Subgroups — C₃×#10610779 #14413114

Order 1920.

968 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	C₂×C₃×D₅×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65407	C₂	1	2	yes	maximal, complemented
5	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65128	C₂	1	8	yes	maximal, complemented
7	#10610779	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×#685	C₄	1	2	yes	—
12	C₂×C₃×#683	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×#681	C₂²	1	2	yes	—
15	C₃×D₅×M₁₆	—	2	8	no	—
16	C₂×C₃×#676	C₂²	1	4	yes	—
17	C₂×C₃×C₈×D₅	C₂²	1	4	yes	—
18	C₃×#4018	C₄	1	4	yes	—
19	C₃×#4018	C₂²	1	4	yes	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	complemented
21	C₃×#3946	C₂²	1	8	yes	complemented
22	C₃×C₄⋊#168	C₂²	1	4	yes	complemented
23	C₃×#2905	—	5	1	no	maximal, Hall
24	C₂×D₅×M₁₆	C₂×C₃	1	2	yes	—
25	C₂×#4018	C₂×C₃	1	1	yes	characteristic
26	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
27	#65407	C₂×C₃	1	2	yes	complemented
28	#65213	C₂×C₃	1	1	yes	characteristic, complemented
29	#65128	C₂×C₃	1	8	yes	complemented
30	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
34	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
37	C₂×C₃×C₄×D₅	Q₈	1	4	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
39	C₃×C₅×M₁₆	—	2	4	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₃×#685	C₂×C₄	1	8	yes	—
44	C₃×#683	C₂×C₄	1	8	yes	—
45	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
46	C₃×C₄×Dic₅	C₂³	1	2	yes	—
47	C₃×#681	—	2	4	no	—
48	C₂×C₃×#168	C₂³	1	4	yes	—
49	C₃×#676	—	2	8	no	—
50	C₃×C₈×D₅	—	2	8	no	—
51	C₂²×C₃×M₁₆	—	5	2	no	—
52	C₂×C₃×#117	—	5	1	no	—
53	C₃×#424	—	5	8	no	—
54	C₃×#423	—	5	2	no	—
55	C₃×#423	—	5	1	no	—
56	C₃×#423	D₅	1	1	yes	characteristic, complemented
57	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×#685	C₃×C₄	1	2	yes	—
61	C₂×#683	C₃×C₄	1	2	yes	—
62	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×#681	C₂²×C₃	1	2	yes	—
64	D₅×M₁₆	—	2	8	no	—
65	C₂×#676	C₂²×C₃	1	4	yes	—
66	C₂×C₈×D₅	C₂²×C₃	1	4	yes	—
67	#4018	C₃×C₄	1	4	yes	—
68	#4018	C₂²×C₃	1	4	yes	—
69	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
70	#3946	C₂²×C₃	1	8	yes	complemented
71	C₄⋊#168	C₂²×C₃	1	4	yes	complemented
72	#2905	—	5	1	no	Sylow
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×D₅	—	4	2	no	—
75	C₂²×C₃×D₅	C₄⋊C₄	1	4	yes	—
76	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₃×Dic₅	—	4	2	no	—
84	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
85	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
86	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
87	C₃×C₄×D₅	C₄⋊C₄	1	8	yes	—
88	C₃×C₄×D₅	C₂×D₄	1	4	yes	—
89	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
90	C₃×C₅×C₈	—	2	4	no	—
91	C₃×#168	—	2	4	no	—
92	C₂³×C₃×C₄	—	5	1	no	—
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₂×C₃×M₁₆	—	5	4	no	—
95	C₂×C₃×M₁₆	—	10	8	no	—
96	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
97	C₂²×C₃×C₈	—	5	4	no	—
98	C₃×#117	—	5	4	no	—
99	C₃×#117	—	5	4	no	—
100	C₂×C₃×C₄⋊C₄	—	5	2	no	—
101	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
102	C₂×C₃×C₄²	—	5	1	no	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₃×C₄⋊C₈	—	5	8	no	—
105	C₃×C₄⋊C₈	—	5	4	no	—
106	C₃×C₄⋊C₈	C₂×D₅	1	4	yes	complemented
107	C₂³×D₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
111	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
114	C₂×C₄×D₅	C₃×Q₈	1	4	yes	—
115	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
116	C₅×M₁₆	—	2	4	no	—
117	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
118	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
119	C₄²×C₅	C₂³×C₃	1	2	yes	—
120	#685	C₂×C₃×C₄	1	8	yes	—
121	#683	C₂×C₃×C₄	1	8	yes	—
122	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
123	C₄×Dic₅	C₂³×C₃	1	2	yes	—
124	#681	—	2	4	no	—
125	C₂×#168	C₂³×C₃	1	4	yes	—
126	#676	—	2	8	no	—
127	C₈×D₅	—	2	8	no	—
128	C₂²×M₁₆	—	5	2	no	—
129	C₂×#117	—	5	1	no	—
130	#424	—	5	8	no	—
131	#423	—	5	2	no	—
132	#423	—	5	1	no	—
133	#423	C₃×D₅	1	1	yes	characteristic, complemented
134	C₂²×C₃×C₅	—	2	2	no	—
135	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	2	4	no	—
138	C₂×C₃×D₅	—	4	4	no	—
139	C₂×C₃×D₅	C₂×C₄⋊C₄	1	4	yes	—
140	C₃×C₄×C₅	—	2	4	no	—
141	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
142	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
144	C₃×Dic₅	—	2	4	no	—
145	C₃×Dic₅	C₂×C₄⋊C₄	1	4	yes	—
146	C₂⁴×C₃	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₂²×C₃×C₄	—	5	2	no	—
149	C₂²×C₃×C₄	—	5	4	no	—
150	C₂²×C₃×C₄	—	5	4	no	—
151	C₂²×C₃×C₄	—	5	2	no	—
152	C₂²×C₃×C₄	—	5	2	no	—
153	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₃×M₁₆	—	2	4	no	—
156	C₃×M₁₆	—	10	8	no	—
157	C₃×M₁₆	—	10	4	no	—
158	C₂×C₃×C₈	—	5	4	no	—
159	C₂×C₃×C₈	—	10	8	no	—
160	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
161	C₃×C₄⋊C₄	—	5	8	no	—
162	C₃×C₂²⋊C₄	—	5	8	no	—
163	C₃×C₄²	—	5	2	no	—
164	C₃×C₄²	—	5	2	no	—
165	C₃×C₄²	C₄×D₅	1	2	yes	—
166	C₃×C₄²	C₂²×D₅	1	2	yes	—
167	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
168	C₂²×D₅	—	4	2	no	—
169	C₂²×D₅	C₃×C₄⋊C₄	1	4	yes	—
170	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
176	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×Dic₅	—	4	2	no	—
178	C₂×Dic₅	C₃×C₄⋊C₄	1	4	yes	—
179	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
180	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
181	C₄×D₅	C₃×C₄⋊C₄	1	8	yes	—
182	C₄×D₅	C₂×C₃×D₄	1	4	yes	—
183	C₄×D₅	C₂×C₃×Q₈	1	4	yes	—
184	C₅×C₈	—	2	4	no	—
185	#168	—	2	4	no	—
186	C₂³×C₄	—	5	1	no	—
187	C₂×M₁₆	—	5	2	no	—
188	C₂×M₁₆	—	5	4	no	—
189	C₂×M₁₆	—	10	8	no	—
190	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
191	C₂²×C₈	—	5	4	no	—
192	#117	—	5	4	no	—
193	#117	—	5	4	no	—
194	C₂×C₄⋊C₄	—	5	2	no	—
195	C₂×C₂²⋊C₄	—	5	2	no	—
196	C₂×C₄²	—	5	1	no	—
197	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
198	C₄⋊C₈	—	5	8	no	—
199	C₄⋊C₈	—	5	4	no	—
200	C₄⋊C₈	C₂×C₃×D₅	1	4	yes	complemented
201	C₂×C₃×C₅	—	2	2	no	—
202	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₅	#568	1	2	yes	—
204	C₃×D₅	—	2	4	no	—
205	C₂³×C₃	—	5	2	no	—
206	C₂³×C₃	—	5	4	no	—
207	C₂³×C₃	—	20	2	no	—
208	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	5	2	no	—
211	C₂×C₃×C₄	—	5	4	no	—
212	C₂×C₃×C₄	—	5	4	no	—
213	C₂×C₃×C₄	—	5	4	no	—
214	C₂×C₃×C₄	—	5	8	no	—
215	C₂×C₃×C₄	—	5	4	no	—
216	C₂×C₃×C₄	—	20	2	no	—
217	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
218	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
220	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
221	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
222	C₃×C₈	—	2	4	no	—
223	C₃×C₈	—	10	4	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
226	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
227	C₂×D₅	—	2	4	no	—
228	C₂×D₅	—	4	4	no	—
229	C₂×D₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
232	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
234	Dic₅	—	2	4	no	—
235	Dic₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
236	C₂⁴	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	5	4	no	—
240	C₂²×C₄	—	5	4	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
244	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
245	M₁₆	—	2	4	no	—
246	M₁₆	—	10	8	no	—
247	M₁₆	—	10	4	no	—
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	10	8	no	—
250	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
251	C₄⋊C₄	—	5	8	no	—
252	C₂²⋊C₄	—	5	8	no	—
253	C₄²	—	5	2	no	—
254	C₄²	—	5	2	no	—
255	C₄²	C₃×C₄×D₅	1	2	yes	—
256	C₄²	C₂²×C₃×D₅	1	2	yes	—
257	C₃×C₅	#2905	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	5	4	no	—
260	C₂²×C₃	—	10	4	no	—
261	C₂²×C₃	—	20	4	no	—
262	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
263	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	5	4	no	—
266	C₃×C₄	—	10	4	no	—
267	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
268	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
269	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₅	C₃×#568	1	2	yes	—
273	D₅	—	2	4	no	—
274	C₂³	—	5	2	no	—
275	C₂³	—	5	4	no	—
276	C₂³	—	20	2	no	—
277	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	4	no	—
281	C₂×C₄	—	5	4	no	—
282	C₂×C₄	—	5	4	no	—
283	C₂×C₄	—	5	8	no	—
284	C₂×C₄	—	5	4	no	—
285	C₂×C₄	—	20	2	no	—
286	C₂×C₄	C₂×C₃×C₄×D₅	1	4	yes	—
287	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
288	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
289	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
290	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	10	4	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	10	4	no	—
295	C₂×C₃	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
296	C₂×C₃	#66079	1	2	yes	—
297	C₅	C₃×#2905	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	5	4	no	—
300	C₂²	—	10	4	no	—
301	C₂²	—	20	4	no	—
302	C₂²	C₃×D₅×C₄⋊C₄	1	2	yes	—
303	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	5	4	no	—
306	C₄	—	10	4	no	—
307	C₄	C₃×D₅×C₄⋊C₄	1	2	yes	—
308	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
309	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
310	C₃	#10610779	1	1	yes	characteristic, direct product factor, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	10	4	no	—
313	C₂	C₂×C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
314	C₂	C₃×#66079	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	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#423	D₅
hypocenter	C₅	C₃×#2905
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#423	C₂	semidirect
4. C₃×#65407	C₂	semidirect
5. C₃×#65213	C₂	semidirect
6. C₃×#65128	C₂	semidirect
7. #10610779	C₃	direct (N × H)
20. C₃×C₅×C₄⋊C₈	C₂²	semidirect
21. C₃×#3946	C₂²	semidirect
22. C₃×C₄⋊#168	C₂²	semidirect
26. C₅×#423	C₂×C₃	semidirect
27. #65407	C₂×C₃	semidirect
28. #65213	C₂×C₃	semidirect
29. #65128	C₂×C₃	semidirect
56. C₃×#423	D₅	semidirect
69. C₅×C₄⋊C₈	C₂²×C₃	semidirect
70. #3946	C₂²×C₃	semidirect
71. C₄⋊#168	C₂²×C₃	semidirect
106. C₃×C₄⋊C₈	C₂×D₅	semidirect
133. #423	C₃×D₅	semidirect
200. C₄⋊C₈	C₂×C₃×D₅	semidirect
257. C₃×C₅	#2905	semidirect
297. C₅	C₃×#2905	semidirect
310. C₃	#10610779	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×D₅×M₁₆	2
C₂×C₃×#4018	2
C₃×C₅×#423	2
C₃×#65407	2
C₃×#65213	2
C₃×#65128	2
#10610779	3
C₃×#2905	5

Subgroups — C3×#10610779 #14413114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610779 #14413114