Subgroups — #13105038

Order 1600.

714 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×#87873	C₂	1	2	yes	maximal, complemented
2	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717029	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717027	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717026	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717024	C₂	1	2	yes	maximal, complemented
10	C₄×#6038	C₂	1	2	yes	maximal, complemented
11	#11716970	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
14	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
19	C₂×#6039	C₂²	1	2	yes	—
20	C₂×#6038	C₂²	1	2	yes	—
21	C₂×#6038	C₂²	1	2	yes	—
22	C₂×#6038	C₂²	1	2	yes	—
23	C₂×#6037	C₂²	1	2	yes	—
24	#87873	—	2	8	no	—
25	C₄²×C₅²	C₂²	1	1	yes	characteristic
26	#87808	C₂²	1	2	yes	—
27	C₄⋊#1039	C₂²	1	1	yes	characteristic
28	C₅×#685	C₂²	1	2	yes	—
29	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
31	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
32	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
33	#87780	C₂²	1	2	yes	—
34	#87779	C₂²	1	2	yes	—
35	#87778	C₂²	1	2	yes	—
36	#65882	—	5	1	no	maximal
37	#65811	—	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×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×#1039	C₂³	1	2	yes	—
45	C₄⋊(C₅⋊D₅)	—	2	2	no	—
46	C₄×C₅⋊D₅	—	2	4	no	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₅×D₂₀	—	2	2	no	—
52	C₄×C₅×D₅	—	2	4	no	—
53	C₄×C₅×D₅	—	2	2	no	—
54	C₄×C₅×D₅	—	2	2	no	—
55	C₄×C₅×D₅	D₄	1	4	yes	complemented
56	#6039	—	2	4	no	—
57	#6038	—	2	4	no	—
58	#6038	—	2	4	no	—
59	#6038	—	2	4	no	—
60	#6037	—	2	4	no	—
61	C₂×#709	—	5	2	no	—
62	C₂×#709	—	5	2	no	—
63	C₄×C₅×D₄	—	5	1	no	—
64	C₂×C₄²×C₅	—	5	1	no	—
65	#4086	—	5	1	no	—
66	#4084	—	5	1	no	—
67	#4076	—	5	1	no	—
68	#4076	—	5	2	no	—
69	#4076	—	5	1	no	—
70	C₄×C₂²⋊D₅	—	5	2	no	—
71	#4041	—	5	1	no	—
72	#4034	—	5	1	no	—
73	#4031	—	5	2	no	—
74	#4030	—	5	2	no	—
75	#4021	—	5	2	no	—
76	#4020	—	5	1	no	—
77	C₄×D₂₀	—	5	2	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	C₄²×D₅	D₅	1	1	yes	characteristic, complemented
83	#4015	—	5	1	no	—
84	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
85	C₂×C₅⋊D₅	—	2	2	no	—
86	C₂×C₅⋊D₅	—	2	2	no	—
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	—	2	2	no	—
92	C₂×C₅×D₅	—	2	2	no	—
93	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
94	C₄×C₅²	—	2	1	no	—
95	C₄×C₅²	C₂×D₄	1	2	yes	—
96	C₄×C₅²	#41	1	2	yes	complemented
97	#1039	—	2	2	no	—
98	C₅×Dic₅	—	2	2	no	—
99	C₅×Dic₅	—	2	1	no	—
100	C₅×Dic₅	C₂×D₄	1	2	yes	—
101	C₅×Dic₅	#41	1	2	yes	complemented
102	C₂×C₅×D₄	—	5	1	no	—
103	C₂²×C₄×C₅	—	5	1	no	—
104	C₂²×C₄×C₅	—	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	C₂×C₂²⋊D₅	—	5	2	no	—
109	C₂×C₂²⋊D₅	—	5	2	no	—
110	C₂×C₂²⋊D₅	—	5	2	no	—
111	C₂×C₂²⋊D₅	—	5	2	no	—
112	#709	—	10	8	no	—
113	#709	—	10	8	no	—
114	C₂×D₂₀	—	5	2	no	—
115	C₂×D₂₀	—	5	2	no	—
116	C₂×D₂₀	—	5	2	no	—
117	C₂×D₂₀	—	5	1	no	—
118	C₂×D₂₀	—	5	1	no	—
119	C₂×D₂₀	—	10	2	no	—
120	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
121	C₂×C₄×D₅	—	5	2	no	—
122	C₂×C₄×D₅	—	5	2	no	—
123	C₂×C₄×D₅	—	10	4	no	—
124	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
125	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
126	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
127	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
128	C₂×Dic₁₀	—	5	2	no	—
129	C₂×Dic₁₀	—	5	2	no	—
130	C₅×C₄⋊C₄	—	5	1	no	—
131	C₅×C₂²⋊C₄	—	5	2	no	—
132	C₄²×C₅	—	2	2	no	—
133	C₄²×C₅	—	5	1	no	—
134	C₄²×C₅	—	5	1	no	—
135	C₄²×C₅	—	5	1	no	—
136	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
137	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
138	C₂²⋊Dic₅	—	5	2	no	—
139	#685	—	5	2	no	—
140	#685	—	5	2	no	—
141	#685	—	5	2	no	—
142	#685	—	5	2	no	—
143	#685	—	5	2	no	—
144	#685	—	10	4	no	—
145	#685	C₂×D₅	1	2	yes	—
146	C₄⋊Dic₅	—	5	1	no	—
147	C₄⋊Dic₅	—	5	1	no	—
148	C₄⋊Dic₅	—	5	2	no	—
149	C₄⋊Dic₅	—	10	2	no	—
150	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
151	#683	—	5	2	no	—
152	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
153	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
154	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
155	#532	—	25	1	no	Sylow
156	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₅²	C₂×#41	1	1	yes	characteristic
158	C₂×C₅²	C₂×#41	1	1	yes	characteristic
159	C₅⋊D₅	—	4	2	no	—
160	C₅×D₅	—	2	2	no	—
161	C₅×D₅	—	4	2	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	—	10	4	no	—
167	C₂²×D₅	C₂²×D₅	1	2	yes	—
168	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
169	C₅×D₄	—	10	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	5	1	no	—
174	C₂×C₄×C₅	—	5	1	no	—
175	C₂×C₄×C₅	—	5	4	no	—
176	C₂×C₄×C₅	—	5	2	no	—
177	C₂×C₄×C₅	—	5	1	no	—
178	C₂×C₄×C₅	—	10	2	no	—
179	C₂×C₄×C₅	—	10	4	no	—
180	C₂×C₄×C₅	—	10	2	no	—
181	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
187	C₂²⋊D₅	—	10	4	no	—
188	C₂²⋊D₅	—	10	4	no	—
189	C₂²⋊D₅	—	10	4	no	—
190	C₂²⋊D₅	—	10	4	no	—
191	C₂²⋊D₅	—	10	4	no	—
192	C₂×Dic₅	—	5	2	no	—
193	C₂×Dic₅	—	5	2	no	—
194	C₂×Dic₅	—	10	4	no	—
195	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
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	D₂₀	—	2	2	no	—
200	D₂₀	—	10	8	no	—
201	D₂₀	—	10	4	no	—
202	D₂₀	—	10	4	no	—
203	D₂₀	—	10	4	no	—
204	D₂₀	—	10	2	no	—
205	D₂₀	—	10	2	no	—
206	C₄×D₅	—	2	2	no	—
207	C₄×D₅	—	2	4	no	—
208	C₄×D₅	—	2	2	no	—
209	C₄×D₅	—	10	4	no	—
210	C₄×D₅	—	10	4	no	—
211	C₄×D₅	—	20	8	no	—
212	C₄×D₅	D₂₀	1	4	yes	complemented
213	Dic₁₀	—	10	4	no	—
214	Dic₁₀	—	10	4	no	—
215	C₂×#41	—	25	2	no	—
216	C₄⋊Q₈	—	25	1	no	—
217	#127	—	25	1	no	—
218	#124	—	25	2	no	—
219	#121	—	25	2	no	—
220	#121	—	25	1	no	—
221	#121	—	25	1	no	—
222	C₄×D₄	—	5	1	no	—
223	C₄×D₄	—	25	2	no	—
224	C₄×D₄	—	25	1	no	—
225	C₂×C₄²	—	5	1	no	—
226	C₅²	#532	1	1	yes	characteristic, complemented, Sylow
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	5	2	no	—
229	C₂²×C₅	—	10	1	no	—
230	C₂²×C₅	—	10	1	no	—
231	C₂²×C₅	—	10	2	no	—
232	C₂²×C₅	—	10	2	no	—
233	C₂²×C₅	—	10	2	no	—
234	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
235	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	2	2	no	—
240	C₂×D₅	—	2	2	no	—
241	C₂×D₅	—	10	8	no	—
242	C₂×D₅	—	10	2	no	—
243	C₂×D₅	—	10	2	no	—
244	C₂×D₅	—	10	2	no	—
245	C₂×D₅	—	10	2	no	—
246	C₂×D₅	—	10	2	no	—
247	C₂×D₅	—	10	2	no	—
248	C₂×D₅	—	20	4	no	—
249	C₂×D₅	—	20	4	no	—
250	C₂×D₅	C₂×D₂₀	1	2	yes	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	4	no	—
254	C₄×C₅	—	2	4	no	—
255	C₄×C₅	—	4	2	no	—
256	C₄×C₅	—	5	2	no	—
257	C₄×C₅	—	5	2	no	—
258	C₄×C₅	—	10	2	no	—
259	C₄×C₅	—	10	1	no	—
260	C₄×C₅	C₂×D₂₀	1	2	yes	—
261	C₄×C₅	#709	1	2	yes	complemented
262	C₄×C₅	#709	1	2	yes	complemented
263	C₄×C₅	D₄×D₅	1	2	yes	—
264	Dic₅	—	2	1	no	—
265	Dic₅	—	2	2	no	—
266	Dic₅	—	10	2	no	—
267	Dic₅	—	10	2	no	—
268	Dic₅	—	20	4	no	—
269	Dic₅	C₂×D₂₀	1	2	yes	—
270	Dic₅	#709	1	2	yes	complemented
271	#41	—	50	8	no	—
272	C₂×Q₈	—	25	2	no	—
273	C₂×D₄	—	5	1	no	—
274	C₂×D₄	—	25	2	no	—
275	C₂×D₄	—	25	2	no	—
276	C₂×D₄	—	25	2	no	—
277	C₂×D₄	—	25	2	no	—
278	C₂×D₄	—	25	1	no	—
279	C₂²×C₄	—	5	1	no	—
280	C₂²×C₄	—	5	1	no	—
281	C₂²×C₄	—	5	1	no	—
282	C₂²×C₄	—	5	2	no	—
283	C₂²×C₄	—	25	2	no	—
284	C₄⋊C₄	—	5	1	no	—
285	C₄⋊C₄	—	25	2	no	—
286	C₄⋊C₄	—	25	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	C₄²	—	5	1	no	—
292	C₄²	—	5	1	no	—
293	C₄²	—	5	1	no	—
294	C₄²	D₅²	1	1	yes	characteristic
295	C₂×C₅	—	2	2	no	—
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	10	2	no	—
299	C₂×C₅	—	20	2	no	—
300	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
301	C₂×C₅	C₂×#709	1	1	yes	characteristic
302	C₂×C₅	C₂×#709	1	1	yes	characteristic
303	C₂×C₅	C₂×#709	1	1	yes	characteristic
304	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
305	C₂×C₅	D₅×#41	1	1	yes	characteristic
306	D₅	—	2	2	no	—
307	D₅	—	4	2	no	—
308	D₅	—	20	8	no	—
309	D₅	—	20	2	no	—
310	D₅	—	20	2	no	—
311	C₂³	—	5	1	no	—
312	C₂³	—	5	2	no	—
313	C₂³	—	25	2	no	—
314	Q₈	—	50	4	no	—
315	D₄	—	10	2	no	—
316	D₄	—	50	4	no	—
317	D₄	—	50	4	no	—
318	D₄	—	50	4	no	—
319	D₄	—	50	4	no	—
320	D₄	—	50	2	no	—
321	C₂×C₄	—	5	1	no	—
322	C₂×C₄	—	5	1	no	—
323	C₂×C₄	—	5	4	no	—
324	C₂×C₄	—	5	1	no	—
325	C₂×C₄	—	5	2	no	—
326	C₂×C₄	—	10	2	no	—
327	C₂×C₄	—	10	2	no	—
328	C₂×C₄	—	10	4	no	—
329	C₂×C₄	—	25	2	no	—
330	C₂×C₄	—	50	4	no	—
331	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
332	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
333	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
334	C₅	—	2	2	no	—
335	C₅	#65811	1	1	yes	characteristic, complemented
336	C₅	#65882	1	1	yes	characteristic, complemented
337	C₂²	—	5	2	no	—
338	C₂²	—	10	1	no	—
339	C₂²	—	10	1	no	—
340	C₂²	—	10	2	no	—
341	C₂²	—	10	2	no	—
342	C₂²	—	10	2	no	—
343	C₂²	—	50	2	no	—
344	C₂²	—	50	2	no	—
345	C₂²	—	50	2	no	—
346	C₂²	C₂²×D₅²	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	5	2	no	—
349	C₄	—	5	2	no	—
350	C₄	—	10	1	no	—
351	C₄	—	10	2	no	—
352	C₄	—	50	2	no	—
353	C₄	#87873	1	2	yes	complemented
354	C₄	D₅×D₂₀	1	2	yes	—
355	C₂	—	10	2	no	—
356	C₂	—	20	2	no	—
357	C₂	—	100	2	no	—
358	C₂	C₂×#87873	1	1	yes	characteristic
359	C₂	C₂×D₅×D₂₀	1	1	yes	characteristic
360	C₂	D₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Frattini	C₂²	C₂²×D₅²
Fitting	C₄²×C₅²	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	#532
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#87873	C₂	semidirect
2. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. C₄×C₅×D₂₀	C₂	semidirect
4. C₄²×C₅×D₅	C₂	semidirect
5. #11717036	C₂	semidirect
6. #11717029	C₂	semidirect
7. #11717027	C₂	semidirect
8. #11717026	C₂	semidirect
9. #11717024	C₂	semidirect
10. C₄×#6038	C₂	semidirect
11. #11716970	C₂	semidirect
55. C₄×C₅×D₅	D₄	semidirect
81. C₄×D₂₀	D₅	semidirect
82. C₄²×D₅	D₅	semidirect
96. C₄×C₅²	#41	semidirect
101. C₅×Dic₅	#41	semidirect
212. C₄×D₅	D₂₀	semidirect
226. C₅²	#532	semidirect
261. C₄×C₅	#709	semidirect
262. C₄×C₅	#709	semidirect
270. Dic₅	#709	semidirect
335. C₅	#65811	semidirect
336. C₅	#65882	semidirect
353. C₄	#87873	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87873	2
C₄×C₄⋊(C₅⋊D₅)	2
C₄×C₅×D₂₀	2
C₄²×C₅×D₅	2
#11717036	2
#11717029	2
#11717027	2
#11717026	2
#11717024	2
C₄×#6038	2
#11716970	2
#65882	5
#65811	5