Subgroups — D₅×#4131 #13107621

Order 1600.

552 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#711	C₂	1	1	yes	maximal, characteristic
2	#11717426	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4131	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717345	C₂	1	2	yes	maximal, complemented
5	D₅×#704	C₂	1	2	yes	maximal, complemented
6	#11717343	C₂	1	2	yes	maximal, complemented
7	C₂×D₅×#170	C₂	1	2	yes	maximal, complemented
8	#11717329	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅×#700	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717327	C₂	1	1	yes	maximal, characteristic, complemented
11	D₅×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#87902	C₄	1	1	yes	characteristic
13	C₅×#711	C₂²	1	1	yes	characteristic
14	C₅×D₄×D₅	C₄	1	1	yes	characteristic
15	D₅×C₂²⋊D₅	C₄	1	2	yes	—
16	#87880	C₄	1	2	yes	—
17	#87879	C₂²	1	2	yes	—
18	C₂×D₅×Dic₅	C₂²	1	2	yes	—
19	C₄×D₅²	C₂²	1	1	yes	characteristic
20	#87871	C₄	1	1	yes	characteristic
21	#87870	C₂²	1	1	yes	characteristic
22	D₅×Dic₁₀	C₄	1	1	yes	characteristic
23	#87853	C₂²	1	2	yes	complemented
24	C₂×#6032	C₂²	1	2	yes	complemented
25	#87849	C₂²	1	1	yes	characteristic, complemented
26	D₅⋊#168	C₂²	1	1	yes	characteristic, complemented
27	C₅×#704	C₂²	1	2	yes	complemented
28	C₂×C₅×#170	C₂²	1	2	yes	complemented
29	C₅×#700	C₂²	1	1	yes	characteristic, complemented
30	C₅×D₅⋊C₈	C₂²	1	1	yes	characteristic, complemented
31	#87829	C₂²	1	2	yes	complemented
32	#87828	C₂²	1	2	yes	complemented
33	#87828	C₂²	1	2	yes	complemented
34	#87827	C₂²	1	1	yes	characteristic, complemented
35	#87827	C₂²	1	1	yes	characteristic, complemented
36	D₅×#170	C₂²	1	1	yes	characteristic
37	D₅×#170	C₂²	1	1	yes	characteristic
38	D₅×#170	C₂²	1	2	yes	—
39	D₅×#170	C₂²	1	2	yes	complemented
40	D₅×#170	C₂²	1	2	yes	complemented
41	C₂×#4131	—	5	1	no	maximal
42	D₅×#131	—	5	1	no	maximal
43	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
44	C₂×D₅²	C₂×C₄	1	1	yes	characteristic
45	C₅²×D₄	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊(C₅⋊D₅)	C₂×C₄	1	2	yes	—
47	C₂×#1039	C₂×C₄	1	2	yes	—
48	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
49	#6045	C₂×C₄	1	1	yes	characteristic
50	C₅×C₂²⋊D₅	C₂×C₄	1	2	yes	—
51	C₅×C₂²⋊D₅	C₂×C₄	1	2	yes	—
52	C₂×C₅×Dic₅	C₂³	1	2	yes	—
53	C₅×D₂₀	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
56	C₅×Dic₁₀	C₂×C₄	1	1	yes	characteristic
57	#6039	C₂×C₄	1	2	yes	—
58	#6038	C₂×C₄	1	2	yes	—
59	#6037	C₂×C₄	1	2	yes	—
60	#6036	C₂³	1	1	yes	characteristic
61	D₅×Dic₅	C₂×C₄	1	2	yes	—
62	D₅×Dic₅	C₂×C₄	1	2	yes	—
63	D₅×Dic₅	C₂×C₄	1	1	yes	characteristic
64	D₅×Dic₅	C₂³	1	1	yes	characteristic
65	D₅×Dic₅	C₂³	1	2	yes	—
66	#6032	C₂³	1	1	yes	characteristic
67	#6032	C₂³	1	2	yes	—
68	#6032	C₂³	1	1	yes	characteristic
69	C₅×#170	C₂³	1	1	yes	characteristic
70	C₅×#170	C₂³	1	2	yes	—
71	C₅×#170	C₂³	1	1	yes	characteristic
72	D₅×#41	—	5	1	no	—
73	C₂×#711	—	5	1	no	—
74	C₂×#704	—	5	2	no	—
75	C₂²×#170	—	5	2	no	—
76	#4131	—	5	1	no	—
77	#4131	—	5	2	no	—
78	#4131	—	5	2	no	—
79	#4131	—	5	1	no	—
80	#4131	—	5	1	no	—
81	#4131	D₅	1	1	yes	characteristic, direct product factor
82	C₂×#700	—	5	1	no	—
83	C₂×D₅⋊C₈	—	5	1	no	—
84	C₅×#131	—	5	1	no	—
85	#4094	—	5	1	no	—
86	#4053	—	5	1	no	—
87	#4053	—	5	2	no	—
88	D₅×M₁₆	—	5	1	no	—
89	D₅×M₁₆	—	5	2	no	—
90	#4047	—	5	1	no	—
91	#4047	—	5	2	no	—
92	C₂×C₈×D₅	—	5	1	no	—
93	C₂×C₈×D₅	—	5	2	no	—
94	C₂²×C₅²	C₂²×C₄	1	2	yes	—
95	C₂×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×D₅	—	2	4	no	—
97	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
99	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
100	D₅²	—	2	2	no	—
101	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
102	#1039	C₂²×C₄	1	1	yes	characteristic
103	#1039	C₂²×C₄	1	2	yes	—
104	C₅×Dic₅	C₂²×C₄	1	2	yes	—
105	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
106	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
107	C₅×#41	—	5	1	no	—
108	C₂×C₅×D₄	—	5	1	no	—
109	C₂×C₂²⋊D₅	—	5	2	no	—
110	C₂²×Dic₅	—	5	2	no	—
111	#713	—	5	1	no	—
112	Q₈×D₅	—	5	1	no	—
113	#711	—	5	1	no	—
114	#711	—	5	1	no	—
115	#711	—	5	2	no	—
116	#711	—	5	2	no	—
117	#711	—	5	2	no	—
118	#711	—	5	1	no	—
119	#711	—	5	1	no	—
120	#711	—	20	1	no	—
121	#711	C₂×D₅	1	1	yes	characteristic
122	D₄×D₅	—	5	2	no	—
123	D₄×D₅	AGL(1,𝔽₅)	1	1	yes	characteristic
124	#709	—	5	2	no	—
125	#709	—	5	1	no	—
126	C₂×C₄×D₅	—	5	2	no	—
127	C₂×C₄×D₅	—	5	1	no	—
128	C₂×C₄×D₅	—	5	1	no	—
129	C₂×Dic₁₀	—	5	1	no	—
130	#704	—	5	2	no	—
131	#704	—	5	2	no	—
132	#704	—	5	2	no	—
133	#704	C₂×D₅	1	2	yes	complemented
134	C₂×#170	—	5	2	no	—
135	C₂×#170	—	5	1	no	—
136	C₂×#170	—	5	1	no	—
137	C₂×#170	—	5	2	no	—
138	C₂×#170	—	5	2	no	—
139	C₂×#170	—	5	2	no	—
140	C₂×#170	C₂×D₅	1	2	yes	complemented
141	#700	—	5	1	no	—
142	#700	—	5	2	no	—
143	#700	C₂×D₅	1	1	yes	characteristic, complemented
144	D₅⋊C₈	—	5	1	no	—
145	D₅⋊C₈	—	5	1	no	—
146	D₅⋊C₈	—	5	1	no	—
147	D₅⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
148	C₅×M₁₆	—	5	1	no	—
149	C₅×M₁₆	—	5	2	no	—
150	C₂×C₅×C₈	—	5	1	no	—
151	C₂×C₅×C₈	—	5	2	no	—
152	#681	—	5	1	no	—
153	#681	—	5	2	no	—
154	C₂×#168	—	5	1	no	—
155	C₂×#168	—	5	2	no	—
156	#676	—	5	2	no	—
157	#676	—	5	2	no	—
158	#676	—	5	2	no	—
159	C₈×D₅	—	5	1	no	—
160	C₈×D₅	—	5	1	no	—
161	C₈×D₅	—	5	2	no	—
162	C₈×D₅	—	5	2	no	—
163	C₈×D₅	—	5	2	no	—
164	C₈×D₅	—	5	1	no	—
165	C₈×D₅	—	5	1	no	—
166	C₂×#131	—	25	1	no	Sylow
167	C₂×C₅²	—	2	2	no	—
168	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
169	C₅⋊D₅	—	2	1	no	—
170	C₅×D₅	—	2	1	no	—
171	C₅×D₅	—	2	2	no	—
172	C₅×D₅	#131	1	2	yes	complemented
173	C₂³×C₅	—	5	2	no	—
174	C₂²×D₅	—	5	1	no	—
175	C₂²×D₅	—	5	1	no	—
176	C₂²×D₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
177	C₅×Q₈	—	5	1	no	—
178	C₅×D₄	—	4	1	no	—
179	C₅×D₄	—	5	2	no	—
180	C₅×D₄	—	5	2	no	—
181	C₅×D₄	—	5	1	no	—
182	C₅×D₄	C₄×D₅	1	1	yes	characteristic
183	C₅×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₂×C₄×C₅	—	5	2	no	—
185	C₂×C₄×C₅	—	5	1	no	—
186	C₂×C₄×C₅	—	5	1	no	—
187	C₂²⋊D₅	—	5	2	no	—
188	C₂²⋊D₅	—	5	2	no	—
189	C₂²⋊D₅	—	5	2	no	—
190	C₂²⋊D₅	—	5	2	no	—
191	C₂²⋊D₅	—	5	2	no	—
192	C₂²⋊D₅	—	20	2	no	—
193	C₂²⋊D₅	C₄×D₅	1	2	yes	—
194	C₂²⋊D₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
195	C₂×Dic₅	—	5	2	no	—
196	C₂×Dic₅	—	5	2	no	—
197	C₂×Dic₅	—	5	2	no	—
198	C₂×Dic₅	—	5	2	no	—
199	C₂×Dic₅	—	5	1	no	—
200	C₂×Dic₅	—	5	1	no	—
201	C₂×Dic₅	—	5	2	no	—
202	C₂×Dic₅	—	20	2	no	—
203	C₂×Dic₅	C₂²×D₅	1	2	yes	—
204	D₂₀	—	5	2	no	—
205	D₂₀	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₄×D₅	—	5	1	no	—
207	C₄×D₅	—	5	2	no	—
208	C₄×D₅	—	5	2	no	—
209	C₄×D₅	—	5	1	no	—
210	C₄×D₅	—	5	2	no	—
211	C₄×D₅	—	5	1	no	—
212	C₄×D₅	—	5	1	no	—
213	C₄×D₅	—	5	1	no	—
214	C₄×D₅	—	5	1	no	—
215	C₄×D₅	—	20	1	no	—
216	C₄×D₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
218	Dic₁₀	—	5	1	no	—
219	Dic₁₀	—	5	1	no	—
220	Dic₁₀	—	5	2	no	—
221	Dic₁₀	—	5	2	no	—
222	Dic₁₀	—	20	1	no	—
223	Dic₁₀	C₄×D₅	1	1	yes	characteristic
224	#170	—	5	1	no	—
225	#170	—	5	1	no	—
226	#170	—	5	2	no	—
227	#170	C₂²×D₅	1	1	yes	characteristic
228	#170	C₂²×D₅	1	1	yes	characteristic
229	#170	C₂²×D₅	1	2	yes	—
230	C₅×C₈	—	5	1	no	—
231	C₅×C₈	—	5	2	no	—
232	C₅×C₈	—	5	1	no	—
233	#168	—	5	1	no	—
234	#168	—	5	2	no	—
235	#168	—	5	1	no	—
236	C₂×#41	—	25	1	no	—
237	#131	—	5	1	no	—
238	#131	—	25	1	no	—
239	#131	—	25	2	no	—
240	#131	—	25	2	no	—
241	#131	—	25	1	no	—
242	#131	—	25	1	no	—
243	C₂×M₁₆	—	25	1	no	—
244	C₂×M₁₆	—	25	2	no	—
245	C₂²×C₈	—	25	1	no	—
246	C₂²×C₈	—	25	2	no	—
247	C₅²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	5	1	no	—
250	C₂²×C₅	—	5	2	no	—
251	C₂²×C₅	—	5	1	no	—
252	C₂²×C₅	—	10	4	no	—
253	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
254	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
255	C₂×D₅	—	2	4	no	—
256	C₂×D₅	—	5	1	no	—
257	C₂×D₅	—	5	1	no	—
258	C₂×D₅	—	10	2	no	—
259	C₂×D₅	—	10	2	no	—
260	C₂×D₅	—	20	1	no	—
261	C₂×D₅	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×D₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
263	C₂×D₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₄×C₅	—	4	1	no	—
265	C₄×C₅	—	5	1	no	—
266	C₄×C₅	—	5	2	no	—
267	C₄×C₅	—	5	1	no	—
268	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
269	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
270	Dic₅	—	5	2	no	—
271	Dic₅	—	5	1	no	—
272	Dic₅	—	5	2	no	—
273	Dic₅	—	5	1	no	—
274	Dic₅	—	20	2	no	—
275	Dic₅	—	20	1	no	—
276	Dic₅	C₂×C₄×D₅	1	2	yes	—
277	Dic₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
278	Dic₅	C₂³×D₅	1	1	yes	characteristic
279	#41	—	5	1	no	—
280	#41	—	25	2	no	—
281	#41	—	25	1	no	—
282	#41	—	25	2	no	—
283	#41	—	25	1	no	—
284	#41	—	25	1	no	—
285	C₂×Q₈	—	25	1	no	—
286	C₂×D₄	—	5	1	no	—
287	C₂×D₄	—	25	2	no	—
288	C₂²×C₄	—	25	1	no	—
289	C₂²×C₄	—	25	2	no	—
290	M₁₆	—	5	1	no	—
291	M₁₆	—	5	2	no	—
292	M₁₆	—	25	2	no	—
293	M₁₆	—	25	2	no	—
294	M₁₆	—	25	2	no	—
295	M₁₆	—	25	1	no	—
296	M₁₆	—	25	2	no	—
297	C₂×C₈	—	5	1	no	—
298	C₂×C₈	—	5	2	no	—
299	C₂×C₈	—	25	1	no	—
300	C₂×C₈	—	25	1	no	—
301	C₂×C₈	—	25	2	no	—
302	C₂×C₈	—	25	1	no	—
303	C₂×C₈	—	25	1	no	—
304	C₂×C₈	—	25	2	no	—
305	C₂×C₈	—	25	2	no	—
306	C₂×C₈	—	25	1	no	—
307	C₂×C₈	—	25	2	no	—
308	C₂×C₅	—	2	2	no	—
309	C₂×C₅	—	2	2	no	—
310	C₂×C₅	—	4	1	no	—
311	C₂×C₅	—	5	2	no	—
312	C₂×C₅	—	8	2	no	—
313	C₂×C₅	—	10	2	no	—
314	C₂×C₅	—	10	1	no	—
315	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
316	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
317	D₅	—	2	1	no	—
318	D₅	—	2	2	no	—
319	D₅	—	10	1	no	—
320	D₅	—	10	1	no	—
321	D₅	—	40	1	no	—
322	D₅	#4131	1	2	yes	direct product factor
323	C₂³	—	5	2	no	—
324	C₂³	—	25	1	no	—
325	Q₈	—	5	1	no	—
326	Q₈	—	25	2	no	—
327	Q₈	—	25	1	no	—
328	D₄	—	5	2	no	—
329	D₄	—	5	2	no	—
330	D₄	—	5	1	no	—
331	D₄	—	25	2	no	—
332	D₄	—	25	2	no	—
333	D₄	—	25	2	no	—
334	D₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
335	C₂×C₄	—	5	2	no	—
336	C₂×C₄	—	5	1	no	—
337	C₂×C₄	—	5	1	no	—
338	C₂×C₄	—	25	1	no	—
339	C₂×C₄	—	25	1	no	—
340	C₂×C₄	—	25	2	no	—
341	C₂×C₄	—	25	2	no	—
342	C₂×C₄	—	25	1	no	—
343	C₂×C₄	—	25	2	no	—
344	C₂×C₄	—	25	2	no	—
345	C₂×C₄	—	25	1	no	—
346	C₈	—	5	1	no	—
347	C₈	—	5	1	no	—
348	C₈	—	5	2	no	—
349	C₈	—	25	1	no	—
350	C₈	—	25	1	no	—
351	C₈	—	25	2	no	—
352	C₅	—	4	1	no	—
353	C₅	D₅×#131	1	1	yes	characteristic, complemented
354	C₅	C₂×#4131	1	1	yes	characteristic, complemented
355	C₂²	—	5	1	no	—
356	C₂²	—	5	1	no	—
357	C₂²	—	5	2	no	—
358	C₂²	—	10	4	no	—
359	C₂²	—	25	1	no	—
360	C₂²	—	50	2	no	—
361	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	2	yes	—
362	C₄	—	5	1	no	—
363	C₄	—	5	1	no	—
364	C₄	—	5	2	no	—
365	C₄	—	25	1	no	—
366	C₄	—	25	2	no	—
367	C₄	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
368	C₂	—	2	2	no	—
369	C₂	—	5	2	no	—
370	C₂	—	10	2	no	—
371	C₂	—	10	1	no	—
372	C₂	—	50	1	no	—
373	C₂	C₂²×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₅×AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂³×C₄
Fitting	C₅²×D₄	C₂×C₄
prime radical	D₅×#711	C₂
cosocle	C₅×Dic₅	C₂⁴
hypercenter	D₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #11717426	C₂	semidirect
3. C₅×#4131	C₂	semidirect
4. #11717345	C₂	semidirect
5. D₅×#704	C₂	semidirect
6. #11717343	C₂	semidirect
7. C₂×D₅×#170	C₂	semidirect
8. #11717329	C₂	semidirect
9. D₅×#700	C₂	semidirect
10. #11717327	C₂	semidirect
11. D₅×D₅⋊C₈	C₂	semidirect
23. #87853	C₂²	semidirect
24. C₂×#6032	C₂²	semidirect
25. #87849	C₂²	semidirect
26. D₅⋊#168	C₂²	semidirect
27. C₅×#704	C₂²	semidirect
28. C₂×C₅×#170	C₂²	semidirect
29. C₅×#700	C₂²	semidirect
30. C₅×D₅⋊C₈	C₂²	semidirect
31. #87829	C₂²	semidirect
32. #87828	C₂²	semidirect
33. #87828	C₂²	semidirect
34. #87827	C₂²	semidirect
35. #87827	C₂²	semidirect
39. D₅×#170	C₂²	semidirect
40. D₅×#170	C₂²	semidirect
81. #4131	D₅	direct (N × H)
133. #704	C₂×D₅	semidirect
140. C₂×#170	C₂×D₅	semidirect
143. #700	C₂×D₅	semidirect
147. D₅⋊C₈	C₂×D₅	semidirect
172. C₅×D₅	#131	semidirect
247. C₅²	C₂×#131	semidirect
322. D₅	#4131	direct (N × H)
353. C₅	D₅×#131	semidirect
354. C₅	C₂×#4131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₅×#711	2
#11717426	2
C₅×#4131	2
#11717345	2
D₅×#704	2
#11717343	2
C₂×D₅×#170	2
#11717329	2
D₅×#700	2
#11717327	2
D₅×D₅⋊C₈	2
C₂×#4131	5
D₅×#131	5

Subgroups — D5×#4131 #13107621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4131 #13107621