Subgroups — C₃×#10599983 #14313931

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3930	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64782	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64729	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64711	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599983	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×D₄×D₅	—	2	1	no	—
10	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#686	—	2	1	no	—
12	C₃×D₅×D₈	—	2	2	no	—
13	C₃×D₅×D₈	C₄	1	4	yes	complemented
14	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
16	C₃×#4038	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
19	C₃×#3950	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃×#672	C₂²	1	1	yes	characteristic, complemented
21	C₃×#3940	—	2	1	no	—
22	C₃×#3930	—	2	2	no	—
23	C₃×#2140	—	5	1	no	maximal, Hall
24	C₂×D₅×D₈	C₂×C₃	1	1	yes	characteristic
25	C₂×#3930	C₂×C₃	1	1	yes	characteristic, complemented
26	#65169	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#357	C₂×C₃	1	1	yes	characteristic, complemented
28	#64782	C₂×C₃	1	1	yes	characteristic, complemented
29	#64729	C₂×C₃	1	1	yes	characteristic, complemented
30	#64711	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂³×C₃×D₅	—	4	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
34	C₃×D₄×D₅	—	2	4	no	—
35	C₃×D₄×D₅	—	4	2	no	—
36	C₂×C₃×D₂₀	—	2	1	no	—
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₃×C₅×D₈	—	2	1	no	—
39	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×#686	—	2	4	no	—
43	C₃×#685	—	4	1	no	—
44	C₃×C₄⋊Dic₅	—	2	1	no	—
45	C₃×C₄×Dic₅	—	4	1	no	—
46	C₂×C₃×#168	D₄	1	1	yes	characteristic
47	C₃×D₄₀	—	2	1	no	—
48	C₃×D₄₀	C₂×C₄	1	2	yes	—
49	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
50	C₃×C₈×D₅	D₄	1	2	yes	—
51	C₃×C₅×C₁₆	—	2	1	no	—
52	C₃×#672	—	2	1	no	—
53	C₂²×C₃×D₈	—	5	1	no	—
54	C₂×C₃×M₃₂	—	5	1	no	—
55	C₃×#427	—	5	1	no	—
56	C₃×#357	—	5	1	no	—
57	C₃×#357	—	5	1	no	—
58	C₃×#357	—	5	1	no	—
59	C₃×#357	D₅	1	1	yes	characteristic, complemented
60	C₂×D₄×D₅	—	2	1	no	—
61	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×#686	—	2	1	no	—
63	D₅×D₈	—	2	2	no	—
64	D₅×D₈	C₃×C₄	1	4	yes	complemented
65	C₂×D₄₀	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
67	#4038	—	2	1	no	—
68	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
70	#3950	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×#672	C₂²×C₃	1	1	yes	characteristic, complemented
72	#3940	—	2	1	no	—
73	#3930	—	2	2	no	—
74	#2140	—	5	1	no	Sylow
75	C₂³×C₃×C₅	—	4	1	no	—
76	C₂²×C₃×D₅	—	4	1	no	—
77	C₂²×C₃×D₅	—	4	4	no	—
78	C₂²×C₃×D₅	—	8	2	no	—
79	C₂²×C₃×D₅	—	8	2	no	—
80	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
81	C₃×C₅×D₄	—	2	2	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₃×C₂²⋊D₅	—	4	4	no	—
86	C₂×C₃×Dic₅	—	4	1	no	—
87	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
88	C₃×D₂₀	—	2	2	no	—
89	C₃×D₂₀	—	4	1	no	—
90	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
91	C₃×C₄×D₅	SD₁₆	1	2	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₃×#168	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₃×D₄	—	10	1	no	—
96	C₂×C₃×D₈	—	5	4	no	—
97	C₂×C₃×D₈	—	5	1	no	—
98	C₂×C₃×D₈	—	10	2	no	—
99	C₂×C₃×D₈	—	10	1	no	—
100	C₂×C₃×D₈	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₃×C₈	—	5	1	no	—
102	C₃×#117	—	10	1	no	—
103	C₃×M₃₂	—	10	2	no	—
104	C₂×C₃×C₁₆	—	5	1	no	—
105	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
106	C₃×#107	—	5	1	no	—
107	C₃×#107	C₂×D₅	1	1	yes	characteristic, complemented
108	C₃×#106	—	10	1	no	—
109	C₂³×D₅	—	4	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₂²⋊D₅	—	4	1	no	—
112	D₄×D₅	—	2	4	no	—
113	D₄×D₅	—	4	2	no	—
114	C₂×D₂₀	—	2	1	no	—
115	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
116	C₅×D₈	—	2	1	no	—
117	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	1	no	—
120	#686	—	2	4	no	—
121	#685	—	4	1	no	—
122	C₄⋊Dic₅	—	2	1	no	—
123	C₄×Dic₅	—	4	1	no	—
124	C₂×#168	C₃×D₄	1	1	yes	characteristic
125	D₄₀	—	2	1	no	—
126	D₄₀	C₂×C₃×C₄	1	2	yes	—
127	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
128	C₈×D₅	C₃×D₄	1	2	yes	—
129	C₅×C₁₆	—	2	1	no	—
130	#672	—	2	1	no	—
131	C₂²×D₈	—	5	1	no	—
132	C₂×M₃₂	—	5	1	no	—
133	#427	—	5	1	no	—
134	#357	—	5	1	no	—
135	#357	—	5	1	no	—
136	#357	—	5	1	no	—
137	#357	C₃×D₅	1	1	yes	characteristic, complemented
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	—	8	1	no	—
140	C₂²×C₃×C₅	—	8	1	no	—
141	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×C₃×D₅	—	2	1	no	—
143	C₂×C₃×D₅	—	2	1	no	—
144	C₂×C₃×D₅	—	4	2	no	—
145	C₂×C₃×D₅	—	8	8	no	—
146	C₂×C₃×D₅	—	8	1	no	—
147	C₂×C₃×D₅	—	8	1	no	—
148	C₂×C₃×D₅	#102	1	2	yes	—
149	C₃×C₄×C₅	—	8	1	no	—
150	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
152	C₃×Dic₅	—	8	1	no	—
153	C₃×Dic₅	#102	1	2	yes	—
154	C₂⁴×C₃	—	20	1	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	10	4	no	—
157	C₂×C₃×D₄	—	10	1	no	—
158	C₂×C₃×D₄	—	20	2	no	—
159	C₂×C₃×D₄	—	20	1	no	—
160	C₂²×C₃×C₄	—	5	1	no	—
161	C₃×D₈	—	2	1	no	—
162	C₃×D₈	—	5	2	no	—
163	C₃×D₈	—	10	4	no	—
164	C₃×D₈	—	10	1	no	—
165	C₃×D₈	C₄×D₅	1	2	yes	—
166	C₂×C₃×C₈	—	5	1	no	—
167	C₂×C₃×C₈	—	5	2	no	—
168	C₂×C₃×C₈	—	5	2	no	—
169	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
170	C₃×C₄⋊C₄	—	2	1	no	—
171	C₃×C₄⋊C₄	—	10	1	no	—
172	C₃×C₂²⋊C₄	—	20	1	no	—
173	C₃×C₄²	—	20	1	no	—
174	C₃×C₁₆	—	2	1	no	—
175	C₃×C₁₆	—	10	1	no	—
176	C₂³×C₅	—	4	1	no	—
177	C₂²×D₅	—	4	1	no	—
178	C₂²×D₅	—	4	4	no	—
179	C₂²×D₅	—	8	2	no	—
180	C₂²×D₅	—	8	2	no	—
181	C₂²×D₅	C₃×D₈	1	1	yes	characteristic
182	C₅×D₄	—	2	2	no	—
183	C₅×D₄	—	4	1	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²⋊D₅	—	4	4	no	—
187	C₂×Dic₅	—	4	1	no	—
188	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
189	D₂₀	—	2	2	no	—
190	D₂₀	—	4	1	no	—
191	C₄×D₅	C₃×C₂²⋊C₄	1	2	yes	—
192	C₄×D₅	C₃×SD₁₆	1	2	yes	—
193	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
194	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
195	#168	C₃×C₂²⋊C₄	1	2	yes	—
196	C₂²×D₄	—	10	1	no	—
197	C₂×D₈	—	5	4	no	—
198	C₂×D₈	—	5	1	no	—
199	C₂×D₈	—	10	2	no	—
200	C₂×D₈	—	10	1	no	—
201	C₂×D₈	C₂×C₃×D₅	1	1	yes	characteristic
202	C₂²×C₈	—	5	1	no	—
203	#117	—	10	1	no	—
204	M₃₂	—	10	2	no	—
205	C₂×C₁₆	—	5	1	no	—
206	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic, complemented
207	#107	—	5	1	no	—
208	#107	C₂×C₃×D₅	1	1	yes	characteristic, complemented
209	#106	—	10	1	no	—
210	C₂×C₃×C₅	—	8	2	no	—
211	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
212	C₂×C₃×C₅	#509	1	1	yes	characteristic
213	C₂×C₃×C₅	#509	1	1	yes	characteristic
214	C₃×D₅	—	2	2	no	—
215	C₃×D₅	—	8	2	no	—
216	C₂³×C₃	—	4	1	no	—
217	C₂³×C₃	—	5	1	no	—
218	C₂³×C₃	—	20	4	no	—
219	C₂³×C₃	—	20	1	no	—
220	C₂³×C₃	—	40	2	no	—
221	C₂³×C₃	—	40	2	no	—
222	C₃×D₄	—	2	2	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	—	10	2	no	—
225	C₃×D₄	—	20	4	no	—
226	C₃×D₄	—	20	1	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	5	1	no	—
229	C₂×C₃×C₄	—	5	2	no	—
230	C₂×C₃×C₄	—	5	2	no	—
231	C₂×C₃×C₄	—	20	1	no	—
232	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
233	C₃×C₈	—	5	2	no	—
234	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
235	C₃×C₈	D₄×D₅	1	1	yes	characteristic
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	—	8	1	no	—
238	C₂²×C₅	—	8	1	no	—
239	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
240	C₂×D₅	—	2	1	no	—
241	C₂×D₅	—	2	1	no	—
242	C₂×D₅	—	4	2	no	—
243	C₂×D₅	—	8	8	no	—
244	C₂×D₅	—	8	1	no	—
245	C₂×D₅	—	8	1	no	—
246	C₂×D₅	C₃×#102	1	2	yes	—
247	C₄×C₅	—	8	1	no	—
248	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
250	Dic₅	—	8	1	no	—
251	Dic₅	C₃×#102	1	2	yes	—
252	C₂⁴	—	20	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	10	4	no	—
255	C₂×D₄	—	10	1	no	—
256	C₂×D₄	—	20	2	no	—
257	C₂×D₄	—	20	1	no	—
258	C₂²×C₄	—	5	1	no	—
259	D₈	—	2	1	no	—
260	D₈	—	5	2	no	—
261	D₈	—	10	4	no	—
262	D₈	—	10	1	no	—
263	D₈	C₃×C₄×D₅	1	2	yes	—
264	C₂×C₈	—	5	1	no	—
265	C₂×C₈	—	5	2	no	—
266	C₂×C₈	—	5	2	no	—
267	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	10	1	no	—
270	C₂²⋊C₄	—	20	1	no	—
271	C₄²	—	20	1	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	10	1	no	—
274	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	—	4	2	no	—
276	C₂²×C₃	—	5	2	no	—
277	C₂²×C₃	—	8	1	no	—
278	C₂²×C₃	—	8	1	no	—
279	C₂²×C₃	—	10	1	no	—
280	C₂²×C₃	—	10	1	no	—
281	C₂²×C₃	—	20	2	no	—
282	C₂²×C₃	—	40	8	no	—
283	C₂²×C₃	—	40	1	no	—
284	C₂²×C₃	—	40	1	no	—
285	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
286	C₃×C₄	—	5	2	no	—
287	C₃×C₄	—	8	1	no	—
288	C₃×C₄	—	40	1	no	—
289	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
290	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
291	C₂×C₅	—	8	2	no	—
292	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
293	C₂×C₅	C₃×#509	1	1	yes	characteristic
294	C₂×C₅	C₃×#509	1	1	yes	characteristic
295	D₅	—	2	2	no	—
296	D₅	—	8	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	5	1	no	—
299	C₂³	—	20	4	no	—
300	C₂³	—	20	1	no	—
301	C₂³	—	40	2	no	—
302	C₂³	—	40	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	1	no	—
305	D₄	—	10	2	no	—
306	D₄	—	20	4	no	—
307	D₄	—	20	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	5	1	no	—
310	C₂×C₄	—	5	2	no	—
311	C₂×C₄	—	5	2	no	—
312	C₂×C₄	—	20	1	no	—
313	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
314	C₈	—	5	2	no	—
315	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
316	C₈	C₃×D₄×D₅	1	1	yes	characteristic
317	C₂×C₃	—	8	2	no	—
318	C₂×C₃	—	10	2	no	—
319	C₂×C₃	—	40	2	no	—
320	C₂×C₃	D₅×#102	1	1	yes	characteristic
321	C₂×C₃	#65200	1	1	yes	characteristic
322	C₂×C₃	#65203	1	1	yes	characteristic
323	C₅	C₃×#2140	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	4	2	no	—
325	C₂²	—	5	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	10	1	no	—
329	C₂²	—	10	1	no	—
330	C₂²	—	20	2	no	—
331	C₂²	—	40	8	no	—
332	C₂²	—	40	1	no	—
333	C₂²	—	40	1	no	—
334	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
335	C₄	—	5	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	40	1	no	—
338	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
339	C₄	C₃×D₅×SD₁₆	1	1	yes	characteristic
340	C₃	#10599983	1	1	yes	characteristic, direct product factor, Sylow
341	C₂	—	8	2	no	—
342	C₂	—	10	2	no	—
343	C₂	—	40	2	no	—
344	C₂	C₃×D₅×#102	1	1	yes	characteristic
345	C₂	C₃×#65200	1	1	yes	characteristic
346	C₂	C₃×#65203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×D₈
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#357	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×D₅×D₈	C₂
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#357	D₅
hypocenter	C₅	C₃×#2140
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#357

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3930	C₂	semidirect
3. C₃×#65169	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×#64782	C₂	semidirect
6. C₃×#64729	C₂	semidirect
7. C₃×#64711	C₂	semidirect
8. #10599983	C₃	direct (N × H)
13. C₃×D₅×D₈	C₄	semidirect
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
18. C₃×C₅×#107	C₂²	semidirect
19. C₃×#3950	C₂²	semidirect
20. C₂×C₃×#672	C₂²	semidirect
25. C₂×#3930	C₂×C₃	semidirect
26. #65169	C₂×C₃	semidirect
27. C₅×#357	C₂×C₃	semidirect
28. #64782	C₂×C₃	semidirect
29. #64729	C₂×C₃	semidirect
30. #64711	C₂×C₃	semidirect
59. C₃×#357	D₅	semidirect
64. D₅×D₈	C₃×C₄	semidirect
68. C₂×C₅×C₁₆	C₂²×C₃	semidirect
69. C₅×#107	C₂²×C₃	semidirect
70. #3950	C₂²×C₃	semidirect
71. C₂×#672	C₂²×C₃	semidirect
105. C₂×C₃×C₁₆	C₂×D₅	semidirect
107. C₃×#107	C₂×D₅	semidirect
137. #357	C₃×D₅	semidirect
206. C₂×C₁₆	C₂×C₃×D₅	semidirect
208. #107	C₂×C₃×D₅	semidirect
274. C₃×C₅	#2140	semidirect
323. C₅	C₃×#2140	semidirect
340. C₃	#10599983	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×D₈	2
C₂×C₃×#3930	2
C₃×#65169	2
C₃×C₅×#357	2
C₃×#64782	2
C₃×#64729	2
C₃×#64711	2
#10599983	3
C₃×#2140	5

Subgroups — C3×#10599983 #14313931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599983 #14313931