Subgroups — C₃×#65945 #11755622

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#711	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4081	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4029	C₂	1	2	yes	maximal, complemented
11	C₃×#4024	C₂	1	2	yes	maximal, complemented
12	#65945	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	1	yes	characteristic
18	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
20	C₃×#711	—	2	8	no	—
21	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
23	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
29	C₃×#685	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×#683	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
37	C₃×#547	—	5	1	no	maximal, Hall
38	C₂×#711	C₂×C₃	1	2	yes	complemented
39	C₅×#121	C₂×C₃	1	1	yes	characteristic, complemented
40	#4084	C₂×C₃	1	1	yes	characteristic, complemented
41	#4081	C₂×C₃	1	2	yes	complemented
42	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
43	C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
45	D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
46	#4034	C₂×C₃	1	1	yes	characteristic, complemented
47	#4029	C₂×C₃	1	2	yes	complemented
48	#4024	C₂×C₃	1	2	yes	complemented
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
52	C₃×C₅×D₄	—	2	4	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₃×C₂²⋊D₅	—	2	4	no	—
59	C₃×C₂²⋊D₅	D₄	1	4	yes	complemented
60	C₂×C₃×Dic₅	—	2	4	no	—
61	C₂×C₃×Dic₅	—	2	4	no	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₂³	1	2	yes	—
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₃×C₄×D₅	—	2	4	no	—
68	C₃×C₄×D₅	—	2	2	no	—
69	C₃×Dic₁₀	—	2	4	no	—
70	C₂×C₃×#41	—	5	2	no	—
71	C₃×C₄⋊Q₈	—	5	1	no	—
72	C₃×#123	—	5	2	no	—
73	C₃×#123	—	5	2	no	—
74	C₃×C₂²⋊Q₈	—	5	2	no	—
75	C₃×#121	—	5	1	no	—
76	C₃×#121	D₅	1	1	yes	characteristic, complemented
77	C₃×C₄×D₄	—	5	1	no	—
78	C₃×C₄×D₄	—	5	1	no	—
79	C₂×C₃×C₄⋊C₄	—	5	1	no	—
80	C₂×C₃×C₄⋊C₄	—	5	1	no	—
81	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
82	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
86	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
87	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
88	#711	—	2	8	no	—
89	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
90	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
91	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
92	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
95	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
96	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
97	#685	C₂²×C₃	1	1	yes	characteristic
98	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
99	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
100	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
101	#683	C₂²×C₃	1	1	yes	characteristic
102	#683	C₂²×C₃	1	2	yes	—
103	#683	C₂²×C₃	1	2	yes	—
104	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
105	#547	—	5	1	no	Sylow
106	C₂²×C₃×C₅	—	2	1	no	—
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
112	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	—	2	1	no	—
115	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	#41	1	2	yes	complemented
119	C₃×Dic₅	—	2	2	no	—
120	C₃×Dic₅	—	2	2	no	—
121	C₃×Dic₅	—	2	1	no	—
122	C₃×Dic₅	—	2	1	no	—
123	C₃×Dic₅	C₂×D₄	1	2	yes	—
124	C₃×#41	—	10	8	no	—
125	C₂×C₃×Q₈	—	5	2	no	—
126	C₂×C₃×D₄	—	5	1	no	—
127	C₂×C₃×D₄	—	5	2	no	—
128	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
130	C₂²×C₃×C₄	—	5	2	no	—
131	C₂²×C₃×C₄	—	5	2	no	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₂²×C₃×C₄	—	5	2	no	—
134	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	5	2	no	—
136	C₃×C₄⋊C₄	—	5	1	no	—
137	C₃×C₄⋊C₄	—	5	2	no	—
138	C₃×C₄⋊C₄	—	5	1	no	—
139	C₃×C₄⋊C₄	—	5	1	no	—
140	C₃×C₄⋊C₄	—	5	2	no	—
141	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	—	5	1	no	—
143	C₃×C₂²⋊C₄	—	5	2	no	—
144	C₃×C₂²⋊C₄	—	5	2	no	—
145	C₃×C₂²⋊C₄	—	5	1	no	—
146	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
147	C₃×C₄²	—	5	1	no	—
148	C₂³×C₅	C₂³×C₃	1	2	yes	—
149	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
150	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
151	C₅×D₄	—	2	4	no	—
152	C₅×D₄	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
157	C₂²⋊D₅	—	2	4	no	—
158	C₂²⋊D₅	C₃×D₄	1	4	yes	complemented
159	C₂×Dic₅	—	2	4	no	—
160	C₂×Dic₅	—	2	4	no	—
161	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
162	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
163	C₂×Dic₅	C₂³×C₃	1	2	yes	—
164	C₂×Dic₅	C₂³×C₃	1	2	yes	—
165	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
166	C₄×D₅	—	2	4	no	—
167	C₄×D₅	—	2	2	no	—
168	Dic₁₀	—	2	4	no	—
169	C₂×#41	—	5	2	no	—
170	C₄⋊Q₈	—	5	1	no	—
171	#123	—	5	2	no	—
172	#123	—	5	2	no	—
173	C₂²⋊Q₈	—	5	2	no	—
174	#121	—	5	1	no	—
175	#121	C₃×D₅	1	1	yes	characteristic, complemented
176	C₄×D₄	—	5	1	no	—
177	C₄×D₄	—	5	1	no	—
178	C₂×C₄⋊C₄	—	5	1	no	—
179	C₂×C₄⋊C₄	—	5	1	no	—
180	C₂×C₃×C₅	—	2	2	no	—
181	C₂×C₃×C₅	—	4	2	no	—
182	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
184	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
185	C₃×D₅	—	2	2	no	—
186	C₂³×C₃	—	5	1	no	—
187	C₂³×C₃	C₂²×D₅	1	2	yes	—
188	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
189	C₃×Q₈	—	10	4	no	—
190	C₃×D₄	—	2	4	no	—
191	C₃×D₄	—	2	2	no	—
192	C₃×D₄	—	5	4	no	—
193	C₃×D₄	—	10	4	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	2	no	—
199	C₂×C₃×C₄	—	5	2	no	—
200	C₂×C₃×C₄	—	10	4	no	—
201	C₂×C₃×C₄	—	10	4	no	—
202	C₂×C₃×C₄	—	10	4	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
206	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
213	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	C₃×#41	1	2	yes	complemented
220	Dic₅	—	2	2	no	—
221	Dic₅	—	2	2	no	—
222	Dic₅	—	2	1	no	—
223	Dic₅	—	2	1	no	—
224	Dic₅	C₂×C₃×D₄	1	2	yes	—
225	#41	—	10	8	no	—
226	C₂×Q₈	—	5	2	no	—
227	C₂×D₄	—	5	1	no	—
228	C₂×D₄	—	5	2	no	—
229	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
230	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
236	C₄⋊C₄	—	5	1	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	1	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
243	C₂²⋊C₄	—	5	1	no	—
244	C₂²⋊C₄	—	5	2	no	—
245	C₂²⋊C₄	—	5	2	no	—
246	C₂²⋊C₄	—	5	1	no	—
247	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
248	C₄²	—	5	1	no	—
249	C₃×C₅	#547	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	5	2	no	—
256	C₂²×C₃	—	10	1	no	—
257	C₂²×C₃	—	10	1	no	—
258	C₂²×C₃	D₄×D₅	1	2	yes	—
259	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	5	2	no	—
263	C₃×C₄	—	10	2	no	—
264	C₃×C₄	—	10	2	no	—
265	C₃×C₄	—	10	1	no	—
266	C₃×C₄	—	10	1	no	—
267	C₃×C₄	#711	1	2	yes	complemented
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	4	2	no	—
270	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
272	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
273	D₅	—	2	2	no	—
274	C₂³	—	5	1	no	—
275	C₂³	C₂²×C₃×D₅	1	2	yes	—
276	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
277	Q₈	—	10	4	no	—
278	D₄	—	2	4	no	—
279	D₄	—	2	2	no	—
280	D₄	—	5	4	no	—
281	D₄	—	10	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	5	1	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	—	10	4	no	—
290	C₂×C₄	—	10	4	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
294	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	4	2	no	—
297	C₂×C₃	—	10	2	no	—
298	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
299	C₂×C₃	C₂×#711	1	1	yes	characteristic
300	C₂×C₃	#4150	1	1	yes	characteristic
301	C₅	C₃×#547	1	1	yes	characteristic, complemented, Sylow
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	5	2	no	—
308	C₂²	—	10	1	no	—
309	C₂²	—	10	1	no	—
310	C₂²	C₃×D₄×D₅	1	2	yes	—
311	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	5	2	no	—
315	C₄	—	10	2	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	1	no	—
318	C₄	—	10	1	no	—
319	C₄	C₃×#711	1	2	yes	complemented
320	C₃	#65945	1	1	yes	characteristic, direct product factor, Sylow
321	C₂	—	2	2	no	—
322	C₂	—	4	2	no	—
323	C₂	—	10	2	no	—
324	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
325	C₂	C₂×C₃×#711	1	1	yes	characteristic
326	C₂	C₃×#4150	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×#547
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. C₂×C₃×#711	C₂	semidirect
2. C₃×C₅×#121	C₂	semidirect
3. C₃×#4084	C₂	semidirect
4. C₃×#4081	C₂	semidirect
5. C₃×D₄×Dic₅	C₂	semidirect
6. C₃×C₄×C₂²⋊D₅	C₂	semidirect
7. C₂×C₃×C₄⋊Dic₅	C₂	semidirect
8. C₃×D₅×C₄⋊C₄	C₂	semidirect
9. C₃×#4034	C₂	semidirect
10. C₃×#4029	C₂	semidirect
11. C₃×#4024	C₂	semidirect
12. #65945	C₃	direct (N × H)
38. C₂×#711	C₂×C₃	semidirect
39. C₅×#121	C₂×C₃	semidirect
40. #4084	C₂×C₃	semidirect
41. #4081	C₂×C₃	semidirect
42. D₄×Dic₅	C₂×C₃	semidirect
43. C₄×C₂²⋊D₅	C₂×C₃	semidirect
44. C₂×C₄⋊Dic₅	C₂×C₃	semidirect
45. D₅×C₄⋊C₄	C₂×C₃	semidirect
46. #4034	C₂×C₃	semidirect
47. #4029	C₂×C₃	semidirect
48. #4024	C₂×C₃	semidirect
59. C₃×C₂²⋊D₅	D₄	semidirect
76. C₃×#121	D₅	semidirect
118. C₃×C₄×C₅	#41	semidirect
158. C₂²⋊D₅	C₃×D₄	semidirect
175. #121	C₃×D₅	semidirect
219. C₄×C₅	C₃×#41	semidirect
249. C₃×C₅	#547	semidirect
267. C₃×C₄	#711	semidirect
301. C₅	C₃×#547	semidirect
319. C₄	C₃×#711	semidirect
320. C₃	#65945	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#711	2
C₃×C₅×#121	2
C₃×#4084	2
C₃×#4081	2
C₃×D₄×Dic₅	2
C₃×C₄×C₂²⋊D₅	2
C₂×C₃×C₄⋊Dic₅	2
C₃×D₅×C₄⋊C₄	2
C₃×#4034	2
C₃×#4029	2
C₃×#4024	2
#65945	3
C₃×#547	5

Subgroups — C3×#65945 #11755622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65945 #11755622