₂×C₃×Q₈	1	2	yes	—
162	C₃²×C₄	—	2	4	no	—
163	C₃²×C₄	—	2	2	no	—
164	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
165	C₃×Dic₃	—	2	4	no	—
166	C₃×Dic₃	C₂×C₃×Q₈	1	2	yes	—
167	C₃×Dic₃	C₃×#41	1	4	yes	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	—	3	2	no	—
170	C₂²×C₉	—	6	2	no	—
171	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	—	2	4	no	—
174	C₄×C₉	—	2	2	no	—
175	C₄×C₉	—	3	2	no	—
176	C₄×C₉	—	3	4	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	4	no	—
179	C₄×C₉	—	6	4	no	—
180	C₄×C₉	D₃×Q₈	1	2	yes	—
181	C₄⋊Q₈	—	3	2	no	—
182	#125	—	3	1	no	—
183	#125	D₃×C₉	1	1	yes	characteristic, complemented
184	C₂²⋊Q₈	—	3	4	no	—
185	C₄×Q₈	—	3	4	no	—
186	#117	—	3	2	no	—
187	C₂×C₄²	—	3	1	no	—
188	C₃×C₉	#533	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₃	—	3	1	no	—
190	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
191	C₃×Q₈	—	6	8	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	3	1	no	—
196	C₂×C₃×C₄	—	3	2	no	—
197	C₂×C₃×C₄	—	3	4	no	—
198	C₂×C₃×C₄	—	3	4	no	—
199	C₂×C₃×C₄	—	6	4	no	—
200	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
201	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
202	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
205	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
206	C₂×Dic₃	C₂³×C₉	1	4	yes	—
207	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
208	C₂×Dic₃	C₂³×C₉	1	2	yes	—
209	C₄×D₃	—	2	4	no	—
210	C₄×D₃	Q₈×C₉	1	4	yes	complemented
211	Dic₆	—	2	8	no	—
212	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
213	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
214	C₃×D₃	—	2	2	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	2	no	—
217	C₂×C₉	—	6	2	no	—
218	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
219	C₂×C₉	D₃×#41	1	2	yes	—
220	C₂×Q₈	—	3	4	no	—
221	C₂²×C₄	—	3	1	no	—
222	C₂²×C₄	—	3	2	no	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	3	4	no	—
225	C₄⋊C₄	—	3	4	no	—
226	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
227	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
228	C₂²⋊C₄	—	3	4	no	—
229	C₄²	—	3	2	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	3	4	no	—
232	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	3	2	no	—
235	C₂²×C₃	—	6	2	no	—
236	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
237	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
238	C₂×D₃	—	2	2	no	—
239	C₂×D₃	C₂×Q₈×C₉	1	2	yes	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	4	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	3	2	no	—
246	C₃×C₄	—	3	4	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	4	4	no	—
249	C₃×C₄	—	6	4	no	—
250	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
251	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
252	Dic₃	—	2	4	no	—
253	Dic₃	C₂×Q₈×C₉	1	2	yes	—
254	Dic₃	C₉×#41	1	4	yes	—
255	C₃²	C₃×#533	1	1	yes	characteristic
256	C₉	—	2	1	no	—
257	C₉	#5688	1	1	yes	characteristic, direct product factor
258	C₂³	—	3	1	no	—
259	Q₈	—	6	8	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	3	4	no	—
263	C₂×C₄	—	3	4	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
266	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
267	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	6	2	no	—
271	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
272	C₂×C₃	C₂×C₉×#41	1	2	yes	—
273	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
274	C₂×C₃	C₃×D₃×#41	1	2	yes	—
275	D₃	—	2	2	no	—
276	C₂²	—	3	2	no	—
277	C₂²	—	6	2	no	—
278	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	3	2	no	—
282	C₄	—	3	4	no	—
283	C₄	—	6	4	no	—
284	C₄	D₃×Q₈×C₉	1	2	yes	—
285	C₃	—	2	1	no	—
286	C₃	C₉×#533	1	1	yes	characteristic, complemented
287	C₃	C₃×#5688	1	1	yes	characteristic
288	C₂	—	6	2	no	—
289	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
290	C₂	D₃×C₉×#41	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×#125	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
4. C₉×#888	C₂	semidirect
5. C₉×#887	C₂	semidirect
7. C₉×#870	C₂	semidirect
33. C₉×#125	D₃	semidirect
46. C₄×D₃×C₉	Q₈	semidirect
49. #5688	C₉	direct (N × H)
91. C₃×#125	C₂×C₉	semidirect
99. #888	C₂×C₉	semidirect
100. #887	C₂×C₉	semidirect
102. #870	C₂×C₉	semidirect
183. #125	D₃×C₉	semidirect
188. C₃×C₉	#533	semidirect
210. C₄×D₃	Q₈×C₉	semidirect
257. C₉	#5688	direct (N × H)
286. C₃	C₉×#533	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₉×#896	2
C₉×#892	2
C₉×#888	2
C₉×#887	2
C₄²×D₃×C₉	2
C₉×#870	2
C₃×#5688	3
C₉×#533	3

Subgroups — C9×#5688 #13153163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5688 #13153163