Subgroups — C₂×C₃×#66087 #14510134

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4150	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66087	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#4055	C₂	1	6	yes	maximal, complemented
5	C₂²×C₃×Dic₂₀	C₂	1	3	yes	maximal, complemented
6	C₂×C₃×#4050	C₂	1	3	yes	maximal, complemented
7	C₂×#66087	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	C₃×#4150	—	2	8	no	—
10	C₃×#4150	C₂²	1	16	yes	complemented
11	C₂×C₃×Q₈×D₅	—	2	2	no	—
12	C₂×C₃×#711	—	2	6	no	—
13	C₂×C₃×#709	C₂²	1	6	yes	—
14	C₂²×C₃×Dic₁₀	C₂²	1	6	yes	—
15	C₃×C₅×#131	C₂²	1	8	yes	complemented
16	C₂×C₃×C₅×M₁₆	C₂²	1	3	yes	—
17	C₂²×C₃×C₅×C₈	C₂²	1	3	yes	—
18	C₃×#4055	C₂²	1	48	yes	complemented
19	C₂×C₃×Dic₂₀	C₂²	1	3	yes	—
20	C₂×C₃×Dic₂₀	C₂²	1	6	yes	—
21	C₂×C₃×Dic₂₀	C₂²	1	24	yes	complemented
22	C₃×#4050	C₂²	1	24	yes	complemented
23	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
24	C₂×C₃×#677	C₂²	1	6	yes	—
25	C₂×C₃×#578	—	5	1	no	maximal, Hall
26	C₂×#4150	C₂×C₃	1	2	yes	complemented
27	C₂×C₅×#131	C₂×C₃	1	1	yes	characteristic, complemented
28	#66087	C₂×C₃	1	16	yes	direct product factor
29	C₂×#4055	C₂×C₃	1	6	yes	complemented
30	C₂²×Dic₂₀	C₂×C₃	1	3	yes	complemented
31	C₂×#4050	C₂×C₃	1	3	yes	complemented
32	C₃×C₅×#41	D₄	1	4	yes	—
33	C₃×C₅×#41	C₂³	1	4	yes	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂×C₃×C₂²⋊D₅	—	2	6	no	—
38	C₂²×C₃×Dic₅	—	2	6	no	—
39	C₃×Q₈×D₅	—	2	16	no	—
40	C₃×#711	—	2	48	no	—
41	C₃×#709	—	2	12	no	—
42	C₃×#709	C₂³	1	24	yes	—
43	C₂×C₃×D₂₀	C₂³	1	2	yes	—
44	C₂×C₃×C₄×D₅	—	2	6	no	—
45	C₂×C₃×Dic₁₀	—	2	6	no	—
46	C₂×C₃×Dic₁₀	—	2	12	no	—
47	C₂×C₃×Dic₁₀	C₂³	1	6	yes	—
48	C₂×C₃×Dic₁₀	C₂³	1	24	yes	—
49	C₃×C₅×M₁₆	C₂³	1	12	yes	—
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
52	C₂×C₃×C₅×C₈	C₂³	1	12	yes	—
53	C₃×Dic₂₀	C₂³	1	12	yes	—
54	C₃×Dic₂₀	C₂³	1	24	yes	—
55	C₃×D₄₀	C₂³	1	4	yes	—
56	C₃×#677	C₂³	1	24	yes	—
57	C₂×C₃×2₋¹⁺⁴	—	5	2	no	—
58	C₃×#578	—	5	16	no	—
59	C₂×C₃×#137	—	5	6	no	—
60	C₂×C₃×#135	—	5	3	no	—
61	C₂²×C₃×Q₁₆	—	5	3	no	—
62	C₂×C₃×#131	D₅	1	1	yes	characteristic, complemented
63	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
64	#4150	—	2	8	no	—
65	#4150	C₂²×C₃	1	16	yes	complemented
66	C₂×Q₈×D₅	—	2	2	no	—
67	C₂×#711	—	2	6	no	—
68	C₂×#709	C₂²×C₃	1	6	yes	—
69	C₂²×Dic₁₀	C₂²×C₃	1	6	yes	—
70	C₅×#131	C₂²×C₃	1	8	yes	complemented
71	C₂×C₅×M₁₆	C₂²×C₃	1	3	yes	—
72	C₂²×C₅×C₈	C₂²×C₃	1	3	yes	—
73	#4055	C₂²×C₃	1	48	yes	complemented
74	C₂×Dic₂₀	C₂²×C₃	1	3	yes	—
75	C₂×Dic₂₀	C₂²×C₃	1	6	yes	—
76	C₂×Dic₂₀	C₂²×C₃	1	24	yes	complemented
77	#4050	C₂²×C₃	1	24	yes	complemented
78	C₂×D₄₀	C₂²×C₃	1	1	yes	characteristic
79	C₂×#677	C₂²×C₃	1	6	yes	—
80	C₂×#578	—	5	1	no	Sylow
81	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
82	C₂²×C₃×D₅	—	2	2	no	—
83	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
84	C₃×C₅×D₄	C₂×D₄	1	12	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
89	C₃×C₂²⋊D₅	—	2	24	no	—
90	C₂×C₃×Dic₅	—	2	6	no	—
91	C₂×C₃×Dic₅	—	2	24	no	—
92	C₃×D₂₀	—	2	2	no	—
93	C₃×D₂₀	C₂⁴	1	4	yes	—
94	C₃×C₄×D₅	—	2	24	no	—
95	C₃×Dic₁₀	—	2	6	no	—
96	C₃×Dic₁₀	—	2	24	no	—
97	C₃×Dic₁₀	C₂⁴	1	12	yes	—
98	C₃×C₅×C₈	C₂⁴	1	6	yes	—
99	C₃×C₅×C₈	C₂⁴	1	2	yes	—
100	C₃×2₋¹⁺⁴	—	5	16	no	—
101	C₃×2₋¹⁺⁴	—	10	8	no	—
102	C₂×C₃×#41	—	5	6	no	—
103	C₂×C₃×#41	—	10	6	no	—
104	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
105	C₂²×C₃×Q₈	—	5	6	no	—
106	C₂²×C₃×Q₈	—	10	2	no	—
107	C₃×#137	—	5	48	no	—
108	C₃×#135	—	5	24	no	—
109	C₂×C₃×Q₁₆	—	5	3	no	—
110	C₂×C₃×Q₁₆	—	5	6	no	—
111	C₂×C₃×Q₁₆	—	5	24	no	—
112	C₂×C₃×SD₁₆	—	5	6	no	—
113	C₂×C₃×D₈	—	5	1	no	—
114	C₃×#131	C₂×D₅	1	8	yes	complemented
115	C₂×C₃×M₁₆	C₂×D₅	1	3	yes	—
116	C₂²×C₃×C₈	C₂×D₅	1	3	yes	—
117	C₅×#41	C₃×D₄	1	4	yes	—
118	C₅×#41	C₂³×C₃	1	4	yes	—
119	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
120	C₂×C₅×D₄	C₃×D₄	1	3	yes	—
121	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
122	C₂×C₂²⋊D₅	—	2	6	no	—
123	C₂²×Dic₅	—	2	6	no	—
124	Q₈×D₅	—	2	16	no	—
125	#711	—	2	48	no	—
126	#709	—	2	12	no	—
127	#709	C₂³×C₃	1	24	yes	—
128	C₂×D₂₀	C₂³×C₃	1	2	yes	—
129	C₂×C₄×D₅	—	2	6	no	—
130	C₂×Dic₁₀	—	2	6	no	—
131	C₂×Dic₁₀	—	2	12	no	—
132	C₂×Dic₁₀	C₂³×C₃	1	6	yes	—
133	C₂×Dic₁₀	C₂³×C₃	1	24	yes	—
134	C₅×M₁₆	C₂³×C₃	1	12	yes	—
135	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₅×C₈	C₂³×C₃	1	3	yes	—
137	C₂×C₅×C₈	C₂³×C₃	1	12	yes	—
138	Dic₂₀	C₂³×C₃	1	12	yes	—
139	Dic₂₀	C₂³×C₃	1	24	yes	—
140	D₄₀	C₂³×C₃	1	4	yes	—
141	#677	C₂³×C₃	1	24	yes	—
142	C₂×2₋¹⁺⁴	—	5	2	no	—
143	#578	—	5	16	no	—
144	C₂×#137	—	5	6	no	—
145	C₂×#135	—	5	3	no	—
146	C₂²×Q₁₆	—	5	3	no	—
147	C₂×#131	C₃×D₅	1	1	yes	characteristic, complemented
148	C₂²×C₃×C₅	—	2	6	no	—
149	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃×D₅	—	2	4	no	—
152	C₂×C₃×D₅	—	4	4	no	—
153	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄×C₅	C₂²×D₄	1	6	yes	—
155	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
156	C₃×Dic₅	—	2	12	no	—
157	C₃×#41	—	5	24	no	—
158	C₃×#41	—	10	48	no	—
159	C₃×#41	—	10	12	no	—
160	C₃×#41	D₂₀	1	4	yes	—
161	C₃×#41	C₂²×D₅	1	4	yes	—
162	C₂×C₃×Q₈	—	5	6	no	—
163	C₂×C₃×Q₈	—	5	24	no	—
164	C₂×C₃×Q₈	—	10	16	no	—
165	C₂×C₃×Q₈	—	10	6	no	—
166	C₂×C₃×Q₈	—	10	12	no	—
167	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
168	C₂×C₃×D₄	—	5	2	no	—
169	C₂×C₃×D₄	—	10	6	no	—
170	C₂×C₃×D₄	D₂₀	1	3	yes	—
171	C₂²×C₃×C₄	—	10	6	no	—
172	C₂²×C₃×C₄	—	10	6	no	—
173	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
174	C₃×Q₁₆	—	5	12	no	—
175	C₃×Q₁₆	—	5	24	no	—
176	C₃×SD₁₆	—	5	24	no	—
177	C₃×D₈	—	5	4	no	—
178	C₃×M₁₆	C₂²×D₅	1	12	yes	—
179	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
181	C₂×C₃×C₈	C₂²×D₅	1	12	yes	—
182	C₂³×C₅	C₂×C₃×D₄	1	3	yes	—
183	C₂²×D₅	—	2	2	no	—
184	C₅×Q₈	C₂×C₃×D₄	1	4	yes	—
185	C₅×D₄	C₂×C₃×D₄	1	12	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
188	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂⁴×C₃	1	6	yes	—
190	C₂²⋊D₅	—	2	24	no	—
191	C₂×Dic₅	—	2	6	no	—
192	C₂×Dic₅	—	2	24	no	—
193	D₂₀	—	2	2	no	—
194	D₂₀	C₂⁴×C₃	1	4	yes	—
195	C₄×D₅	—	2	24	no	—
196	Dic₁₀	—	2	6	no	—
197	Dic₁₀	—	2	24	no	—
198	Dic₁₀	C₂⁴×C₃	1	12	yes	—
199	C₅×C₈	C₂⁴×C₃	1	6	yes	—
200	C₅×C₈	C₂⁴×C₃	1	2	yes	—
201	2₋¹⁺⁴	—	5	16	no	—
202	2₋¹⁺⁴	—	10	8	no	—
203	C₂×#41	—	5	6	no	—
204	C₂×#41	—	10	6	no	—
205	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
206	C₂²×Q₈	—	5	6	no	—
207	C₂²×Q₈	—	10	2	no	—
208	#137	—	5	48	no	—
209	#135	—	5	24	no	—
210	C₂×Q₁₆	—	5	3	no	—
211	C₂×Q₁₆	—	5	6	no	—
212	C₂×Q₁₆	—	5	24	no	—
213	C₂×SD₁₆	—	5	6	no	—
214	C₂×D₈	—	5	1	no	—
215	#131	C₂×C₃×D₅	1	8	yes	complemented
216	C₂×M₁₆	C₂×C₃×D₅	1	3	yes	—
217	C₂²×C₈	C₂×C₃×D₅	1	3	yes	—
218	C₂×C₃×C₅	—	2	6	no	—
219	C₂×C₃×C₅	#578	1	2	yes	complemented
220	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
221	C₃×D₅	—	4	4	no	—
222	C₂³×C₃	—	10	2	no	—
223	C₂³×C₃	C₂×D₂₀	1	3	yes	—
224	C₃×Q₈	—	5	12	no	—
225	C₃×Q₈	—	10	24	no	—
226	C₃×Q₈	—	10	6	no	—
227	C₃×Q₈	C₂×D₂₀	1	4	yes	—
228	C₃×D₄	—	5	4	no	—
229	C₃×D₄	—	10	24	no	—
230	C₃×D₄	—	10	2	no	—
231	C₃×D₄	C₂×D₂₀	1	12	yes	—
232	C₂×C₃×C₄	—	10	24	no	—
233	C₂×C₃×C₄	—	10	6	no	—
234	C₂×C₃×C₄	—	10	24	no	—
235	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
236	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
237	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
238	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
239	C₃×C₈	C₂³×D₅	1	2	yes	—
240	C₃×C₈	C₂³×D₅	1	6	yes	—
241	C₂²×C₅	—	2	6	no	—
242	C₂²×C₅	C₂²×C₃×D₄	1	6	yes	—
243	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
244	C₂×D₅	—	2	4	no	—
245	C₂×D₅	—	4	4	no	—
246	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
247	C₄×C₅	C₂²×C₃×D₄	1	6	yes	—
248	C₄×C₅	C₂⁵×C₃	1	1	yes	characteristic
249	Dic₅	—	2	12	no	—
250	#41	—	5	24	no	—
251	#41	—	10	48	no	—
252	#41	—	10	12	no	—
253	#41	C₃×D₂₀	1	4	yes	—
254	#41	C₂²×C₃×D₅	1	4	yes	—
255	C₂×Q₈	—	5	6	no	—
256	C₂×Q₈	—	5	24	no	—
257	C₂×Q₈	—	10	16	no	—
258	C₂×Q₈	—	10	6	no	—
259	C₂×Q₈	—	10	12	no	—
260	C₂×Q₈	C₃×D₂₀	1	1	yes	characteristic
261	C₂×D₄	—	5	2	no	—
262	C₂×D₄	—	10	6	no	—
263	C₂×D₄	C₃×D₂₀	1	3	yes	—
264	C₂²×C₄	—	10	6	no	—
265	C₂²×C₄	—	10	6	no	—
266	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
267	Q₁₆	—	5	12	no	—
268	Q₁₆	—	5	24	no	—
269	SD₁₆	—	5	24	no	—
270	D₈	—	5	4	no	—
271	M₁₆	C₂²×C₃×D₅	1	12	yes	—
272	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
273	C₂×C₈	C₂²×C₃×D₅	1	3	yes	—
274	C₂×C₈	C₂²×C₃×D₅	1	12	yes	—
275	C₃×C₅	C₂×#578	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	6	no	—
277	C₂²×C₃	—	10	4	no	—
278	C₂²×C₃	—	20	4	no	—
279	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
280	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
281	C₃×C₄	—	10	12	no	—
282	C₃×C₄	C₂²×D₂₀	1	6	yes	—
283	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
284	C₃×C₄	C₂⁴×D₅	1	1	yes	characteristic
285	C₂×C₅	—	2	6	no	—
286	C₂×C₅	C₃×#578	1	2	yes	complemented
287	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
288	D₅	—	4	4	no	—
289	C₂³	—	10	2	no	—
290	C₂³	C₂×C₃×D₂₀	1	3	yes	—
291	Q₈	—	5	12	no	—
292	Q₈	—	10	24	no	—
293	Q₈	—	10	6	no	—
294	Q₈	C₂×C₃×D₂₀	1	4	yes	—
295	D₄	—	5	4	no	—
296	D₄	—	10	24	no	—
297	D₄	—	10	2	no	—
298	D₄	C₂×C₃×D₂₀	1	12	yes	—
299	C₂×C₄	—	10	6	no	—
300	C₂×C₄	—	10	24	no	—
301	C₂×C₄	—	10	24	no	—
302	C₂×C₄	C₂×C₃×D₂₀	1	3	yes	—
303	C₂×C₄	C₂×C₃×D₂₀	1	6	yes	—
304	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂³×C₃×D₅	1	6	yes	—
306	C₈	C₂³×C₃×D₅	1	2	yes	—
307	C₈	C₂³×C₃×D₅	1	6	yes	—
308	C₂×C₃	—	2	6	no	—
309	C₂×C₃	—	20	4	no	—
310	C₂×C₃	#66087	1	2	yes	direct product factor
311	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
312	C₅	C₂×C₃×#578	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	6	no	—
314	C₂²	—	10	4	no	—
315	C₂²	—	20	4	no	—
316	C₂²	C₂²×C₃×D₂₀	1	6	yes	—
317	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
318	C₄	—	10	12	no	—
319	C₄	C₂²×C₃×D₂₀	1	6	yes	—
320	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
321	C₄	C₂⁴×C₃×D₅	1	1	yes	characteristic
322	C₃	C₂×#66087	1	1	yes	characteristic, direct product factor, Sylow
323	C₂	—	2	6	no	—
324	C₂	—	20	4	no	—
325	C₂	C₃×#66087	1	2	yes	direct product factor
326	C₂	C₂³×C₃×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₂₀
commutator, cosocle	C₄×C₅	C₂⁵×C₃
Frattini	C₄	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₃×#131	D₅
hypocenter	C₅	C₂×C₃×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4150	C₂	semidirect
2. C₂×C₃×C₅×#131	C₂	semidirect
3. C₃×#66087	C₂	direct (N × H)
4. C₂×C₃×#4055	C₂	semidirect
5. C₂²×C₃×Dic₂₀	C₂	semidirect
6. C₂×C₃×#4050	C₂	semidirect
7. C₂×#66087	C₃	direct (N × H)
10. C₃×#4150	C₂²	semidirect
15. C₃×C₅×#131	C₂²	semidirect
18. C₃×#4055	C₂²	semidirect
21. C₂×C₃×Dic₂₀	C₂²	semidirect
22. C₃×#4050	C₂²	semidirect
26. C₂×#4150	C₂×C₃	semidirect
27. C₂×C₅×#131	C₂×C₃	semidirect
28. #66087	C₂×C₃	direct (N × H)
29. C₂×#4055	C₂×C₃	semidirect
30. C₂²×Dic₂₀	C₂×C₃	semidirect
31. C₂×#4050	C₂×C₃	semidirect
62. C₂×C₃×#131	D₅	semidirect
65. #4150	C₂²×C₃	semidirect
70. C₅×#131	C₂²×C₃	semidirect
73. #4055	C₂²×C₃	semidirect
76. C₂×Dic₂₀	C₂²×C₃	semidirect
77. #4050	C₂²×C₃	semidirect
114. C₃×#131	C₂×D₅	semidirect
147. C₂×#131	C₃×D₅	semidirect
215. #131	C₂×C₃×D₅	semidirect
219. C₂×C₃×C₅	#578	semidirect
275. C₃×C₅	C₂×#578	semidirect
286. C₂×C₅	C₃×#578	semidirect
310. C₂×C₃	#66087	direct (N × H)
312. C₅	C₂×C₃×#578	semidirect
322. C₃	C₂×#66087	direct (N × H)
325. C₂	C₃×#66087	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×#4150	2
C₂×C₃×C₅×#131	2
C₃×#66087	2
C₂×C₃×#4055	2
C₂²×C₃×Dic₂₀	2
C₂×C₃×#4050	2
C₂×#66087	3
C₂×C₃×#578	5

Subgroups — C2×C3×#66087 #14510134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66087 #14510134