Subgroups — #13105759

Order 1600.

578 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#709	C₂	1	1	yes	maximal, characteristic
2	#11717221	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716935	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716932	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716923	C₂	1	2	yes	maximal, complemented
9	#11716922	C₂	1	2	yes	maximal, complemented
10	#11716909	C₂	1	2	yes	maximal, complemented
11	D₅×#676	C₂	1	2	yes	maximal, complemented
12	#87900	C₂²	1	1	yes	characteristic
13	C₅×#709	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	D₅×C₂²⋊D₅	C₄	1	2	yes	—
16	#87879	C₄	1	2	yes	—
17	D₅×D₂₀	C₄	1	1	yes	characteristic
18	C₄×D₅²	C₂²	1	2	yes	—
19	#87874	C₄	1	1	yes	characteristic
20	#87873	C₂²	1	2	yes	—
21	#87870	C₄	1	1	yes	characteristic
22	D₅×Dic₁₀	C₄	1	1	yes	characteristic
23	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
24	#87804	C₂²	1	1	yes	characteristic, complemented
25	#87799	C₂²	1	2	yes	complemented
26	C₈×C₅⋊D₅	C₂²	1	2	yes	complemented
27	C₅×#681	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
29	C₅×#676	C₂²	1	1	yes	characteristic
30	C₅×#676	C₂²	1	1	yes	characteristic
31	C₅×#676	C₂²	1	1	yes	characteristic, complemented
32	C₅×#676	C₂²	1	1	yes	characteristic, complemented
33	C₅×#676	C₂²	1	2	yes	complemented
34	C₅×C₈×D₅	C₂²	1	2	yes	complemented
35	#87769	C₂²	1	2	yes	complemented
36	#87768	C₂²	1	2	yes	—
37	#87768	C₂²	1	2	yes	complemented
38	#87767	C₂²	1	2	yes	complemented
39	D₅×#168	C₂²	1	2	yes	complemented
40	D₅×#168	C₂²	1	2	yes	complemented
41	#66084	—	5	1	no	maximal
42	#66079	—	5	1	no	maximal
43	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×D₅²	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
46	C₂²⋊(C₅⋊D₅)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₅⋊D₅)	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅⋊D₅	C₂³	1	2	yes	—
49	#6045	C₂×C₄	1	1	yes	characteristic
50	C₅×C₂²⋊D₅	C₂×C₄	1	2	yes	—
51	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₅×D₂₀	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×D₅	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	C₄×C₅×D₅	C₂³	1	2	yes	—
57	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
58	C₅×Dic₁₀	C₂×C₄	1	1	yes	characteristic
59	#6039	C₂×C₄	1	2	yes	—
60	#6038	C₂×C₄	1	2	yes	—
61	#6038	C₂×C₄	1	2	yes	—
62	#6037	C₂×C₄	1	2	yes	—
63	#6036	C₂×C₄	1	2	yes	—
64	D₅×Dic₅	C₂×C₄	1	2	yes	—
65	D₅×Dic₅	C₂×C₄	1	2	yes	—
66	C₅²×C₈	C₂³	1	1	yes	characteristic
67	C₅²×C₈	C₂³	1	1	yes	characteristic
68	#6029	C₂³	1	2	yes	—
69	C₅×#168	C₂³	1	1	yes	characteristic
70	C₅×#168	C₂³	1	2	yes	—
71	C₅×#168	C₂³	1	1	yes	characteristic
72	D₅×#41	—	5	1	no	—
73	C₂×#709	—	5	1	no	—
74	C₅×#131	—	5	1	no	—
75	C₂×C₅×M₁₆	—	5	1	no	—
76	#4094	—	5	1	no	—
77	C₂×#681	—	5	1	no	—
78	#4053	—	5	1	no	—
79	#4053	—	5	2	no	—
80	#4053	—	5	2	no	—
81	#4053	—	5	2	no	—
82	D₅×M₁₆	—	5	1	no	—
83	D₅×M₁₆	—	5	2	no	—
84	D₅×M₁₆	—	5	2	no	—
85	D₅×M₁₆	—	5	2	no	—
86	#4047	—	5	1	no	—
87	#4047	—	5	1	no	—
88	#4047	—	5	1	no	—
89	#4047	—	5	1	no	—
90	#4047	—	5	2	no	—
91	#4047	—	10	2	no	—
92	#4047	D₅	1	1	yes	characteristic, complemented
93	C₂×#676	—	5	2	no	—
94	C₂×#676	D₅	1	1	yes	characteristic, complemented
95	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
97	C₂×C₅×D₅	—	4	1	no	—
98	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
100	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
101	D₅²	—	4	2	no	—
102	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
103	C₄×C₅²	C₂⁴	1	1	yes	characteristic
104	#1039	C₂²×C₄	1	2	yes	—
105	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
106	C₅×Dic₅	C₂²×C₄	1	2	yes	—
107	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
108	C₅×#41	—	5	1	no	—
109	C₂²×C₄×C₅	—	5	1	no	—
110	C₂×C₂²⋊D₅	—	5	2	no	—
111	#713	—	5	1	no	—
112	Q₈×D₅	—	5	1	no	—
113	#711	—	5	2	no	—
114	#711	—	5	1	no	—
115	D₄×D₅	—	5	2	no	—
116	D₄×D₅	—	5	1	no	—
117	#709	—	5	1	no	—
118	#709	—	5	1	no	—
119	#709	—	5	2	no	—
120	#709	—	5	1	no	—
121	#709	—	5	2	no	—
122	#709	—	5	2	no	—
123	#709	—	5	1	no	—
124	#709	—	10	2	no	—
125	#709	C₂×D₅	1	1	yes	characteristic
126	C₂×D₂₀	—	5	1	no	—
127	C₂×C₄×D₅	—	5	2	no	—
128	C₂×C₄×D₅	—	5	2	no	—
129	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
130	C₂×Dic₁₀	—	5	1	no	—
131	C₅×M₁₆	—	5	1	no	—
132	C₅×M₁₆	—	5	1	no	—
133	C₅×M₁₆	—	5	1	no	—
134	C₅×M₁₆	—	5	1	no	—
135	C₅×M₁₆	—	5	1	no	—
136	C₅×M₁₆	—	5	2	no	—
137	C₂×C₅×C₈	—	2	2	no	—
138	C₂×C₅×C₈	—	5	1	no	—
139	C₂×C₅×C₈	—	5	2	no	—
140	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
141	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
142	#681	—	5	1	no	—
143	#681	—	5	1	no	—
144	#681	—	5	2	no	—
145	#681	—	5	2	no	—
146	#681	—	10	2	no	—
147	#681	C₂×D₅	1	1	yes	characteristic, complemented
148	C₂×#168	—	5	2	no	—
149	C₂×#168	—	5	2	no	—
150	C₂×#168	C₂×D₅	1	1	yes	characteristic, complemented
151	#676	—	5	2	no	—
152	#676	—	5	2	no	—
153	#676	—	5	2	no	—
154	#676	—	5	2	no	—
155	#676	—	5	2	no	—
156	#676	—	5	2	no	—
157	#676	—	10	4	no	—
158	#676	C₂×D₅	1	1	yes	characteristic
159	#676	C₂×D₅	1	1	yes	characteristic
160	#676	C₂×D₅	1	1	yes	characteristic, complemented
161	#676	C₂×D₅	1	1	yes	characteristic, complemented
162	#676	C₂×D₅	1	2	yes	complemented
163	C₈×D₅	—	5	2	no	—
164	C₈×D₅	—	5	2	no	—
165	C₈×D₅	—	5	2	no	—
166	C₈×D₅	—	5	2	no	—
167	C₈×D₅	—	5	2	no	—
168	C₈×D₅	—	5	2	no	—
169	C₈×D₅	—	10	4	no	—
170	C₈×D₅	C₂×D₅	1	2	yes	complemented
171	#568	—	25	1	no	Sylow
172	C₂×C₅²	—	2	1	no	—
173	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
174	C₅⋊D₅	—	2	2	no	—
175	C₅×D₅	—	2	1	no	—
176	C₅×D₅	—	2	2	no	—
177	C₅×D₅	—	2	1	no	—
178	C₂³×C₅	—	5	1	no	—
179	C₂²×D₅	—	5	2	no	—
180	C₂²×D₅	—	5	2	no	—
181	C₂²×D₅	C₄×D₅	1	1	yes	characteristic
182	C₅×Q₈	—	5	1	no	—
183	C₅×D₄	—	5	2	no	—
184	C₅×D₄	—	5	1	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	5	1	no	—
187	C₂×C₄×C₅	—	5	1	no	—
188	C₂×C₄×C₅	—	5	1	no	—
189	C₂×C₄×C₅	—	5	1	no	—
190	C₂×C₄×C₅	—	5	2	no	—
191	C₂×C₄×C₅	—	5	1	no	—
192	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
194	C₂²⋊D₅	—	5	2	no	—
195	C₂²⋊D₅	—	5	2	no	—
196	C₂²⋊D₅	—	5	2	no	—
197	C₂²⋊D₅	—	5	2	no	—
198	C₂²⋊D₅	—	5	2	no	—
199	C₂²⋊D₅	—	5	2	no	—
200	C₂²⋊D₅	—	10	4	no	—
201	C₂²⋊D₅	C₄×D₅	1	2	yes	—
202	C₂×Dic₅	—	5	2	no	—
203	C₂×Dic₅	—	5	2	no	—
204	C₂×Dic₅	C₄×D₅	1	1	yes	characteristic
205	D₂₀	—	5	2	no	—
206	D₂₀	—	5	2	no	—
207	D₂₀	—	5	1	no	—
208	D₂₀	—	5	1	no	—
209	D₂₀	—	10	2	no	—
210	D₂₀	C₄×D₅	1	1	yes	characteristic
211	C₄×D₅	—	5	2	no	—
212	C₄×D₅	—	5	2	no	—
213	C₄×D₅	—	5	2	no	—
214	C₄×D₅	—	5	2	no	—
215	C₄×D₅	—	5	2	no	—
216	C₄×D₅	—	5	2	no	—
217	C₄×D₅	—	10	4	no	—
218	C₄×D₅	C₄×D₅	1	1	yes	characteristic
219	C₄×D₅	C₄×D₅	1	1	yes	characteristic
220	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
221	C₄×D₅	C₂²×D₅	1	2	yes	—
222	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
223	Dic₁₀	—	5	1	no	—
224	Dic₁₀	—	5	1	no	—
225	Dic₁₀	—	5	2	no	—
226	Dic₁₀	—	5	2	no	—
227	Dic₁₀	—	10	2	no	—
228	Dic₁₀	C₄×D₅	1	1	yes	characteristic
229	C₅×C₈	—	2	2	no	—
230	C₅×C₈	—	2	2	no	—
231	C₅×C₈	—	5	1	no	—
232	C₅×C₈	—	5	2	no	—
233	C₅×C₈	—	5	1	no	—
234	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
235	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
236	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
237	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
238	#168	—	5	2	no	—
239	#168	—	5	2	no	—
240	#168	—	10	4	no	—
241	#168	C₂²×D₅	1	1	yes	characteristic
242	#168	C₂²×D₅	1	2	yes	—
243	#168	C₂²×D₅	1	1	yes	characteristic
244	C₂×#41	—	25	1	no	—
245	#131	—	5	1	no	—
246	#131	—	25	1	no	—
247	#131	—	25	2	no	—
248	#131	—	25	1	no	—
249	#131	—	25	2	no	—
250	#131	—	25	1	no	—
251	C₂×M₁₆	—	5	1	no	—
252	C₂×M₁₆	—	25	2	no	—
253	C₂×M₁₆	—	25	2	no	—
254	C₂×M₁₆	—	25	1	no	—
255	C₅²	#568	1	1	yes	characteristic, complemented, Sylow
256	C₂²×C₅	—	2	2	no	—
257	C₂²×C₅	—	5	1	no	—
258	C₂²×C₅	—	5	2	no	—
259	C₂²×C₅	—	5	1	no	—
260	C₂²×C₅	—	20	1	no	—
261	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
263	C₂×D₅	—	4	1	no	—
264	C₂×D₅	—	5	2	no	—
265	C₂×D₅	—	5	2	no	—
266	C₂×D₅	—	10	4	no	—
267	C₂×D₅	—	20	2	no	—
268	C₂×D₅	—	20	2	no	—
269	C₂×D₅	C₂×C₄×D₅	1	1	yes	characteristic
270	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
271	C₂×D₅	C₂×C₄×D₅	1	1	yes	characteristic
272	C₄×C₅	—	2	2	no	—
273	C₄×C₅	—	2	2	no	—
274	C₄×C₅	—	5	1	no	—
275	C₄×C₅	—	5	2	no	—
276	C₄×C₅	—	5	1	no	—
277	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
278	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
279	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
280	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
281	Dic₅	—	5	2	no	—
282	Dic₅	—	5	2	no	—
283	Dic₅	—	10	4	no	—
284	Dic₅	C₂×C₄×D₅	1	1	yes	characteristic
285	Dic₅	C₂×C₄×D₅	1	2	yes	—
286	Dic₅	C₂×C₄×D₅	1	1	yes	characteristic
287	#41	—	5	1	no	—
288	#41	—	25	2	no	—
289	#41	—	25	2	no	—
290	#41	—	25	1	no	—
291	#41	—	25	1	no	—
292	#41	—	25	1	no	—
293	C₂×Q₈	—	25	1	no	—
294	C₂×D₄	—	25	1	no	—
295	C₂×D₄	—	25	2	no	—
296	C₂²×C₄	—	5	1	no	—
297	C₂²×C₄	—	25	2	no	—
298	M₁₆	—	5	1	no	—
299	M₁₆	—	5	1	no	—
300	M₁₆	—	5	1	no	—
301	M₁₆	—	5	1	no	—
302	M₁₆	—	5	1	no	—
303	M₁₆	—	5	2	no	—
304	M₁₆	—	25	2	no	—
305	M₁₆	—	25	2	no	—
306	M₁₆	—	25	2	no	—
307	M₁₆	—	25	1	no	—
308	M₁₆	—	25	2	no	—
309	C₂×C₈	—	5	1	no	—
310	C₂×C₈	—	5	2	no	—
311	C₂×C₈	—	25	2	no	—
312	C₂×C₈	—	25	2	no	—
313	C₂×C₈	—	25	2	no	—
314	C₂×C₈	—	25	2	no	—
315	C₂×C₈	D₅²	1	1	yes	characteristic, complemented
316	C₂×C₅	—	2	1	no	—
317	C₂×C₅	—	2	1	no	—
318	C₂×C₅	—	2	2	no	—
319	C₂×C₅	—	4	2	no	—
320	C₂×C₅	—	10	2	no	—
321	C₂×C₅	—	10	1	no	—
322	C₂×C₅	—	10	1	no	—
323	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
324	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
325	D₅	—	2	1	no	—
326	D₅	—	2	1	no	—
327	D₅	—	2	2	no	—
328	D₅	—	10	2	no	—
329	D₅	—	10	2	no	—
330	D₅	—	20	4	no	—
331	C₂³	—	5	1	no	—
332	C₂³	—	25	2	no	—
333	Q₈	—	5	1	no	—
334	Q₈	—	25	2	no	—
335	Q₈	—	25	1	no	—
336	D₄	—	5	2	no	—
337	D₄	—	5	1	no	—
338	D₄	—	25	2	no	—
339	D₄	—	25	2	no	—
340	D₄	—	25	2	no	—
341	D₄	—	25	2	no	—
342	D₄	—	25	1	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	5	1	no	—
346	C₂×C₄	—	5	2	no	—
347	C₂×C₄	—	5	1	no	—
348	C₂×C₄	—	5	1	no	—
349	C₂×C₄	—	25	2	no	—
350	C₂×C₄	—	25	2	no	—
351	C₂×C₄	—	25	2	no	—
352	C₂×C₄	—	25	2	no	—
353	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
354	C₈	—	5	1	no	—
355	C₈	—	5	1	no	—
356	C₈	—	5	2	no	—
357	C₈	—	25	2	no	—
358	C₈	C₂×D₅²	1	1	yes	characteristic
359	C₈	C₂×D₅²	1	1	yes	characteristic
360	C₅	—	2	2	no	—
361	C₅	#66079	1	1	yes	characteristic, complemented
362	C₅	#66084	1	1	yes	characteristic, complemented
363	C₂²	—	5	1	no	—
364	C₂²	—	5	1	no	—
365	C₂²	—	5	2	no	—
366	C₂²	—	20	1	no	—
367	C₂²	—	25	2	no	—
368	C₂²	—	100	2	no	—
369	C₂²	C₄×D₅²	1	1	yes	characteristic
370	C₄	—	5	1	no	—
371	C₄	—	5	1	no	—
372	C₄	—	5	2	no	—
373	C₄	—	25	2	no	—
374	C₄	C₄×D₅²	1	1	yes	characteristic
375	C₄	C₂²×D₅²	1	1	yes	characteristic
376	C₂	—	2	1	no	—
377	C₂	—	10	1	no	—
378	C₂	—	10	1	no	—
379	C₂	—	10	2	no	—
380	C₂	—	50	2	no	—
381	C₂	C₂×C₄×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₅²
commutator, socle	C₂×C₅²	C₂³×C₄
Fitting	C₂×C₅²×C₈	C₂²
prime radical	D₅×#709	C₂
cosocle	C₄×C₅²	C₂⁴
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#568
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #11717221	C₂	semidirect
3. C₅×#4047	C₂	semidirect
4. C₂×C₅×#676	C₂	semidirect
5. #11716935	C₂	semidirect
6. #11716932	C₂	semidirect
7. D₅×#681	C₂	semidirect
8. #11716923	C₂	semidirect
9. #11716922	C₂	semidirect
10. #11716909	C₂	semidirect
11. D₅×#676	C₂	semidirect
23. C₂×C₅²×C₈	C₂²	semidirect
24. #87804	C₂²	semidirect
25. #87799	C₂²	semidirect
26. C₈×C₅⋊D₅	C₂²	semidirect
27. C₅×#681	C₂²	semidirect
28. C₂×C₅×#168	C₂²	semidirect
31. C₅×#676	C₂²	semidirect
32. C₅×#676	C₂²	semidirect
33. C₅×#676	C₂²	semidirect
34. C₅×C₈×D₅	C₂²	semidirect
35. #87769	C₂²	semidirect
37. #87768	C₂²	semidirect
38. #87767	C₂²	semidirect
39. D₅×#168	C₂²	semidirect
40. D₅×#168	C₂²	semidirect
92. #4047	D₅	semidirect
94. C₂×#676	D₅	semidirect
140. C₂×C₅×C₈	C₂×D₅	semidirect
141. C₂×C₅×C₈	C₂×D₅	semidirect
147. #681	C₂×D₅	semidirect
150. C₂×#168	C₂×D₅	semidirect
160. #676	C₂×D₅	semidirect
161. #676	C₂×D₅	semidirect
162. #676	C₂×D₅	semidirect
170. C₈×D₅	C₂×D₅	semidirect
255. C₅²	#568	semidirect
315. C₂×C₈	D₅²	semidirect
361. C₅	#66079	semidirect
362. C₅	#66084	semidirect

Maximal subgroups

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

subgroup	index
D₅×#709	2
#11717221	2
C₅×#4047	2
C₂×C₅×#676	2
#11716935	2
#11716932	2
D₅×#681	2
#11716923	2
#11716922	2
#11716909	2
D₅×#676	2
#66084	5
#66079	5