Subgroups — C₂×C₃×#65480 #14438839

Order 1920.

1184 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×#688	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic
5	C₃×#65480	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65480	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
10	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×#4093	—	2	4	no	—
14	C₃×#4093	C₂²	1	8	yes	complemented
15	C₂×C₃×#688	—	2	8	no	—
16	C₂×C₃×#688	C₂²	1	8	yes	complemented
17	C₂×C₃×#685	—	2	4	no	—
18	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₄₀	—	2	4	no	—
21	C₂×C₃×D₄₀	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	4	yes	complemented
23	C₂×C₃×#495	—	5	1	no	maximal, Hall
24	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#4093	C₂×C₃	1	2	yes	complemented
26	C₂²×#688	C₂×C₃	1	2	yes	complemented
27	C₂²×D₄₀	C₂×C₃	1	1	yes	characteristic
28	#65480	C₂×C₃	1	8	yes	direct product factor
29	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×D₅	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	—	2	4	no	—
33	C₂×C₃×C₅×Q₈	—	2	4	no	—
34	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₃×Dic₅	D₄	1	2	yes	—
38	C₂×C₃×D₂₀	—	2	4	no	—
39	C₂×C₃×D₂₀	—	2	4	no	—
40	C₂×C₃×D₂₀	—	2	2	no	—
41	C₂×C₃×D₂₀	C₂³	1	4	yes	—
42	C₃×C₅×Q₁₆	—	2	8	no	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₃×#688	—	2	16	no	—
46	C₃×#685	—	2	16	no	—
47	C₃×C₄×Dic₅	D₄	1	2	yes	—
48	C₃×C₄×Dic₅	C₂³	1	2	yes	—
49	C₂×C₃×#168	D₄	1	4	yes	—
50	C₂×C₃×#168	C₂³	1	2	yes	—
51	C₃×D₄₀	—	2	8	no	—
52	C₂²×C₃×Q₁₆	D₅	1	1	yes	characteristic, complemented
53	C₂²×C₃×SD₁₆	—	5	2	no	—
54	C₂²×C₃×D₈	—	5	1	no	—
55	C₂×C₃×#124	—	5	2	no	—
56	C₃×#495	—	5	8	no	—
57	C₂×C₃×C₄×C₈	—	5	1	no	—
58	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
59	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
60	C₂×C₅×Q₁₆	—	2	4	no	—
61	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	complemented
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	#4093	—	2	4	no	—
64	#4093	C₂²×C₃	1	8	yes	complemented
65	C₂×#688	—	2	8	no	—
66	C₂×#688	C₂²×C₃	1	8	yes	complemented
67	C₂×#685	—	2	4	no	—
68	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
70	C₂×D₄₀	—	2	4	no	—
71	C₂×D₄₀	C₂²×C₃	1	4	yes	—
72	C₈×Dic₅	C₂²×C₃	1	4	yes	complemented
73	C₂×#495	—	5	1	no	Sylow
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×D₅	—	2	4	no	—
76	C₂²×C₃×D₅	—	4	4	no	—
77	C₂²×C₃×D₅	—	4	8	no	—
78	C₃×C₅×Q₈	—	2	4	no	—
79	C₃×C₅×Q₈	—	2	4	no	—
80	C₃×C₅×Q₈	—	4	4	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	4	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×C₃×Dic₅	—	2	4	no	—
88	C₂×C₃×Dic₅	C₂×D₄	1	4	yes	—
89	C₃×D₂₀	—	2	4	no	—
90	C₃×D₂₀	—	2	4	no	—
91	C₃×D₂₀	—	4	4	no	—
92	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
93	C₃×#168	C₂×D₄	1	4	yes	—
94	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
95	C₂²×C₃×D₄	—	5	2	no	—
96	C₂×C₃×Q₁₆	—	2	4	no	—
97	C₂×C₃×Q₁₆	C₂×D₅	1	4	yes	complemented
98	C₂×C₃×SD₁₆	—	5	8	no	—
99	C₂×C₃×SD₁₆	—	10	8	no	—
100	C₂×C₃×D₈	—	5	4	no	—
101	C₂×C₃×D₈	—	10	4	no	—
102	C₂²×C₃×C₈	—	5	1	no	—
103	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
104	C₃×#124	—	5	8	no	—
105	C₃×#124	—	10	4	no	—
106	C₂×C₃×C₂²⋊C₄	—	10	4	no	—
107	C₂×C₃×C₄²	—	5	1	no	—
108	C₃×C₄×C₈	—	5	4	no	—
109	C₂³×D₅	—	2	2	no	—
110	C₂×C₅×Q₈	—	2	2	no	—
111	C₂×C₅×Q₈	—	2	4	no	—
112	C₂×C₅×Q₈	—	2	4	no	—
113	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
114	C₂²×C₄×C₅	—	2	2	no	—
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×Dic₅	C₃×D₄	1	2	yes	—
117	C₂×D₂₀	—	2	4	no	—
118	C₂×D₂₀	—	2	4	no	—
119	C₂×D₂₀	—	2	2	no	—
120	C₂×D₂₀	C₂³×C₃	1	4	yes	—
121	C₅×Q₁₆	—	2	8	no	—
122	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
123	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
124	#688	—	2	16	no	—
125	#685	—	2	16	no	—
126	C₄×Dic₅	C₃×D₄	1	2	yes	—
127	C₄×Dic₅	C₂³×C₃	1	2	yes	—
128	C₂×#168	C₃×D₄	1	4	yes	—
129	C₂×#168	C₂³×C₃	1	2	yes	—
130	D₄₀	—	2	8	no	—
131	C₂²×Q₁₆	C₃×D₅	1	1	yes	characteristic, complemented
132	C₂²×SD₁₆	—	5	2	no	—
133	C₂²×D₈	—	5	1	no	—
134	C₂×#124	—	5	2	no	—
135	#495	—	5	8	no	—
136	C₂×C₄×C₈	—	5	1	no	—
137	C₂²×C₃×C₅	#127	1	2	yes	—
138	C₂²×C₃×C₅	#135	1	4	yes	—
139	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₃×D₅	—	4	4	no	—
141	C₂×C₃×D₅	—	4	8	no	—
142	C₂×C₃×D₅	—	8	8	no	—
143	C₃×C₄×C₅	—	4	4	no	—
144	C₃×C₄×C₅	#127	1	2	yes	—
145	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₃×Dic₅	—	2	4	no	—
148	C₂⁴×C₃	—	10	2	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	—	2	4	no	—
151	C₂×C₃×Q₈	—	2	4	no	—
152	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
153	C₂×C₃×D₄	—	5	4	no	—
154	C₂×C₃×D₄	—	10	4	no	—
155	C₂×C₃×D₄	—	10	2	no	—
156	C₂×C₃×D₄	—	10	4	no	—
157	C₂²×C₃×C₄	—	2	2	no	—
158	C₂²×C₃×C₄	—	5	2	no	—
159	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₃×Q₁₆	—	2	8	no	—
161	C₃×SD₁₆	—	10	16	no	—
162	C₃×D₈	—	10	8	no	—
163	C₂×C₃×C₈	—	5	4	no	—
164	C₂×C₃×C₈	—	5	2	no	—
165	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
166	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
167	C₃×C₂²⋊C₄	—	10	16	no	—
168	C₃×C₄²	—	5	2	no	—
169	C₃×C₄²	—	5	2	no	—
170	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂²×D₅	—	2	4	no	—
172	C₂²×D₅	—	4	4	no	—
173	C₂²×D₅	—	4	8	no	—
174	C₅×Q₈	—	2	4	no	—
175	C₅×Q₈	—	2	4	no	—
176	C₅×Q₈	—	4	4	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	4	4	no	—
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
183	C₂×Dic₅	—	2	4	no	—
184	C₂×Dic₅	C₂×C₃×D₄	1	4	yes	—
185	D₂₀	—	2	4	no	—
186	D₂₀	—	2	4	no	—
187	D₂₀	—	4	4	no	—
188	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
189	#168	C₂×C₃×D₄	1	4	yes	—
190	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
191	C₂²×D₄	—	5	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×Q₁₆	C₂×C₃×D₅	1	4	yes	complemented
194	C₂×SD₁₆	—	5	8	no	—
195	C₂×SD₁₆	—	10	8	no	—
196	C₂×D₈	—	5	4	no	—
197	C₂×D₈	—	10	4	no	—
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
200	#124	—	5	8	no	—
201	#124	—	10	4	no	—
202	C₂×C₂²⋊C₄	—	10	4	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₄×C₈	—	5	4	no	—
205	C₂×C₃×C₅	#495	1	4	yes	complemented
206	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#135	1	2	yes	—
208	C₃×D₅	—	8	4	no	—
209	C₂³×C₃	—	10	4	no	—
210	C₂³×C₃	—	20	8	no	—
211	C₂³×C₃	—	20	4	no	—
212	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
213	C₃×Q₈	—	2	4	no	—
214	C₃×Q₈	—	2	4	no	—
215	C₃×Q₈	—	4	4	no	—
216	C₃×D₄	—	10	4	no	—
217	C₃×D₄	—	10	4	no	—
218	C₃×D₄	—	20	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	5	4	no	—
222	C₂×C₃×C₄	—	10	4	no	—
223	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
224	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
226	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
227	C₃×C₈	—	5	4	no	—
228	C₃×C₈	C₂×C₂²⋊D₅	1	4	yes	—
229	C₂²×C₅	C₃×#127	1	2	yes	—
230	C₂²×C₅	C₃×#135	1	4	yes	—
231	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₂×D₅	—	4	8	no	—
233	C₂×D₅	—	4	4	no	—
234	C₂×D₅	—	8	8	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	C₃×#127	1	2	yes	—
237	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
238	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
239	Dic₅	—	2	4	no	—
240	C₂⁴	—	10	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
245	C₂×D₄	—	5	4	no	—
246	C₂×D₄	—	10	4	no	—
247	C₂×D₄	—	10	2	no	—
248	C₂×D₄	—	10	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	5	2	no	—
251	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
252	Q₁₆	—	2	8	no	—
253	SD₁₆	—	10	16	no	—
254	D₈	—	10	8	no	—
255	C₂×C₈	—	5	4	no	—
256	C₂×C₈	—	5	2	no	—
257	C₂×C₈	C₃×C₂²⋊D₅	1	4	yes	—
258	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
259	C₂²⋊C₄	—	10	16	no	—
260	C₄²	—	5	2	no	—
261	C₄²	—	5	2	no	—
262	C₃×C₅	C₂×#495	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	20	8	no	—
264	C₂²×C₃	—	20	4	no	—
265	C₂²×C₃	—	40	8	no	—
266	C₂²×C₃	#4065	1	4	yes	—
267	C₂²×C₃	#4086	1	2	yes	—
268	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
269	C₃×C₄	—	4	4	no	—
270	C₃×C₄	—	10	4	no	—
271	C₃×C₄	#4086	1	2	yes	—
272	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
273	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
274	C₂×C₅	C₃×#495	1	4	yes	complemented
275	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
276	C₂×C₅	C₂×C₃×#135	1	2	yes	—
277	D₅	—	8	4	no	—
278	C₂³	—	10	4	no	—
279	C₂³	—	20	4	no	—
280	C₂³	—	20	8	no	—
281	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	4	no	—
285	D₄	—	10	4	no	—
286	D₄	—	10	4	no	—
287	D₄	—	20	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	5	4	no	—
291	C₂×C₄	—	10	4	no	—
292	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
293	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
294	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
295	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
296	C₈	—	5	4	no	—
297	C₈	C₂×C₃×C₂²⋊D₅	1	4	yes	—
298	C₂×C₃	—	40	4	no	—
299	C₂×C₃	#65480	1	4	yes	direct product factor
300	C₂×C₃	C₂×#4065	1	2	yes	—
301	C₂×C₃	C₂×#4086	1	1	yes	characteristic
302	C₅	C₂×C₃×#495	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	20	8	no	—
304	C₂²	—	20	4	no	—
305	C₂²	—	40	8	no	—
306	C₂²	C₃×#4065	1	4	yes	—
307	C₂²	C₃×#4086	1	2	yes	—
308	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
309	C₄	—	4	4	no	—
310	C₄	—	10	4	no	—
311	C₄	C₃×#4086	1	2	yes	—
312	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
313	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
314	C₃	C₂×#65480	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	40	4	no	—
316	C₂	C₃×#65480	1	4	yes	direct product factor
317	C₂	C₂×C₃×#4065	1	2	yes	—
318	C₂	C₂×C₃×#4086	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₅×Q₁₆	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₄₀	C₂
hypercenter	C₂²×C₃×Q₁₆	D₅
hypocenter	C₅	C₂×C₃×#495
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₁₆	C₂	semidirect
2. C₂×C₃×#4093	C₂	semidirect
3. C₂²×C₃×#688	C₂	semidirect
5. C₃×#65480	C₂	direct (N × H)
6. C₂×C₃×C₈×Dic₅	C₂	semidirect
7. C₂×#65480	C₃	direct (N × H)
11. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
14. C₃×#4093	C₂²	semidirect
16. C₂×C₃×#688	C₂²	semidirect
22. C₃×C₈×Dic₅	C₂²	semidirect
24. C₂²×C₅×Q₁₆	C₂×C₃	semidirect
25. C₂×#4093	C₂×C₃	semidirect
26. C₂²×#688	C₂×C₃	semidirect
28. #65480	C₂×C₃	direct (N × H)
29. C₂×C₈×Dic₅	C₂×C₃	semidirect
52. C₂²×C₃×Q₁₆	D₅	semidirect
61. C₂×C₅×Q₁₆	C₂²×C₃	semidirect
64. #4093	C₂²×C₃	semidirect
66. C₂×#688	C₂²×C₃	semidirect
72. C₈×Dic₅	C₂²×C₃	semidirect
97. C₂×C₃×Q₁₆	C₂×D₅	semidirect
131. C₂²×Q₁₆	C₃×D₅	semidirect
193. C₂×Q₁₆	C₂×C₃×D₅	semidirect
205. C₂×C₃×C₅	#495	semidirect
262. C₃×C₅	C₂×#495	semidirect
274. C₂×C₅	C₃×#495	semidirect
299. C₂×C₃	#65480	direct (N × H)
302. C₅	C₂×C₃×#495	semidirect
314. C₃	C₂×#65480	direct (N × H)
316. C₂	C₃×#65480	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×C₅×Q₁₆	2
C₂×C₃×#4093	2
C₂²×C₃×#688	2
C₂²×C₃×D₄₀	2
C₃×#65480	2
C₂×C₃×C₈×Dic₅	2
C₂×#65480	3
C₂×C₃×#495	5

Subgroups — C2×C3×#65480 #14438839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65480 #14438839