Subgroups — C₂×C₃×#65198 #14478390

Order 1920.

720 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	C₂²×C₃×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65198	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×Dic₄₀	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#3932	C₂	1	2	yes	maximal, complemented
7	C₂×#65198	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×#709	—	2	1	no	—
9	C₂²×C₃×Dic₁₀	—	2	1	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₂₀	—	2	2	no	—
12	C₂×C₃×Dic₂₀	—	2	1	no	—
13	C₂×C₃×Dic₂₀	C₂²	1	2	yes	—
14	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₂₀	C₂²	1	4	yes	complemented
16	C₃×#4050	—	2	2	no	—
17	C₃×#4050	C₂²	1	4	yes	—
18	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
19	C₂×C₃×#677	—	2	1	no	—
20	C₃×C₅×M₃₂	C₂²	1	4	yes	complemented
21	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
22	C₃×Dic₄₀	C₂²	1	8	yes	complemented
23	C₃×#3932	C₂²	1	8	yes	complemented
24	C₂×C₃×#510	—	5	1	no	maximal, Hall
25	C₂²×Dic₂₀	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#4050	C₂×C₃	1	1	yes	characteristic
27	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
28	#65198	C₂×C₃	1	8	yes	direct product factor
29	C₂×Dic₄₀	C₂×C₃	1	2	yes	complemented
30	C₂×#3932	C₂×C₃	1	2	yes	complemented
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
33	C₂²×C₃×Dic₅	—	4	1	no	—
34	C₃×#709	—	2	4	no	—
35	C₃×#709	—	4	2	no	—
36	C₂×C₃×D₂₀	—	2	1	no	—
37	C₂×C₃×C₄×D₅	—	4	1	no	—
38	C₂×C₃×Dic₁₀	—	2	2	no	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×Dic₁₀	—	2	1	no	—
41	C₂×C₃×Dic₁₀	—	4	1	no	—
42	C₂×C₃×Dic₁₀	—	4	2	no	—
43	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
47	C₃×Dic₂₀	—	2	1	no	—
48	C₃×Dic₂₀	—	2	2	no	—
49	C₃×Dic₂₀	—	2	4	no	—
50	C₃×Dic₂₀	C₂³	1	4	yes	—
51	C₃×Dic₂₀	C₂³	1	2	yes	—
52	C₃×D₄₀	—	2	1	no	—
53	C₃×D₄₀	C₂³	1	2	yes	—
54	C₃×#677	—	2	4	no	—
55	C₃×C₅×C₁₆	C₂³	1	4	yes	—
56	C₂×C₃×#135	—	5	1	no	—
57	C₂²×C₃×Q₁₆	—	5	1	no	—
58	C₃×#510	—	5	8	no	—
59	C₂×C₃×Q₃₂	—	5	2	no	—
60	C₂×C₃×SD₃₂	—	5	2	no	—
61	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
62	C₂×#709	—	2	1	no	—
63	C₂²×Dic₁₀	—	2	1	no	—
64	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×Dic₂₀	—	2	2	no	—
66	C₂×Dic₂₀	—	2	1	no	—
67	C₂×Dic₂₀	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₂₀	C₂²×C₃	1	2	yes	—
69	C₂×Dic₂₀	C₂²×C₃	1	4	yes	complemented
70	#4050	—	2	2	no	—
71	#4050	C₂²×C₃	1	4	yes	—
72	C₂×D₄₀	C₂²×C₃	1	1	yes	characteristic
73	C₂×#677	—	2	1	no	—
74	C₅×M₃₂	C₂²×C₃	1	4	yes	complemented
75	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
76	Dic₄₀	C₂²×C₃	1	8	yes	complemented
77	#3932	C₂²×C₃	1	8	yes	complemented
78	C₂×#510	—	5	1	no	Sylow
79	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₃×D₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₃×C₂²⋊D₅	—	4	4	no	—
86	C₂×C₃×Dic₅	—	4	4	no	—
87	C₂×C₃×Dic₅	—	4	2	no	—
88	C₂×C₃×Dic₅	—	4	1	no	—
89	C₃×D₂₀	—	2	2	no	—
90	C₃×D₂₀	—	4	1	no	—
91	C₃×C₄×D₅	—	4	4	no	—
92	C₃×Dic₁₀	—	2	2	no	—
93	C₃×Dic₁₀	—	2	4	no	—
94	C₃×Dic₁₀	—	4	2	no	—
95	C₃×Dic₁₀	—	4	4	no	—
96	C₃×Dic₁₀	—	4	1	no	—
97	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×#41	—	10	1	no	—
101	C₂²×C₃×Q₈	—	10	1	no	—
102	C₃×#135	—	5	4	no	—
103	C₃×#135	—	10	2	no	—
104	C₂×C₃×Q₁₆	—	5	2	no	—
105	C₂×C₃×Q₁₆	—	5	1	no	—
106	C₂×C₃×Q₁₆	—	5	4	no	—
107	C₂×C₃×Q₁₆	—	10	1	no	—
108	C₂×C₃×Q₁₆	—	10	2	no	—
109	C₂×C₃×SD₁₆	—	10	1	no	—
110	C₂×C₃×D₈	—	5	1	no	—
111	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
112	C₃×Q₃₂	—	5	8	no	—
113	C₃×SD₃₂	—	5	8	no	—
114	C₃×M₃₂	C₂×D₅	1	4	yes	complemented
115	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
116	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
117	C₂×C₂²⋊D₅	—	4	1	no	—
118	C₂²×Dic₅	—	4	1	no	—
119	#709	—	2	4	no	—
120	#709	—	4	2	no	—
121	C₂×D₂₀	—	2	1	no	—
122	C₂×C₄×D₅	—	4	1	no	—
123	C₂×Dic₁₀	—	2	2	no	—
124	C₂×Dic₁₀	—	2	4	no	—
125	C₂×Dic₁₀	—	2	1	no	—
126	C₂×Dic₁₀	—	4	1	no	—
127	C₂×Dic₁₀	—	4	2	no	—
128	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
129	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
130	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
132	Dic₂₀	—	2	1	no	—
133	Dic₂₀	—	2	2	no	—
134	Dic₂₀	—	2	4	no	—
135	Dic₂₀	C₂³×C₃	1	4	yes	—
136	Dic₂₀	C₂³×C₃	1	2	yes	—
137	D₄₀	—	2	1	no	—
138	D₄₀	C₂³×C₃	1	2	yes	—
139	#677	—	2	4	no	—
140	C₅×C₁₆	C₂³×C₃	1	4	yes	—
141	C₂×#135	—	5	1	no	—
142	C₂²×Q₁₆	—	5	1	no	—
143	#510	—	5	8	no	—
144	C₂×Q₃₂	—	5	2	no	—
145	C₂×SD₃₂	—	5	2	no	—
146	C₂×M₃₂	C₃×D₅	1	1	yes	characteristic, complemented
147	C₂²×C₃×C₅	—	2	2	no	—
148	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
149	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₃×D₅	—	4	2	no	—
151	C₂×C₃×D₅	—	8	2	no	—
152	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
153	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₃×Dic₅	—	4	4	no	—
156	C₃×Dic₅	—	4	2	no	—
157	C₃×#41	—	10	4	no	—
158	C₃×#41	—	20	2	no	—
159	C₂×C₃×Q₈	—	10	1	no	—
160	C₂×C₃×Q₈	—	10	4	no	—
161	C₂×C₃×Q₈	—	10	2	no	—
162	C₂×C₃×Q₈	—	20	2	no	—
163	C₂×C₃×Q₈	—	20	1	no	—
164	C₂×C₃×D₄	—	10	1	no	—
165	C₂×C₃×D₄	—	20	1	no	—
166	C₂²×C₃×C₄	—	20	1	no	—
167	C₂²×C₃×C₄	—	20	1	no	—
168	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
169	C₃×Q₁₆	—	5	4	no	—
170	C₃×Q₁₆	—	5	2	no	—
171	C₃×Q₁₆	—	10	2	no	—
172	C₃×Q₁₆	—	10	4	no	—
173	C₃×Q₁₆	—	10	1	no	—
174	C₃×SD₁₆	—	10	4	no	—
175	C₃×D₈	—	5	2	no	—
176	C₃×D₈	—	10	1	no	—
177	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
178	C₂×C₃×C₈	D₂₀	1	2	yes	—
179	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
181	C₃×C₁₆	C₂²×D₅	1	4	yes	—
182	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
183	C₂²×D₅	—	4	1	no	—
184	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
188	C₂²⋊D₅	—	4	4	no	—
189	C₂×Dic₅	—	4	4	no	—
190	C₂×Dic₅	—	4	2	no	—
191	C₂×Dic₅	—	4	1	no	—
192	D₂₀	—	2	2	no	—
193	D₂₀	—	4	1	no	—
194	C₄×D₅	—	4	4	no	—
195	Dic₁₀	—	2	2	no	—
196	Dic₁₀	—	2	4	no	—
197	Dic₁₀	—	4	2	no	—
198	Dic₁₀	—	4	4	no	—
199	Dic₁₀	—	4	1	no	—
200	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
201	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
202	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
203	C₂×#41	—	10	1	no	—
204	C₂²×Q₈	—	10	1	no	—
205	#135	—	5	4	no	—
206	#135	—	10	2	no	—
207	C₂×Q₁₆	—	5	2	no	—
208	C₂×Q₁₆	—	5	4	no	—
209	C₂×Q₁₆	—	5	1	no	—
210	C₂×Q₁₆	—	10	1	no	—
211	C₂×Q₁₆	—	10	2	no	—
212	C₂×SD₁₆	—	10	1	no	—
213	C₂×D₈	—	5	1	no	—
214	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
215	Q₃₂	—	5	8	no	—
216	SD₃₂	—	5	8	no	—
217	M₃₂	C₂×C₃×D₅	1	4	yes	complemented
218	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	—
219	C₂×C₃×C₅	—	2	2	no	—
220	C₂×C₃×C₅	#510	1	2	yes	complemented
221	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
222	C₃×D₅	—	8	2	no	—
223	C₂³×C₃	—	20	1	no	—
224	C₂³×C₃	D₄₀	1	1	yes	characteristic
225	C₃×Q₈	—	10	2	no	—
226	C₃×Q₈	—	10	4	no	—
227	C₃×Q₈	—	20	4	no	—
228	C₃×Q₈	—	20	2	no	—
229	C₃×Q₈	—	20	1	no	—
230	C₃×D₄	—	10	2	no	—
231	C₃×D₄	—	20	4	no	—
232	C₃×D₄	—	20	1	no	—
233	C₂×C₃×C₄	—	20	4	no	—
234	C₂×C₃×C₄	—	20	1	no	—
235	C₂×C₃×C₄	—	20	2	no	—
236	C₂×C₃×C₄	—	20	4	no	—
237	C₂×C₃×C₄	D₄₀	1	2	yes	—
238	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
240	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
241	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
242	C₃×C₈	C₂×D₂₀	1	2	yes	—
243	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
246	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
247	C₂×D₅	—	4	2	no	—
248	C₂×D₅	—	8	2	no	—
249	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
250	C₄×C₅	C₂×C₃×D₈	1	2	yes	—
251	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
252	Dic₅	—	4	4	no	—
253	Dic₅	—	4	2	no	—
254	#41	—	10	4	no	—
255	#41	—	20	2	no	—
256	C₂×Q₈	—	10	1	no	—
257	C₂×Q₈	—	10	2	no	—
258	C₂×Q₈	—	10	4	no	—
259	C₂×Q₈	—	20	1	no	—
260	C₂×Q₈	—	20	2	no	—
261	C₂×D₄	—	10	1	no	—
262	C₂×D₄	—	20	1	no	—
263	C₂²×C₄	—	20	1	no	—
264	C₂²×C₄	—	20	1	no	—
265	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
266	Q₁₆	—	5	4	no	—
267	Q₁₆	—	5	2	no	—
268	Q₁₆	—	10	2	no	—
269	Q₁₆	—	10	4	no	—
270	Q₁₆	—	10	1	no	—
271	SD₁₆	—	10	4	no	—
272	D₈	—	5	2	no	—
273	D₈	—	10	1	no	—
274	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
275	C₂×C₈	C₃×D₂₀	1	2	yes	—
276	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
277	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
278	C₁₆	C₂²×C₃×D₅	1	4	yes	—
279	C₃×C₅	C₂×#510	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	20	2	no	—
282	C₂²×C₃	—	40	2	no	—
283	C₂²×C₃	C₂×D₄₀	1	2	yes	—
284	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
285	C₃×C₄	—	20	2	no	—
286	C₃×C₄	—	20	4	no	—
287	C₃×C₄	C₂×D₄₀	1	2	yes	—
288	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
289	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	C₃×#510	1	2	yes	complemented
292	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
293	D₅	—	8	2	no	—
294	C₂³	—	20	1	no	—
295	C₂³	C₃×D₄₀	1	1	yes	characteristic
296	Q₈	—	10	2	no	—
297	Q₈	—	10	4	no	—
298	Q₈	—	20	4	no	—
299	Q₈	—	20	2	no	—
300	Q₈	—	20	1	no	—
301	D₄	—	10	2	no	—
302	D₄	—	20	4	no	—
303	D₄	—	20	1	no	—
304	C₂×C₄	—	20	1	no	—
305	C₂×C₄	—	20	4	no	—
306	C₂×C₄	—	20	2	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	C₃×D₄₀	1	1	yes	characteristic
309	C₂×C₄	C₃×D₄₀	1	2	yes	—
310	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
312	C₈	C₂×C₃×D₂₀	1	1	yes	characteristic
313	C₈	C₂×C₃×D₂₀	1	2	yes	—
314	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	40	2	no	—
317	C₂×C₃	#65198	1	2	yes	direct product factor
318	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
319	C₅	C₂×C₃×#510	1	1	yes	characteristic, complemented, Sylow
320	C₂²	—	2	2	no	—
321	C₂²	—	20	2	no	—
322	C₂²	—	40	2	no	—
323	C₂²	C₂×C₃×D₄₀	1	2	yes	—
324	C₂²	C₂×C₃×D₄₀	1	1	yes	characteristic
325	C₄	—	20	2	no	—
326	C₄	—	20	4	no	—
327	C₄	C₂×C₃×D₄₀	1	2	yes	—
328	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
329	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
330	C₃	C₂×#65198	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	40	2	no	—
333	C₂	C₃×#65198	1	2	yes	direct product factor
334	C₂	C₂²×C₃×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₄₀
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×#4050	C₂
hypercenter	C₂×C₃×M₃₂	D₅
hypocenter	C₅	C₂×C₃×#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Dic₂₀	C₂	semidirect
3. C₂×C₃×C₅×M₃₂	C₂	semidirect
4. C₃×#65198	C₂	direct (N × H)
5. C₂×C₃×Dic₄₀	C₂	semidirect
6. C₂×C₃×#3932	C₂	semidirect
7. C₂×#65198	C₃	direct (N × H)
15. C₂×C₃×Dic₂₀	C₂²	semidirect
20. C₃×C₅×M₃₂	C₂²	semidirect
22. C₃×Dic₄₀	C₂²	semidirect
23. C₃×#3932	C₂²	semidirect
25. C₂²×Dic₂₀	C₂×C₃	semidirect
27. C₂×C₅×M₃₂	C₂×C₃	semidirect
28. #65198	C₂×C₃	direct (N × H)
29. C₂×Dic₄₀	C₂×C₃	semidirect
30. C₂×#3932	C₂×C₃	semidirect
61. C₂×C₃×M₃₂	D₅	semidirect
69. C₂×Dic₂₀	C₂²×C₃	semidirect
74. C₅×M₃₂	C₂²×C₃	semidirect
76. Dic₄₀	C₂²×C₃	semidirect
77. #3932	C₂²×C₃	semidirect
114. C₃×M₃₂	C₂×D₅	semidirect
146. C₂×M₃₂	C₃×D₅	semidirect
217. M₃₂	C₂×C₃×D₅	semidirect
220. C₂×C₃×C₅	#510	semidirect
279. C₃×C₅	C₂×#510	semidirect
291. C₂×C₅	C₃×#510	semidirect
317. C₂×C₃	#65198	direct (N × H)
319. C₅	C₂×C₃×#510	semidirect
330. C₃	C₂×#65198	direct (N × H)
333. C₂	C₃×#65198	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×Dic₂₀	2
C₂×C₃×#4050	2
C₂×C₃×C₅×M₃₂	2
C₃×#65198	2
C₂×C₃×Dic₄₀	2
C₂×C₃×#3932	2
C₂×#65198	3
C₂×C₃×#510	5

Subgroups — C2×C3×#65198 #14478390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65198 #14478390