Subgroups — C₃×#65899 #11755576

Order 960.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4085	C₂	1	2	yes	maximal, complemented
3	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4044	C₂	1	2	yes	maximal, complemented
8	C₃×#4029	C₂	1	2	yes	maximal, complemented
9	C₃×#4027	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
12	#65899	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
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	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
19	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
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	1	yes	characteristic
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	2	yes	—
34	C₃×#683	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#553	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
38	#4085	C₂×C₃	1	2	yes	complemented
39	#4076	C₂×C₃	1	1	yes	characteristic, complemented
40	#4076	C₂×C₃	1	1	yes	characteristic, complemented
41	#4072	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
43	#4044	C₂×C₃	1	2	yes	complemented
44	#4029	C₂×C₃	1	2	yes	complemented
45	#4027	C₂×C₃	1	2	yes	complemented
46	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
47	#4016	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₅	C₂³	1	2	yes	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₃×C₂²⋊D₅	—	2	4	no	—
60	C₃×C₂²⋊D₅	—	4	2	no	—
61	C₂×C₃×Dic₅	—	4	2	no	—
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₂×C₃×Dic₅	C₂³	1	2	yes	—
64	C₂×C₃×Dic₅	C₂³	1	2	yes	—
65	C₃×D₂₀	—	2	2	no	—
66	C₃×C₄×D₅	—	4	2	no	—
67	C₃×#126	—	5	2	no	—
68	C₃×#125	—	5	1	no	—
69	C₃×#123	—	5	2	no	—
70	C₃×#121	—	5	2	no	—
71	C₃×#121	—	5	1	no	—
72	C₃×#121	—	5	1	no	—
73	C₃×C₄×D₄	—	5	2	no	—
74	C₃×C₄×D₄	—	5	1	no	—
75	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
76	C₃×#117	—	5	1	no	—
77	C₂×C₃×C₄⋊C₄	—	5	1	no	—
78	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
82	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
83	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
84	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
86	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
90	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
91	#685	C₂²×C₃	1	2	yes	—
92	#685	C₂²×C₃	1	2	yes	—
93	#685	C₂²×C₃	1	2	yes	—
94	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
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	#683	C₂²×C₃	1	2	yes	—
99	#683	C₂²×C₃	1	2	yes	—
100	C₄×Dic₅	C₂²×C₃	1	2	yes	—
101	#553	—	5	1	no	Sylow
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	—	2	1	no	—
106	C₂²×C₃×C₅	—	2	1	no	—
107	C₂²×C₃×C₅	#41	1	2	yes	—
108	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
109	C₂×C₃×D₅	—	2	2	no	—
110	C₂×C₃×D₅	—	2	2	no	—
111	C₂×C₃×D₅	—	2	2	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	1	no	—
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	#41	1	2	yes	—
117	C₃×Dic₅	—	2	2	no	—
118	C₃×Dic₅	—	2	2	no	—
119	C₃×Dic₅	—	2	2	no	—
120	C₂×C₃×D₄	—	5	2	no	—
121	C₂×C₃×D₄	—	5	2	no	—
122	C₂×C₃×D₄	—	5	1	no	—
123	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	2	no	—
125	C₂²×C₃×C₄	—	5	2	no	—
126	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
128	C₃×C₄⋊C₄	—	5	2	no	—
129	C₃×C₄⋊C₄	—	5	2	no	—
130	C₃×C₄⋊C₄	—	5	1	no	—
131	C₃×C₄⋊C₄	—	5	1	no	—
132	C₃×C₄⋊C₄	—	5	2	no	—
133	C₃×C₄⋊C₄	—	5	1	no	—
134	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
135	C₃×C₂²⋊C₄	—	5	2	no	—
136	C₃×C₂²⋊C₄	—	5	2	no	—
137	C₃×C₂²⋊C₄	—	5	2	no	—
138	C₃×C₂²⋊C₄	—	5	2	no	—
139	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
140	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
141	C₃×C₄²	—	5	2	no	—
142	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
143	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
145	C₂²×D₅	C₂³×C₃	1	2	yes	—
146	C₅×D₄	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
154	C₂²⋊D₅	—	2	4	no	—
155	C₂²⋊D₅	—	4	2	no	—
156	C₂×Dic₅	—	4	2	no	—
157	C₂×Dic₅	C₂³×C₃	1	2	yes	—
158	C₂×Dic₅	C₂³×C₃	1	2	yes	—
159	C₂×Dic₅	C₂³×C₃	1	2	yes	—
160	D₂₀	—	2	2	no	—
161	C₄×D₅	—	4	2	no	—
162	#126	—	5	2	no	—
163	#125	—	5	1	no	—
164	#123	—	5	2	no	—
165	#121	—	5	2	no	—
166	#121	—	5	1	no	—
167	#121	—	5	1	no	—
168	C₄×D₄	—	5	2	no	—
169	C₄×D₄	—	5	1	no	—
170	C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
171	#117	—	5	1	no	—
172	C₂×C₄⋊C₄	—	5	1	no	—
173	C₂×C₃×C₅	—	2	2	no	—
174	C₂×C₃×C₅	—	4	1	no	—
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
177	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
178	C₃×D₅	—	4	2	no	—
179	C₂³×C₃	—	5	2	no	—
180	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
181	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
182	C₃×D₄	—	2	2	no	—
183	C₃×D₄	—	10	4	no	—
184	C₃×D₄	—	10	2	no	—
185	C₃×D₄	—	20	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	5	2	no	—
190	C₂×C₃×C₄	—	5	2	no	—
191	C₂×C₃×C₄	—	20	2	no	—
192	C₂×C₃×C₄	—	20	2	no	—
193	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	C₃×#41	1	2	yes	—
204	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	C₃×#41	1	2	yes	—
213	Dic₅	—	2	2	no	—
214	Dic₅	—	2	2	no	—
215	Dic₅	—	2	2	no	—
216	C₂×D₄	—	5	2	no	—
217	C₂×D₄	—	5	2	no	—
218	C₂×D₄	—	5	1	no	—
219	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
220	C₂²×C₄	—	5	2	no	—
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
224	C₄⋊C₄	—	5	1	no	—
225	C₄⋊C₄	—	5	1	no	—
226	C₄⋊C₄	—	5	2	no	—
227	C₄⋊C₄	—	5	1	no	—
228	C₄⋊C₄	—	5	2	no	—
229	C₄⋊C₄	—	5	2	no	—
230	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
231	C₂²⋊C₄	—	5	2	no	—
232	C₂²⋊C₄	—	5	2	no	—
233	C₂²⋊C₄	—	5	2	no	—
234	C₂²⋊C₄	—	5	2	no	—
235	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
236	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
237	C₄²	—	5	2	no	—
238	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
239	C₃×C₅	#553	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	10	2	no	—
246	C₂²×C₃	—	10	2	no	—
247	C₂²×C₃	—	10	2	no	—
248	C₂²×C₃	#709	1	2	yes	—
249	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	10	2	no	—
255	C₃×C₄	—	10	2	no	—
256	C₃×C₄	—	10	2	no	—
257	C₃×C₄	#711	1	2	yes	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	4	1	no	—
260	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
261	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
262	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
263	D₅	—	4	2	no	—
264	C₂³	—	5	2	no	—
265	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
266	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
267	D₄	—	2	2	no	—
268	D₄	—	10	4	no	—
269	D₄	—	10	2	no	—
270	D₄	—	20	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	20	2	no	—
277	C₂×C₄	—	20	2	no	—
278	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	20	2	no	—
286	C₂×C₃	C₂×#709	1	1	yes	characteristic
287	C₂×C₃	C₂×#711	1	1	yes	characteristic
288	C₂×C₃	#4149	1	1	yes	characteristic
289	C₅	C₃×#553	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	10	2	no	—
296	C₂²	—	10	2	no	—
297	C₂²	—	10	2	no	—
298	C₂²	C₃×#709	1	2	yes	—
299	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	10	2	no	—
305	C₄	—	10	2	no	—
306	C₄	—	10	2	no	—
307	C₄	C₃×#711	1	2	yes	—
308	C₃	#65899	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	2	2	no	—
310	C₂	—	4	1	no	—
311	C₂	—	20	2	no	—
312	C₂	C₂×C₃×#709	1	1	yes	characteristic
313	C₂	C₂×C₃×#711	1	1	yes	characteristic
314	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×#553
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×D₄	C₂	semidirect
2. C₃×#4085	C₂	semidirect
3. C₃×#4076	C₂	semidirect
4. C₃×#4076	C₂	semidirect
5. C₃×#4072	C₂	semidirect
6. C₂×C₃×C₄⋊Dic₅	C₂	semidirect
7. C₃×#4044	C₂	semidirect
8. C₃×#4029	C₂	semidirect
9. C₃×#4027	C₂	semidirect
10. C₃×C₄×D₂₀	C₂	semidirect
11. C₃×#4016	C₂	semidirect
12. #65899	C₃	direct (N × H)
37. C₄×C₅×D₄	C₂×C₃	semidirect
38. #4085	C₂×C₃	semidirect
39. #4076	C₂×C₃	semidirect
40. #4076	C₂×C₃	semidirect
41. #4072	C₂×C₃	semidirect
42. C₂×C₄⋊Dic₅	C₂×C₃	semidirect
43. #4044	C₂×C₃	semidirect
44. #4029	C₂×C₃	semidirect
45. #4027	C₂×C₃	semidirect
46. C₄×D₂₀	C₂×C₃	semidirect
47. #4016	C₂×C₃	semidirect
75. C₃×C₄×D₄	D₅	semidirect
170. C₄×D₄	C₃×D₅	semidirect
239. C₃×C₅	#553	semidirect
289. C₅	C₃×#553	semidirect
308. C₃	#65899	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×C₄×C₅×D₄	2
C₃×#4085	2
C₃×#4076	2
C₃×#4072	2
C₂×C₃×C₄⋊Dic₅	2
C₃×#4044	2
C₃×#4029	2
C₃×#4027	2
C₃×C₄×D₂₀	2
C₃×#4016	2
#65899	3
C₃×#553	5

Subgroups — C3×#65899 #11755576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65899 #11755576