Subgroups — #13105085

Order 1600.

770 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×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4018	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717039	C₂	1	2	yes	maximal, complemented
6	D₅×#685	C₂	1	2	yes	maximal, complemented
7	#11717022	C₂	1	2	yes	maximal, complemented
8	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
9	Dic₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716999	C₂	1	2	yes	maximal, complemented
11	#11716959	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅²	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	complemented
15	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
18	C₂×#6038	C₂²	1	2	yes	—
19	C₂×#6036	C₂²	1	2	yes	complemented
20	C₂×D₅×Dic₅	C₂²	1	2	yes	complemented
21	D₅×D₂₀	C₄	1	8	yes	complemented
22	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
23	#87808	C₂²	1	2	yes	complemented
24	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
25	C₅×#685	C₂²	1	1	yes	characteristic
26	C₅×#685	C₂²	1	1	yes	characteristic
27	C₅×#685	C₂²	1	2	yes	complemented
28	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
29	C₅×#683	C₂²	1	1	yes	characteristic, complemented
30	C₅×#683	C₂²	1	1	yes	characteristic, complemented
31	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
32	#87779	C₂²	1	2	yes	complemented
33	#87778	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	complemented
35	Dic₅²	C₂²	1	2	yes	complemented
36	#65879	—	5	1	no	maximal
37	#65855	—	5	1	no	maximal
38	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
39	C₂²×C₅×D₅	C₂³	1	2	yes	—
40	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
41	C₂×D₅²	—	4	2	no	—
42	C₂×D₅²	—	4	2	no	—
43	C₂×D₅²	C₂×C₄	1	8	yes	complemented
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
47	C₂×#1039	C₂³	1	2	yes	—
48	C₄⋊(C₅⋊D₅)	C₂×C₄	1	4	yes	complemented
49	C₄×C₅⋊D₅	—	4	2	no	—
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
52	C₂×C₅×Dic₅	C₂³	1	2	yes	—
53	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
54	C₅×D₂₀	C₂×C₄	1	4	yes	complemented
55	C₄×C₅×D₅	—	4	2	no	—
56	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
57	#6038	C₂×C₄	1	8	yes	complemented
58	#6036	—	4	2	no	—
59	D₅×Dic₅	—	4	2	no	—
60	C₂×D₄×D₅	—	5	1	no	—
61	C₂²×D₂₀	—	5	1	no	—
62	C₄×C₅×D₄	—	5	1	no	—
63	C₅×#117	—	5	1	no	—
64	D₄×Dic₅	—	5	1	no	—
65	C₄×C₂²⋊D₅	—	5	2	no	—
66	C₂×#685	—	5	2	no	—
67	#4072	—	5	1	no	—
68	#4039	—	5	2	no	—
69	#4039	—	5	2	no	—
70	#4039	—	5	1	no	—
71	#4038	—	5	1	no	—
72	#4027	—	5	2	no	—
73	D₅×C₂²⋊C₄	—	5	2	no	—
74	D₅×C₂²⋊C₄	—	5	2	no	—
75	D₅×C₂²⋊C₄	—	5	2	no	—
76	C₄×D₂₀	—	5	1	no	—
77	C₄×D₂₀	—	5	1	no	—
78	C₄×D₂₀	—	5	1	no	—
79	C₄×D₂₀	—	5	1	no	—
80	C₄×D₂₀	—	10	2	no	—
81	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
82	#4018	D₅	1	1	yes	characteristic, complemented
83	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
84	C₂×C₅⋊D₅	—	2	2	no	—
85	C₂×C₅⋊D₅	—	2	2	no	—
86	C₂×C₅⋊D₅	C₂²×C₄	1	4	yes	—
87	C₂×C₅×D₅	—	2	2	no	—
88	C₂×C₅×D₅	—	2	2	no	—
89	C₂×C₅×D₅	—	2	1	no	—
90	C₂×C₅×D₅	—	2	1	no	—
91	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
92	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
93	D₅²	—	4	8	no	—
94	C₄×C₅²	—	2	1	no	—
95	C₄×C₅²	—	2	1	no	—
96	C₄×C₅²	C₂²×C₄	1	2	yes	—
97	#1039	—	2	2	no	—
98	C₅×Dic₅	—	2	1	no	—
99	C₅×Dic₅	—	2	1	no	—
100	C₅×Dic₅	—	2	2	no	—
101	C₅×Dic₅	C₂²×C₄	1	2	yes	—
102	C₂³×D₅	—	5	2	no	—
103	C₂³×D₅	—	5	2	no	—
104	C₂×C₅×D₄	—	5	1	no	—
105	C₂²×C₄×C₅	—	5	1	no	—
106	C₂²×C₄×C₅	—	5	2	no	—
107	C₂×C₂²⋊D₅	—	5	2	no	—
108	C₂²×Dic₅	—	5	2	no	—
109	D₄×D₅	—	5	8	no	—
110	C₂×D₂₀	—	5	2	no	—
111	C₂×D₂₀	—	5	1	no	—
112	C₂×D₂₀	—	5	1	no	—
113	C₂×D₂₀	—	5	8	no	—
114	C₂×D₂₀	—	10	2	no	—
115	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄×D₅	—	5	2	no	—
117	C₂×C₄×D₅	—	5	2	no	—
118	C₂×C₄×D₅	—	5	2	no	—
119	C₂×C₄×D₅	—	5	2	no	—
120	C₂×C₄×D₅	—	5	2	no	—
121	C₂×C₄×D₅	—	10	4	no	—
122	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
123	C₂×C₄×D₅	C₂×D₅	1	2	yes	complemented
124	C₅×C₄⋊C₄	—	5	1	no	—
125	C₅×C₄⋊C₄	—	5	1	no	—
126	C₅×C₄⋊C₄	—	5	1	no	—
127	C₅×C₂²⋊C₄	—	5	1	no	—
128	C₅×C₂²⋊C₄	—	5	1	no	—
129	C₅×C₂²⋊C₄	—	5	2	no	—
130	C₄²×C₅	—	2	2	no	—
131	C₄²×C₅	—	5	1	no	—
132	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
133	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
134	C₂²⋊Dic₅	—	5	2	no	—
135	C₂²⋊Dic₅	—	5	2	no	—
136	#685	—	5	2	no	—
137	#685	—	5	2	no	—
138	#685	—	5	2	no	—
139	#685	—	5	2	no	—
140	#685	—	5	2	no	—
141	#685	—	5	2	no	—
142	#685	—	10	4	no	—
143	#685	C₂×D₅	1	1	yes	characteristic
144	#685	C₂×D₅	1	1	yes	characteristic
145	#685	C₂×D₅	1	2	yes	complemented
146	C₄⋊Dic₅	—	5	1	no	—
147	C₄⋊Dic₅	—	5	1	no	—
148	C₄⋊Dic₅	—	10	2	no	—
149	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
150	#683	C₂×D₅	1	1	yes	characteristic, complemented
151	#683	C₂×D₅	1	1	yes	characteristic, complemented
152	C₄×Dic₅	—	5	2	no	—
153	C₄×Dic₅	—	5	2	no	—
154	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
155	#518	—	25	1	no	Sylow
156	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
159	C₅⋊D₅	—	2	4	no	—
160	C₅×D₅	—	2	2	no	—
161	C₅×D₅	—	2	4	no	—
162	C₂³×C₅	—	5	1	no	—
163	C₂³×C₅	—	5	2	no	—
164	C₂²×D₅	—	5	2	no	—
165	C₂²×D₅	—	5	2	no	—
166	C₂²×D₅	—	5	8	no	—
167	C₂²×D₅	—	5	8	no	—
168	C₂²×D₅	—	10	4	no	—
169	C₂²×D₅	—	20	2	no	—
170	C₂²×D₅	—	20	2	no	—
171	C₂²×D₅	—	20	2	no	—
172	C₂²×D₅	—	20	2	no	—
173	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
174	C₂²×D₅	C₂²×D₅	1	2	yes	—
175	C₅×D₄	—	5	4	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	5	1	no	—
180	C₂×C₄×C₅	—	5	4	no	—
181	C₂×C₄×C₅	—	5	2	no	—
182	C₂×C₄×C₅	—	5	1	no	—
183	C₂×C₄×C₅	—	5	1	no	—
184	C₂×C₄×C₅	—	20	2	no	—
185	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
191	C₂²⋊D₅	—	5	8	no	—
192	C₂×Dic₅	—	5	2	no	—
193	C₂×Dic₅	—	5	2	no	—
194	C₂×Dic₅	—	10	4	no	—
195	C₂×Dic₅	—	20	2	no	—
196	C₂×Dic₅	C₂²×D₅	1	2	yes	—
197	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
198	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
199	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
200	D₂₀	—	5	8	no	—
201	D₂₀	—	5	4	no	—
202	D₂₀	—	5	4	no	—
203	D₂₀	—	10	8	no	—
204	D₂₀	C₄×D₅	1	4	yes	complemented
205	C₄×D₅	—	4	2	no	—
206	C₄×D₅	—	20	2	no	—
207	C₄×D₅	—	20	2	no	—
208	C₄×D₅	—	20	2	no	—
209	C₄×D₅	—	20	2	no	—
210	C₄×D₅	—	20	2	no	—
211	C₄×D₅	—	20	8	no	—
212	C₄×D₅	C₄×D₅	1	4	yes	complemented
213	C₂²×D₄	—	25	1	no	—
214	C₄×D₄	—	5	1	no	—
215	C₄×D₄	—	25	1	no	—
216	C₄×D₄	—	25	1	no	—
217	C₄×D₄	—	25	2	no	—
218	C₄×D₄	—	25	2	no	—
219	C₄×D₄	—	25	1	no	—
220	#117	—	5	1	no	—
221	#117	—	25	1	no	—
222	C₂×C₂²⋊C₄	—	25	2	no	—
223	C₂×C₂²⋊C₄	—	25	2	no	—
224	C₅²	#518	1	1	yes	characteristic, complemented, Sylow
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	5	4	no	—
227	C₂²×C₅	—	5	2	no	—
228	C₂²×C₅	—	10	1	no	—
229	C₂²×C₅	—	10	2	no	—
230	C₂²×C₅	—	10	2	no	—
231	C₂²×C₅	—	10	1	no	—
232	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
233	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
234	C₂×D₅	—	2	2	no	—
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	—	5	4	no	—
239	C₂×D₅	—	5	4	no	—
240	C₂×D₅	—	10	8	no	—
241	C₂×D₅	—	10	2	no	—
242	C₂×D₅	—	10	2	no	—
243	C₂×D₅	—	10	2	no	—
244	C₂×D₅	—	10	2	no	—
245	C₂×D₅	—	20	4	no	—
246	C₂×D₅	—	20	4	no	—
247	C₂×D₅	—	20	8	no	—
248	C₂×D₅	—	20	8	no	—
249	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
250	C₂×D₅	C₂×C₄×D₅	1	4	yes	—
251	C₄×C₅	—	2	4	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	4	no	—
257	C₄×C₅	—	4	2	no	—
258	C₄×C₅	—	5	2	no	—
259	C₄×C₅	—	10	1	no	—
260	C₄×C₅	—	10	1	no	—
261	C₄×C₅	—	10	2	no	—
262	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
263	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
264	Dic₅	—	2	2	no	—
265	Dic₅	—	2	1	no	—
266	Dic₅	—	2	1	no	—
267	Dic₅	—	10	2	no	—
268	Dic₅	—	10	2	no	—
269	Dic₅	—	20	4	no	—
270	Dic₅	C₂×C₄×D₅	1	2	yes	—
271	C₂⁴	—	25	2	no	—
272	C₂×D₄	—	5	1	no	—
273	C₂×D₄	—	25	8	no	—
274	C₂×D₄	—	25	2	no	—
275	C₂×D₄	—	25	1	no	—
276	C₂²×C₄	—	5	1	no	—
277	C₂²×C₄	—	5	2	no	—
278	C₂²×C₄	—	25	2	no	—
279	C₂²×C₄	—	25	2	no	—
280	C₂²×C₄	—	25	2	no	—
281	C₄⋊C₄	—	5	1	no	—
282	C₄⋊C₄	—	5	1	no	—
283	C₄⋊C₄	—	5	1	no	—
284	C₄⋊C₄	—	25	1	no	—
285	C₂²⋊C₄	—	5	1	no	—
286	C₂²⋊C₄	—	5	1	no	—
287	C₂²⋊C₄	—	5	2	no	—
288	C₂²⋊C₄	—	25	2	no	—
289	C₂²⋊C₄	—	25	2	no	—
290	C₂²⋊C₄	—	25	2	no	—
291	C₂²⋊C₄	—	25	2	no	—
292	C₄²	—	5	1	no	—
293	C₄²	—	25	2	no	—
294	C₄²	D₅²	1	1	yes	characteristic, complemented
295	C₂×C₅	—	2	2	no	—
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	10	4	no	—
299	C₂×C₅	—	10	2	no	—
300	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
301	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
302	C₂×C₅	#4144	1	1	yes	characteristic
303	C₂×C₅	#4149	1	1	yes	characteristic
304	C₂×C₅	#4149	1	1	yes	characteristic
305	C₂×C₅	#4149	1	1	yes	characteristic
306	D₅	—	2	2	no	—
307	D₅	—	2	4	no	—
308	D₅	—	10	4	no	—
309	D₅	—	10	4	no	—
310	D₅	—	20	8	no	—
311	C₂³	—	5	1	no	—
312	C₂³	—	5	2	no	—
313	C₂³	—	25	8	no	—
314	C₂³	—	25	2	no	—
315	C₂³	—	100	2	no	—
316	C₂³	—	100	2	no	—
317	D₄	—	5	4	no	—
318	D₄	—	25	8	no	—
319	D₄	—	25	4	no	—
320	C₂×C₄	—	5	1	no	—
321	C₂×C₄	—	5	4	no	—
322	C₂×C₄	—	5	1	no	—
323	C₂×C₄	—	5	1	no	—
324	C₂×C₄	—	5	2	no	—
325	C₂×C₄	—	20	2	no	—
326	C₂×C₄	—	25	2	no	—
327	C₂×C₄	—	100	2	no	—
328	C₂×C₄	—	100	2	no	—
329	C₂×C₄	—	100	2	no	—
330	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
331	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
332	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
333	C₅	—	2	2	no	—
334	C₅	#65855	1	1	yes	characteristic, complemented
335	C₅	#65879	1	1	yes	characteristic, complemented
336	C₂²	—	5	2	no	—
337	C₂²	—	5	4	no	—
338	C₂²	—	10	2	no	—
339	C₂²	—	10	1	no	—
340	C₂²	—	10	2	no	—
341	C₂²	—	10	1	no	—
342	C₂²	—	25	4	no	—
343	C₂²	—	50	2	no	—
344	C₂²	—	50	2	no	—
345	C₂²	—	100	8	no	—
346	C₂²	C₂²×D₅²	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	5	2	no	—
350	C₄	—	10	2	no	—
351	C₄	—	10	1	no	—
352	C₄	—	10	1	no	—
353	C₄	—	50	2	no	—
354	C₄	C₄×D₅²	1	2	yes	—
355	C₂	—	10	2	no	—
356	C₂	—	10	4	no	—
357	C₂	—	50	4	no	—
358	C₂	C₂×C₄×D₅²	1	1	yes	characteristic
359	C₂	#11717472	1	1	yes	characteristic
360	C₂	#11717473	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₄²×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×D₅×D₂₀	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	#518
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. C₄×C₅×D₂₀	C₂	semidirect
4. C₅×#4018	C₂	semidirect
5. #11717039	C₂	semidirect
6. D₅×#685	C₂	semidirect
7. #11717022	C₂	semidirect
8. #11717013	C₂	semidirect
9. Dic₅×D₂₀	C₂	semidirect
10. #11716999	C₂	semidirect
11. #11716959	C₂	semidirect
14. C₂×C₄×C₅⋊D₅	C₂²	semidirect
17. C₂×C₄×C₅×D₅	C₂²	semidirect
19. C₂×#6036	C₂²	semidirect
20. C₂×D₅×Dic₅	C₂²	semidirect
21. D₅×D₂₀	C₄	semidirect
22. C₄²×C₅²	C₂²	semidirect
23. #87808	C₂²	semidirect
24. C₄⋊#1039	C₂²	semidirect
27. C₅×#685	C₂²	semidirect
28. C₅×C₄⋊Dic₅	C₂²	semidirect
29. C₅×#683	C₂²	semidirect
30. C₅×#683	C₂²	semidirect
31. C₄×C₅×Dic₅	C₂²	semidirect
32. #87779	C₂²	semidirect
34. #87778	C₂²	semidirect
35. Dic₅²	C₂²	semidirect
43. C₂×D₅²	C₂×C₄	semidirect
48. C₄⋊(C₅⋊D₅)	C₂×C₄	semidirect
54. C₅×D₂₀	C₂×C₄	semidirect
56. C₄×C₅×D₅	C₂×C₄	semidirect
57. #6038	C₂×C₄	semidirect
81. C₄×D₂₀	D₅	semidirect
82. #4018	D₅	semidirect
123. C₂×C₄×D₅	C₂×D₅	semidirect
132. C₄²×C₅	C₂×D₅	semidirect
133. C₄²×C₅	C₂×D₅	semidirect
145. #685	C₂×D₅	semidirect
149. C₄⋊Dic₅	C₂×D₅	semidirect
150. #683	C₂×D₅	semidirect
151. #683	C₂×D₅	semidirect
154. C₄×Dic₅	C₂×D₅	semidirect
204. D₂₀	C₄×D₅	semidirect
212. C₄×D₅	C₄×D₅	semidirect
224. C₅²	#518	semidirect
294. C₄²	D₅²	semidirect
334. C₅	#65855	semidirect
335. C₅	#65879	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×D₂₀	2
C₄×C₄⋊(C₅⋊D₅)	2
C₄×C₅×D₂₀	2
C₅×#4018	2
#11717039	2
D₅×#685	2
#11717022	2
#11717013	2
Dic₅×D₂₀	2
#11716999	2
#11716959	2
#65879	5
#65855	5