Subgroups — #13138398

Order 1728.

434 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	#11720959	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63375	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720275	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590909	—	3	1	no	maximal
9	#10588797	—	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	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
18	(C₂×D₃)⋊Dic₉	C₄	1	2	yes	—
19	Dic₃⋊Dic₉	C₄	1	2	yes	—
20	D₃×C₉⋊C₈	C₄	1	4	yes	complemented
21	C₂×C₃×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic
22	#63855	—	3	1	no	—
23	C₂×D₃×C₃⋊C₈	—	3	1	no	—
24	C₈×C₃⋊Dic₃	—	3	1	no	—
25	C₃×#815	—	3	1	no	—
26	#63555	—	3	1	no	—
27	Dic₃×C₃⋊C₈	—	3	1	no	—
28	C₂×C₉×M₁₆	—	3	1	no	—
29	#63488	—	3	1	no	—
30	C₂²×C₉⋊C₈	—	3	1	no	—
31	#63375	—	3	1	no	—
32	#63375	D₃	1	1	yes	characteristic, complemented
33	C₈×Dic₉	—	3	1	no	—
34	C₈×Dic₉	—	3	1	no	—
35	C₈×Dic₉	—	6	1	no	—
36	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
39	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
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⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
45	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
46	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	—
47	#5122	—	9	1	no	—
48	#4708	—	9	1	no	—
49	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄×C₃⋊Dic₃	—	3	1	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
53	C₃×C₄×Dic₃	—	3	1	no	—
54	C₂×C₃×C₃⋊C₈	—	3	1	no	—
55	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×ΓL(1,𝔽₂₅)	Dic₃	1	4	yes	—
57	Dic₃⋊Dic₃	—	3	2	no	—
58	(C₂×D₃)⋊Dic₃	—	3	2	no	—
59	D₃×C₃⋊C₈	—	3	4	no	—
60	C₂²×C₄×C₉	—	3	1	no	—
61	C₉×M₁₆	—	3	4	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	—	3	1	no	—
64	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
65	#3676	—	3	2	no	—
66	C₄⋊Dic₉	—	3	2	no	—
67	C₄×Dic₉	—	3	1	no	—
68	C₄×Dic₉	—	6	1	no	—
69	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
70	C₂×C₉⋊C₈	—	3	1	no	—
71	C₂×C₉⋊C₈	—	3	4	no	—
72	C₂×C₉⋊C₈	—	6	1	no	—
73	C₂×C₉⋊C₈	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	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃⋊Dic₉	—	2	2	no	—
80	C₉×Dic₃	C₄²	1	2	yes	—
81	C₃×Dic₉	—	2	2	no	—
82	C₂×C₃×M₁₆	—	3	1	no	—
83	#926	—	9	1	no	—
84	C₂²×C₃⋊C₈	—	9	1	no	—
85	C₂×ΓL(1,𝔽₂₅)	D₉	1	1	yes	characteristic
86	C₂×D₃×C₈	—	9	1	no	—
87	#872	—	9	1	no	—
88	C₃×#97	—	9	1	no	—
89	#815	—	3	1	no	—
90	#815	—	9	1	no	—
91	#815	—	9	1	no	—
92	C₈×Dic₃	—	9	1	no	—
93	C₈×Dic₃	—	9	1	no	—
94	C₈×Dic₃	—	9	1	no	—
95	C₈×Dic₃	—	18	1	no	—
96	C₄×C₃⋊C₈	—	9	1	no	—
97	C₂²×C₃×D₃	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	—	3	2	no	—
100	C₂×C₃×Dic₃	—	3	2	no	—
101	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
102	C₃×C₄×D₃	C₄×D₃	1	2	yes	—
103	C₃×C₄×D₃	C₂×Dic₃	1	2	yes	—
104	C₂³×C₉	—	3	1	no	—
105	C₃²×C₈	C₂×Dic₃	1	2	yes	—
106	C₃⋊(C₃⋊C₈)	—	3	2	no	—
107	C₃×C₃⋊C₈	—	3	2	no	—
108	C₃×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	3	1	no	—
111	C₂×C₄×C₉	—	3	2	no	—
112	C₂×C₄×C₉	—	3	2	no	—
113	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
114	C₂×Dic₉	—	3	2	no	—
115	C₂×Dic₉	—	6	2	no	—
116	C₂×Dic₉	C₄×D₃	1	2	yes	—
117	C₈×C₉	—	2	2	no	—
118	C₈×C₉	—	3	2	no	—
119	C₈×C₉	C₄×D₃	1	2	yes	—
120	C₉⋊C₈	—	3	2	no	—
121	C₉⋊C₈	—	6	2	no	—
122	C₉⋊C₈	C₄×D₃	1	2	yes	—
123	#405	—	27	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	C₂×D₉	1	1	yes	characteristic
129	C₃×M₁₆	—	3	4	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	3	1	no	—
132	C₂×C₃×C₈	—	9	1	no	—
133	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
134	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
135	C₃×C₄²	—	9	1	no	—
136	#217	—	9	2	no	—
137	(C₂×D₃)⋊C₄	—	9	2	no	—
138	C₄⋊Dic₃	—	9	2	no	—
139	Dic₃⋊C₄	—	9	2	no	—
140	C₄×Dic₃	—	3	1	no	—
141	C₄×Dic₃	—	9	1	no	—
142	C₄×Dic₃	—	9	1	no	—
143	C₄×Dic₃	—	18	1	no	—
144	C₂×C₃⋊C₈	—	3	1	no	—
145	C₂×C₃⋊C₈	—	9	4	no	—
146	C₂×C₃⋊C₈	—	9	1	no	—
147	C₂×C₃⋊C₈	—	9	1	no	—
148	C₂×C₃⋊C₈	—	18	1	no	—
149	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
150	ΓL(1,𝔽₂₅)	Dic₉	1	4	yes	complemented
151	D₃×C₈	—	9	4	no	—
152	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×D₃	—	2	2	no	—
154	C₂×C₃×D₃	C₄×Dic₃	1	2	yes	—
155	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
156	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
157	C₃⋊Dic₃	—	6	2	no	—
158	C₃×Dic₃	—	6	2	no	—
159	C₃×Dic₃	C₄×Dic₃	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	1	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	3	2	no	—
167	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
168	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
169	Dic₉	—	2	2	no	—
170	Dic₉	—	6	4	no	—
171	Dic₉	—	6	2	no	—
172	C₂×M₁₆	—	3	1	no	—
173	C₂²×C₈	—	27	1	no	—
174	#117	—	27	1	no	—
175	#97	—	9	1	no	—
176	#97	—	27	1	no	—
177	C₄×C₈	—	27	1	no	—
178	C₄×C₈	—	27	1	no	—
179	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	3	1	no	—
181	C₂²×D₃	C₄×D₉	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	3	1	no	—
184	C₂×C₃×C₄	—	3	2	no	—
185	C₂×C₃×C₄	—	3	2	no	—
186	C₂×C₃×C₄	—	9	2	no	—
187	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
189	C₂×Dic₃	—	3	2	no	—
190	C₂×Dic₃	—	9	2	no	—
191	C₂×Dic₃	—	9	2	no	—
192	C₂×Dic₃	—	18	2	no	—
193	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
194	C₄×D₃	C₄×D₉	1	2	yes	—
195	C₄×D₃	C₂×Dic₉	1	2	yes	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	3	2	no	—
198	C₃×C₈	—	9	2	no	—
199	C₃×C₈	C₂×Dic₉	1	2	yes	—
200	C₃×C₈	D₃×Dic₃	1	2	yes	—
201	C₃⋊C₈	—	3	2	no	—
202	C₃⋊C₈	—	9	2	no	—
203	C₃⋊C₈	—	9	2	no	—
204	C₃⋊C₈	—	18	2	no	—
205	C₃⋊C₈	C₂×Dic₉	1	2	yes	—
206	C₂×C₃²	#907	1	2	yes	—
207	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
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	—	9	1	no	—
218	C₂×C₈	—	27	4	no	—
219	C₂×C₈	—	27	1	no	—
220	C₂×C₈	D₃×D₉	1	1	yes	characteristic
221	C₄⋊C₄	—	27	2	no	—
222	C₂²⋊C₄	—	27	2	no	—
223	C₄²	—	9	1	no	—
224	C₄²	—	27	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	3	2	no	—
227	C₂²×C₃	—	6	2	no	—
228	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
229	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	C₄×Dic₉	1	2	yes	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	3	2	no	—
235	C₃×C₄	—	18	2	no	—
236	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
237	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
238	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
239	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
240	Dic₃	—	6	2	no	—
241	Dic₃	—	18	4	no	—
242	Dic₃	—	18	2	no	—
243	Dic₃	—	18	2	no	—
244	Dic₃	C₄×Dic₉	1	2	yes	—
245	C₃²	#5122	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	#4708	1	1	yes	characteristic, complemented
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	—	9	2	no	—
253	C₂×C₄	—	27	2	no	—
254	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
255	C₈	—	3	2	no	—
256	C₈	—	9	2	no	—
257	C₈	—	27	2	no	—
258	C₈	D₃×Dic₉	1	2	yes	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	6	2	no	—
262	C₂×C₃	#63469	1	2	yes	—
263	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
264	C₂×C₃	#63805	1	2	yes	—
265	C₂×C₃	C₄×D₃×Dic₃	1	1	yes	characteristic
266	D₃	—	2	2	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
270	C₄	—	3	2	no	—
271	C₄	—	18	2	no	—
272	C₄	—	54	2	no	—
273	C₄	C₄×D₃×D₉	1	1	yes	characteristic
274	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
275	C₃	—	2	1	no	—
276	C₃	#10588797	1	1	yes	characteristic, complemented
277	C₃	#10590909	1	1	yes	characteristic
278	C₂	—	6	2	no	—
279	C₂	#11720809	1	2	yes	—
280	C₂	C₄×D₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₃	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	#405
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊Dic₉	C₂	semidirect
5. C₃×#63375	C₂	semidirect
6. #11720275	C₂	semidirect
7. C₃⋊C₈×Dic₉	C₂	semidirect
17. C₉×ΓL(1,𝔽₂₅)	C₄	semidirect
20. D₃×C₉⋊C₈	C₄	semidirect
32. #63375	D₃	semidirect
150. ΓL(1,𝔽₂₅)	Dic₉	semidirect
179. C₃×C₉	#405	semidirect
247. C₉	#4708	semidirect
276. C₃	#10588797	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
#11720959	2
C₂×D₃×C₉⋊C₈	2
C₈×C₃⋊Dic₉	2
C₃×#63375	2
#11720275	2
C₃⋊C₈×Dic₉	2
#10590909	3
#10588797	3