Subgroups — D₅×#2110 #12251955

Order 1280.

738 conjugacy classes of subgroups.

The subgroup lattice has 362 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	2	yes	maximal
2	D₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₅×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602218	C₂	1	2	yes	maximal, complemented
5	#10600677	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599880	C₂	1	4	yes	maximal, complemented
7	D₅×#347	C₂	1	4	yes	maximal, complemented
8	C₂×D₅×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
10	D₅×#117	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#405	C₂²	1	1	yes	characteristic
13	#65411	C₄	1	1	yes	characteristic
14	#65390	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	#65286	C₄	1	1	yes	characteristic
17	D₅×M₃₂	C₄	1	4	yes	complemented
18	D₅×M₃₂	C₄	1	4	yes	complemented
19	D₅×M₃₂	C₂²	1	4	yes	—
20	D₅×M₃₂	C₂²	1	4	yes	—
21	C₂×#3930	C₄	1	4	yes	complemented
22	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
23	#64997	C₂²	1	2	yes	—
24	D₅×#97	C₂²	1	2	yes	—
25	#64978	C₄	1	2	yes	—
26	C₄×C₈×D₅	C₄	1	2	yes	—
27	C₅×#347	C₂²	1	4	yes	complemented
28	#64732	C₂²	1	8	yes	complemented
29	#64686	C₂²	1	4	yes	complemented
30	C₂×#2110	—	5	1	no	maximal, Sylow
31	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×#117	C₂×C₄	1	1	yes	characteristic
35	#4072	C₂×C₄	1	1	yes	characteristic
36	C₂×#681	C₂×C₄	1	1	yes	characteristic
37	C₂²×#168	C₂×C₄	1	1	yes	characteristic
38	D₅×M₁₆	—	2	2	no	—
39	D₅×M₁₆	—	2	2	no	—
40	C₂×#676	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
50	#4038	—	2	1	no	—
51	D₅×C₄⋊C₄	—	2	1	no	—
52	D₅×C₂²⋊C₄	—	2	1	no	—
53	#4023	—	2	1	no	—
54	#4018	C₂×C₄	1	2	yes	—
55	C₄²×D₅	C₂×C₄	1	2	yes	—
56	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
57	C₅×M₃₂	C₂³	1	4	yes	—
58	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
59	C₅×#97	C₂³	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
61	#3947	C₂×C₄	1	2	yes	—
62	#3947	C₂³	1	2	yes	—
63	C₈×Dic₅	C₂×C₄	1	2	yes	—
64	C₈×Dic₅	C₂×C₄	1	2	yes	—
65	#3944	C₂×C₄	1	4	yes	complemented
66	#3944	C₂³	1	4	yes	—
67	C₂×#672	C₂×C₄	1	4	yes	complemented
68	#3935	C₂×C₄	1	2	yes	—
69	C₄×#168	C₂×C₄	1	2	yes	—
70	#3930	C₂×C₄	1	8	yes	—
71	#3930	C₂×C₄	1	8	yes	complemented
72	D₅×C₁₆	C₂×C₄	1	8	yes	—
73	D₅×C₁₆	C₂×C₄	1	8	yes	complemented
74	C₂²×M₃₂	—	5	2	no	—
75	C₂×#405	—	5	1	no	—
76	#2110	—	5	2	no	—
77	#2110	—	5	1	no	—
78	#2110	—	5	4	no	—
79	#2110	D₅	1	1	yes	characteristic, direct product factor
80	C₂×#347	—	5	4	no	—
81	C₂³×D₅	M₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₅	M₁₆	1	1	yes	characteristic
84	C₂×C₄×D₅	—	2	2	no	—
85	C₂×C₄×D₅	C₄²	1	2	yes	—
86	C₂×C₄×D₅	C₄²	1	2	yes	—
87	C₂×C₄×D₅	M₁₆	1	1	yes	characteristic
88	C₂×C₄×D₅	M₁₆	1	1	yes	characteristic
89	C₂×C₄×D₅	M₁₆	1	2	yes	—
90	C₂×C₄×D₅	M₁₆	1	2	yes	—
91	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
93	C₅×M₁₆	—	2	2	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	1	no	—
100	C₅×C₂²⋊C₄	—	2	1	no	—
101	C₄²×C₅	C₂²×C₄	1	2	yes	—
102	C₂²⋊Dic₅	—	2	1	no	—
103	#685	—	2	2	no	—
104	C₄⋊Dic₅	—	2	1	no	—
105	#683	—	2	2	no	—
106	C₄×Dic₅	—	2	2	no	—
107	C₄×Dic₅	C₂²×C₄	1	2	yes	—
108	#681	—	2	2	no	—
109	C₂×#168	C₄²	1	2	yes	—
110	C₂×#168	C₄²	1	2	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	C₂×#168	C₂²×C₄	1	1	yes	characteristic
113	C₂×#168	C₂²×C₄	1	1	yes	characteristic
114	#676	—	2	4	no	—
115	C₈×D₅	—	2	2	no	—
116	C₈×D₅	—	2	2	no	—
117	C₈×D₅	C₄²	1	1	yes	characteristic
118	C₈×D₅	C₄²	1	1	yes	characteristic
119	C₈×D₅	C₄²	1	2	yes	—
120	C₈×D₅	C₄²	1	1	yes	characteristic
121	C₈×D₅	C₄²	1	1	yes	characteristic
122	C₈×D₅	C₄²	1	2	yes	—
123	C₈×D₅	C₂²×C₄	1	2	yes	—
124	C₈×D₅	C₂²×C₄	1	2	yes	—
125	C₈×D₅	C₂²×C₄	1	2	yes	—
126	C₈×D₅	C₂²×C₄	1	2	yes	—
127	C₅×C₁₆	C₂²×C₄	1	8	yes	—
128	#672	C₂²×C₄	1	8	yes	—
129	C₂²×M₁₆	—	5	1	no	—
130	C₂³×C₈	—	5	1	no	—
131	C₂×#117	—	5	1	no	—
132	C₂×M₃₂	—	5	2	no	—
133	C₂×M₃₂	—	5	4	no	—
134	C₂×M₃₂	—	5	4	no	—
135	C₂×M₃₂	—	5	4	no	—
136	C₂×M₃₂	—	5	4	no	—
137	C₂×M₃₂	—	5	4	no	—
138	C₂×M₃₂	C₂×D₅	1	2	yes	—
139	C₂²×C₁₆	—	5	4	no	—
140	#405	—	5	1	no	—
141	#405	—	5	1	no	—
142	#405	—	5	1	no	—
143	#405	—	5	2	no	—
144	#405	—	5	2	no	—
145	#405	C₂×D₅	1	1	yes	characteristic
146	C₂×#97	—	5	2	no	—
147	C₂×C₄×C₈	—	5	2	no	—
148	#347	—	5	8	no	—
149	#347	—	5	4	no	—
150	#347	C₂×D₅	1	4	yes	complemented
151	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
152	C₂²×D₅	—	4	2	no	—
153	C₂²×D₅	#97	1	2	yes	—
154	C₂²×D₅	#97	1	2	yes	—
155	C₂²×D₅	C₂×M₁₆	1	1	yes	characteristic
156	C₂²×D₅	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
162	C₂×Dic₅	—	2	2	no	—
163	C₂×Dic₅	#97	1	2	yes	—
164	C₂×Dic₅	#97	1	2	yes	—
165	C₂×Dic₅	C₂×M₁₆	1	1	yes	characteristic
166	C₂×Dic₅	C₂×M₁₆	1	1	yes	characteristic
167	C₄×D₅	—	4	4	no	—
168	C₄×D₅	#97	1	2	yes	—
169	C₄×D₅	#97	1	2	yes	—
170	C₄×D₅	#97	1	2	yes	—
171	C₄×D₅	#97	1	2	yes	—
172	C₄×D₅	C₂×C₄²	1	2	yes	—
173	C₄×D₅	C₂×C₄²	1	1	yes	characteristic
174	C₄×D₅	C₂×C₄²	1	1	yes	characteristic
175	C₄×D₅	C₂×M₁₆	1	1	yes	characteristic
176	C₄×D₅	C₂×M₁₆	1	1	yes	characteristic
177	C₄×D₅	C₂×M₁₆	1	2	yes	—
178	C₅×C₈	—	2	2	no	—
179	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
180	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
181	C₅×C₈	C₂³×C₄	1	2	yes	—
182	#168	—	2	2	no	—
183	#168	C₂×C₄²	1	2	yes	—
184	#168	C₂×C₄²	1	1	yes	characteristic
185	#168	C₂×C₄²	1	1	yes	characteristic
186	C₂³×C₄	—	5	1	no	—
187	C₂×M₁₆	—	5	1	no	—
188	C₂×M₁₆	—	5	2	no	—
189	C₂×M₁₆	—	10	2	no	—
190	C₂×M₁₆	—	10	2	no	—
191	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
192	C₂²×C₈	—	5	1	no	—
193	C₂²×C₈	—	5	1	no	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	—	5	2	no	—
196	C₂²×C₈	—	5	2	no	—
197	C₂²×C₈	—	5	2	no	—
198	C₂²×C₈	—	5	2	no	—
199	C₂²×C₈	—	5	1	no	—
200	C₂²×C₈	—	5	1	no	—
201	C₂²×C₈	—	5	2	no	—
202	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
203	#117	—	5	1	no	—
204	#117	—	5	2	no	—
205	#117	—	10	1	no	—
206	#117	—	10	1	no	—
207	#117	C₄×D₅	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	10	1	no	—
209	C₂×C₂²⋊C₄	—	10	1	no	—
210	C₂×C₄²	—	5	2	no	—
211	M₃₂	—	5	4	no	—
212	M₃₂	—	5	4	no	—
213	M₃₂	—	5	8	no	—
214	M₃₂	—	5	8	no	—
215	M₃₂	C₄×D₅	1	4	yes	complemented
216	M₃₂	C₂²×D₅	1	4	yes	—
217	C₂×C₁₆	—	5	4	no	—
218	C₂×C₁₆	—	5	8	no	—
219	C₂×C₁₆	—	5	8	no	—
220	C₂×C₁₆	C₄×D₅	1	4	yes	complemented
221	#97	—	5	2	no	—
222	#97	—	5	2	no	—
223	#97	—	5	2	no	—
224	#97	C₂²×D₅	1	2	yes	—
225	C₄×C₈	—	5	2	no	—
226	C₄×C₈	—	5	2	no	—
227	C₄×C₈	—	5	2	no	—
228	C₄×C₈	C₄×D₅	1	2	yes	—
229	C₂²×C₅	—	4	1	no	—
230	C₂²×C₅	C₂×#97	1	2	yes	—
231	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	—	2	4	no	—
234	C₂×D₅	—	4	2	no	—
235	C₂×D₅	—	4	1	no	—
236	C₂×D₅	C₂×#97	1	1	yes	characteristic
237	C₂×D₅	C₂×#97	1	2	yes	—
238	C₂×D₅	C₂×#97	1	1	yes	characteristic
239	C₄×C₅	—	4	2	no	—
240	C₄×C₅	C₂×#97	1	2	yes	—
241	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
242	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
243	Dic₅	—	4	2	no	—
244	Dic₅	C₂×#97	1	1	yes	characteristic
245	Dic₅	C₂×#97	1	1	yes	characteristic
246	Dic₅	C₂×#97	1	2	yes	—
247	C₂⁴	—	5	1	no	—
248	C₂²×C₄	—	5	1	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	2	no	—
252	C₂²×C₄	—	5	2	no	—
253	C₂²×C₄	—	5	1	no	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	5	2	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	—	10	2	no	—
258	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
259	M₁₆	—	2	2	no	—
260	M₁₆	—	10	4	no	—
261	M₁₆	—	10	2	no	—
262	C₂×C₈	—	5	1	no	—
263	C₂×C₈	—	5	1	no	—
264	C₂×C₈	—	5	2	no	—
265	C₂×C₈	—	5	2	no	—
266	C₂×C₈	—	5	2	no	—
267	C₂×C₈	—	5	2	no	—
268	C₂×C₈	—	5	2	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	1	no	—
271	C₂×C₈	—	5	1	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	—	5	1	no	—
274	C₂×C₈	—	5	1	no	—
275	C₂×C₈	—	5	2	no	—
276	C₂×C₈	—	5	2	no	—
277	C₂×C₈	—	10	2	no	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
280	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
281	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
282	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	10	2	no	—
286	C₄⋊C₄	—	10	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	10	2	no	—
289	C₂²⋊C₄	—	10	1	no	—
290	C₄²	—	5	2	no	—
291	C₄²	—	10	2	no	—
292	C₄²	C₂×C₄×D₅	1	2	yes	—
293	C₁₆	—	5	8	no	—
294	C₁₆	C₂×C₄×D₅	1	8	yes	—
295	C₂×C₅	—	2	1	no	—
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	C₂²×#97	1	1	yes	characteristic
298	D₅	—	2	1	no	—
299	D₅	—	2	2	no	—
300	D₅	#2110	1	2	yes	direct product factor
301	C₂³	—	5	1	no	—
302	C₂³	—	5	2	no	—
303	C₂³	—	5	1	no	—
304	C₂³	—	5	2	no	—
305	C₂³	—	20	2	no	—
306	C₂³	D₅×M₁₆	1	1	yes	characteristic
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	5	1	no	—
310	C₂×C₄	—	5	2	no	—
311	C₂×C₄	—	5	2	no	—
312	C₂×C₄	—	5	1	no	—
313	C₂×C₄	—	5	2	no	—
314	C₂×C₄	—	5	1	no	—
315	C₂×C₄	—	5	1	no	—
316	C₂×C₄	—	5	1	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	—	10	2	no	—
323	C₂×C₄	—	20	4	no	—
324	C₂×C₄	C₄²×D₅	1	2	yes	—
325	C₂×C₄	D₅×M₁₆	1	2	yes	—
326	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	5	1	no	—
330	C₈	—	5	1	no	—
331	C₈	—	5	2	no	—
332	C₈	—	10	2	no	—
333	C₈	C₄²×D₅	1	1	yes	characteristic
334	C₈	C₄²×D₅	1	1	yes	characteristic
335	C₈	C₂²×C₄×D₅	1	2	yes	—
336	C₅	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	4	1	no	—
338	C₂²	—	5	1	no	—
339	C₂²	—	5	1	no	—
340	C₂²	—	5	2	no	—
341	C₂²	—	10	2	no	—
342	C₂²	—	10	4	no	—
343	C₂²	—	20	1	no	—
344	C₂²	—	20	2	no	—
345	C₂²	D₅×#97	1	2	yes	—
346	C₂²	C₂×D₅×M₁₆	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	5	1	no	—
349	C₄	—	5	1	no	—
350	C₄	—	5	2	no	—
351	C₄	—	20	2	no	—
352	C₄	D₅×#97	1	2	yes	—
353	C₄	C₂×C₄²×D₅	1	1	yes	characteristic
354	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	5	2	no	—
358	C₂	—	10	1	no	—
359	C₂	—	10	2	no	—
360	C₂	C₂×D₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2110	C₂
socle	C₂×C₅	C₂²×#97
prime radical	C₂³×D₅	M₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2110	D₅
hypocenter	C₅	C₂×#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. C₅×#2110	C₂	semidirect
4. #10602218	C₂	semidirect
5. #10600677	C₂	semidirect
6. #10599880	C₂	semidirect
7. D₅×#347	C₂	semidirect
17. D₅×M₃₂	C₄	semidirect
18. D₅×M₃₂	C₄	semidirect
21. C₂×#3930	C₄	semidirect
22. C₂×D₅×C₁₆	C₄	semidirect
27. C₅×#347	C₂²	semidirect
28. #64732	C₂²	semidirect
29. #64686	C₂²	semidirect
56. C₅×M₃₂	C₂×C₄	semidirect
58. C₂×C₅×C₁₆	C₂×C₄	semidirect
65. #3944	C₂×C₄	semidirect
67. C₂×#672	C₂×C₄	semidirect
71. #3930	C₂×C₄	semidirect
73. D₅×C₁₆	C₂×C₄	semidirect
79. #2110	D₅	direct (N × H)
150. #347	C₂×D₅	semidirect
215. M₃₂	C₄×D₅	semidirect
220. C₂×C₁₆	C₄×D₅	semidirect
300. D₅	#2110	direct (N × H)
336. C₅	C₂×#2110	semidirect

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
D₅×#405	2
C₅×#2110	2
#10602218	2
#10600677	2
#10599880	2
D₅×#347	2
C₂×#2110	5

Subgroups — D5×#2110 #12251955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#2110 #12251955