Subgroups — C₉×#5717 #13153268

Order 1728.

1392 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₄×C₉	C₂	1	4	yes	maximal, complemented
2	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#938	C₂	1	4	yes	maximal, complemented
4	C₉×#937	C₂	1	4	yes	maximal, complemented
5	C₄×C₉×D₁₂	C₂	1	2	yes	maximal, complemented
6	C₃×#5717	C₃	1	1	yes	maximal, characteristic
7	D₄²×C₉	—	3	1	no	maximal
8	C₂³×D₃×C₉	C₂²	1	4	yes	complemented
9	C₂×C₃×D₄×C₉	C₂²	1	4	yes	complemented
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	complemented
11	C₂×C₉×#67	C₂²	1	8	yes	complemented
12	D₃×D₄×C₉	—	2	16	no	—
13	C₂×C₉×D₁₂	C₂²	1	2	yes	—
14	C₂×C₄×D₃×C₉	C₂²	1	4	yes	complemented
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
16	C₉×#217	C₂²	1	4	yes	complemented
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
18	C₉×C₄⋊Dic₃	C₂²	1	2	yes	complemented
19	C₂×C₃×D₃×D₄	C₂×C₃	1	4	yes	—
20	C₃²×#127	C₂×C₃	1	1	yes	characteristic
21	C₃×#938	C₂×C₃	1	4	yes	—
22	C₃×#937	C₂×C₃	1	4	yes	—
23	C₃×C₄×D₁₂	C₂×C₃	1	2	yes	—
24	C₂²×D₄×C₉	—	3	4	no	—
25	C₉×#127	—	2	1	no	—
26	C₉×#127	D₃	1	1	yes	characteristic, complemented
27	C₉×#121	—	3	4	no	—
28	C₉×#120	—	3	4	no	—
29	C₄×D₄×C₉	—	3	2	no	—
30	C₂³×C₃×C₉	C₂³	1	4	yes	—
31	C₂²×D₃×C₉	—	2	16	no	—
32	C₂²×D₃×C₉	—	4	4	no	—
33	C₂²×D₃×C₉	C₂³	1	4	yes	—
34	C₃×D₄×C₉	—	2	8	no	—
35	C₃×D₄×C₉	—	4	2	no	—
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
38	C₉×#67	—	2	16	no	—
39	C₂×C₉×Dic₃	C₂³	1	4	yes	—
40	C₉×D₁₂	D₄	1	8	yes	complemented
41	C₄×D₃×C₉	—	2	8	no	—
42	C₃×D₄²	—	3	1	no	—
43	#5717	C₉	1	1	yes	characteristic, direct product factor
44	C₂³×C₃×D₃	C₂²×C₃	1	4	yes	—
45	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
46	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×#67	C₂²×C₃	1	8	yes	—
48	C₃×D₃×D₄	—	2	16	no	—
49	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
51	C₂⁴×C₉	—	3	4	no	—
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×#217	C₂²×C₃	1	4	yes	—
54	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
56	C₂×D₄×C₉	—	2	4	no	—
57	C₂×D₄×C₉	—	2	2	no	—
58	C₂×D₄×C₉	—	3	8	no	—
59	C₂×D₄×C₉	—	3	2	no	—
60	C₂×D₄×C₉	—	6	16	no	—
61	C₂×D₄×C₉	C₂×D₃	1	4	yes	complemented
62	C₂×D₄×C₉	C₂×D₃	1	2	yes	complemented
63	C₂²×C₄×C₉	—	3	4	no	—
64	C₉×C₄⋊C₄	—	3	2	no	—
65	C₉×C₂²⋊C₄	—	3	4	no	—
66	C₉×C₂²⋊C₄	—	3	4	no	—
67	C₄²×C₉	—	2	1	no	—
68	C₄²×C₉	C₂×D₃	1	1	yes	characteristic, complemented
69	C₂²×C₃×C₉	—	2	8	no	—
70	C₂²×C₃×C₉	—	2	4	no	—
71	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
72	C₂×D₃×C₉	—	2	4	no	—
73	C₂×D₃×C₉	—	2	4	no	—
74	C₂×D₃×C₉	—	4	16	no	—
75	C₂×D₃×C₉	C₂×D₄	1	8	yes	complemented
76	C₃×C₄×C₉	—	2	1	no	—
77	C₃×C₄×C₉	C₂×D₄	1	4	yes	complemented
78	C₉×Dic₃	—	2	4	no	—
79	C₂²×C₃×D₄	—	3	4	no	—
80	C₂×D₃×D₄	C₂×C₉	1	4	yes	complemented
81	C₃×#127	—	2	1	no	—
82	C₃×#127	C₂×C₉	1	1	yes	characteristic, complemented
83	C₃×#127	C₃×D₃	1	1	yes	characteristic
84	C₃×#121	—	3	4	no	—
85	C₃×#120	—	3	4	no	—
86	C₃×C₄×D₄	—	3	2	no	—
87	#938	C₂×C₉	1	4	yes	complemented
88	#937	C₂×C₉	1	4	yes	complemented
89	C₄×D₁₂	C₂×C₉	1	2	yes	complemented
90	C₂³×C₃²	C₂³×C₃	1	4	yes	—
91	C₂²×C₃×D₃	—	2	16	no	—
92	C₂²×C₃×D₃	—	4	4	no	—
93	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
94	C₃²×D₄	—	2	8	no	—
95	C₃²×D₄	—	4	2	no	—
96	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
98	C₃×#67	—	2	16	no	—
99	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
100	C₃×D₁₂	C₃×D₄	1	8	yes	—
101	C₃×C₄×D₃	—	2	8	no	—
102	C₂³×C₉	—	2	4	no	—
103	C₂³×C₉	—	3	4	no	—
104	C₂³×C₉	—	6	16	no	—
105	C₂³×C₉	—	12	4	no	—
106	C₂³×C₉	C₂²×D₃	1	4	yes	—
107	D₄×C₉	—	2	8	no	—
108	D₄×C₉	—	3	8	no	—
109	D₄×C₉	—	4	8	no	—
110	D₄×C₉	—	4	2	no	—
111	D₄×C₉	—	4	4	no	—
112	D₄×C₉	—	6	16	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	3	4	no	—
116	C₂×C₄×C₉	—	6	8	no	—
117	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
119	D₄²	—	3	1	no	Sylow
120	C₂×C₃×C₉	—	4	4	no	—
121	C₂×C₃×C₉	C₂²×D₄	1	2	yes	—
122	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
123	D₃×C₉	—	2	8	no	—
124	C₂⁴×C₃	—	3	4	no	—
125	C₂³×D₃	C₂²×C₉	1	4	yes	complemented
126	C₂×C₃×D₄	—	2	4	no	—
127	C₂×C₃×D₄	—	2	2	no	—
128	C₂×C₃×D₄	—	3	8	no	—
129	C₂×C₃×D₄	—	3	2	no	—
130	C₂×C₃×D₄	—	6	16	no	—
131	C₂×C₃×D₄	C₂²×C₉	1	4	yes	complemented
132	C₂×C₃×D₄	C₂²×C₉	1	2	yes	complemented
133	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
134	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
135	C₂²×C₃×C₄	—	3	4	no	—
136	C₂×#67	C₂²×C₉	1	8	yes	complemented
137	D₃×D₄	—	2	16	no	—
138	C₂×D₁₂	C₂²×C₉	1	2	yes	—
139	C₂×C₄×D₃	C₂²×C₉	1	4	yes	complemented
140	C₃×C₄⋊C₄	—	3	2	no	—
141	C₃×C₂²⋊C₄	—	3	4	no	—
142	C₃×C₂²⋊C₄	—	3	4	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic, complemented
145	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
146	#217	C₂²×C₉	1	4	yes	complemented
147	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	complemented
148	C₄⋊Dic₃	C₂²×C₉	1	2	yes	complemented
149	C₂²×C₃²	—	2	8	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
152	C₂×C₃×D₃	—	2	4	no	—
153	C₂×C₃×D₃	—	2	4	no	—
154	C₂×C₃×D₃	—	4	16	no	—
155	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
156	C₃²×C₄	—	2	1	no	—
157	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
158	C₃×Dic₃	—	2	4	no	—
159	C₂²×C₉	—	2	8	no	—
160	C₂²×C₉	—	2	4	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	3	8	no	—
163	C₂²×C₉	—	4	8	no	—
164	C₂²×C₉	—	4	4	no	—
165	C₂²×C₉	—	6	4	no	—
166	C₂²×C₉	—	6	4	no	—
167	C₂²×C₉	—	12	16	no	—
168	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	4	no	—
172	C₄×C₉	—	6	4	no	—
173	C₄×C₉	D₃×D₄	1	4	yes	complemented
174	C₂²×D₄	—	3	4	no	—
175	#127	D₃×C₉	1	1	yes	characteristic, complemented
176	#121	—	3	4	no	—
177	#120	—	3	4	no	—
178	C₄×D₄	—	3	2	no	—
179	C₃×C₉	D₄²	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	3	4	no	—
182	C₂³×C₃	—	6	16	no	—
183	C₂³×C₃	—	12	4	no	—
184	C₂³×C₃	C₂³×C₉	1	4	yes	—
185	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
186	C₂²×D₃	—	2	16	no	—
187	C₂²×D₃	—	4	4	no	—
188	C₂²×D₃	C₂³×C₉	1	4	yes	—
189	C₃×D₄	—	2	8	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	—	3	8	no	—
192	C₃×D₄	—	4	8	no	—
193	C₃×D₄	—	4	2	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	4	4	no	—
196	C₃×D₄	—	6	16	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	3	4	no	—
200	C₂×C₃×C₄	—	6	8	no	—
201	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
205	#67	—	2	16	no	—
206	C₂×Dic₃	C₂³×C₉	1	4	yes	—
207	D₁₂	D₄×C₉	1	8	yes	complemented
208	C₄×D₃	—	2	8	no	—
209	C₂×C₃²	—	4	4	no	—
210	C₂×C₃²	C₂²×C₃×D₄	1	2	yes	—
211	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
212	C₃×D₃	—	2	8	no	—
213	C₂×C₉	—	2	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	4	4	no	—
216	C₂×C₉	—	6	8	no	—
217	C₂×C₉	—	8	4	no	—
218	C₂×C₉	C₂×D₃×D₄	1	2	yes	—
219	C₂×C₉	#1004	1	1	yes	characteristic
220	C₂⁴	—	3	4	no	—
221	C₂×D₄	—	3	8	no	—
222	C₂×D₄	—	3	2	no	—
223	C₂×D₄	—	6	16	no	—
224	C₂×D₄	C₂×D₃×C₉	1	4	yes	complemented
225	C₂×D₄	C₂×D₃×C₉	1	2	yes	complemented
226	C₂²×C₄	—	3	4	no	—
227	C₄⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	4	no	—
229	C₂²⋊C₄	—	3	4	no	—
230	C₄²	C₂×D₃×C₉	1	1	yes	characteristic, complemented
231	C₂²×C₃	—	2	8	no	—
232	C₂²×C₃	—	2	4	no	—
233	C₂²×C₃	—	2	8	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	3	8	no	—
237	C₂²×C₃	—	4	8	no	—
238	C₂²×C₃	—	4	4	no	—
239	C₂²×C₃	—	6	4	no	—
240	C₂²×C₃	—	6	4	no	—
241	C₂²×C₃	—	12	16	no	—
242	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
243	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
244	C₂×D₃	—	2	4	no	—
245	C₂×D₃	—	2	4	no	—
246	C₂×D₃	—	4	16	no	—
247	C₂×D₃	C₂×D₄×C₉	1	8	yes	complemented
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	6	4	no	—
253	C₃×C₄	C₂×D₄×C₉	1	4	yes	complemented
254	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
255	Dic₃	—	2	4	no	—
256	C₃²	C₃×D₄²	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	#5717	1	1	yes	characteristic, direct product factor
259	C₂³	—	3	4	no	—
260	C₂³	—	6	16	no	—
261	C₂³	—	12	4	no	—
262	C₂³	C₂²×D₃×C₉	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	3	8	no	—
265	D₄	—	4	2	no	—
266	D₄	—	6	16	no	—
267	C₂×C₄	—	3	4	no	—
268	C₂×C₄	—	6	8	no	—
269	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	4	4	no	—
274	C₂×C₃	—	4	4	no	—
275	C₂×C₃	—	6	8	no	—
276	C₂×C₃	—	8	4	no	—
277	C₂×C₃	C₂²×D₄×C₉	1	2	yes	—
278	C₂×C₃	C₉×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂×C₃	C₂×C₃×D₃×D₄	1	2	yes	—
280	C₂×C₃	C₃×#1004	1	1	yes	characteristic
281	D₃	—	2	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	3	8	no	—
285	C₂²	—	6	4	no	—
286	C₂²	—	6	4	no	—
287	C₂²	—	12	16	no	—
288	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	6	4	no	—
291	C₄	D₃×D₄×C₉	1	4	yes	complemented
292	C₃	—	2	1	no	—
293	C₃	D₄²×C₉	1	1	yes	characteristic, complemented
294	C₃	C₃×#5717	1	1	yes	characteristic
295	C₂	—	4	4	no	—
296	C₂	—	6	8	no	—
297	C₂	C₂×D₃×D₄×C₉	1	2	yes	—
298	C₂	C₉×#1004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5717	C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	D₄²×C₉
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#127

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₄×C₉	C₂	semidirect
2. C₃×C₉×#127	C₂	semidirect
3. C₉×#938	C₂	semidirect
4. C₉×#937	C₂	semidirect
5. C₄×C₉×D₁₂	C₂	semidirect
8. C₂³×D₃×C₉	C₂²	semidirect
9. C₂×C₃×D₄×C₉	C₂²	semidirect
10. C₂×C₃×D₄×C₉	C₂²	semidirect
11. C₂×C₉×#67	C₂²	semidirect
14. C₂×C₄×D₃×C₉	C₂²	semidirect
15. C₃×C₄²×C₉	C₂²	semidirect
16. C₉×#217	C₂²	semidirect
17. C₉×(C₂×D₃)⋊C₄	C₂²	semidirect
18. C₉×C₄⋊Dic₃	C₂²	semidirect
26. C₉×#127	D₃	semidirect
40. C₉×D₁₂	D₄	semidirect
43. #5717	C₉	direct (N × H)
61. C₂×D₄×C₉	C₂×D₃	semidirect
62. C₂×D₄×C₉	C₂×D₃	semidirect
68. C₄²×C₉	C₂×D₃	semidirect
75. C₂×D₃×C₉	C₂×D₄	semidirect
77. C₃×C₄×C₉	C₂×D₄	semidirect
80. C₂×D₃×D₄	C₂×C₉	semidirect
82. C₃×#127	C₂×C₉	semidirect
87. #938	C₂×C₉	semidirect
88. #937	C₂×C₉	semidirect
89. C₄×D₁₂	C₂×C₉	semidirect
125. C₂³×D₃	C₂²×C₉	semidirect
131. C₂×C₃×D₄	C₂²×C₉	semidirect
132. C₂×C₃×D₄	C₂²×C₉	semidirect
136. C₂×#67	C₂²×C₉	semidirect
139. C₂×C₄×D₃	C₂²×C₉	semidirect
144. C₃×C₄²	C₂²×C₉	semidirect
146. #217	C₂²×C₉	semidirect
147. (C₂×D₃)⋊C₄	C₂²×C₉	semidirect
148. C₄⋊Dic₃	C₂²×C₉	semidirect
173. C₄×C₉	D₃×D₄	semidirect
175. #127	D₃×C₉	semidirect
179. C₃×C₉	D₄²	semidirect
207. D₁₂	D₄×C₉	semidirect
224. C₂×D₄	C₂×D₃×C₉	semidirect
225. C₂×D₄	C₂×D₃×C₉	semidirect
230. C₄²	C₂×D₃×C₉	semidirect
247. C₂×D₃	C₂×D₄×C₉	semidirect
253. C₃×C₄	C₂×D₄×C₉	semidirect
258. C₉	#5717	direct (N × H)
291. C₄	D₃×D₄×C₉	semidirect
293. C₃	D₄²×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₄×C₉	2
C₃×C₉×#127	2
C₉×#938	2
C₉×#937	2
C₄×C₉×D₁₂	2
C₃×#5717	3
D₄²×C₉	3

Subgroups — C9×#5717 #13153268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5717 #13153268