Subgroups — C₉×#4708 #13141513

Order 1728.

430 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#872	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4708	C₃	1	1	yes	maximal, characteristic
9	C₉×#405	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
15	C₉×Dic₃⋊C₄	C₄	1	2	yes	—
16	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
20	D₃×C₈×C₉	C₄	1	4	yes	complemented
21	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×D₃×C₈	C₂×C₃	1	1	yes	characteristic
23	C₃×#872	C₂×C₃	1	1	yes	characteristic
24	C₃²×#97	C₂×C₃	1	1	yes	characteristic
25	C₃×#815	C₂×C₃	1	1	yes	characteristic
26	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₂×C₉×M₁₆	—	3	1	no	—
29	C₂²×C₈×C₉	—	3	1	no	—
30	C₉×#117	—	3	1	no	—
31	C₉×#97	—	2	1	no	—
32	C₉×#97	—	3	1	no	—
33	C₉×#97	D₃	1	1	yes	characteristic, complemented
34	C₄×C₈×C₉	—	3	1	no	—
35	C₄×C₈×C₉	—	3	1	no	—
36	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
42	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
43	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
44	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
45	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
46	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
47	C₃×#405	—	3	1	no	—
48	#4708	C₉	1	1	yes	characteristic, direct product factor
49	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×(C₂×D₃)⋊C₄	C₃×C₄	1	2	yes	—
54	C₃×Dic₃⋊C₄	C₃×C₄	1	2	yes	—
55	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	—
59	C₃×D₃×C₈	C₃×C₄	1	4	yes	—
60	C₂²×C₄×C₉	—	3	1	no	—
61	C₉×M₁₆	—	3	4	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	—	2	1	no	—
64	C₂×C₈×C₉	—	3	1	no	—
65	C₂×C₈×C₉	—	3	1	no	—
66	C₂×C₈×C₉	—	3	4	no	—
67	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
68	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
69	C₉×C₄⋊C₄	—	3	2	no	—
70	C₉×C₂²⋊C₄	—	3	2	no	—
71	C₄²×C₉	—	2	1	no	—
72	C₄²×C₉	—	3	1	no	—
73	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×D₃×C₉	—	2	2	no	—
76	C₂×D₃×C₉	C₄²	1	2	yes	—
77	C₃×C₄×C₉	—	2	2	no	—
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₉×Dic₃	—	2	2	no	—
81	C₉×Dic₃	C₄²	1	2	yes	—
82	C₂×C₃×M₁₆	—	3	1	no	—
83	C₂²×C₃×C₈	—	3	1	no	—
84	C₃×#117	—	3	1	no	—
85	C₂×ΓL(1,𝔽₂₅)	C₂×C₉	1	1	yes	characteristic
86	C₂×D₃×C₈	C₂×C₉	1	1	yes	characteristic
87	#872	C₂×C₉	1	1	yes	characteristic
88	C₃×#97	—	2	1	no	—
89	C₃×#97	—	3	1	no	—
90	C₃×#97	C₂×C₉	1	1	yes	characteristic, complemented
91	C₃×#97	C₃×D₃	1	1	yes	characteristic
92	C₃×C₄×C₈	—	3	1	no	—
93	C₃×C₄×C₈	—	3	1	no	—
94	#815	C₂×C₉	1	1	yes	characteristic, complemented
95	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
96	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
97	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
102	C₃×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
103	C₃×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
104	C₂³×C₉	—	3	1	no	—
105	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
106	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
107	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
108	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	3	1	no	—
112	C₂×C₄×C₉	—	3	2	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	—	3	2	no	—
115	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
116	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
117	C₈×C₉	—	2	2	no	—
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	—	3	2	no	—
120	C₈×C₉	—	3	2	no	—
121	C₈×C₉	C₄×D₃	1	2	yes	—
122	C₈×C₉	C₄×D₃	1	2	yes	—
123	#405	—	3	1	no	Sylow
124	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₃×C₉	#131	1	2	yes	—
126	D₃×C₉	—	2	2	no	—
127	C₂²×C₃×C₄	—	3	1	no	—
128	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
129	C₃×M₁₆	—	3	4	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	—	3	4	no	—
135	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	3	2	no	—
140	C₃×C₂²⋊C₄	—	3	2	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	3	1	no	—
143	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
144	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
145	(C₂×D₃)⋊C₄	C₄×C₉	1	2	yes	—
146	Dic₃⋊C₄	C₄×C₉	1	2	yes	—
147	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
149	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
150	ΓL(1,𝔽₂₅)	C₄×C₉	1	4	yes	complemented
151	D₃×C₈	C₄×C₉	1	4	yes	complemented
152	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×C₃×D₃	—	2	2	no	—
154	C₂×C₃×D₃	C₃×C₄²	1	2	yes	—
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₃×Dic₃	—	2	2	no	—
159	C₃×Dic₃	C₃×C₄²	1	2	yes	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	3	2	no	—
162	C₂²×C₉	—	6	2	no	—
163	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	3	2	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	—	6	2	no	—
170	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×M₁₆	—	3	1	no	—
173	C₂²×C₈	—	3	1	no	—
174	#117	—	3	1	no	—
175	#97	—	3	1	no	—
176	#97	D₃×C₉	1	1	yes	characteristic, complemented
177	C₄×C₈	—	3	1	no	—
178	C₄×C₈	—	3	1	no	—
179	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	3	1	no	—
181	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	3	1	no	—
185	C₂×C₃×C₄	—	3	2	no	—
186	C₂×C₃×C₄	—	3	2	no	—
187	C₂×C₃×C₄	—	3	2	no	—
188	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
189	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
191	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
192	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
193	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
194	C₄×D₃	C₂×C₄×C₉	1	2	yes	—
195	C₄×D₃	C₂×C₄×C₉	1	2	yes	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	3	2	no	—
199	C₃×C₈	—	3	2	no	—
200	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
201	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
202	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
203	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
204	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	—
205	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	—
206	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
207	C₂×C₃²	C₃×#131	1	2	yes	—
208	C₃×D₃	—	2	2	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	6	2	no	—
212	C₂×C₉	C₄²×D₃	1	1	yes	characteristic
213	C₂×C₉	#907	1	2	yes	—
214	C₂²×C₄	—	3	1	no	—
215	M₁₆	—	3	4	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	—	3	4	no	—
219	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
220	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
221	C₄⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₄²	—	3	1	no	—
224	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	3	2	no	—
227	C₂²×C₃	—	6	2	no	—
228	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	C₄²×C₉	1	2	yes	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	3	2	no	—
237	C₃×C₄	—	4	2	no	—
238	C₃×C₄	—	6	2	no	—
239	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
240	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
242	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
243	Dic₃	—	2	2	no	—
244	Dic₃	C₄²×C₉	1	2	yes	—
245	C₃²	C₃×#405	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	#4708	1	1	yes	characteristic, direct product factor
248	C₂³	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
254	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
255	C₈	—	3	2	no	—
256	C₈	—	3	2	no	—
257	C₈	C₄×D₃×C₉	1	2	yes	—
258	C₈	C₄×D₃×C₉	1	2	yes	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	6	2	no	—
262	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
263	C₂×C₃	C₉×#131	1	2	yes	—
264	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
265	C₂×C₃	C₃×#907	1	2	yes	—
266	D₃	—	2	2	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	3	2	no	—
272	C₄	—	6	2	no	—
273	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
274	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
275	C₃	—	2	1	no	—
276	C₃	C₉×#405	1	1	yes	characteristic, complemented
277	C₃	C₃×#4708	1	1	yes	characteristic
278	C₂	—	6	2	no	—
279	C₂	C₄²×D₃×C₉	1	1	yes	characteristic
280	C₂	C₉×#907	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂²×C₃×D₃	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#97	D₃
hypocenter	C₃	C₉×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₉×#97

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×#97	C₂	semidirect
5. C₉×#815	C₂	semidirect
6. C₈×C₉×Dic₃	C₂	semidirect
7. C₄×C₉×C₃⋊C₈	C₂	semidirect
19. C₉×ΓL(1,𝔽₂₅)	C₄	semidirect
20. D₃×C₈×C₉	C₄	semidirect
33. C₉×#97	D₃	semidirect
48. #4708	C₉	direct (N × H)
90. C₃×#97	C₂×C₉	semidirect
94. #815	C₂×C₉	semidirect
95. C₈×Dic₃	C₂×C₉	semidirect
96. C₄×C₃⋊C₈	C₂×C₉	semidirect
150. ΓL(1,𝔽₂₅)	C₄×C₉	semidirect
151. D₃×C₈	C₄×C₉	semidirect
176. #97	D₃×C₉	semidirect
179. C₃×C₉	#405	semidirect
247. C₉	#4708	direct (N × H)
276. C₃	C₉×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×ΓL(1,𝔽₂₅)	2
C₂×D₃×C₈×C₉	2
C₉×#872	2
C₃×C₉×#97	2
C₉×#815	2
C₈×C₉×Dic₃	2
C₄×C₉×C₃⋊C₈	2
C₃×#4708	3
C₉×#405	3

Subgroups — C9×#4708 #13141513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4708 #13141513