Subgroups — C₃×#65930 #11755607

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#4086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4085	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	C₂×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4039	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4031	C₂	1	2	yes	maximal, complemented
11	C₃×#4027	C₂	1	2	yes	maximal, complemented
12	#65930	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	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
22	C₃×#711	—	2	8	no	—
23	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
25	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
26	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
29	C₃×#685	C₂²	1	2	yes	—
30	C₃×#685	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
37	C₃×#532	—	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	#4086	C₂×C₃	1	1	yes	characteristic, complemented
41	#4085	C₂×C₃	1	2	yes	complemented
42	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
43	#4076	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×C₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
45	#4039	C₂×C₃	1	1	yes	characteristic, complemented
46	#4034	C₂×C₃	1	1	yes	characteristic, complemented
47	#4031	C₂×C₃	1	2	yes	complemented
48	#4027	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	2	yes	—
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	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₃×C₂²⋊D₅	—	2	4	no	—
59	C₃×C₂²⋊D₅	—	2	4	no	—
60	C₃×C₂²⋊D₅	—	2	4	no	—
61	C₂×C₃×Dic₅	—	2	4	no	—
62	C₂×C₃×Dic₅	—	2	2	no	—
63	C₂×C₃×Dic₅	D₄	1	4	yes	complemented
64	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
69	C₃×D₂₀	—	2	2	no	—
70	C₃×C₄×D₅	—	2	4	no	—
71	C₃×Dic₁₀	—	2	4	no	—
72	C₂×C₃×#41	—	5	2	no	—
73	C₃×C₄⋊Q₈	—	5	1	no	—
74	C₃×#127	—	5	1	no	—
75	C₃×#124	—	5	2	no	—
76	C₃×#121	—	5	2	no	—
77	C₃×#121	—	5	1	no	—
78	C₃×#121	D₅	1	1	yes	characteristic, complemented
79	C₃×C₄×D₄	—	5	1	no	—
80	C₃×C₄×D₄	—	5	1	no	—
81	C₃×C₄×D₄	—	5	2	no	—
82	C₂×C₃×C₄²	—	5	1	no	—
83	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
84	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
88	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
89	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
90	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
91	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
92	#711	—	2	8	no	—
93	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
95	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
96	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
99	#685	C₂²×C₃	1	2	yes	—
100	#685	C₂²×C₃	1	2	yes	—
101	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
102	#683	C₂²×C₃	1	2	yes	—
103	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
104	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
105	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
106	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
107	#532	—	5	1	no	Sylow
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	2	2	no	—
113	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
114	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	—	2	2	no	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	#41	1	2	yes	complemented
121	C₃×Dic₅	—	2	2	no	—
122	C₃×Dic₅	—	2	1	no	—
123	C₃×Dic₅	C₂×D₄	1	2	yes	—
124	C₃×Dic₅	C₂×D₄	1	2	yes	—
125	C₃×Dic₅	#41	1	2	yes	complemented
126	C₃×#41	—	10	8	no	—
127	C₂×C₃×Q₈	—	5	2	no	—
128	C₂×C₃×D₄	—	5	2	no	—
129	C₂×C₃×D₄	—	5	2	no	—
130	C₂×C₃×D₄	—	5	2	no	—
131	C₂×C₃×D₄	—	5	1	no	—
132	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
133	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
134	C₂²×C₃×C₄	—	5	1	no	—
135	C₂²×C₃×C₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	2	no	—
137	C₂²×C₃×C₄	—	5	2	no	—
138	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	5	2	no	—
140	C₃×C₄⋊C₄	—	5	1	no	—
141	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	—	5	2	no	—
143	C₃×C₂²⋊C₄	—	5	2	no	—
144	C₃×C₂²⋊C₄	—	5	2	no	—
145	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
146	C₃×C₄²	—	5	1	no	—
147	C₃×C₄²	—	5	1	no	—
148	C₃×C₄²	—	5	1	no	—
149	C₃×C₄²	—	5	1	no	—
150	C₂³×C₅	C₂³×C₃	1	2	yes	—
151	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
152	C₂²×D₅	C₂³×C₃	1	2	yes	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
158	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
159	C₂²⋊D₅	—	2	4	no	—
160	C₂²⋊D₅	—	2	4	no	—
161	C₂²⋊D₅	—	2	4	no	—
162	C₂×Dic₅	—	2	4	no	—
163	C₂×Dic₅	—	2	2	no	—
164	C₂×Dic₅	C₃×D₄	1	4	yes	complemented
165	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
166	C₂×Dic₅	C₂³×C₃	1	2	yes	—
167	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
168	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
169	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
170	D₂₀	—	2	2	no	—
171	C₄×D₅	—	2	4	no	—
172	Dic₁₀	—	2	4	no	—
173	C₂×#41	—	5	2	no	—
174	C₄⋊Q₈	—	5	1	no	—
175	#127	—	5	1	no	—
176	#124	—	5	2	no	—
177	#121	—	5	2	no	—
178	#121	—	5	1	no	—
179	#121	C₃×D₅	1	1	yes	characteristic, complemented
180	C₄×D₄	—	5	1	no	—
181	C₄×D₄	—	5	1	no	—
182	C₄×D₄	—	5	2	no	—
183	C₂×C₄²	—	5	1	no	—
184	C₂×C₃×C₅	—	2	2	no	—
185	C₂×C₃×C₅	—	4	2	no	—
186	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
189	C₃×D₅	—	4	2	no	—
190	C₂³×C₃	—	5	2	no	—
191	C₂³×C₃	C₂²×D₅	1	2	yes	—
192	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
193	C₃×Q₈	—	10	4	no	—
194	C₃×D₄	—	2	4	no	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	10	4	no	—
197	C₃×D₄	—	10	4	no	—
198	C₃×D₄	—	10	4	no	—
199	C₃×D₄	—	10	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	5	4	no	—
202	C₂×C₃×C₄	—	5	1	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	10	4	no	—
208	C₂×C₃×C₄	—	10	2	no	—
209	C₂×C₃×C₄	—	10	4	no	—
210	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
212	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
219	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	2	2	no	—
222	C₂×D₅	—	2	2	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	C₃×#41	1	2	yes	complemented
226	Dic₅	—	2	2	no	—
227	Dic₅	—	2	1	no	—
228	Dic₅	C₂×C₃×D₄	1	2	yes	—
229	Dic₅	C₂×C₃×D₄	1	2	yes	—
230	Dic₅	C₃×#41	1	2	yes	complemented
231	#41	—	10	8	no	—
232	C₂×Q₈	—	5	2	no	—
233	C₂×D₄	—	5	2	no	—
234	C₂×D₄	—	5	2	no	—
235	C₂×D₄	—	5	2	no	—
236	C₂×D₄	—	5	1	no	—
237	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
238	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
244	C₄⋊C₄	—	5	2	no	—
245	C₄⋊C₄	—	5	1	no	—
246	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
247	C₂²⋊C₄	—	5	2	no	—
248	C₂²⋊C₄	—	5	2	no	—
249	C₂²⋊C₄	—	5	2	no	—
250	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
251	C₄²	—	5	1	no	—
252	C₄²	—	5	1	no	—
253	C₄²	—	5	1	no	—
254	C₄²	—	5	1	no	—
255	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	10	2	no	—
262	C₂²×C₃	—	10	2	no	—
263	C₂²×C₃	—	10	2	no	—
264	C₂²×C₃	D₄×D₅	1	2	yes	—
265	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	5	2	no	—
269	C₃×C₄	—	5	2	no	—
270	C₃×C₄	—	5	2	no	—
271	C₃×C₄	—	10	2	no	—
272	C₃×C₄	—	10	1	no	—
273	C₃×C₄	#711	1	2	yes	complemented
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	4	2	no	—
276	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
278	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
279	D₅	—	4	2	no	—
280	C₂³	—	5	2	no	—
281	C₂³	C₂²×C₃×D₅	1	2	yes	—
282	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
283	Q₈	—	10	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	10	4	no	—
287	D₄	—	10	4	no	—
288	D₄	—	10	4	no	—
289	D₄	—	10	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	5	4	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
302	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	4	2	no	—
305	C₂×C₃	—	20	2	no	—
306	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
307	C₂×C₃	C₂×#711	1	1	yes	characteristic
308	C₂×C₃	D₅×#41	1	1	yes	characteristic
309	C₅	C₃×#532	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	10	2	no	—
316	C₂²	—	10	2	no	—
317	C₂²	—	10	2	no	—
318	C₂²	C₃×D₄×D₅	1	2	yes	—
319	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	5	2	no	—
323	C₄	—	5	2	no	—
324	C₄	—	5	2	no	—
325	C₄	—	10	2	no	—
326	C₄	—	10	1	no	—
327	C₄	C₃×#711	1	2	yes	complemented
328	C₃	#65930	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	4	2	no	—
331	C₂	—	20	2	no	—
332	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
333	C₂	C₂×C₃×#711	1	1	yes	characteristic
334	C₂	C₃×D₅×#41	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×#532
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×#4086	C₂	semidirect
4. C₃×#4085	C₂	semidirect
5. C₃×D₄×Dic₅	C₂	semidirect
6. C₃×#4076	C₂	semidirect
7. C₂×C₃×C₄×Dic₅	C₂	semidirect
8. C₃×#4039	C₂	semidirect
9. C₃×#4034	C₂	semidirect
10. C₃×#4031	C₂	semidirect
11. C₃×#4027	C₂	semidirect
12. #65930	C₃	direct (N × H)
38. C₂×#711	C₂×C₃	semidirect
39. C₅×#121	C₂×C₃	semidirect
40. #4086	C₂×C₃	semidirect
41. #4085	C₂×C₃	semidirect
42. D₄×Dic₅	C₂×C₃	semidirect
43. #4076	C₂×C₃	semidirect
44. C₂×C₄×Dic₅	C₂×C₃	semidirect
45. #4039	C₂×C₃	semidirect
46. #4034	C₂×C₃	semidirect
47. #4031	C₂×C₃	semidirect
48. #4027	C₂×C₃	semidirect
63. C₂×C₃×Dic₅	D₄	semidirect
78. C₃×#121	D₅	semidirect
120. C₃×C₄×C₅	#41	semidirect
125. C₃×Dic₅	#41	semidirect
164. C₂×Dic₅	C₃×D₄	semidirect
179. #121	C₃×D₅	semidirect
225. C₄×C₅	C₃×#41	semidirect
230. Dic₅	C₃×#41	semidirect
255. C₃×C₅	#532	semidirect
273. C₃×C₄	#711	semidirect
309. C₅	C₃×#532	semidirect
327. C₄	C₃×#711	semidirect
328. C₃	#65930	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×#4086	2
C₃×#4085	2
C₃×D₄×Dic₅	2
C₃×#4076	2
C₂×C₃×C₄×Dic₅	2
C₃×#4039	2
C₃×#4034	2
C₃×#4031	2
C₃×#4027	2
#65930	3
C₃×#532	5

Subgroups — C3×#65930 #11755607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65930 #11755607