Subgroups — #13105053

Order 1600.

570 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4016	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717027	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717027	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717005	C₂	1	2	yes	maximal, complemented
8	#11716999	C₂	1	2	yes	maximal, complemented
9	#11716971	C₂	1	2	yes	maximal, complemented
10	#11716966	C₂	1	2	yes	maximal, complemented
11	#11716959	C₂	1	1	yes	maximal, characteristic, 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	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×#6038	C₂²	1	2	yes	—
18	C₂×#6038	C₂²	1	2	yes	—
19	C₂×D₅×Dic₅	C₂²	1	2	yes	—
20	C₄²×C₅²	C₂²	1	1	yes	characteristic
21	#87808	C₂²	1	2	yes	—
22	C₄⋊#1039	C₂²	1	1	yes	characteristic
23	C₅×#685	C₂²	1	1	yes	characteristic
24	C₅×#685	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₅×#683	C₂²	1	2	yes	—
29	C₅×#683	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	#87779	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#66029	—	5	1	no	maximal
36	#65899	—	5	1	no	maximal
37	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
38	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	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×#1039	C₂³	1	2	yes	—
44	C₄⋊(C₅⋊D₅)	—	2	2	no	—
45	C₄×C₅⋊D₅	—	4	2	no	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
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	C₅×D₂₀	—	2	2	no	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	#6038	—	2	4	no	—
54	#6038	—	4	2	no	—
55	D₅×Dic₅	—	4	2	no	—
56	C₅×#125	—	5	1	no	—
57	C₄×C₅×D₄	—	5	1	no	—
58	#4085	—	5	2	no	—
59	#4076	—	5	1	no	—
60	#4076	—	5	1	no	—
61	#4072	—	5	1	no	—
62	C₂×C₄⋊Dic₅	—	5	1	no	—
63	#4044	—	5	2	no	—
64	#4044	—	5	2	no	—
65	#4041	—	5	2	no	—
66	#4041	—	5	1	no	—
67	#4041	—	5	1	no	—
68	#4040	—	5	2	no	—
69	#4039	—	5	2	no	—
70	#4038	—	5	1	no	—
71	D₅×C₄⋊C₄	—	5	1	no	—
72	#4029	—	5	2	no	—
73	#4027	—	5	2	no	—
74	C₄×D₂₀	—	5	1	no	—
75	C₄×D₂₀	—	5	1	no	—
76	C₄×D₂₀	—	10	2	no	—
77	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
78	#4016	D₅	1	1	yes	characteristic, complemented
79	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
80	C₂×C₅⋊D₅	—	2	2	no	—
81	C₂×C₅⋊D₅	—	2	2	no	—
82	C₂×C₅⋊D₅	—	2	2	no	—
83	C₂×C₅×D₅	—	2	1	no	—
84	C₂×C₅×D₅	—	2	1	no	—
85	C₂×C₅×D₅	—	2	1	no	—
86	C₂×C₅×D₅	—	2	1	no	—
87	C₂×C₅×D₅	—	2	1	no	—
88	C₂×C₅×D₅	#41	1	2	yes	—
89	C₄×C₅²	—	2	1	no	—
90	C₄×C₅²	—	2	1	no	—
91	C₄×C₅²	#41	1	2	yes	—
92	#1039	—	2	2	no	—
93	C₅×Dic₅	—	2	2	no	—
94	C₅×Dic₅	—	2	1	no	—
95	C₅×Dic₅	—	2	1	no	—
96	C₅×Dic₅	—	2	2	no	—
97	C₂×C₅×D₄	—	5	1	no	—
98	C₂²×C₄×C₅	—	5	1	no	—
99	C₂²×C₄×C₅	—	5	1	no	—
100	C₂×C₂²⋊D₅	—	5	2	no	—
101	C₂×C₂²⋊D₅	—	5	2	no	—
102	C₂²×Dic₅	—	5	2	no	—
103	C₂×D₂₀	—	5	2	no	—
104	C₂×D₂₀	—	5	1	no	—
105	C₂×D₂₀	—	5	1	no	—
106	C₂×D₂₀	—	5	2	no	—
107	C₂×D₂₀	—	10	2	no	—
108	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
109	C₂×C₄×D₅	—	5	2	no	—
110	C₂×C₄×D₅	—	5	2	no	—
111	C₂×C₄×D₅	—	5	2	no	—
112	C₂×C₄×D₅	—	10	4	no	—
113	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
114	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	5	1	no	—
116	C₅×C₄⋊C₄	—	5	1	no	—
117	C₅×C₄⋊C₄	—	5	1	no	—
118	C₅×C₄⋊C₄	—	5	2	no	—
119	C₅×C₄⋊C₄	—	5	2	no	—
120	C₅×C₂²⋊C₄	—	5	1	no	—
121	C₅×C₂²⋊C₄	—	5	1	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
124	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
125	C₂²⋊Dic₅	—	5	2	no	—
126	#685	—	5	2	no	—
127	#685	—	5	2	no	—
128	#685	—	5	2	no	—
129	#685	—	5	2	no	—
130	#685	—	5	2	no	—
131	#685	—	5	2	no	—
132	#685	—	5	2	no	—
133	#685	—	10	4	no	—
134	#685	C₂×D₅	1	1	yes	characteristic
135	#685	C₂×D₅	1	1	yes	characteristic
136	C₄⋊Dic₅	—	5	1	no	—
137	C₄⋊Dic₅	—	5	1	no	—
138	C₄⋊Dic₅	—	5	2	no	—
139	C₄⋊Dic₅	—	10	2	no	—
140	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
141	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
142	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
143	#683	—	5	2	no	—
144	#683	C₂×D₅	1	2	yes	—
145	#683	C₂×D₅	1	2	yes	—
146	C₄×Dic₅	—	5	2	no	—
147	C₄×Dic₅	—	5	2	no	—
148	#553	—	25	1	no	Sylow
149	C₂×C₅²	C₂×#41	1	1	yes	characteristic
150	C₂×C₅²	C₂×#41	1	1	yes	characteristic
151	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
152	C₅⋊D₅	—	4	2	no	—
153	C₅×D₅	—	2	2	no	—
154	C₅×D₅	—	4	1	no	—
155	C₂³×C₅	—	5	1	no	—
156	C₂³×C₅	—	5	1	no	—
157	C₂²×D₅	—	5	2	no	—
158	C₂²×D₅	—	5	2	no	—
159	C₂²×D₅	—	10	4	no	—
160	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
161	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
162	C₅×D₄	—	10	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	5	2	no	—
168	C₂×C₄×C₅	—	5	2	no	—
169	C₂×C₄×C₅	—	5	1	no	—
170	C₂×C₄×C₅	—	10	2	no	—
171	C₂×C₄×C₅	—	10	2	no	—
172	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
178	C₂²⋊D₅	—	10	4	no	—
179	C₂²⋊D₅	—	20	2	no	—
180	C₂×Dic₅	—	5	2	no	—
181	C₂×Dic₅	—	5	2	no	—
182	C₂×Dic₅	—	10	4	no	—
183	C₂×Dic₅	—	20	2	no	—
184	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×Dic₅	C₂²×D₅	1	2	yes	—
186	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₅	C₂²×D₅	1	2	yes	—
188	D₂₀	—	2	2	no	—
189	D₂₀	—	10	8	no	—
190	D₂₀	—	10	4	no	—
191	D₂₀	—	10	2	no	—
192	D₂₀	—	10	2	no	—
193	D₂₀	—	20	2	no	—
194	C₄×D₅	—	2	2	no	—
195	C₄×D₅	—	2	2	no	—
196	C₄×D₅	—	20	2	no	—
197	C₄×D₅	—	20	2	no	—
198	C₄×D₅	—	20	2	no	—
199	C₄×D₅	—	20	8	no	—
200	#126	—	25	2	no	—
201	#125	—	5	1	no	—
202	#123	—	25	2	no	—
203	#121	—	25	1	no	—
204	#121	—	25	1	no	—
205	#121	—	25	2	no	—
206	C₄×D₄	—	5	1	no	—
207	C₄×D₄	—	25	2	no	—
208	C₄×D₄	—	25	1	no	—
209	#117	—	25	1	no	—
210	C₂×C₄⋊C₄	—	25	1	no	—
211	C₅²	#553	1	1	yes	characteristic, complemented, Sylow
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	5	2	no	—
214	C₂²×C₅	—	10	1	no	—
215	C₂²×C₅	—	10	1	no	—
216	C₂²×C₅	—	10	1	no	—
217	C₂²×C₅	—	10	1	no	—
218	C₂²×C₅	—	10	1	no	—
219	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
220	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	—	2	1	no	—
223	C₂×D₅	—	2	1	no	—
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	—	2	1	no	—
226	C₂×D₅	—	10	8	no	—
227	C₂×D₅	—	10	2	no	—
228	C₂×D₅	—	10	2	no	—
229	C₂×D₅	—	10	2	no	—
230	C₂×D₅	—	10	2	no	—
231	C₂×D₅	—	10	2	no	—
232	C₂×D₅	—	10	2	no	—
233	C₂×D₅	—	20	4	no	—
234	C₂×D₅	—	20	4	no	—
235	C₂×D₅	#709	1	2	yes	—
236	C₄×C₅	—	2	4	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	4	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	10	2	no	—
244	C₄×C₅	—	10	1	no	—
245	C₄×C₅	—	10	1	no	—
246	C₄×C₅	—	10	2	no	—
247	C₄×C₅	#711	1	2	yes	—
248	C₄×C₅	#713	1	2	yes	—
249	Dic₅	—	2	1	no	—
250	Dic₅	—	2	1	no	—
251	Dic₅	—	2	2	no	—
252	Dic₅	—	2	2	no	—
253	Dic₅	—	10	2	no	—
254	Dic₅	—	10	2	no	—
255	Dic₅	—	20	4	no	—
256	C₂×D₄	—	5	1	no	—
257	C₂×D₄	—	25	2	no	—
258	C₂×D₄	—	25	2	no	—
259	C₂×D₄	—	25	1	no	—
260	C₂²×C₄	—	5	1	no	—
261	C₂²×C₄	—	5	1	no	—
262	C₂²×C₄	—	25	2	no	—
263	C₂²×C₄	—	25	2	no	—
264	C₄⋊C₄	—	5	1	no	—
265	C₄⋊C₄	—	5	1	no	—
266	C₄⋊C₄	—	5	1	no	—
267	C₄⋊C₄	—	5	2	no	—
268	C₄⋊C₄	—	5	2	no	—
269	C₄⋊C₄	—	25	2	no	—
270	C₄⋊C₄	—	25	1	no	—
271	C₂²⋊C₄	—	5	1	no	—
272	C₂²⋊C₄	—	5	1	no	—
273	C₂²⋊C₄	—	25	2	no	—
274	C₂²⋊C₄	—	25	2	no	—
275	C₂²⋊C₄	—	25	2	no	—
276	C₂²⋊C₄	—	25	2	no	—
277	C₄²	—	25	2	no	—
278	C₄²	D₅²	1	1	yes	characteristic
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	10	2	no	—
283	C₂×C₅	—	20	1	no	—
284	C₂×C₅	C₂×#709	1	1	yes	characteristic
285	C₂×C₅	C₂×#711	1	1	yes	characteristic
286	C₂×C₅	C₂×#713	1	1	yes	characteristic
287	C₂×C₅	D₅×#41	1	1	yes	characteristic
288	C₂×C₅	#4149	1	1	yes	characteristic
289	C₂×C₅	#4149	1	1	yes	characteristic
290	D₅	—	2	2	no	—
291	D₅	—	4	1	no	—
292	D₅	—	20	8	no	—
293	D₅	—	20	2	no	—
294	D₅	—	20	2	no	—
295	C₂³	—	5	1	no	—
296	C₂³	—	5	1	no	—
297	C₂³	—	25	2	no	—
298	D₄	—	10	2	no	—
299	D₄	—	50	4	no	—
300	D₄	—	50	2	no	—
301	D₄	—	100	2	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	5	2	no	—
306	C₂×C₄	—	10	2	no	—
307	C₂×C₄	—	10	2	no	—
308	C₂×C₄	—	25	2	no	—
309	C₂×C₄	—	100	2	no	—
310	C₂×C₄	—	100	2	no	—
311	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
312	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
314	C₅	—	2	2	no	—
315	C₅	#65899	1	1	yes	characteristic, complemented
316	C₅	#66029	1	1	yes	characteristic, complemented
317	C₂²	—	5	2	no	—
318	C₂²	—	10	1	no	—
319	C₂²	—	10	1	no	—
320	C₂²	—	10	1	no	—
321	C₂²	—	10	1	no	—
322	C₂²	—	10	1	no	—
323	C₂²	—	50	2	no	—
324	C₂²	—	50	2	no	—
325	C₂²	—	50	2	no	—
326	C₂²	C₂²×D₅²	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	10	1	no	—
330	C₄	—	10	1	no	—
331	C₄	—	10	2	no	—
332	C₄	—	10	2	no	—
333	C₄	—	50	2	no	—
334	C₄	#87871	1	2	yes	—
335	C₂	—	10	2	no	—
336	C₂	—	20	1	no	—
337	C₂	—	100	2	no	—
338	C₂	C₂×#87871	1	1	yes	characteristic
339	C₂	D₅×#709	1	1	yes	characteristic
340	C₂	#11717473	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	#553
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
2. C₄×C₅×D₂₀	C₂	semidirect
3. C₅×#4016	C₂	semidirect
4. #11717027	C₂	semidirect
5. #11717027	C₂	semidirect
6. D₅×C₄⋊Dic₅	C₂	semidirect
7. #11717005	C₂	semidirect
8. #11716999	C₂	semidirect
9. #11716971	C₂	semidirect
10. #11716966	C₂	semidirect
11. #11716959	C₂	semidirect
77. C₄×D₂₀	D₅	semidirect
78. #4016	D₅	semidirect
211. C₅²	#553	semidirect
315. C₅	#65899	semidirect
316. C₅	#66029	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₅⋊D₅)	2
C₄×C₅×D₂₀	2
C₅×#4016	2
#11717027	2
D₅×C₄⋊Dic₅	2
#11717005	2
#11716999	2
#11716971	2
#11716966	2
#11716959	2
#66029	5
#65899	5