Subgroups — C₂×D₅×#365 #12252305

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×D₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64733	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64714	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64707	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×#365	C₂	1	8	yes	maximal, direct product factor
9	C₂×D₅×M₁₆	—	2	1	no	—
10	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₂×D₅×C₄⋊C₄	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	1	yes	characteristic
14	C₂×C₅×#106	C₂²	1	1	yes	characteristic
15	C₂×#3954	—	2	1	no	—
16	C₂×#3951	C₂²	1	1	yes	characteristic
17	C₂×#3949	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	1	yes	characteristic
19	C₂×#3940	—	2	1	no	—
20	D₅×M₃₂	C₄	1	4	yes	complemented
21	D₅×M₃₂	C₂²	1	4	yes	—
22	C₂×#3930	C₄	1	2	yes	complemented
23	C₂×D₅×C₁₆	C₄	1	2	yes	complemented
24	D₅×#108	—	2	2	no	—
25	D₅×#108	C₂²	1	4	yes	—
26	D₅×#106	—	2	2	no	—
27	D₅×#106	C₂²	1	4	yes	—
28	C₅×#365	C₂²	1	4	yes	complemented
29	#64733	C₂²	1	4	yes	complemented
30	#64714	C₂²	1	4	yes	complemented
31	#64707	C₂²	1	4	yes	complemented
32	C₂²×#365	—	5	1	no	maximal, Sylow
33	C₂²×C₄×D₅	—	4	1	no	—
34	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	—	2	1	no	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊Dic₅	—	2	1	no	—
39	C₂×#683	—	4	1	no	—
40	C₂×#681	—	2	1	no	—
41	C₂²×#168	D₄	1	1	yes	characteristic
42	D₅×M₁₆	—	2	4	no	—
43	D₅×M₁₆	—	4	2	no	—
44	C₂×#676	—	4	1	no	—
45	C₂×C₈×D₅	—	4	1	no	—
46	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₅	D₄	1	2	yes	—
50	C₂×C₈×D₅	Q₈	1	2	yes	—
51	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
52	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
53	C₂×C₈×D₅	C₂³	1	2	yes	—
54	D₅×C₄⋊C₄	—	2	4	no	—
55	D₅×C₄⋊C₄	—	4	2	no	—
56	C₅×M₃₂	C₂×C₄	1	2	yes	complemented
57	C₅×M₃₂	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
59	C₅×#108	—	2	1	no	—
60	C₅×#108	C₂³	1	2	yes	—
61	C₅×#106	—	2	1	no	—
62	C₅×#106	C₂³	1	2	yes	—
63	#3954	—	2	4	no	—
64	#3951	—	2	1	no	—
65	#3951	C₂³	1	2	yes	—
66	#3949	—	2	1	no	—
67	#3949	C₂³	1	2	yes	—
68	#3944	C₂×C₄	1	2	yes	complemented
69	#3944	C₂³	1	2	yes	—
70	C₂×#672	C₂×C₄	1	2	yes	complemented
71	#3940	—	2	4	no	—
72	#3930	C₂×C₄	1	8	yes	complemented
73	D₅×C₁₆	C₂×C₄	1	8	yes	complemented
74	C₂²×M₃₂	—	5	1	no	—
75	C₂²×#108	—	5	1	no	—
76	C₂²×#106	—	5	1	no	—
77	C₂×#365	—	5	1	no	—
78	C₂×#365	—	5	1	no	—
79	C₂×#365	—	5	1	no	—
80	C₂×#365	—	5	8	no	—
81	C₂×#365	D₅	1	1	yes	characteristic, direct product factor
82	C₂³×D₅	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	4	1	no	—
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₅	—	4	1	no	—
86	C₂²×Dic₅	SD₁₆	1	1	yes	characteristic
87	C₂×C₄×D₅	—	4	4	no	—
88	C₂×C₄×D₅	—	8	4	no	—
89	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅	SD₁₆	1	2	yes	—
93	C₂×C₄×D₅	SD₁₆	1	1	yes	characteristic
94	C₂×C₄×D₅	SD₁₆	1	1	yes	characteristic
95	C₂×C₄×D₅	SD₁₆	1	2	yes	—
96	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	4	1	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₄⋊Dic₅	—	2	2	no	—
109	C₄⋊Dic₅	—	4	1	no	—
110	#683	—	4	4	no	—
111	#681	—	2	2	no	—
112	#681	—	4	1	no	—
113	C₂×#168	—	4	1	no	—
114	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
115	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
116	C₂×#168	C₄⋊C₄	1	2	yes	—
117	C₂×#168	C₂×D₄	1	2	yes	—
118	#676	—	4	4	no	—
119	C₈×D₅	—	4	4	no	—
120	C₈×D₅	C₄⋊C₄	1	2	yes	—
121	C₈×D₅	C₄⋊C₄	1	2	yes	—
122	C₈×D₅	C₄⋊C₄	1	2	yes	—
123	C₈×D₅	C₄⋊C₄	1	2	yes	—
124	C₈×D₅	C₂²×C₄	1	2	yes	—
125	C₈×D₅	C₂²×C₄	1	2	yes	—
126	C₈×D₅	C₂×Q₈	1	2	yes	—
127	C₈×D₅	C₂×Q₈	1	2	yes	—
128	C₅×C₁₆	C₂²×C₄	1	4	yes	complemented
129	#672	C₂²×C₄	1	4	yes	complemented
130	C₂²×M₁₆	—	10	1	no	—
131	C₂³×C₈	—	5	1	no	—
132	C₂²×C₄⋊C₄	—	10	1	no	—
133	C₂×M₃₂	—	5	1	no	—
134	C₂×M₃₂	—	5	4	no	—
135	C₂×M₃₂	—	5	4	no	—
136	C₂×M₃₂	—	5	2	no	—
137	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
138	C₂²×C₁₆	—	5	2	no	—
139	C₂×#108	—	5	1	no	—
140	C₂×#108	—	5	4	no	—
141	C₂×#108	—	10	2	no	—
142	C₂×#108	—	10	1	no	—
143	C₂×#108	C₂×D₅	1	1	yes	characteristic
144	C₂×#106	—	5	1	no	—
145	C₂×#106	—	5	4	no	—
146	C₂×#106	—	10	2	no	—
147	C₂×#106	—	10	1	no	—
148	C₂×#106	C₂×D₅	1	1	yes	characteristic
149	#365	—	5	4	no	—
150	#365	—	5	4	no	—
151	#365	—	5	4	no	—
152	#365	C₂×D₅	1	4	yes	direct product factor
153	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
154	C₂²×D₅	—	2	4	no	—
155	C₂²×D₅	#106	1	2	yes	—
156	C₂²×D₅	#106	1	1	yes	characteristic
157	C₂²×D₅	#106	1	1	yes	characteristic
158	C₂²×D₅	C₂×SD₁₆	1	2	yes	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	8	2	no	—
161	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
167	C₂×Dic₅	—	4	2	no	—
168	C₂×Dic₅	—	8	2	no	—
169	C₂×Dic₅	#106	1	2	yes	—
170	C₂×Dic₅	#106	1	1	yes	characteristic
171	C₂×Dic₅	#106	1	1	yes	characteristic
172	C₂×Dic₅	C₂×SD₁₆	1	2	yes	—
173	C₄×D₅	—	8	8	no	—
174	C₄×D₅	#106	1	2	yes	—
175	C₄×D₅	#106	1	2	yes	—
176	C₄×D₅	#106	1	2	yes	—
177	C₄×D₅	#106	1	2	yes	—
178	C₄×D₅	C₂×C₄⋊C₄	1	2	yes	—
179	C₄×D₅	C₂×C₄⋊C₄	1	2	yes	—
180	C₄×D₅	C₂×SD₁₆	1	2	yes	—
181	C₄×D₅	C₂×SD₁₆	1	2	yes	—
182	C₅×C₈	—	4	2	no	—
183	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
186	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
187	#168	—	4	2	no	—
188	#168	C₂×C₄⋊C₄	1	2	yes	—
189	#168	C₂×C₄⋊C₄	1	2	yes	—
190	C₂³×C₄	—	5	1	no	—
191	C₂³×C₄	—	20	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	10	4	no	—
194	C₂×M₁₆	—	10	1	no	—
195	C₂×M₁₆	—	20	1	no	—
196	C₂×M₁₆	—	20	2	no	—
197	C₂²×C₈	—	5	1	no	—
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	—	5	2	no	—
200	C₂²×C₈	—	5	1	no	—
201	C₂²×C₈	—	5	2	no	—
202	C₂²×C₈	—	5	2	no	—
203	C₂²×C₈	—	5	1	no	—
204	C₂²×C₈	—	5	1	no	—
205	C₂²×C₈	—	5	2	no	—
206	C₂²×C₈	—	20	1	no	—
207	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	10	4	no	—
210	C₂×C₄⋊C₄	—	10	1	no	—
211	C₂×C₄⋊C₄	—	20	2	no	—
212	C₂×C₄⋊C₄	—	20	1	no	—
213	M₃₂	—	5	2	no	—
214	M₃₂	—	5	2	no	—
215	M₃₂	—	5	8	no	—
216	M₃₂	C₄×D₅	1	2	yes	complemented
217	M₃₂	C₂²×D₅	1	2	yes	—
218	C₂×C₁₆	—	5	2	no	—
219	C₂×C₁₆	—	5	8	no	—
220	C₂×C₁₆	C₄×D₅	1	2	yes	complemented
221	#108	—	2	1	no	—
222	#108	—	5	2	no	—
223	#108	—	10	4	no	—
224	#108	—	10	1	no	—
225	#108	C₂²×D₅	1	2	yes	—
226	#106	—	2	1	no	—
227	#106	—	5	2	no	—
228	#106	—	10	4	no	—
229	#106	—	10	1	no	—
230	#106	C₂²×D₅	1	2	yes	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	C₂×#106	1	1	yes	characteristic
233	C₂²×C₅	C₂×#106	1	1	yes	characteristic
234	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	2	4	no	—
237	C₂×D₅	—	2	4	no	—
238	C₂×D₅	#365	1	4	yes	direct product factor
239	C₂×D₅	C₂×#106	1	2	yes	—
240	C₂×D₅	C₂×#106	1	2	yes	—
241	C₄×C₅	—	8	2	no	—
242	C₄×C₅	C₂×#106	1	1	yes	characteristic
243	C₄×C₅	C₂×#106	1	1	yes	characteristic
244	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
245	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
246	Dic₅	—	8	2	no	—
247	Dic₅	C₂×#106	1	2	yes	—
248	Dic₅	C₂×#106	1	2	yes	—
249	C₂⁴	—	5	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	5	2	no	—
253	C₂²×C₄	—	5	2	no	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	5	1	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	2	no	—
259	C₂²×C₄	—	5	1	no	—
260	C₂²×C₄	—	20	4	no	—
261	C₂²×C₄	—	20	1	no	—
262	C₂²×C₄	—	40	4	no	—
263	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	10	2	no	—
267	M₁₆	—	20	4	no	—
268	M₁₆	—	20	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	5	2	no	—
271	C₂×C₈	—	5	2	no	—
272	C₂×C₈	—	5	1	no	—
273	C₂×C₈	—	5	1	no	—
274	C₂×C₈	—	5	2	no	—
275	C₂×C₈	—	5	2	no	—
276	C₂×C₈	—	5	2	no	—
277	C₂×C₈	—	5	2	no	—
278	C₂×C₈	—	5	2	no	—
279	C₂×C₈	—	5	2	no	—
280	C₂×C₈	—	5	2	no	—
281	C₂×C₈	—	5	2	no	—
282	C₂×C₈	—	20	4	no	—
283	C₂×C₈	—	20	1	no	—
284	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂×C₈	D₄×D₅	1	1	yes	characteristic
287	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
288	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	10	2	no	—
293	C₄⋊C₄	—	20	4	no	—
294	C₄⋊C₄	—	20	1	no	—
295	C₁₆	—	5	4	no	—
296	C₁₆	C₂×C₄×D₅	1	4	yes	complemented
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	C₂×#365	1	2	yes	complemented
300	C₂×C₅	C₂²×#106	1	1	yes	characteristic
301	D₅	—	2	2	no	—
302	D₅	C₂×#365	1	4	yes	direct product factor
303	C₂³	—	5	1	no	—
304	C₂³	—	5	2	no	—
305	C₂³	—	5	2	no	—
306	C₂³	—	5	1	no	—
307	C₂³	—	10	4	no	—
308	C₂³	D₅×SD₁₆	1	1	yes	characteristic
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	5	2	no	—
311	C₂×C₄	—	5	2	no	—
312	C₂×C₄	—	5	1	no	—
313	C₂×C₄	—	5	1	no	—
314	C₂×C₄	—	5	2	no	—
315	C₂×C₄	—	5	2	no	—
316	C₂×C₄	—	5	2	no	—
317	C₂×C₄	—	5	2	no	—
318	C₂×C₄	—	5	2	no	—
319	C₂×C₄	—	5	2	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	5	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	20	2	no	—
324	C₂×C₄	—	40	8	no	—
325	C₂×C₄	—	40	2	no	—
326	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
328	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
329	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
330	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
332	C₈	—	4	2	no	—
333	C₈	—	5	2	no	—
334	C₈	—	5	2	no	—
335	C₈	—	20	2	no	—
336	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
337	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
339	C₈	C₂×Q₈×D₅	1	1	yes	characteristic
340	C₅	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	2	no	—
342	C₂²	—	5	2	no	—
343	C₂²	—	5	4	no	—
344	C₂²	—	5	2	no	—
345	C₂²	—	10	4	no	—
346	C₂²	—	10	4	no	—
347	C₂²	—	10	2	no	—
348	C₂²	D₅×#106	1	1	yes	characteristic
349	C₂²	D₅×#106	1	1	yes	characteristic
350	C₂²	C₂×D₅×SD₁₆	1	1	yes	characteristic
351	C₄	—	5	2	no	—
352	C₄	—	5	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	40	2	no	—
355	C₄	D₅×#106	1	1	yes	characteristic
356	C₄	D₅×#106	1	1	yes	characteristic
357	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
358	C₄	C₂×D₅×SD₁₆	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	5	4	no	—
362	C₂	—	10	2	no	—
363	C₂	D₅×#365	1	2	yes	direct product factor
364	C₂	C₂×D₅×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×#106
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#365	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂³×D₅	SD₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#365	D₅
hypocenter	C₅	C₂²×#365
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#365	C₂	semidirect
5. C₂×#64733	C₂	semidirect
6. C₂×#64714	C₂	semidirect
7. C₂×#64707	C₂	semidirect
8. D₅×#365	C₂	direct (N × H)
20. D₅×M₃₂	C₄	semidirect
22. C₂×#3930	C₄	semidirect
23. C₂×D₅×C₁₆	C₄	semidirect
28. C₅×#365	C₂²	semidirect
29. #64733	C₂²	semidirect
30. #64714	C₂²	semidirect
31. #64707	C₂²	semidirect
56. C₅×M₃₂	C₂×C₄	semidirect
58. C₂×C₅×C₁₆	C₂×C₄	semidirect
68. #3944	C₂×C₄	semidirect
70. C₂×#672	C₂×C₄	semidirect
72. #3930	C₂×C₄	semidirect
73. D₅×C₁₆	C₂×C₄	semidirect
81. C₂×#365	D₅	direct (N × H)
128. C₅×C₁₆	C₂²×C₄	semidirect
129. #672	C₂²×C₄	semidirect
152. #365	C₂×D₅	direct (N × H)
216. M₃₂	C₄×D₅	semidirect
220. C₂×C₁₆	C₄×D₅	semidirect
238. C₂×D₅	#365	direct (N × H)
296. C₁₆	C₂×C₄×D₅	semidirect
299. C₂×C₅	C₂×#365	semidirect
302. D₅	C₂×#365	direct (N × H)
340. C₅	C₂²×#365	semidirect
363. C₂	D₅×#365	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×D₅×M₃₂	2
C₂×D₅×#108	2
C₂×D₅×#106	2
C₂×C₅×#365	2
C₂×#64733	2
C₂×#64714	2
C₂×#64707	2
D₅×#365	2
C₂²×#365	5

Subgroups — C2×D5×#365 #12252305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#365 #12252305