Subgroups — C₉×#4700 #13141441

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#904	C₂	1	1	yes	maximal, characteristic
2	C₉×#901	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#805	C₂	1	2	yes	maximal, complemented
6	C₃×#4700	C₃	1	1	yes	maximal, characteristic
7	C₉×#443	—	3	1	no	maximal
8	C₉×#235	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×#208	C₂²	1	2	yes	—
13	C₉×Dic₁₂	C₄	1	1	yes	characteristic, complemented
14	C₉×D₂₄	C₄	1	1	yes	characteristic, complemented
15	C₉×#204	C₄	1	2	yes	complemented
16	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
17	D₃×C₈×C₉	—	2	2	no	—
18	C₃×#904	C₂×C₃	1	1	yes	characteristic
19	C₃×#901	C₂×C₃	1	2	yes	—
20	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
21	C₃×#819	C₂×C₃	1	1	yes	characteristic
22	C₃×#805	C₂×C₃	1	2	yes	—
23	C₉×#135	—	3	1	no	—
24	C₉×#131	—	3	2	no	—
25	C₉×#108	—	3	1	no	—
26	C₉×#104	—	3	2	no	—
27	C₄×C₈×C₉	—	2	1	no	—
28	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×#67	—	2	2	no	—
32	C₉×D₁₂	C₂×C₄	1	2	yes	—
33	C₄×D₃×C₉	—	2	2	no	—
34	C₉×Dic₆	C₂×C₄	1	2	yes	—
35	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×C₉	D₄	1	1	yes	characteristic
38	C₃×C₈×C₉	D₄	1	1	yes	characteristic
39	C₉×C₃⋊C₈	—	2	2	no	—
40	C₃×#443	—	3	1	no	—
41	#4700	C₉	1	1	yes	characteristic, direct product factor
42	C₃×#235	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×#208	C₂²×C₃	1	2	yes	—
47	C₃×Dic₁₂	C₃×C₄	1	1	yes	characteristic
48	C₃×D₂₄	C₃×C₄	1	1	yes	characteristic
49	C₃×#204	C₃×C₄	1	2	yes	—
50	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
51	C₃×D₃×C₈	—	2	2	no	—
52	C₉×#41	—	3	2	no	—
53	C₉×Q₁₆	—	3	1	no	—
54	C₉×SD₁₆	—	3	2	no	—
55	C₉×D₈	—	3	1	no	—
56	C₉×M₁₆	—	3	2	no	—
57	C₉×M₁₆	—	6	2	no	—
58	C₂×C₈×C₉	—	2	1	no	—
59	C₂×C₈×C₉	—	2	1	no	—
60	C₂×C₈×C₉	—	6	2	no	—
61	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
63	C₄²×C₉	—	2	1	no	—
64	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₃×C₉	#41	1	1	yes	characteristic
66	C₂×D₃×C₉	—	2	2	no	—
67	C₃×C₄×C₉	—	2	2	no	—
68	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₉×Dic₃	—	2	2	no	—
71	C₃×#135	—	3	1	no	—
72	C₃×#131	—	3	2	no	—
73	#904	C₂×C₉	1	1	yes	characteristic
74	#901	C₂×C₉	1	2	yes	complemented
75	C₃×#108	—	3	1	no	—
76	C₃×#104	—	3	2	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
79	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	#819	C₂×C₉	1	1	yes	characteristic, complemented
81	#805	C₂×C₉	1	2	yes	complemented
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₃×#67	—	2	2	no	—
85	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
86	C₃×C₄×D₃	—	2	2	no	—
87	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
88	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
89	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
90	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
91	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
92	C₃×C₃⋊C₈	—	2	2	no	—
93	Q₈×C₉	—	3	2	no	—
94	D₄×C₉	—	3	2	no	—
95	D₄×C₉	—	6	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	6	2	no	—
100	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
101	C₈×C₉	—	2	1	no	—
102	C₈×C₉	—	2	1	no	—
103	C₈×C₉	—	2	1	no	—
104	C₈×C₉	—	2	1	no	—
105	C₈×C₉	—	6	2	no	—
106	C₈×C₉	C₄×D₃	1	1	yes	characteristic
107	C₈×C₉	C₄×D₃	1	1	yes	characteristic
108	C₈×C₉	D₁₂	1	1	yes	characteristic
109	C₈×C₉	D₁₂	1	1	yes	characteristic
110	#443	—	3	1	no	Sylow
111	C₂×C₃×C₉	—	2	1	no	—
112	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
113	D₃×C₉	—	4	2	no	—
114	C₃×#41	—	3	2	no	—
115	#235	C₂²×C₉	1	2	yes	—
116	C₃×Q₁₆	—	3	1	no	—
117	C₃×SD₁₆	—	3	2	no	—
118	C₃×D₈	—	3	1	no	—
119	C₃×M₁₆	—	3	2	no	—
120	C₃×M₁₆	—	6	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	6	2	no	—
124	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
131	#208	C₂²×C₉	1	2	yes	—
132	Dic₁₂	C₄×C₉	1	1	yes	characteristic, complemented
133	D₂₄	C₄×C₉	1	1	yes	characteristic, complemented
134	#204	C₄×C₉	1	2	yes	complemented
135	ΓL(1,𝔽₂₅)	—	2	2	no	—
136	D₃×C₈	—	2	2	no	—
137	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
138	C₂×C₃×D₃	—	2	2	no	—
139	C₃²×C₄	—	2	2	no	—
140	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₃×Dic₃	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	6	2	no	—
145	C₂²×C₉	#235	1	1	yes	characteristic
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	6	2	no	—
151	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
152	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
153	#135	—	3	1	no	—
154	#131	—	3	2	no	—
155	#108	—	3	1	no	—
156	#104	—	3	2	no	—
157	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
158	C₃×C₉	#443	1	1	yes	characteristic, complemented, Sylow
159	C₃×Q₈	—	3	2	no	—
160	C₃×D₄	—	3	2	no	—
161	C₃×D₄	—	6	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
169	#67	—	2	2	no	—
170	D₁₂	C₂×C₄×C₉	1	2	yes	—
171	C₄×D₃	—	2	2	no	—
172	Dic₆	C₂×C₄×C₉	1	2	yes	—
173	C₃×C₈	—	2	1	no	—
174	C₃×C₈	—	2	1	no	—
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	6	2	no	—
178	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
179	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
180	C₃×C₈	D₄×C₉	1	1	yes	characteristic
181	C₃×C₈	D₄×C₉	1	1	yes	characteristic
182	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
183	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
184	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
185	C₃×C₈	C₃×D₁₂	1	1	yes	characteristic
186	C₃⋊C₈	—	2	2	no	—
187	C₂×C₃²	—	2	1	no	—
188	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
189	C₃×D₃	—	4	2	no	—
190	C₂×C₉	—	2	2	no	—
191	C₂×C₉	—	2	1	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	12	2	no	—
194	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
195	#41	—	3	2	no	—
196	Q₁₆	—	3	1	no	—
197	SD₁₆	—	3	2	no	—
198	D₈	—	3	1	no	—
199	M₁₆	—	3	2	no	—
200	M₁₆	—	6	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
203	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
204	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	6	2	no	—
207	C₂²×C₃	C₉×#41	1	1	yes	characteristic
208	C₂²×C₃	C₃×#235	1	1	yes	characteristic
209	C₂×D₃	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	6	2	no	—
216	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
217	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
218	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
219	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
220	Dic₃	—	2	2	no	—
221	C₃²	C₃×#443	1	1	yes	characteristic
222	C₉	—	2	1	no	—
223	C₉	#4700	1	1	yes	characteristic, direct product factor
224	Q₈	—	3	2	no	—
225	D₄	—	3	2	no	—
226	D₄	—	6	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
230	C₈	—	6	2	no	—
231	C₈	C₄×D₃×C₉	1	1	yes	characteristic
232	C₈	C₄×D₃×C₉	1	1	yes	characteristic
233	C₈	C₉×D₁₂	1	1	yes	characteristic
234	C₈	C₉×D₁₂	1	1	yes	characteristic
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	12	2	no	—
240	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
241	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
242	D₃	—	4	2	no	—
243	C₂²	—	6	2	no	—
244	C₂²	C₉×#235	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	6	2	no	—
247	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
248	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
249	C₃	—	2	1	no	—
250	C₃	C₉×#443	1	1	yes	characteristic, complemented
251	C₃	C₃×#4700	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	12	2	no	—
254	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₉	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂×C₃²	C₃×C₄×D₄
prime radical	C₃×#904	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₉×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈×C₉ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₉×#901	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₉×#819	C₂	semidirect
5. C₉×#805	C₂	semidirect
13. C₉×Dic₁₂	C₄	semidirect
14. C₉×D₂₄	C₄	semidirect
15. C₉×#204	C₄	semidirect
28. C₄×C₈×C₉	D₃	semidirect
41. #4700	C₉	direct (N × H)
74. #901	C₂×C₉	semidirect
78. C₃×C₄×C₈	C₂×C₉	semidirect
80. #819	C₂×C₉	semidirect
81. #805	C₂×C₉	semidirect
132. Dic₁₂	C₄×C₉	semidirect
133. D₂₄	C₄×C₉	semidirect
134. #204	C₄×C₉	semidirect
157. C₄×C₈	D₃×C₉	semidirect
158. C₃×C₉	#443	semidirect
223. C₉	#4700	direct (N × H)
250. C₃	C₉×#443	semidirect

Maximal subgroups

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

subgroup	index
C₉×#904	2
C₉×#901	2
C₃×C₄×C₈×C₉	2
C₉×#819	2
C₉×#805	2
C₃×#4700	3
C₉×#443	3

Subgroups — C9×#4700 #13141441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4700 #13141441