Subgroups — C₃×#65953 #11755630

Order 960.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4085	C₂	1	2	yes	maximal, complemented
4	C₃×#4084	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4039	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4032	C₂	1	2	yes	maximal, complemented
11	C₃×#4029	C₂	1	2	yes	maximal, complemented
12	#65953	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
20	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
25	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#685	C₂²	1	2	yes	—
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
33	C₃×#683	C₂²	1	2	yes	—
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×#540	—	5	1	no	maximal, Hall
37	C₅×#121	C₂×C₃	1	1	yes	characteristic, complemented
38	#4086	C₂×C₃	1	1	yes	characteristic, complemented
39	#4085	C₂×C₃	1	2	yes	complemented
40	#4084	C₂×C₃	1	2	yes	complemented
41	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
42	#4076	C₂×C₃	1	1	yes	characteristic, complemented
43	#4072	C₂×C₃	1	1	yes	characteristic, complemented
44	#4039	C₂×C₃	1	1	yes	characteristic, complemented
45	#4036	C₂×C₃	1	1	yes	characteristic, complemented
46	#4032	C₂×C₃	1	2	yes	complemented
47	#4029	C₂×C₃	1	2	yes	complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂²×C₃×D₅	C₂³	1	2	yes	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₃×C₅×D₄	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊D₅	—	4	2	no	—
58	C₃×C₂²⋊D₅	—	4	2	no	—
59	C₃×C₂²⋊D₅	—	4	2	no	—
60	C₂×C₃×Dic₅	—	4	2	no	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₃×D₂₀	—	2	2	no	—
67	C₃×C₄×D₅	—	4	2	no	—
68	C₃×#127	—	5	1	no	—
69	C₃×#125	—	5	1	no	—
70	C₃×#123	—	5	2	no	—
71	C₃×#123	—	5	2	no	—
72	C₃×#121	—	5	2	no	—
73	C₃×#121	—	5	2	no	—
74	C₃×#121	—	5	1	no	—
75	C₃×#121	D₅	1	1	yes	characteristic, complemented
76	C₃×C₄×D₄	—	5	1	no	—
77	C₃×C₄×D₄	—	5	1	no	—
78	C₃×#117	—	5	1	no	—
79	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
85	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
86	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
88	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
91	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
92	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
93	#685	C₂²×C₃	1	2	yes	—
94	#685	C₂²×C₃	1	2	yes	—
95	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
96	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
97	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
98	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
99	#683	C₂²×C₃	1	2	yes	—
100	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
101	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
102	#540	—	5	1	no	Sylow
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	1	no	—
107	C₂²×C₃×C₅	—	2	2	no	—
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
110	C₂×C₃×D₅	—	2	2	no	—
111	C₂×C₃×D₅	—	2	2	no	—
112	C₂×C₃×D₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	#41	1	2	yes	—
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	—	2	1	no	—
118	C₃×Dic₅	—	2	2	no	—
119	C₃×Dic₅	—	2	1	no	—
120	C₃×Dic₅	—	2	1	no	—
121	C₂×C₃×D₄	—	5	2	no	—
122	C₂×C₃×D₄	—	5	2	no	—
123	C₂×C₃×D₄	—	5	2	no	—
124	C₂×C₃×D₄	—	5	1	no	—
125	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
126	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
127	C₂²×C₃×C₄	—	5	2	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	5	2	no	—
131	C₃×C₄⋊C₄	—	5	1	no	—
132	C₃×C₄⋊C₄	—	5	1	no	—
133	C₃×C₄⋊C₄	—	5	1	no	—
134	C₃×C₄⋊C₄	—	5	2	no	—
135	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	—	5	1	no	—
137	C₃×C₂²⋊C₄	—	5	1	no	—
138	C₃×C₂²⋊C₄	—	5	2	no	—
139	C₃×C₂²⋊C₄	—	5	2	no	—
140	C₃×C₂²⋊C₄	—	5	2	no	—
141	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
142	C₃×C₄²	—	5	1	no	—
143	C₃×C₄²	—	5	1	no	—
144	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
145	C₂³×C₅	C₂³×C₃	1	2	yes	—
146	C₂²×D₅	C₂³×C₃	1	2	yes	—
147	C₅×D₄	—	2	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
151	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
153	C₂²⋊D₅	—	4	2	no	—
154	C₂²⋊D₅	—	4	2	no	—
155	C₂²⋊D₅	—	4	2	no	—
156	C₂×Dic₅	—	4	2	no	—
157	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
158	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
159	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
160	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
161	C₂×Dic₅	C₂³×C₃	1	2	yes	—
162	D₂₀	—	2	2	no	—
163	C₄×D₅	—	4	2	no	—
164	#127	—	5	1	no	—
165	#125	—	5	1	no	—
166	#123	—	5	2	no	—
167	#123	—	5	2	no	—
168	#121	—	5	2	no	—
169	#121	—	5	2	no	—
170	#121	—	5	1	no	—
171	#121	C₃×D₅	1	1	yes	characteristic, complemented
172	C₄×D₄	—	5	1	no	—
173	C₄×D₄	—	5	1	no	—
174	#117	—	5	1	no	—
175	C₂×C₃×C₅	—	4	1	no	—
176	C₂×C₃×C₅	—	4	2	no	—
177	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
178	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
180	C₃×D₅	—	4	2	no	—
181	C₂³×C₃	—	5	2	no	—
182	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
183	C₂³×C₃	C₂²×D₅	1	2	yes	—
184	C₃×D₄	—	2	2	no	—
185	C₃×D₄	—	4	2	no	—
186	C₃×D₄	—	10	2	no	—
187	C₃×D₄	—	20	2	no	—
188	C₃×D₄	—	20	2	no	—
189	C₃×D₄	—	20	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	2	no	—
196	C₂×C₃×C₄	—	20	2	no	—
197	C₂×C₃×C₄	—	20	2	no	—
198	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
200	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×D₅	—	2	2	no	—
209	C₂×D₅	—	2	2	no	—
210	C₂×D₅	—	2	2	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	C₃×#41	1	2	yes	—
214	Dic₅	—	2	1	no	—
215	Dic₅	—	2	1	no	—
216	Dic₅	—	2	2	no	—
217	Dic₅	—	2	1	no	—
218	Dic₅	—	2	1	no	—
219	C₂×D₄	—	5	2	no	—
220	C₂×D₄	—	5	2	no	—
221	C₂×D₄	—	5	2	no	—
222	C₂×D₄	—	5	1	no	—
223	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
224	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
225	C₂²×C₄	—	5	2	no	—
226	C₂²×C₄	—	5	2	no	—
227	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
228	C₄⋊C₄	—	5	1	no	—
229	C₄⋊C₄	—	5	1	no	—
230	C₄⋊C₄	—	5	1	no	—
231	C₄⋊C₄	—	5	2	no	—
232	C₄⋊C₄	—	5	2	no	—
233	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
234	C₂²⋊C₄	—	5	1	no	—
235	C₂²⋊C₄	—	5	1	no	—
236	C₂²⋊C₄	—	5	2	no	—
237	C₂²⋊C₄	—	5	2	no	—
238	C₂²⋊C₄	—	5	2	no	—
239	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
240	C₄²	—	5	1	no	—
241	C₄²	—	5	1	no	—
242	C₃×C₅	#540	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	10	2	no	—
250	C₂²×C₃	—	10	2	no	—
251	C₂²×C₃	—	10	2	no	—
252	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	10	1	no	—
256	C₃×C₄	—	10	1	no	—
257	C₃×C₄	—	10	1	no	—
258	C₃×C₄	—	10	2	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	#711	1	2	yes	—
261	C₂×C₅	—	4	1	no	—
262	C₂×C₅	—	4	2	no	—
263	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
264	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
266	D₅	—	4	2	no	—
267	C₂³	—	5	2	no	—
268	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
269	C₂³	C₂²×C₃×D₅	1	2	yes	—
270	D₄	—	2	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	10	2	no	—
273	D₄	—	20	2	no	—
274	D₄	—	20	2	no	—
275	D₄	—	20	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	1	no	—
279	C₂×C₄	—	5	1	no	—
280	C₂×C₄	—	5	1	no	—
281	C₂×C₄	—	5	2	no	—
282	C₂×C₄	—	20	2	no	—
283	C₂×C₄	—	20	2	no	—
284	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
286	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
287	C₂×C₃	—	4	1	no	—
288	C₂×C₃	—	4	2	no	—
289	C₂×C₃	—	20	2	no	—
290	C₂×C₃	C₂×#711	1	1	yes	characteristic
291	C₂×C₃	#4144	1	1	yes	characteristic
292	C₂×C₃	#4149	1	1	yes	characteristic
293	C₅	C₃×#540	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	10	2	no	—
302	C₂²	—	10	2	no	—
303	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	10	1	no	—
307	C₄	—	10	1	no	—
308	C₄	—	10	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	1	no	—
311	C₄	C₃×#711	1	2	yes	—
312	C₃	#65953	1	1	yes	characteristic, direct product factor, Sylow
313	C₂	—	4	1	no	—
314	C₂	—	4	2	no	—
315	C₂	—	20	2	no	—
316	C₂	C₂×C₃×#711	1	1	yes	characteristic
317	C₂	C₃×#4144	1	1	yes	characteristic
318	C₂	C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#121	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#121	D₅
hypocenter	C₅	C₃×#540
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#121

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#121	C₂	semidirect
2. C₃×#4086	C₂	semidirect
3. C₃×#4085	C₂	semidirect
4. C₃×#4084	C₂	semidirect
5. C₃×D₄×Dic₅	C₂	semidirect
6. C₃×#4076	C₂	semidirect
7. C₃×#4072	C₂	semidirect
8. C₃×#4039	C₂	semidirect
9. C₃×#4036	C₂	semidirect
10. C₃×#4032	C₂	semidirect
11. C₃×#4029	C₂	semidirect
12. #65953	C₃	direct (N × H)
37. C₅×#121	C₂×C₃	semidirect
38. #4086	C₂×C₃	semidirect
39. #4085	C₂×C₃	semidirect
40. #4084	C₂×C₃	semidirect
41. D₄×Dic₅	C₂×C₃	semidirect
42. #4076	C₂×C₃	semidirect
43. #4072	C₂×C₃	semidirect
44. #4039	C₂×C₃	semidirect
45. #4036	C₂×C₃	semidirect
46. #4032	C₂×C₃	semidirect
47. #4029	C₂×C₃	semidirect
75. C₃×#121	D₅	semidirect
171. #121	C₃×D₅	semidirect
242. C₃×C₅	#540	semidirect
293. C₅	C₃×#540	semidirect
312. C₃	#65953	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#121	2
C₃×#4086	2
C₃×#4085	2
C₃×#4084	2
C₃×D₄×Dic₅	2
C₃×#4076	2
C₃×#4072	2
C₃×#4039	2
C₃×#4036	2
C₃×#4032	2
C₃×#4029	2
#65953	3
C₃×#540	5

Subgroups — C3×#65953 #11755630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65953 #11755630