Subgroups — C₃×#10600576 #14296365

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3947	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3947	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3946	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×#683	C₂	1	1	yes	maximal, characteristic
6	C₃×#64770	C₂	1	1	yes	maximal, characteristic
7	C₃×#64744	C₂	1	1	yes	maximal, characteristic
8	#10600576	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×#683	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	—	2	2	no	—
18	C₃×#3963	C₄	1	2	yes	—
19	C₃×#3947	—	2	2	no	—
20	C₃×#3947	C₄	1	4	yes	complemented
21	C₃×#3946	C₄	1	4	yes	complemented
22	C₃×#1771	—	5	1	no	maximal, Hall
23	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
24	C₂×#3947	C₂×C₃	1	1	yes	characteristic
25	C₂×#3947	C₂×C₃	1	1	yes	characteristic
26	C₂×#3946	C₂×C₃	1	1	yes	characteristic
27	C₄×#683	C₂×C₃	1	1	yes	characteristic
28	#64770	C₂×C₃	1	1	yes	characteristic
29	#64744	C₂×C₃	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
41	C₃×C₄²×C₅	—	2	1	no	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×#683	—	2	4	no	—
44	C₃×C₄×Dic₅	—	2	1	no	—
45	C₃×C₄×Dic₅	—	2	1	no	—
46	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
47	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
48	C₂×C₃×#168	—	2	1	no	—
49	C₂×C₃×#168	—	2	1	no	—
50	C₂×C₃×#168	—	2	1	no	—
51	C₂×C₃×#168	—	2	1	no	—
52	C₂×C₃×#168	C₂×C₄	1	2	yes	—
53	C₂×C₃×#168	C₂×C₄	1	2	yes	—
54	C₂×C₃×C₄⋊C₈	—	5	1	no	—
55	C₂×C₃×#97	—	5	1	no	—
56	C₂×C₃×#97	—	5	1	no	—
57	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
58	C₃×C₄×C₄⋊C₄	—	5	1	no	—
59	C₃×#336	—	5	1	no	—
60	C₃×#336	—	5	1	no	—
61	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×#683	C₃×C₄	1	2	yes	—
65	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
67	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
68	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
69	C₄×C₅×C₈	—	2	2	no	—
70	#3963	C₃×C₄	1	2	yes	—
71	#3947	—	2	2	no	—
72	#3947	C₃×C₄	1	4	yes	complemented
73	#3946	C₃×C₄	1	4	yes	complemented
74	#1771	—	5	1	no	Sylow
75	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
85	C₂×C₃×Dic₅	—	2	2	no	—
86	C₂×C₃×Dic₅	—	2	2	no	—
87	C₂×C₃×Dic₅	—	2	2	no	—
88	C₂×C₃×Dic₅	—	2	2	no	—
89	C₂×C₃×Dic₅	—	2	2	no	—
90	C₂×C₃×Dic₅	C₄²	1	2	yes	—
91	C₂×C₃×Dic₅	C₄²	1	2	yes	—
92	C₃×C₅×C₈	—	2	2	no	—
93	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
94	C₃×#168	—	2	2	no	—
95	C₃×#168	—	2	2	no	—
96	C₂²×C₃×C₈	—	5	1	no	—
97	C₂²×C₃×C₈	—	5	1	no	—
98	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
100	C₂×C₃×C₄⋊C₄	—	5	2	no	—
101	C₂×C₃×C₄²	—	5	1	no	—
102	C₂×C₃×C₄²	—	5	1	no	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₃×C₄⋊C₈	—	5	4	no	—
105	C₃×#97	—	5	4	no	—
106	C₃×#97	—	10	2	no	—
107	C₃×C₄×C₈	—	2	2	no	—
108	C₃×#95	—	5	2	no	—
109	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
112	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
119	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
120	C₄²×C₅	—	2	1	no	—
121	C₄²×C₅	—	2	1	no	—
122	#683	—	2	4	no	—
123	C₄×Dic₅	—	2	1	no	—
124	C₄×Dic₅	—	2	1	no	—
125	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
126	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
127	C₂×#168	—	2	1	no	—
128	C₂×#168	—	2	1	no	—
129	C₂×#168	—	2	1	no	—
130	C₂×#168	—	2	1	no	—
131	C₂×#168	C₂×C₃×C₄	1	2	yes	—
132	C₂×#168	C₂×C₃×C₄	1	2	yes	—
133	C₂×C₄⋊C₈	—	5	1	no	—
134	C₂×#97	—	5	1	no	—
135	C₂×#97	—	5	1	no	—
136	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
137	C₄×C₄⋊C₄	—	5	1	no	—
138	#336	—	5	1	no	—
139	#336	—	5	1	no	—
140	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
141	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
143	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
144	C₂²×C₃×C₅	#131	1	2	yes	—
145	C₃×C₄×C₅	—	2	4	no	—
146	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	#117	1	1	yes	characteristic
148	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
149	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
150	C₃×Dic₅	—	2	2	no	—
151	C₃×Dic₅	—	2	2	no	—
152	C₃×Dic₅	—	4	2	no	—
153	C₂²×C₃×C₄	—	5	1	no	—
154	C₂²×C₃×C₄	—	5	1	no	—
155	C₂²×C₃×C₄	—	5	2	no	—
156	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
157	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	—	2	1	no	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	—	5	2	no	—
161	C₂×C₃×C₈	—	5	2	no	—
162	C₂×C₃×C₈	—	10	1	no	—
163	C₂×C₃×C₈	—	10	1	no	—
164	C₂×C₃×C₈	—	10	1	no	—
165	C₂×C₃×C₈	—	10	1	no	—
166	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
167	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
168	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
169	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
170	C₃×C₄⋊C₄	—	10	4	no	—
171	C₃×C₄²	—	2	1	no	—
172	C₃×C₄²	—	2	1	no	—
173	C₃×C₄²	—	5	2	no	—
174	C₃×C₄²	—	5	2	no	—
175	C₃×C₄²	—	10	1	no	—
176	C₃×C₄²	—	10	1	no	—
177	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	—
181	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
186	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
187	C₂×Dic₅	—	2	2	no	—
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	—	2	2	no	—
190	C₂×Dic₅	—	2	2	no	—
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	C₃×C₄²	1	2	yes	—
193	C₂×Dic₅	C₃×C₄²	1	2	yes	—
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	C₃×C₄⋊C₄	1	4	yes	complemented
196	#168	—	2	2	no	—
197	#168	—	2	2	no	—
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	—	5	1	no	—
200	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
201	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	5	2	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₂×C₄²	—	5	1	no	—
205	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
206	C₄⋊C₈	—	5	4	no	—
207	#97	—	5	4	no	—
208	#97	—	10	2	no	—
209	C₄×C₈	—	2	2	no	—
210	#95	—	5	2	no	—
211	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
213	C₂×C₃×C₅	#405	1	1	yes	characteristic
214	C₂×C₃×C₅	#435	1	2	yes	—
215	C₂×C₃×C₅	#446	1	2	yes	—
216	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	2	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	10	2	no	—
225	C₂×C₃×C₄	—	10	2	no	—
226	C₂×C₃×C₄	#676	1	4	yes	—
227	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₃×C₄	#709	1	1	yes	characteristic
231	C₂×C₃×C₄	#709	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₃×C₈	—	2	2	no	—
234	C₃×C₈	—	10	2	no	—
235	C₃×C₈	—	10	2	no	—
236	C₃×C₈	#683	1	4	yes	complemented
237	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
238	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
239	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
240	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
241	C₂²×C₅	C₃×#131	1	2	yes	—
242	C₄×C₅	—	2	4	no	—
243	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₄×C₅	C₃×#117	1	1	yes	characteristic
245	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
246	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
247	Dic₅	—	2	2	no	—
248	Dic₅	—	2	2	no	—
249	Dic₅	—	4	2	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	5	2	no	—
253	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	10	1	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	1	no	—
262	C₂×C₈	—	10	1	no	—
263	C₂×C₈	C₃×Dic₁₀	1	2	yes	—
264	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
265	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
266	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
267	C₄⋊C₄	—	10	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	5	2	no	—
271	C₄²	—	5	2	no	—
272	C₄²	—	10	1	no	—
273	C₄²	—	10	1	no	—
274	C₃×C₅	#1771	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	C₄×Dic₁₀	1	1	yes	characteristic
276	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
277	C₂²×C₃	C₂×#676	1	2	yes	—
278	C₂²×C₃	#4047	1	2	yes	—
279	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
280	C₃×C₄	—	2	4	no	—
281	C₃×C₄	—	10	2	no	—
282	C₃×C₄	—	10	2	no	—
283	C₃×C₄	—	20	2	no	—
284	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
285	C₃×C₄	#4018	1	1	yes	characteristic
286	C₃×C₄	C₂×#683	1	1	yes	characteristic
287	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
288	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
289	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
290	C₂×C₅	C₃×#405	1	1	yes	characteristic
291	C₂×C₅	C₃×#435	1	2	yes	—
292	C₂×C₅	C₃×#446	1	2	yes	—
293	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	C₃×#676	1	4	yes	—
304	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
305	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
307	C₂×C₄	C₃×#709	1	1	yes	characteristic
308	C₂×C₄	C₃×#709	1	1	yes	characteristic
309	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	10	2	no	—
312	C₈	—	10	2	no	—
313	C₈	C₃×#683	1	4	yes	complemented
314	C₂×C₃	#64968	1	2	yes	—
315	C₂×C₃	C₄×#676	1	1	yes	characteristic
316	C₂×C₃	#64978	1	1	yes	characteristic
317	C₂×C₃	C₄×#683	1	1	yes	characteristic
318	C₂×C₃	#65400	1	2	yes	—
319	C₅	C₃×#1771	1	1	yes	characteristic, complemented, Sylow
320	C₂²	C₃×C₄×Dic₁₀	1	1	yes	characteristic
321	C₂²	C₃×C₄²×D₅	1	1	yes	characteristic
322	C₂²	C₂×C₃×#676	1	2	yes	—
323	C₂²	C₃×#4047	1	2	yes	—
324	C₂²	C₃×C₄×C₂²⋊D₅	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	2	no	—
328	C₄	—	20	2	no	—
329	C₄	C₃×C₄×Dic₁₀	1	1	yes	characteristic
330	C₄	C₃×#4018	1	1	yes	characteristic
331	C₄	C₂×C₃×#683	1	1	yes	characteristic
332	C₄	C₃×C₄×C₂²⋊D₅	1	1	yes	characteristic
333	C₃	#10600576	1	1	yes	characteristic, direct product factor, Sylow
334	C₂	C₃×#64968	1	2	yes	—
335	C₂	C₃×C₄×#676	1	1	yes	characteristic
336	C₂	C₃×#64978	1	1	yes	characteristic
337	C₂	C₃×C₄×#683	1	1	yes	characteristic
338	C₂	C₃×#65400	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×#1771
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #10600576	C₃	direct (N × H)
20. C₃×#3947	C₄	semidirect
21. C₃×#3946	C₄	semidirect
72. #3947	C₃×C₄	semidirect
73. #3946	C₃×C₄	semidirect
93. C₃×C₅×C₈	C₄⋊C₄	semidirect
195. C₅×C₈	C₃×C₄⋊C₄	semidirect
236. C₃×C₈	#683	semidirect
274. C₃×C₅	#1771	semidirect
313. C₈	C₃×#683	semidirect
319. C₅	C₃×#1771	semidirect
333. C₃	#10600576	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×C₄×C₅×C₈	2
C₂×C₃×#3947	2
C₂×C₃×#3946	2
C₃×C₄×#683	2
C₃×#64770	2
C₃×#64744	2
#10600576	3
C₃×#1771	5

Subgroups — C3×#10600576 #14296365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600576 #14296365