Subgroups — C₂×C₅×#4909 #14478389

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×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4909	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×Dic₂₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#800	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#510	—	3	1	no	maximal, Hall
8	C₂×C₅×#235	—	2	1	no	—
9	C₂²×C₅×Dic₆	—	2	1	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₁₂	—	2	2	no	—
12	C₂×C₅×Dic₁₂	—	2	1	no	—
13	C₂×C₅×Dic₁₂	C₂²	1	2	yes	—
14	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₂	C₂²	1	4	yes	complemented
16	C₅×#904	—	2	2	no	—
17	C₅×#904	C₂²	1	4	yes	—
18	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×#204	—	2	1	no	—
20	C₃×C₅×M₃₂	C₂²	1	4	yes	complemented
21	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
22	C₅×Dic₂₄	C₂²	1	8	yes	complemented
23	C₅×#800	C₂²	1	8	yes	complemented
24	C₂×#4909	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#135	—	3	1	no	—
26	C₂²×C₅×Q₁₆	—	3	1	no	—
27	C₅×#510	—	3	8	no	—
28	C₂×C₅×Q₃₂	—	3	2	no	—
29	C₂×C₅×SD₃₂	—	3	2	no	—
30	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂×C₅×#67	—	4	1	no	—
33	C₂²×C₅×Dic₃	—	4	1	no	—
34	C₅×#235	—	2	4	no	—
35	C₅×#235	—	4	2	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₄×C₅×D₃	—	4	1	no	—
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₅×Dic₆	—	2	2	no	—
40	C₂×C₅×Dic₆	—	2	4	no	—
41	C₂×C₅×Dic₆	—	4	1	no	—
42	C₂×C₅×Dic₆	—	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	C₅×Dic₁₂	—	2	1	no	—
48	C₅×Dic₁₂	—	2	2	no	—
49	C₅×Dic₁₂	—	2	4	no	—
50	C₅×Dic₁₂	C₂³	1	2	yes	—
51	C₅×Dic₁₂	C₂³	1	4	yes	—
52	C₅×D₂₄	—	2	1	no	—
53	C₅×D₂₄	C₂³	1	2	yes	—
54	C₅×#204	—	2	4	no	—
55	C₃×C₅×C₁₆	C₂³	1	4	yes	—
56	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#904	C₂×C₅	1	1	yes	characteristic
58	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
59	#4909	C₂×C₅	1	8	yes	direct product factor
60	C₂×Dic₂₄	C₂×C₅	1	2	yes	complemented
61	C₂×#800	C₂×C₅	1	2	yes	complemented
62	C₂×C₅×#41	—	6	1	no	—
63	C₂²×C₅×Q₈	—	6	1	no	—
64	C₅×#135	—	3	4	no	—
65	C₅×#135	—	6	2	no	—
66	C₂×C₅×Q₁₆	—	3	2	no	—
67	C₂×C₅×Q₁₆	—	3	1	no	—
68	C₂×C₅×Q₁₆	—	3	4	no	—
69	C₂×C₅×Q₁₆	—	6	1	no	—
70	C₂×C₅×Q₁₆	—	6	2	no	—
71	C₂×C₅×SD₁₆	—	6	1	no	—
72	C₂×C₅×D₈	—	3	1	no	—
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×Q₃₂	—	3	8	no	—
75	C₅×SD₃₂	—	3	8	no	—
76	C₅×M₃₂	C₂×D₃	1	4	yes	complemented
77	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
78	C₂×#510	—	3	1	no	Sylow
79	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	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	C₅×#67	—	4	4	no	—
86	C₂×C₅×Dic₃	—	4	4	no	—
87	C₂×C₅×Dic₃	—	4	2	no	—
88	C₂×C₅×Dic₃	—	4	1	no	—
89	C₅×D₁₂	—	2	2	no	—
90	C₅×D₁₂	—	4	1	no	—
91	C₄×C₅×D₃	—	4	4	no	—
92	C₅×Dic₆	—	2	2	no	—
93	C₅×Dic₆	—	2	4	no	—
94	C₅×Dic₆	—	4	2	no	—
95	C₅×Dic₆	—	4	4	no	—
96	C₅×Dic₆	—	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×#235	—	2	1	no	—
101	C₂²×Dic₆	—	2	1	no	—
102	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×Dic₁₂	—	2	2	no	—
104	C₂×Dic₁₂	—	2	1	no	—
105	C₂×Dic₁₂	C₂²×C₅	1	2	yes	—
106	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
107	C₂×Dic₁₂	C₂²×C₅	1	4	yes	complemented
108	#904	—	2	2	no	—
109	#904	C₂²×C₅	1	4	yes	—
110	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
111	C₂×#204	—	2	1	no	—
112	C₃×M₃₂	C₂²×C₅	1	4	yes	complemented
113	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
114	Dic₂₄	C₂²×C₅	1	8	yes	complemented
115	#800	C₂²×C₅	1	8	yes	complemented
116	C₅×#41	—	6	4	no	—
117	C₅×#41	—	12	2	no	—
118	C₂×C₅×Q₈	—	6	1	no	—
119	C₂×C₅×Q₈	—	6	4	no	—
120	C₂×C₅×Q₈	—	6	2	no	—
121	C₂×C₅×Q₈	—	12	2	no	—
122	C₂×C₅×Q₈	—	12	1	no	—
123	C₂×C₅×D₄	—	6	1	no	—
124	C₂×C₅×D₄	—	12	1	no	—
125	C₂²×C₄×C₅	—	12	1	no	—
126	C₂²×C₄×C₅	—	12	1	no	—
127	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
128	C₅×Q₁₆	—	3	4	no	—
129	C₅×Q₁₆	—	3	2	no	—
130	C₅×Q₁₆	—	6	2	no	—
131	C₅×Q₁₆	—	6	4	no	—
132	C₅×Q₁₆	—	6	1	no	—
133	C₅×SD₁₆	—	6	4	no	—
134	C₅×D₈	—	3	2	no	—
135	C₅×D₈	—	6	1	no	—
136	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
137	C₂×C₅×C₈	D₁₂	1	2	yes	—
138	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
140	C₅×C₁₆	C₂²×D₃	1	4	yes	—
141	C₂×#135	—	3	1	no	—
142	C₂²×Q₁₆	—	3	1	no	—
143	#510	—	3	8	no	—
144	C₂×Q₃₂	—	3	2	no	—
145	C₂×SD₃₂	—	3	2	no	—
146	C₂×M₃₂	C₅×D₃	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×C₅×D₃	—	4	2	no	—
151	C₂×C₅×D₃	—	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	C₅×Dic₃	—	4	4	no	—
156	C₅×Dic₃	—	4	2	no	—
157	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
158	C₂×#67	—	4	1	no	—
159	C₂²×Dic₃	—	4	1	no	—
160	#235	—	2	4	no	—
161	#235	—	4	2	no	—
162	C₂×D₁₂	—	2	1	no	—
163	C₂×C₄×D₃	—	4	1	no	—
164	C₂×Dic₆	—	2	1	no	—
165	C₂×Dic₆	—	2	2	no	—
166	C₂×Dic₆	—	2	4	no	—
167	C₂×Dic₆	—	4	1	no	—
168	C₂×Dic₆	—	4	2	no	—
169	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
171	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
172	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
173	Dic₁₂	—	2	1	no	—
174	Dic₁₂	—	2	2	no	—
175	Dic₁₂	—	2	4	no	—
176	Dic₁₂	C₂³×C₅	1	4	yes	—
177	Dic₁₂	C₂³×C₅	1	2	yes	—
178	D₂₄	—	2	1	no	—
179	D₂₄	C₂³×C₅	1	2	yes	—
180	#204	—	2	4	no	—
181	C₃×C₁₆	C₂³×C₅	1	4	yes	—
182	C₂³×C₅	—	12	1	no	—
183	C₂³×C₅	D₂₄	1	1	yes	characteristic
184	C₅×Q₈	—	6	2	no	—
185	C₅×Q₈	—	6	4	no	—
186	C₅×Q₈	—	12	4	no	—
187	C₅×Q₈	—	12	2	no	—
188	C₅×Q₈	—	12	1	no	—
189	C₅×D₄	—	6	2	no	—
190	C₅×D₄	—	12	4	no	—
191	C₅×D₄	—	12	1	no	—
192	C₂×C₄×C₅	—	12	4	no	—
193	C₂×C₄×C₅	—	12	1	no	—
194	C₂×C₄×C₅	—	12	2	no	—
195	C₂×C₄×C₅	—	12	4	no	—
196	C₂×C₄×C₅	D₂₄	1	2	yes	—
197	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
199	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
200	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
201	C₅×C₈	C₂×D₁₂	1	2	yes	—
202	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
203	C₂×#41	—	6	1	no	—
204	C₂²×Q₈	—	6	1	no	—
205	#135	—	3	4	no	—
206	#135	—	6	2	no	—
207	C₂×Q₁₆	—	3	2	no	—
208	C₂×Q₁₆	—	3	4	no	—
209	C₂×Q₁₆	—	3	1	no	—
210	C₂×Q₁₆	—	6	1	no	—
211	C₂×Q₁₆	—	6	2	no	—
212	C₂×SD₁₆	—	6	1	no	—
213	C₂×D₈	—	3	1	no	—
214	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
215	Q₃₂	—	3	8	no	—
216	SD₃₂	—	3	8	no	—
217	M₃₂	C₂×C₅×D₃	1	4	yes	complemented
218	C₂×C₁₆	C₂×C₅×D₃	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	C₅×D₃	—	8	2	no	—
223	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
224	C₂²×D₃	—	4	1	no	—
225	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
229	#67	—	4	4	no	—
230	C₂×Dic₃	—	4	4	no	—
231	C₂×Dic₃	—	4	2	no	—
232	C₂×Dic₃	—	4	1	no	—
233	D₁₂	—	2	2	no	—
234	D₁₂	—	4	1	no	—
235	C₄×D₃	—	4	4	no	—
236	Dic₆	—	2	2	no	—
237	Dic₆	—	2	4	no	—
238	Dic₆	—	4	2	no	—
239	Dic₆	—	4	4	no	—
240	Dic₆	—	4	1	no	—
241	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
242	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
243	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	—	12	2	no	—
246	C₂²×C₅	—	24	2	no	—
247	C₂²×C₅	C₂×D₂₄	1	2	yes	—
248	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
249	C₄×C₅	—	12	2	no	—
250	C₄×C₅	—	12	4	no	—
251	C₄×C₅	C₂×D₂₄	1	2	yes	—
252	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
253	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
254	#41	—	6	4	no	—
255	#41	—	12	2	no	—
256	C₂×Q₈	—	6	1	no	—
257	C₂×Q₈	—	6	2	no	—
258	C₂×Q₈	—	6	4	no	—
259	C₂×Q₈	—	12	1	no	—
260	C₂×Q₈	—	12	2	no	—
261	C₂×D₄	—	6	1	no	—
262	C₂×D₄	—	12	1	no	—
263	C₂²×C₄	—	12	1	no	—
264	C₂²×C₄	—	12	1	no	—
265	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
266	Q₁₆	—	3	4	no	—
267	Q₁₆	—	3	2	no	—
268	Q₁₆	—	6	2	no	—
269	Q₁₆	—	6	4	no	—
270	Q₁₆	—	6	1	no	—
271	SD₁₆	—	6	4	no	—
272	D₈	—	3	2	no	—
273	D₈	—	6	1	no	—
274	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
275	C₂×C₈	C₅×D₁₂	1	2	yes	—
276	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
277	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
278	C₁₆	C₂²×C₅×D₃	1	4	yes	—
279	C₃×C₅	C₂×#510	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
282	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂×D₃	—	4	2	no	—
284	C₂×D₃	—	8	2	no	—
285	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
286	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
287	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
288	Dic₃	—	4	4	no	—
289	Dic₃	—	4	2	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	24	2	no	—
292	C₂×C₅	#4909	1	2	yes	direct product factor
293	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
294	C₂³	—	12	1	no	—
295	C₂³	C₅×D₂₄	1	1	yes	characteristic
296	Q₈	—	6	2	no	—
297	Q₈	—	6	4	no	—
298	Q₈	—	12	4	no	—
299	Q₈	—	12	2	no	—
300	Q₈	—	12	1	no	—
301	D₄	—	6	2	no	—
302	D₄	—	12	4	no	—
303	D₄	—	12	1	no	—
304	C₂×C₄	—	12	1	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	—	12	2	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
309	C₂×C₄	C₅×D₂₄	1	2	yes	—
310	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
312	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
313	C₈	C₂×C₅×D₁₂	1	2	yes	—
314	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	C₅×#510	1	2	yes	complemented
317	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
318	D₃	—	8	2	no	—
319	C₅	C₂×#4909	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	2	2	no	—
321	C₂²	—	12	2	no	—
322	C₂²	—	24	2	no	—
323	C₂²	C₂×C₅×D₂₄	1	2	yes	—
324	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
325	C₄	—	12	2	no	—
326	C₄	—	12	4	no	—
327	C₄	C₂×C₅×D₂₄	1	2	yes	—
328	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
329	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
330	C₃	C₂×C₅×#510	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	24	2	no	—
333	C₂	C₅×#4909	1	2	yes	direct product factor
334	C₂	C₂²×C₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×#904	C₂
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₂×C₅×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Dic₁₂	C₂	semidirect
3. C₂×C₃×C₅×M₃₂	C₂	semidirect
4. C₅×#4909	C₂	direct (N × H)
5. C₂×C₅×Dic₂₄	C₂	semidirect
6. C₂×C₅×#800	C₂	semidirect
15. C₂×C₅×Dic₁₂	C₂²	semidirect
20. C₃×C₅×M₃₂	C₂²	semidirect
22. C₅×Dic₂₄	C₂²	semidirect
23. C₅×#800	C₂²	semidirect
24. C₂×#4909	C₅	direct (N × H)
30. C₂×C₅×M₃₂	D₃	semidirect
56. C₂²×Dic₁₂	C₂×C₅	semidirect
58. C₂×C₃×M₃₂	C₂×C₅	semidirect
59. #4909	C₂×C₅	direct (N × H)
60. C₂×Dic₂₄	C₂×C₅	semidirect
61. C₂×#800	C₂×C₅	semidirect
76. C₅×M₃₂	C₂×D₃	semidirect
107. C₂×Dic₁₂	C₂²×C₅	semidirect
112. C₃×M₃₂	C₂²×C₅	semidirect
114. Dic₂₄	C₂²×C₅	semidirect
115. #800	C₂²×C₅	semidirect
146. C₂×M₃₂	C₅×D₃	semidirect
217. M₃₂	C₂×C₅×D₃	semidirect
220. C₂×C₃×C₅	#510	semidirect
279. C₃×C₅	C₂×#510	semidirect
292. C₂×C₅	#4909	direct (N × H)
316. C₂×C₃	C₅×#510	semidirect
319. C₅	C₂×#4909	direct (N × H)
330. C₃	C₂×C₅×#510	semidirect
333. C₂	C₅×#4909	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×C₅×Dic₁₂	2
C₂×C₅×#904	2
C₂×C₃×C₅×M₃₂	2
C₅×#4909	2
C₂×C₅×Dic₂₄	2
C₂×C₅×#800	2
C₂×C₅×#510	3
C₂×#4909	5

Subgroups — C2×C5×#4909 #14478389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4909 #14478389