Subgroups — C₃×#10598874 #14286980

Order 1920.

364 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	C₃×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₃×#64977	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₈×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64679	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64679	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64677	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598874	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×D₄₀	C₄	1	1	yes	characteristic
10	C₂×C₃×#676	—	2	1	no	—
11	C₂×C₃×C₈×D₅	—	2	1	no	—
12	C₃×C₄×D₂₀	C₂²	1	1	yes	characteristic
13	C₃×C₄×D₂₀	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#3953	C₄	1	1	yes	characteristic
18	C₃×#3953	C₄	1	1	yes	characteristic
19	C₃×#3952	—	2	1	no	—
20	C₃×#3952	—	2	1	no	—
21	C₃×#3950	C₄	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	C₃×#1586	—	5	1	no	maximal, Hall
25	C₄×D₄₀	C₂×C₃	1	1	yes	characteristic
26	#64977	C₂×C₃	1	1	yes	characteristic, complemented
27	C₈×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#322	C₂×C₃	1	1	yes	characteristic, complemented
29	#64679	C₂×C₃	1	1	yes	characteristic, complemented
30	#64679	C₂×C₃	1	1	yes	characteristic, complemented
31	#64677	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃×C₄×D₅	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₃×#685	—	2	1	no	—
43	C₃×#685	—	2	1	no	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	—	2	1	no	—
47	C₂×C₃×#168	—	2	1	no	—
48	C₃×D₄₀	—	2	2	no	—
49	C₃×#676	—	4	2	no	—
50	C₃×C₈×D₅	—	4	2	no	—
51	C₃×C₄×D₈	—	5	1	no	—
52	C₃×#436	—	5	1	no	—
53	C₃×C₈×D₄	—	5	1	no	—
54	C₃×#334	—	5	1	no	—
55	C₃×D₄⋊C₈	—	5	1	no	—
56	C₃×D₄⋊C₈	—	5	1	no	—
57	C₃×#322	D₅	1	1	yes	characteristic, complemented
58	C₂×D₄₀	C₃×C₄	1	1	yes	characteristic
59	C₂×#676	—	2	1	no	—
60	C₂×C₈×D₅	—	2	1	no	—
61	C₄×D₂₀	C₂²×C₃	1	1	yes	characteristic
62	C₄×D₂₀	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	#3953	C₃×C₄	1	1	yes	characteristic
67	#3953	C₃×C₄	1	1	yes	characteristic
68	#3952	—	2	1	no	—
69	#3952	—	2	1	no	—
70	#3950	C₃×C₄	1	1	yes	characteristic
71	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
72	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
73	#1586	—	5	1	no	Sylow
74	C₂²×C₃×D₅	—	2	1	no	—
75	C₂²×C₃×D₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	1	no	—
80	C₂×C₃×Dic₅	—	2	1	no	—
81	C₃×D₂₀	—	2	1	no	—
82	C₃×D₂₀	—	2	1	no	—
83	C₃×D₂₀	—	2	1	no	—
84	C₃×D₂₀	M₁₆	1	2	yes	complemented
85	C₃×C₄×D₅	—	4	1	no	—
86	C₃×C₄×D₅	—	4	1	no	—
87	C₃×C₄×D₅	—	4	1	no	—
88	C₃×C₄×D₅	—	4	1	no	—
89	C₃×C₅×C₈	—	2	2	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	2	1	no	—
92	C₃×C₅×C₈	—	2	1	no	—
93	C₃×C₅×C₈	—	2	1	no	—
94	C₃×#168	—	4	1	no	—
95	C₃×#168	—	4	1	no	—
96	C₂×C₃×D₈	—	5	1	no	—
97	C₂×C₃×M₁₆	—	10	1	no	—
98	C₂²×C₃×C₈	—	10	1	no	—
99	C₃×C₄×D₄	—	5	1	no	—
100	C₃×C₄×D₄	—	5	1	no	—
101	C₃×#107	—	5	1	no	—
102	C₃×C₄⋊C₈	—	5	1	no	—
103	C₃×C₄⋊C₈	—	5	1	no	—
104	C₃×#102	—	5	1	no	—
105	C₃×#102	—	5	1	no	—
106	C₃×C₂²⋊C₈	—	10	1	no	—
107	C₃×C₂²⋊C₈	—	10	1	no	—
108	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
110	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅	—	2	1	no	—
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
121	#685	—	2	1	no	—
122	#685	—	2	1	no	—
123	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
124	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×#168	—	2	1	no	—
126	C₂×#168	—	2	1	no	—
127	D₄₀	—	2	2	no	—
128	#676	—	4	2	no	—
129	C₈×D₅	—	4	2	no	—
130	C₄×D₈	—	5	1	no	—
131	#436	—	5	1	no	—
132	C₈×D₄	—	5	1	no	—
133	#334	—	5	1	no	—
134	D₄⋊C₈	—	5	1	no	—
135	D₄⋊C₈	—	5	1	no	—
136	#322	C₃×D₅	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₃×D₅	—	2	2	no	—
139	C₂×C₃×D₅	—	4	1	no	—
140	C₂×C₃×D₅	—	4	1	no	—
141	C₂×C₃×D₅	—	4	1	no	—
142	C₂×C₃×D₅	—	4	1	no	—
143	C₂×C₃×D₅	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#131	1	1	yes	characteristic
147	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₃×Dic₅	—	4	1	no	—
150	C₃×Dic₅	—	4	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	—	5	1	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₃×D₈	—	10	2	no	—
156	C₃×M₁₆	—	20	2	no	—
157	C₂×C₃×C₈	—	2	1	no	—
158	C₂×C₃×C₈	—	10	1	no	—
159	C₂×C₃×C₈	—	10	1	no	—
160	C₂×C₃×C₈	—	20	2	no	—
161	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
162	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
163	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
164	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	5	1	no	—
166	C₃×C₄⋊C₄	—	5	1	no	—
167	C₃×C₂²⋊C₄	—	10	1	no	—
168	C₃×C₂²⋊C₄	—	10	1	no	—
169	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
170	C₂²×D₅	—	2	1	no	—
171	C₂²×D₅	—	2	1	no	—
172	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
175	C₂×Dic₅	—	2	1	no	—
176	C₂×Dic₅	—	2	1	no	—
177	D₂₀	—	2	1	no	—
178	D₂₀	—	2	1	no	—
179	D₂₀	—	2	1	no	—
180	D₂₀	C₃×M₁₆	1	2	yes	complemented
181	C₄×D₅	—	4	1	no	—
182	C₄×D₅	—	4	1	no	—
183	C₄×D₅	—	4	1	no	—
184	C₄×D₅	—	4	1	no	—
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	#168	—	4	1	no	—
191	#168	—	4	1	no	—
192	C₂×D₈	—	5	1	no	—
193	C₂×M₁₆	—	10	1	no	—
194	C₂²×C₈	—	10	1	no	—
195	C₄×D₄	—	5	1	no	—
196	C₄×D₄	—	5	1	no	—
197	#107	—	5	1	no	—
198	C₄⋊C₈	—	5	1	no	—
199	C₄⋊C₈	—	5	1	no	—
200	#102	—	5	1	no	—
201	#102	—	5	1	no	—
202	C₂²⋊C₈	—	10	1	no	—
203	C₂²⋊C₈	—	10	1	no	—
204	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
205	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
206	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
207	C₂×C₃×C₅	#435	1	1	yes	characteristic
208	C₂×C₃×C₅	#442	1	1	yes	characteristic
209	C₂×C₃×C₅	#443	1	1	yes	characteristic
210	C₃×D₅	—	4	2	no	—
211	C₃×D₅	—	8	1	no	—
212	C₂³×C₃	—	10	1	no	—
213	C₂³×C₃	—	10	1	no	—
214	C₃×D₄	—	5	2	no	—
215	C₃×D₄	—	10	1	no	—
216	C₃×D₄	—	10	1	no	—
217	C₃×D₄	—	10	1	no	—
218	C₂×C₃×C₄	—	10	1	no	—
219	C₂×C₃×C₄	—	10	1	no	—
220	C₂×C₃×C₄	—	20	1	no	—
221	C₂×C₃×C₄	—	20	1	no	—
222	C₂×C₃×C₄	—	20	1	no	—
223	C₂×C₃×C₄	—	20	1	no	—
224	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₃×C₄	#709	1	1	yes	characteristic
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	20	1	no	—
233	C₃×C₈	—	20	1	no	—
234	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	4	1	no	—
237	C₂×D₅	—	4	1	no	—
238	C₂×D₅	—	4	1	no	—
239	C₂×D₅	—	4	1	no	—
240	C₂×D₅	—	4	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
243	C₄×C₅	C₃×#131	1	1	yes	characteristic
244	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
246	Dic₅	—	4	1	no	—
247	Dic₅	—	4	1	no	—
248	C₂×D₄	—	5	1	no	—
249	C₂×D₄	—	5	1	no	—
250	C₂²×C₄	—	10	1	no	—
251	C₂²×C₄	—	10	1	no	—
252	D₈	—	10	2	no	—
253	M₁₆	—	20	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	10	1	no	—
256	C₂×C₈	—	10	1	no	—
257	C₂×C₈	—	20	2	no	—
258	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
259	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
260	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
261	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	10	1	no	—
265	C₂²⋊C₄	—	10	1	no	—
266	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
267	C₃×C₅	#1586	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	10	2	no	—
269	C₂²×C₃	—	20	1	no	—
270	C₂²×C₃	—	20	1	no	—
271	C₂²×C₃	—	20	1	no	—
272	C₂²×C₃	—	20	1	no	—
273	C₂²×C₃	—	20	1	no	—
274	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	20	1	no	—
277	C₃×C₄	—	20	1	no	—
278	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
279	C₃×C₄	#4053	1	1	yes	characteristic
280	C₃×C₄	#4054	1	1	yes	characteristic
281	C₃×C₄	#4054	1	1	yes	characteristic
282	C₂×C₅	C₃×#435	1	1	yes	characteristic
283	C₂×C₅	C₃×#442	1	1	yes	characteristic
284	C₂×C₅	C₃×#443	1	1	yes	characteristic
285	D₅	—	4	2	no	—
286	D₅	—	8	1	no	—
287	C₂³	—	10	1	no	—
288	C₂³	—	10	1	no	—
289	D₄	—	5	2	no	—
290	D₄	—	10	1	no	—
291	D₄	—	10	1	no	—
292	D₄	—	10	1	no	—
293	C₂×C₄	—	10	1	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	20	1	no	—
296	C₂×C₄	—	20	1	no	—
297	C₂×C₄	—	20	1	no	—
298	C₂×C₄	—	20	1	no	—
299	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
301	C₂×C₄	C₃×#709	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	20	1	no	—
308	C₈	—	20	1	no	—
309	C₂×C₃	—	20	2	no	—
310	C₂×C₃	—	40	1	no	—
311	C₂×C₃	#64989	1	1	yes	characteristic
312	C₂×C₃	#64995	1	1	yes	characteristic
313	C₂×C₃	#65000	1	1	yes	characteristic
314	C₅	C₃×#1586	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	10	2	no	—
316	C₂²	—	20	1	no	—
317	C₂²	—	20	1	no	—
318	C₂²	—	20	1	no	—
319	C₂²	—	20	1	no	—
320	C₂²	—	20	1	no	—
321	C₂²	C₃×C₄×D₂₀	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	20	1	no	—
324	C₄	—	20	1	no	—
325	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
326	C₄	C₃×#4053	1	1	yes	characteristic
327	C₄	C₃×#4054	1	1	yes	characteristic
328	C₄	C₃×#4054	1	1	yes	characteristic
329	C₃	#10598874	1	1	yes	characteristic, direct product factor, Sylow
330	C₂	—	20	2	no	—
331	C₂	—	40	1	no	—
332	C₂	C₃×#64989	1	1	yes	characteristic
333	C₂	C₃×#64995	1	1	yes	characteristic
334	C₂	C₃×#65000	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₄₀	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#322	D₅
hypocenter	C₅	C₃×#1586
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
2. C₃×#64977	C₂	semidirect
3. C₃×C₈×D₂₀	C₂	semidirect
4. C₃×C₅×#322	C₂	semidirect
5. C₃×#64679	C₂	semidirect
6. C₃×#64679	C₂	semidirect
7. C₃×#64677	C₂	semidirect
8. #10598874	C₃	direct (N × H)
26. #64977	C₂×C₃	semidirect
27. C₈×D₂₀	C₂×C₃	semidirect
28. C₅×#322	C₂×C₃	semidirect
29. #64679	C₂×C₃	semidirect
30. #64679	C₂×C₃	semidirect
31. #64677	C₂×C₃	semidirect
57. C₃×#322	D₅	semidirect
84. C₃×D₂₀	M₁₆	semidirect
136. #322	C₃×D₅	semidirect
180. D₂₀	C₃×M₁₆	semidirect
267. C₃×C₅	#1586	semidirect
314. C₅	C₃×#1586	semidirect
329. C₃	#10598874	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×C₄×D₄₀	2
C₃×#64977	2
C₃×C₈×D₂₀	2
C₃×C₅×#322	2
C₃×#64679	2
C₃×#64677	2
#10598874	3
C₃×#1586	5

Subgroups — C3×#10598874 #14286980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598874 #14286980