Subgroups — C₃×#10615007 #14415343

Order 1920.

968 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4094	C₂	1	2	yes	maximal, complemented
3	C₃×#65523	C₂	1	2	yes	maximal, complemented
4	C₃×#65300	C₂	1	8	yes	maximal, complemented
5	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
7	#10615007	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃×C₅×#127	C₄	1	1	yes	characteristic
11	C₃×C₅×#124	C₄	1	2	yes	—
12	C₃×C₅×#121	C₄	1	4	yes	—
13	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×#4094	—	2	8	no	—
16	C₂×C₃×#681	C₂²	1	4	yes	—
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₂²×C₃×#168	C₂²	1	4	yes	—
19	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
20	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
21	C₃×C₂²⋊#168	C₂²	1	8	yes	—
22	C₃×C₄⋊#168	C₂²	1	4	yes	—
23	C₃×#3935	C₂²	1	4	yes	—
24	C₃×#2932	—	5	1	no	maximal, Hall
25	C₅×#532	C₂×C₃	1	1	yes	characteristic
26	C₂×#4094	C₂×C₃	1	2	yes	complemented
27	#65523	C₂×C₃	1	2	yes	complemented
28	#65300	C₂×C₃	1	8	yes	complemented
29	#65213	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#3935	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×C₅×#41	—	2	4	no	—
32	C₃×C₅×#41	—	2	4	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₃×#681	—	2	4	no	—
46	C₃×#681	—	2	8	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#168	—	2	8	no	—
49	C₂×C₃×#168	D₄	1	4	yes	complemented
50	C₂×C₃×#168	C₂³	1	2	yes	—
51	C₂×C₃×#168	C₂³	1	4	yes	—
52	C₂×C₃×#168	C₂³	1	2	yes	—
53	C₂×C₃×#131	—	5	2	no	—
54	C₃×#532	D₅	1	1	yes	characteristic
55	C₃×#435	—	5	8	no	—
56	C₃×#423	—	5	1	no	—
57	C₃×#408	—	5	2	no	—
58	C₂×C₃×#97	—	5	1	no	—
59	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
60	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
61	C₅×#127	C₃×C₄	1	1	yes	characteristic
62	C₅×#124	C₃×C₄	1	2	yes	—
63	C₅×#121	C₃×C₄	1	4	yes	—
64	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
65	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
66	#4094	—	2	8	no	—
67	C₂×#681	C₂²×C₃	1	4	yes	—
68	C₂×#681	C₂²×C₃	1	2	yes	—
69	C₂²×#168	C₂²×C₃	1	4	yes	—
70	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
71	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
72	C₂²⋊#168	C₂²×C₃	1	8	yes	—
73	C₄⋊#168	C₂²×C₃	1	4	yes	—
74	#3935	C₂²×C₃	1	4	yes	—
75	#2932	—	5	1	no	Sylow
76	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
77	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₅×Q₈	—	2	4	no	—
79	C₃×C₅×D₄	—	2	8	no	—
80	C₃×C₅×D₄	—	2	4	no	—
81	C₃×C₅×D₄	—	4	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	2	4	no	—
84	C₂×C₃×C₄×C₅	—	4	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	#41	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₃×#168	—	2	4	no	—
93	C₃×#168	—	2	2	no	—
94	C₃×#168	C₂×D₄	1	4	yes	—
95	C₂×C₃×#41	C₂×D₅	1	2	yes	—
96	C₃×#131	—	10	8	no	—
97	C₂×C₃×M₁₆	—	5	2	no	—
98	C₂×C₃×M₁₆	—	5	4	no	—
99	C₂²×C₃×C₈	—	5	1	no	—
100	C₂²×C₃×C₈	—	5	4	no	—
101	C₂²×C₃×C₈	—	5	1	no	—
102	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
103	C₃×#127	Dic₅	1	1	yes	characteristic
104	C₃×#124	Dic₅	1	2	yes	—
105	C₃×#121	Dic₅	1	4	yes	—
106	C₃×C₄×D₄	C₂×D₅	1	4	yes	—
107	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
108	C₃×C₄⋊C₈	—	5	4	no	—
109	C₃×C₂²⋊C₈	—	5	8	no	—
110	C₃×#97	—	5	4	no	—
111	C₅×#41	—	2	4	no	—
112	C₅×#41	—	2	4	no	—
113	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
115	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
116	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
117	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
118	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
119	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
120	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
121	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
122	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
123	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
124	C₄²×C₅	C₂³×C₃	1	2	yes	—
125	#681	—	2	4	no	—
126	#681	—	2	8	no	—
127	C₂×#168	—	2	2	no	—
128	C₂×#168	—	2	8	no	—
129	C₂×#168	C₃×D₄	1	4	yes	complemented
130	C₂×#168	C₂³×C₃	1	2	yes	—
131	C₂×#168	C₂³×C₃	1	4	yes	—
132	C₂×#168	C₂³×C₃	1	2	yes	—
133	C₂×#131	—	5	2	no	—
134	#532	C₃×D₅	1	1	yes	characteristic
135	#435	—	5	8	no	—
136	#423	—	5	1	no	—
137	#408	—	5	2	no	—
138	C₂×#97	—	5	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	—	2	4	no	—
141	C₂²×C₃×C₅	—	2	8	no	—
142	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
143	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₃×C₄×C₅	—	2	4	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
148	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
150	C₃×#41	—	2	4	no	—
151	C₃×#41	—	2	4	no	—
152	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
153	C₂×C₃×D₄	C₂×Dic₅	1	4	yes	—
154	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
155	C₂×C₃×D₄	C₂×Dic₅	1	4	yes	—
156	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
157	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₃×M₁₆	—	10	8	no	—
160	C₃×M₁₆	—	10	4	no	—
161	C₂×C₃×C₈	—	5	4	no	—
162	C₂×C₃×C₈	—	5	2	no	—
163	C₂×C₃×C₈	—	5	2	no	—
164	C₂×C₃×C₈	—	5	4	no	—
165	C₂×C₃×C₈	—	10	8	no	—
166	C₂×C₃×C₈	—	10	2	no	—
167	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
168	C₃×C₂²⋊C₄	C₂×Dic₅	1	8	yes	—
169	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
170	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
171	C₃×C₄²	C₂²×D₅	1	2	yes	—
172	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
173	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₅×Q₈	—	2	4	no	—
175	C₅×D₄	—	2	8	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	4	4	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
182	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
183	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
184	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	C₃×#41	1	2	yes	—
187	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
188	#168	—	2	4	no	—
189	#168	—	2	2	no	—
190	#168	C₂×C₃×D₄	1	4	yes	—
191	C₂×#41	C₂×C₃×D₅	1	2	yes	—
192	#131	—	10	8	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	—	5	4	no	—
195	C₂²×C₈	—	5	1	no	—
196	C₂²×C₈	—	5	4	no	—
197	C₂²×C₈	—	5	1	no	—
198	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic
199	#127	C₃×Dic₅	1	1	yes	characteristic
200	#124	C₃×Dic₅	1	2	yes	—
201	#121	C₃×Dic₅	1	4	yes	—
202	C₄×D₄	C₂×C₃×D₅	1	4	yes	—
203	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	5	4	no	—
205	C₂²⋊C₈	—	5	8	no	—
206	#97	—	5	4	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	—	4	4	no	—
209	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂×C₃×C₅	#568	1	2	yes	—
211	C₂³×C₃	C₂²×Dic₅	1	4	yes	—
212	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
213	C₃×Q₈	—	2	4	no	—
214	C₃×D₄	—	2	8	no	—
215	C₃×D₄	—	2	4	no	—
216	C₃×D₄	—	4	4	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
221	C₂×C₃×C₄	#711	1	2	yes	—
222	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
223	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
224	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
225	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
227	C₃×C₈	—	5	4	no	—
228	C₃×C₈	—	10	4	no	—
229	C₃×C₈	—	10	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	2	8	no	—
232	C₂²×C₅	—	2	4	no	—
233	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
234	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	4	no	—
238	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
239	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
240	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
241	#41	—	2	4	no	—
242	#41	—	2	4	no	—
243	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
244	C₂×D₄	C₂×C₃×Dic₅	1	4	yes	—
245	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
246	C₂×D₄	C₂×C₃×Dic₅	1	4	yes	—
247	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
249	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
250	M₁₆	—	10	8	no	—
251	M₁₆	—	10	4	no	—
252	C₂×C₈	—	5	4	no	—
253	C₂×C₈	—	5	2	no	—
254	C₂×C₈	—	5	2	no	—
255	C₂×C₈	—	5	4	no	—
256	C₂×C₈	—	10	8	no	—
257	C₂×C₈	—	10	2	no	—
258	C₄⋊C₄	C₂×C₃×Dic₅	1	4	yes	—
259	C₂²⋊C₄	C₂×C₃×Dic₅	1	8	yes	—
260	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
261	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
262	C₄²	C₂²×C₃×D₅	1	2	yes	—
263	C₃×C₅	#2932	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	2	8	no	—
267	C₂²×C₃	D₄×Dic₅	1	2	yes	—
268	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	D₄×Dic₅	1	2	yes	—
273	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
274	C₃×C₄	C₂×#711	1	1	yes	characteristic
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	4	4	no	—
277	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
278	C₂×C₅	C₃×#568	1	2	yes	—
279	C₂³	C₂²×C₃×Dic₅	1	4	yes	—
280	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	D₄	—	2	8	no	—
283	D₄	—	2	4	no	—
284	D₄	—	4	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
289	C₂×C₄	C₃×#711	1	2	yes	—
290	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
291	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
292	C₂×C₄	C₂²×C₃×Dic₅	1	4	yes	—
293	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
294	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
295	C₈	—	5	4	no	—
296	C₈	—	10	4	no	—
297	C₈	—	10	2	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	4	4	no	—
300	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
301	C₂×C₃	#66153	1	2	yes	—
302	C₅	C₃×#2932	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	8	no	—
306	C₂²	C₃×D₄×Dic₅	1	2	yes	—
307	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	C₃×D₄×Dic₅	1	2	yes	—
312	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
313	C₄	C₂×C₃×#711	1	1	yes	characteristic
314	C₃	#10615007	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	2	2	no	—
316	C₂	—	4	4	no	—
317	C₂	C₂×C₃×D₄×Dic₅	1	1	yes	characteristic
318	C₂	C₃×#66153	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#2932
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4094	C₂	semidirect
3. C₃×#65523	C₂	semidirect
4. C₃×#65300	C₂	semidirect
5. C₃×#65213	C₂	semidirect
6. C₂×C₃×#3935	C₂	semidirect
7. #10615007	C₃	direct (N × H)
26. C₂×#4094	C₂×C₃	semidirect
27. #65523	C₂×C₃	semidirect
28. #65300	C₂×C₃	semidirect
29. #65213	C₂×C₃	semidirect
30. C₂×#3935	C₂×C₃	semidirect
49. C₂×C₃×#168	D₄	semidirect
129. C₂×#168	C₃×D₄	semidirect
263. C₃×C₅	#2932	semidirect
302. C₅	C₃×#2932	semidirect
314. C₃	#10615007	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×#532	2
C₂×C₃×#4094	2
C₃×#65523	2
C₃×#65300	2
C₃×#65213	2
C₂×C₃×#3935	2
#10615007	3
C₃×#2932	5

Subgroups — C3×#10615007 #14415343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615007 #14415343