Subgroups — C₉×#4973 #13142564

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₉×C₃⋊#95	C₂	1	4	yes	maximal
5	C₃×#4973	C₃	1	1	yes	maximal, characteristic
6	C₉×#391	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
11	C₂×C₉×Dic₆	—	2	2	no	—
12	C₂×C₉×Dic₆	—	2	2	no	—
13	C₂×C₉×Dic₆	C₄	1	4	yes	complemented
14	C₃×C₉×C₄⋊C₄	—	2	2	no	—
15	C₄×C₉×Dic₃	—	2	2	no	—
16	C₂²×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₂²×Q₈×C₉	—	3	1	no	—
20	C₃×C₃⋊#95	C₂×C₃	1	4	yes	—
21	C₂×C₉×C₄⋊C₄	—	2	1	no	—
22	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
23	C₂×C₄²×C₉	—	3	1	no	—
24	C₉×#95	—	3	4	no	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₂×C₉×Dic₃	—	2	2	no	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
36	C₂×C₉×Dic₃	Q₈	1	4	yes	complemented
37	C₉×Dic₆	—	2	8	no	—
38	C₃×#391	—	3	1	no	—
39	#4973	C₉	1	1	yes	characteristic, direct product factor
40	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
43	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
44	C₂×C₃×Dic₆	—	2	2	no	—
45	C₂×C₃×Dic₆	—	2	2	no	—
46	C₂×C₃×Dic₆	C₃×C₄	1	4	yes	—
47	C₃²×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄×Dic₃	—	2	2	no	—
49	C₂×Q₈×C₉	—	3	4	no	—
50	C₂×Q₈×C₉	—	6	2	no	—
51	C₂×Q₈×C₉	—	6	2	no	—
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	—	3	2	no	—
55	C₂²×C₄×C₉	—	3	2	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	2	4	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₄²×C₉	—	6	2	no	—
61	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₃×C₉	#41	1	2	yes	—
66	C₃×C₄×C₉	—	4	2	no	—
67	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	—
68	C₉×Dic₃	—	2	4	no	—
69	C₉×Dic₃	—	2	4	no	—
70	C₂²×C₃×Q₈	—	3	1	no	—
71	C₂²×Dic₆	C₂×C₉	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	—	2	1	no	—
73	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
75	C₂×C₃×C₄²	—	3	1	no	—
76	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic
77	C₃×#95	—	3	4	no	—
78	C₃⋊#95	C₂×C₉	1	4	yes	—
79	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
84	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
85	C₂×C₃×Dic₃	—	2	2	no	—
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂×C₃×Dic₃	—	2	2	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
91	C₃×Dic₆	—	2	8	no	—
92	C₂³×C₉	—	2	1	no	—
93	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
94	Q₈×C₉	—	6	8	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	2	4	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	3	4	no	—
102	C₂×C₄×C₉	—	3	4	no	—
103	C₂×C₄×C₉	—	4	4	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	—	6	2	no	—
106	C₂×C₄×C₉	—	6	2	no	—
107	C₂×C₄×C₉	—	6	2	no	—
108	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
109	C₂×C₄×C₉	D₁₂	1	2	yes	—
110	C₂×C₄×C₉	#67	1	2	yes	—
111	#391	—	3	1	no	Sylow
112	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
114	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₃×C₉	#124	1	1	yes	characteristic
116	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	3	4	no	—
118	C₂×C₃×Q₈	—	6	2	no	—
119	C₂×C₃×Q₈	—	6	2	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	3	2	no	—
123	C₂²×C₃×C₄	—	3	2	no	—
124	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
125	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
127	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
129	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
130	C₂×Dic₆	—	2	2	no	—
131	C₂×Dic₆	—	2	2	no	—
132	C₂×Dic₆	C₄×C₉	1	4	yes	complemented
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄²	—	6	2	no	—
137	C₄×Dic₃	—	2	2	no	—
138	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
141	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
142	C₂²×C₃²	C₃×#41	1	2	yes	—
143	C₃²×C₄	—	4	2	no	—
144	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
145	C₃×Dic₃	—	2	4	no	—
146	C₃×Dic₃	—	2	4	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
154	C₂²×C₉	#237	1	2	yes	—
155	C₂²×C₉	D₃×Q₈	1	2	yes	—
156	C₂²×C₉	C₂×#67	1	1	yes	characteristic
157	C₄×C₉	—	2	4	no	—
158	C₄×C₉	—	4	4	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	—	6	4	no	—
161	C₄×C₉	—	6	4	no	—
162	C₄×C₉	(C₂×D₃)⋊C₄	1	4	yes	—
163	C₂²×Q₈	—	3	1	no	—
164	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
165	C₂×C₄²	—	3	1	no	—
166	#95	—	3	4	no	—
167	C₃×C₉	#391	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₃	—	2	1	no	—
169	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
170	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₃×Q₈	—	6	8	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	3	4	no	—
181	C₂×C₃×C₄	—	3	4	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	—	6	2	no	—
184	C₂×C₃×C₄	—	6	2	no	—
185	C₂×C₃×C₄	—	6	2	no	—
186	C₂×C₃×C₄	—	6	2	no	—
187	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
188	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
189	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
190	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
191	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
192	C₂×C₃×C₄	C₃×#67	1	2	yes	—
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	2	2	no	—
195	C₂×Dic₃	—	2	2	no	—
196	C₂×Dic₃	—	2	2	no	—
197	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
198	C₂×Dic₃	Q₈×C₉	1	4	yes	complemented
199	Dic₆	—	2	8	no	—
200	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₃²	C₃×C₄×Q₈	1	2	yes	—
202	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
203	C₂×C₃²	C₃×#124	1	1	yes	characteristic
204	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
205	C₂×C₉	—	2	2	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	#887	1	2	yes	—
211	C₂×C₉	#897	1	2	yes	—
212	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×C₉	#936	1	1	yes	characteristic
214	C₂×C₉	#944	1	1	yes	characteristic
215	C₂×Q₈	—	3	4	no	—
216	C₂×Q₈	—	6	2	no	—
217	C₂×Q₈	—	6	2	no	—
218	C₂²×C₄	—	3	2	no	—
219	C₂²×C₄	—	3	2	no	—
220	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
221	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄²	—	6	2	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
230	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
231	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
232	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
233	C₂²×C₃	C₉×#41	1	2	yes	—
234	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
236	C₂²×C₃	C₃×#237	1	2	yes	—
237	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
238	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	4	2	no	—
243	C₃×C₄	—	6	4	no	—
244	C₃×C₄	—	6	4	no	—
245	C₃×C₄	C₉×C₂²⋊C₄	1	4	yes	—
246	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
247	Dic₃	—	2	4	no	—
248	Dic₃	—	2	4	no	—
249	C₃²	C₃×#391	1	1	yes	characteristic
250	C₉	—	2	1	no	—
251	C₉	#4973	1	1	yes	characteristic, direct product factor
252	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
253	Q₈	—	6	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	3	4	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	—	6	2	no	—
262	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
263	C₂×C₄	C₉×D₁₂	1	2	yes	—
264	C₂×C₄	C₉×#67	1	2	yes	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₃	C₄×Q₈×C₉	1	2	yes	—
272	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₃	C₉×#124	1	1	yes	characteristic
274	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
275	C₂×C₃	C₃×#887	1	2	yes	—
276	C₂×C₃	C₃×#897	1	2	yes	—
277	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
278	C₂×C₃	C₃×#936	1	1	yes	characteristic
279	C₂×C₃	C₃×#944	1	1	yes	characteristic
280	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
281	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
282	C₂²	C₉×#237	1	2	yes	—
283	C₂²	D₃×Q₈×C₉	1	2	yes	—
284	C₂²	C₂×C₉×#67	1	1	yes	characteristic
285	C₄	—	4	2	no	—
286	C₄	—	6	4	no	—
287	C₄	—	6	4	no	—
288	C₄	C₉×(C₂×D₃)⋊C₄	1	4	yes	—
289	C₃	—	2	1	no	—
290	C₃	C₉×#391	1	1	yes	characteristic, complemented
291	C₃	C₃×#4973	1	1	yes	characteristic
292	C₂	C₉×#887	1	2	yes	—
293	C₂	C₉×#897	1	2	yes	—
294	C₂	C₂×C₉×(C₂×D₃)⋊C₄	1	1	yes	characteristic
295	C₂	C₉×#936	1	1	yes	characteristic
296	C₂	C₉×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#391
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₂×C₉×Dic₆	C₄	semidirect
36. C₂×C₉×Dic₃	Q₈	semidirect
39. #4973	C₉	direct (N × H)
132. C₂×Dic₆	C₄×C₉	semidirect
167. C₃×C₉	#391	semidirect
198. C₂×Dic₃	Q₈×C₉	semidirect
251. C₉	#4973	direct (N × H)
290. C₃	C₉×#391	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Dic₆	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₄×C₉×Dic₃	2
C₉×C₃⋊#95	2
C₃×#4973	3
C₉×#391	3

Subgroups — C9×#4973 #13142564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4973 #13142564