Subgroups — C₉×#5720 #13153270

Order 1728.

864 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#939	C₂	1	6	yes	maximal, complemented
3	C₉×#937	C₂	1	3	yes	maximal, complemented
4	C₉×#935	C₂	1	3	yes	maximal, complemented
5	C₉×#876	C₂	1	2	yes	maximal, complemented
6	C₃×#5720	C₃	1	1	yes	maximal, characteristic
7	C₉×#560	—	3	1	no	maximal
8	C₂³×D₃×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄×C₉	C₂²	1	3	yes	—
10	C₂×C₃×D₄×C₉	C₂²	1	3	yes	complemented
11	C₂×C₉×#67	C₂²	1	6	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	3	yes	complemented
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
14	C₉×#217	C₂²	1	3	yes	—
15	C₉×#217	C₂²	1	3	yes	complemented
16	C₉×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
17	C₉×Dic₃⋊C₄	C₂²	1	6	yes	complemented
18	C₃²×#127	C₂×C₃	1	1	yes	characteristic
19	C₃×#939	C₂×C₃	1	6	yes	—
20	C₃×#937	C₂×C₃	1	3	yes	—
21	C₃×#935	C₂×C₃	1	3	yes	—
22	C₃×#876	C₂×C₃	1	2	yes	—
23	C₉×#127	—	2	1	no	—
24	C₉×#127	D₃	1	1	yes	characteristic, complemented
25	C₉×#126	—	3	2	no	—
26	C₉×#123	—	3	3	no	—
27	C₉×#121	—	3	6	no	—
28	C₉×#120	—	3	3	no	—
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂³×C₃×C₉	C₂³	1	3	yes	—
31	C₂²×D₃×C₉	—	4	3	no	—
32	C₂²×D₃×C₉	C₂³	1	2	yes	—
33	C₃×D₄×C₉	—	4	3	no	—
34	C₃×D₄×C₉	—	4	3	no	—
35	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
36	C₉×#67	—	4	6	no	—
37	C₂×C₉×Dic₃	—	4	3	no	—
38	C₂×C₉×Dic₃	C₂³	1	6	yes	—
39	C₃×#560	—	3	1	no	—
40	#5720	C₉	1	1	yes	characteristic, direct product factor
41	C₂³×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃²×D₄	C₂²×C₃	1	3	yes	—
43	C₂×C₃²×D₄	C₂²×C₃	1	3	yes	—
44	C₂×C₃×#67	C₂²×C₃	1	6	yes	—
45	C₂²×C₃×Dic₃	C₂²×C₃	1	3	yes	—
46	C₂⁴×C₉	—	3	1	no	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₃×#217	C₂²×C₃	1	3	yes	—
49	C₃×#217	C₂²×C₃	1	3	yes	—
50	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	6	yes	—
51	C₃×Dic₃⋊C₄	C₂²×C₃	1	6	yes	—
52	C₂×D₄×C₉	—	2	3	no	—
53	C₂×D₄×C₉	—	2	3	no	—
54	C₂×D₄×C₉	—	3	6	no	—
55	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
56	C₂×D₄×C₉	C₂×D₃	1	3	yes	complemented
57	C₂²×C₄×C₉	—	3	3	no	—
58	C₉×C₄⋊C₄	—	3	6	no	—
59	C₉×C₂²⋊C₄	—	3	3	no	—
60	C₉×C₂²⋊C₄	—	3	3	no	—
61	C₉×C₂²⋊C₄	—	3	6	no	—
62	C₄²×C₉	—	2	1	no	—
63	C₄²×C₉	C₂×D₃	1	1	yes	characteristic, complemented
64	C₂²×C₃×C₉	—	2	3	no	—
65	C₂²×C₃×C₉	—	2	6	no	—
66	C₂²×C₃×C₉	—	2	3	no	—
67	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
68	C₂×D₃×C₉	—	2	6	no	—
69	C₂×D₃×C₉	—	4	4	no	—
70	C₃×C₄×C₉	—	2	3	no	—
71	C₉×Dic₃	—	2	6	no	—
72	C₃×#127	—	2	1	no	—
73	C₃×#127	C₂×C₉	1	1	yes	characteristic, complemented
74	C₃×#127	C₃×D₃	1	1	yes	characteristic
75	C₃×#126	—	3	2	no	—
76	C₃×#123	—	3	3	no	—
77	C₃×#121	—	3	6	no	—
78	C₃×#120	—	3	3	no	—
79	#939	C₂×C₉	1	6	yes	complemented
80	#937	C₂×C₉	1	3	yes	complemented
81	#935	C₂×C₉	1	3	yes	complemented
82	#876	C₂×C₉	1	2	yes	complemented
83	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
84	C₂³×C₃²	C₂³×C₃	1	3	yes	—
85	C₂²×C₃×D₃	—	4	3	no	—
86	C₂²×C₃×D₃	C₂³×C₃	1	2	yes	—
87	C₃²×D₄	—	4	3	no	—
88	C₃²×D₄	—	4	3	no	—
89	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
90	C₃×#67	—	4	6	no	—
91	C₂×C₃×Dic₃	—	4	3	no	—
92	C₂×C₃×Dic₃	C₂³×C₃	1	6	yes	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	—	2	3	no	—
95	C₂³×C₉	—	3	2	no	—
96	C₂³×C₉	—	12	3	no	—
97	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
98	C₂³×C₉	C₂²×D₃	1	3	yes	—
99	D₄×C₉	—	4	3	no	—
100	D₄×C₉	—	4	3	no	—
101	D₄×C₉	—	4	6	no	—
102	D₄×C₉	—	4	6	no	—
103	D₄×C₉	—	12	6	no	—
104	C₂×C₄×C₉	—	2	3	no	—
105	C₂×C₄×C₉	—	3	6	no	—
106	C₂×C₄×C₉	—	12	3	no	—
107	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
108	#560	—	3	1	no	Sylow
109	C₂×C₃×C₉	—	4	1	no	—
110	C₂×C₃×C₉	—	4	3	no	—
111	C₂×C₃×C₉	H(5,𝔽₂)	1	3	yes	—
112	D₃×C₉	—	4	2	no	—
113	C₂⁴×C₃	—	3	1	no	—
114	C₂³×D₃	C₂²×C₉	1	1	yes	characteristic
115	C₂×C₃×D₄	—	2	3	no	—
116	C₂×C₃×D₄	—	2	3	no	—
117	C₂×C₃×D₄	—	3	6	no	—
118	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
119	C₂×C₃×D₄	C₂²×C₉	1	3	yes	complemented
120	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
121	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
122	C₂²×C₃×C₄	—	3	3	no	—
123	C₂×#67	C₂²×C₉	1	6	yes	—
124	C₂²×Dic₃	C₂²×C₉	1	3	yes	complemented
125	C₃×C₄⋊C₄	—	3	6	no	—
126	C₃×C₂²⋊C₄	—	3	3	no	—
127	C₃×C₂²⋊C₄	—	3	3	no	—
128	C₃×C₂²⋊C₄	—	3	6	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic, complemented
131	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	#217	C₂²×C₉	1	3	yes	—
133	#217	C₂²×C₉	1	3	yes	complemented
134	(C₂×D₃)⋊C₄	C₂²×C₉	1	6	yes	—
135	Dic₃⋊C₄	C₂²×C₉	1	6	yes	complemented
136	C₂²×C₃²	—	2	3	no	—
137	C₂²×C₃²	—	2	6	no	—
138	C₂²×C₃²	—	2	3	no	—
139	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
140	C₂×C₃×D₃	—	2	6	no	—
141	C₂×C₃×D₃	—	4	4	no	—
142	C₃²×C₄	—	2	3	no	—
143	C₃×Dic₃	—	2	6	no	—
144	C₂²×C₉	—	2	3	no	—
145	C₂²×C₉	—	2	6	no	—
146	C₂²×C₉	—	2	3	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	4	3	no	—
149	C₂²×C₉	—	4	6	no	—
150	C₂²×C₉	—	4	3	no	—
151	C₂²×C₉	—	6	6	no	—
152	C₂²×C₉	—	12	4	no	—
153	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
154	C₄×C₉	—	2	6	no	—
155	C₄×C₉	—	2	3	no	—
156	C₄×C₉	—	6	6	no	—
157	#127	D₃×C₉	1	1	yes	characteristic, complemented
158	#126	—	3	2	no	—
159	#123	—	3	3	no	—
160	#121	—	3	6	no	—
161	#120	—	3	3	no	—
162	C₃×C₉	#560	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	—	2	3	no	—
165	C₂³×C₃	—	3	2	no	—
166	C₂³×C₃	—	12	3	no	—
167	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
168	C₂³×C₃	C₂³×C₉	1	3	yes	—
169	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
171	C₂²×D₃	—	4	3	no	—
172	C₂²×D₃	C₂³×C₉	1	2	yes	—
173	C₃×D₄	—	4	3	no	—
174	C₃×D₄	—	4	3	no	—
175	C₃×D₄	—	4	3	no	—
176	C₃×D₄	—	4	3	no	—
177	C₃×D₄	—	4	6	no	—
178	C₃×D₄	—	4	6	no	—
179	C₃×D₄	—	12	6	no	—
180	C₂×C₃×C₄	—	2	3	no	—
181	C₂×C₃×C₄	—	3	6	no	—
182	C₂×C₃×C₄	—	12	3	no	—
183	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
184	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
185	#67	—	4	6	no	—
186	C₂×Dic₃	—	4	3	no	—
187	C₂×Dic₃	C₂³×C₉	1	6	yes	—
188	C₂×C₃²	—	4	1	no	—
189	C₂×C₃²	—	4	3	no	—
190	C₂×C₃²	C₃×H(5,𝔽₂)	1	3	yes	—
191	C₃×D₃	—	4	2	no	—
192	C₂×C₉	—	2	3	no	—
193	C₂×C₉	—	4	3	no	—
194	C₂×C₉	—	4	1	no	—
195	C₂×C₉	—	8	1	no	—
196	C₂×C₉	—	8	3	no	—
197	C₂×C₉	—	12	2	no	—
198	C₂×C₉	#1004	1	3	yes	—
199	C₂⁴	—	3	1	no	—
200	C₂×D₄	—	3	6	no	—
201	C₂×D₄	C₂×D₃×C₉	1	3	yes	—
202	C₂×D₄	C₂×D₃×C₉	1	3	yes	complemented
203	C₂²×C₄	—	3	3	no	—
204	C₄⋊C₄	—	3	6	no	—
205	C₂²⋊C₄	—	3	3	no	—
206	C₂²⋊C₄	—	3	3	no	—
207	C₂²⋊C₄	—	3	6	no	—
208	C₄²	C₂×D₃×C₉	1	1	yes	characteristic, complemented
209	C₂²×C₃	—	2	3	no	—
210	C₂²×C₃	—	2	6	no	—
211	C₂²×C₃	—	2	3	no	—
212	C₂²×C₃	—	2	3	no	—
213	C₂²×C₃	—	2	6	no	—
214	C₂²×C₃	—	2	3	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	4	3	no	—
217	C₂²×C₃	—	4	6	no	—
218	C₂²×C₃	—	4	3	no	—
219	C₂²×C₃	—	6	6	no	—
220	C₂²×C₃	—	12	4	no	—
221	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
222	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
223	C₂×D₃	—	2	6	no	—
224	C₂×D₃	—	4	4	no	—
225	C₃×C₄	—	2	6	no	—
226	C₃×C₄	—	2	3	no	—
227	C₃×C₄	—	2	3	no	—
228	C₃×C₄	—	6	6	no	—
229	Dic₃	—	2	6	no	—
230	C₃²	C₃×#560	1	1	yes	characteristic
231	C₉	—	2	1	no	—
232	C₉	#5720	1	1	yes	characteristic, direct product factor
233	C₂³	—	3	2	no	—
234	C₂³	—	12	3	no	—
235	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
236	C₂³	C₂²×D₃×C₉	1	3	yes	—
237	D₄	—	4	3	no	—
238	D₄	—	4	3	no	—
239	D₄	—	12	6	no	—
240	C₂×C₄	—	3	6	no	—
241	C₂×C₄	—	12	3	no	—
242	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
243	C₂×C₃	—	2	3	no	—
244	C₂×C₃	—	4	1	no	—
245	C₂×C₃	—	4	3	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	—	4	3	no	—
248	C₂×C₃	—	8	1	no	—
249	C₂×C₃	—	8	3	no	—
250	C₂×C₃	—	12	2	no	—
251	C₂×C₃	C₉×H(5,𝔽₂)	1	3	yes	—
252	C₂×C₃	C₃×#1004	1	3	yes	—
253	D₃	—	4	2	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	6	6	no	—
258	C₂²	—	12	4	no	—
259	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
260	C₄	—	2	3	no	—
261	C₄	—	6	6	no	—
262	C₃	—	2	1	no	—
263	C₃	C₉×#560	1	1	yes	characteristic, complemented
264	C₃	C₃×#5720	1	1	yes	characteristic
265	C₂	—	4	1	no	—
266	C₂	—	4	3	no	—
267	C₂	—	12	2	no	—
268	C₂	C₉×#1004	1	3	yes	—

	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×#5720	C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₉×#560
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. C₃×C₉×#127	C₂	semidirect
2. C₉×#939	C₂	semidirect
3. C₉×#937	C₂	semidirect
4. C₉×#935	C₂	semidirect
5. C₉×#876	C₂	semidirect
10. C₂×C₃×D₄×C₉	C₂²	semidirect
12. C₂²×C₉×Dic₃	C₂²	semidirect
13. C₃×C₄²×C₉	C₂²	semidirect
15. C₉×#217	C₂²	semidirect
17. C₉×Dic₃⋊C₄	C₂²	semidirect
24. C₉×#127	D₃	semidirect
40. #5720	C₉	direct (N × H)
56. C₂×D₄×C₉	C₂×D₃	semidirect
63. C₄²×C₉	C₂×D₃	semidirect
73. C₃×#127	C₂×C₉	semidirect
79. #939	C₂×C₉	semidirect
80. #937	C₂×C₉	semidirect
81. #935	C₂×C₉	semidirect
82. #876	C₂×C₉	semidirect
119. C₂×C₃×D₄	C₂²×C₉	semidirect
124. C₂²×Dic₃	C₂²×C₉	semidirect
130. C₃×C₄²	C₂²×C₉	semidirect
133. #217	C₂²×C₉	semidirect
135. Dic₃⋊C₄	C₂²×C₉	semidirect
157. #127	D₃×C₉	semidirect
162. C₃×C₉	#560	semidirect
202. C₂×D₄	C₂×D₃×C₉	semidirect
208. C₄²	C₂×D₃×C₉	semidirect
232. C₉	#5720	direct (N × H)
263. C₃	C₉×#560	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#127	2
C₉×#939	2
C₉×#937	2
C₉×#935	2
C₉×#876	2
C₃×#5720	3
C₉×#560	3

Subgroups — C9×#5720 #13153270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5720 #13153270