Subgroups — #13105065

Order 1600.

574 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#4018	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717030	C₂	1	2	yes	maximal, complemented
5	#11717029	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717027	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717026	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717024	C₂	1	2	yes	maximal, complemented
9	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716971	C₂	1	2	yes	maximal, complemented
11	#11716968	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	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
17	C₂×#6038	C₂²	1	2	yes	—
18	C₂×#6038	C₂²	1	2	yes	—
19	C₂×#6038	C₂²	1	2	yes	—
20	C₂×#6037	C₂²	1	2	yes	—
21	C₄²×C₅²	C₂²	1	1	yes	characteristic
22	#87808	C₂²	1	2	yes	—
23	C₄⋊#1039	C₂²	1	1	yes	characteristic
24	C₅×#685	C₂²	1	2	yes	—
25	C₅×#685	C₂²	1	1	yes	characteristic
26	C₅×#685	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₅×#683	C₂²	1	1	yes	characteristic
29	C₅×#683	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
31	#87781	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87779	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	#65896	—	5	1	no	maximal
36	#65869	—	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	2	yes	—
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	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
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	—	4	2	no	—
53	#6038	—	4	2	no	—
54	#6038	—	4	2	no	—
55	#6038	—	4	2	no	—
56	#6037	—	4	2	no	—
57	C₄×C₅×D₄	—	5	1	no	—
58	C₅×#117	—	5	1	no	—
59	#4086	—	5	1	no	—
60	#4084	—	5	1	no	—
61	#4076	—	5	1	no	—
62	#4076	—	5	2	no	—
63	#4076	—	5	1	no	—
64	#4075	—	5	2	no	—
65	#4041	—	5	1	no	—
66	#4040	—	5	2	no	—
67	#4040	—	5	2	no	—
68	#4036	—	5	1	no	—
69	#4030	—	5	2	no	—
70	#4030	—	5	2	no	—
71	#4030	—	5	2	no	—
72	#4029	—	5	2	no	—
73	#4020	—	5	1	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	#4018	D₅	1	1	yes	characteristic, complemented
79	#4016	—	5	1	no	—
80	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
81	C₂×C₅⋊D₅	—	2	2	no	—
82	C₂×C₅⋊D₅	—	2	2	no	—
83	C₂×C₅⋊D₅	—	2	2	no	—
84	C₂×C₅×D₅	—	2	1	no	—
85	C₂×C₅×D₅	—	2	2	no	—
86	C₂×C₅×D₅	—	2	1	no	—
87	C₂×C₅×D₅	—	2	2	no	—
88	C₂×C₅×D₅	—	2	1	no	—
89	C₂×C₅×D₅	—	2	2	no	—
90	C₄×C₅²	—	2	1	no	—
91	C₄×C₅²	—	2	1	no	—
92	C₄×C₅²	#41	1	2	yes	—
93	#1039	—	2	2	no	—
94	C₅×Dic₅	—	2	1	no	—
95	C₅×Dic₅	—	2	1	no	—
96	C₅×Dic₅	—	2	1	no	—
97	C₅×Dic₅	—	2	2	no	—
98	C₂×C₅×D₄	—	5	1	no	—
99	C₂²×C₄×C₅	—	5	1	no	—
100	C₂²×C₄×C₅	—	5	2	no	—
101	C₂×C₂²⋊D₅	—	5	2	no	—
102	C₂×C₂²⋊D₅	—	5	2	no	—
103	C₂×C₂²⋊D₅	—	5	2	no	—
104	C₂×C₂²⋊D₅	—	5	2	no	—
105	C₂×C₂²⋊D₅	—	5	2	no	—
106	C₂×D₂₀	—	5	1	no	—
107	C₂×D₂₀	—	5	1	no	—
108	C₂×D₂₀	—	5	2	no	—
109	C₂×D₂₀	—	5	2	no	—
110	C₂×D₂₀	—	5	2	no	—
111	C₂×D₂₀	—	10	2	no	—
112	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
113	C₂×C₄×D₅	—	5	2	no	—
114	C₂×C₄×D₅	—	5	2	no	—
115	C₂×C₄×D₅	—	10	4	no	—
116	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
117	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
118	C₅×C₄⋊C₄	—	5	1	no	—
119	C₅×C₄⋊C₄	—	5	1	no	—
120	C₅×C₄⋊C₄	—	5	1	no	—
121	C₅×C₂²⋊C₄	—	5	1	no	—
122	C₅×C₂²⋊C₄	—	5	1	no	—
123	C₅×C₂²⋊C₄	—	5	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	5	1	no	—
126	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
128	C₂²⋊Dic₅	—	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	—	5	2	no	—
134	#685	—	10	4	no	—
135	#685	C₂×D₅	1	1	yes	characteristic
136	#685	C₂×D₅	1	1	yes	characteristic
137	#685	C₂×D₅	1	2	yes	—
138	C₄⋊Dic₅	—	5	1	no	—
139	C₄⋊Dic₅	—	5	1	no	—
140	C₄⋊Dic₅	—	5	2	no	—
141	C₄⋊Dic₅	—	10	2	no	—
142	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
143	#683	—	5	2	no	—
144	#683	—	5	2	no	—
145	#683	—	5	2	no	—
146	#683	C₂×D₅	1	1	yes	characteristic
147	#683	C₂×D₅	1	1	yes	characteristic
148	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
149	#540	—	25	1	no	Sylow
150	C₂×C₅²	C₂×#41	1	1	yes	characteristic
151	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
153	C₅⋊D₅	—	4	2	no	—
154	C₅×D₅	—	4	1	no	—
155	C₅×D₅	—	4	2	no	—
156	C₂³×C₅	—	5	2	no	—
157	C₂³×C₅	—	5	1	no	—
158	C₂²×D₅	—	5	2	no	—
159	C₂²×D₅	—	5	2	no	—
160	C₂²×D₅	—	10	4	no	—
161	C₂²×D₅	C₂²×D₅	1	2	yes	—
162	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
163	C₅×D₄	—	10	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	5	1	no	—
168	C₂×C₄×C₅	—	5	1	no	—
169	C₂×C₄×C₅	—	5	2	no	—
170	C₂×C₄×C₅	—	5	1	no	—
171	C₂×C₄×C₅	—	10	2	no	—
172	C₂×C₄×C₅	—	20	2	no	—
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	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
179	C₂²⋊D₅	—	20	2	no	—
180	C₂²⋊D₅	—	20	2	no	—
181	C₂²⋊D₅	—	20	2	no	—
182	C₂²⋊D₅	—	20	2	no	—
183	C₂²⋊D₅	—	20	2	no	—
184	C₂×Dic₅	—	5	2	no	—
185	C₂×Dic₅	—	5	2	no	—
186	C₂×Dic₅	—	10	4	no	—
187	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
188	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
189	C₂×Dic₅	C₂²×D₅	1	2	yes	—
190	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
191	D₂₀	—	2	2	no	—
192	D₂₀	—	10	8	no	—
193	D₂₀	—	10	2	no	—
194	D₂₀	—	10	2	no	—
195	D₂₀	—	20	2	no	—
196	D₂₀	—	20	2	no	—
197	D₂₀	—	20	2	no	—
198	C₄×D₅	—	2	2	no	—
199	C₄×D₅	—	4	2	no	—
200	C₄×D₅	—	20	2	no	—
201	C₄×D₅	—	20	2	no	—
202	C₄×D₅	—	20	8	no	—
203	#127	—	25	1	no	—
204	#125	—	25	1	no	—
205	#123	—	25	2	no	—
206	#123	—	25	2	no	—
207	#121	—	25	2	no	—
208	#121	—	25	2	no	—
209	#121	—	25	1	no	—
210	#121	—	25	1	no	—
211	C₄×D₄	—	5	1	no	—
212	C₄×D₄	—	25	1	no	—
213	#117	—	5	1	no	—
214	C₅²	#540	1	1	yes	characteristic, complemented, Sylow
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	10	1	no	—
217	C₂²×C₅	—	10	2	no	—
218	C₂²×C₅	—	10	1	no	—
219	C₂²×C₅	—	10	2	no	—
220	C₂²×C₅	—	10	1	no	—
221	C₂²×C₅	—	10	2	no	—
222	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
223	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	—	2	2	no	—
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	2	1	no	—
228	C₂×D₅	—	2	2	no	—
229	C₂×D₅	—	2	1	no	—
230	C₂×D₅	—	10	8	no	—
231	C₂×D₅	—	10	2	no	—
232	C₂×D₅	—	10	2	no	—
233	C₂×D₅	—	10	2	no	—
234	C₂×D₅	—	10	2	no	—
235	C₂×D₅	—	10	2	no	—
236	C₂×D₅	—	10	2	no	—
237	C₂×D₅	—	20	4	no	—
238	C₂×D₅	—	20	4	no	—
239	C₄×C₅	—	2	4	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	4	no	—
245	C₄×C₅	—	4	2	no	—
246	C₄×C₅	—	10	1	no	—
247	C₄×C₅	—	10	1	no	—
248	C₄×C₅	—	10	1	no	—
249	C₄×C₅	—	10	2	no	—
250	C₄×C₅	#709	1	2	yes	—
251	C₄×C₅	#709	1	2	yes	—
252	Dic₅	—	2	1	no	—
253	Dic₅	—	2	2	no	—
254	Dic₅	—	2	1	no	—
255	Dic₅	—	2	1	no	—
256	Dic₅	—	10	2	no	—
257	Dic₅	—	10	2	no	—
258	Dic₅	—	20	4	no	—
259	C₂×D₄	—	5	1	no	—
260	C₂×D₄	—	25	2	no	—
261	C₂×D₄	—	25	2	no	—
262	C₂×D₄	—	25	2	no	—
263	C₂×D₄	—	25	2	no	—
264	C₂×D₄	—	25	1	no	—
265	C₂²×C₄	—	5	1	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	—	25	2	no	—
268	C₄⋊C₄	—	5	1	no	—
269	C₄⋊C₄	—	5	1	no	—
270	C₄⋊C₄	—	5	1	no	—
271	C₄⋊C₄	—	25	2	no	—
272	C₄⋊C₄	—	25	2	no	—
273	C₄⋊C₄	—	25	1	no	—
274	C₂²⋊C₄	—	5	1	no	—
275	C₂²⋊C₄	—	5	1	no	—
276	C₂²⋊C₄	—	5	2	no	—
277	C₂²⋊C₄	—	25	2	no	—
278	C₂²⋊C₄	—	25	2	no	—
279	C₂²⋊C₄	—	25	2	no	—
280	C₄²	—	5	1	no	—
281	C₄²	D₅²	1	1	yes	characteristic
282	C₂×C₅	—	2	2	no	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	2	2	no	—
285	C₂×C₅	—	20	2	no	—
286	C₂×C₅	—	20	1	no	—
287	C₂×C₅	C₂×#709	1	1	yes	characteristic
288	C₂×C₅	C₂×#709	1	1	yes	characteristic
289	C₂×C₅	#4144	1	1	yes	characteristic
290	C₂×C₅	#4149	1	1	yes	characteristic
291	C₂×C₅	#4149	1	1	yes	characteristic
292	C₂×C₅	#4149	1	1	yes	characteristic
293	D₅	—	4	1	no	—
294	D₅	—	4	2	no	—
295	D₅	—	20	8	no	—
296	D₅	—	20	2	no	—
297	D₅	—	20	2	no	—
298	C₂³	—	5	1	no	—
299	C₂³	—	5	2	no	—
300	C₂³	—	25	2	no	—
301	D₄	—	10	2	no	—
302	D₄	—	50	2	no	—
303	D₄	—	100	2	no	—
304	D₄	—	100	2	no	—
305	D₄	—	100	2	no	—
306	D₄	—	100	2	no	—
307	C₂×C₄	—	5	1	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	5	2	no	—
310	C₂×C₄	—	5	1	no	—
311	C₂×C₄	—	10	2	no	—
312	C₂×C₄	—	20	2	no	—
313	C₂×C₄	—	25	2	no	—
314	C₂×C₄	—	100	2	no	—
315	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
316	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
317	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
318	C₅	—	2	2	no	—
319	C₅	#65869	1	1	yes	characteristic, complemented
320	C₅	#65896	1	1	yes	characteristic, complemented
321	C₂²	—	10	1	no	—
322	C₂²	—	10	2	no	—
323	C₂²	—	10	2	no	—
324	C₂²	—	10	1	no	—
325	C₂²	—	10	2	no	—
326	C₂²	—	10	1	no	—
327	C₂²	—	50	2	no	—
328	C₂²	—	50	2	no	—
329	C₂²	—	50	2	no	—
330	C₂²	C₂²×D₅²	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	10	1	no	—
334	C₄	—	10	2	no	—
335	C₄	—	10	1	no	—
336	C₄	—	10	1	no	—
337	C₄	—	50	2	no	—
338	C₄	#87873	1	2	yes	—
339	C₂	—	20	1	no	—
340	C₂	—	20	2	no	—
341	C₂	—	100	2	no	—
342	C₂	C₂×#87873	1	1	yes	characteristic
343	C₂	#11717472	1	1	yes	characteristic
344	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	#540
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×#4018	C₂	semidirect
4. #11717030	C₂	semidirect
5. #11717029	C₂	semidirect
6. #11717027	C₂	semidirect
7. #11717026	C₂	semidirect
8. #11717024	C₂	semidirect
9. #11716988	C₂	semidirect
10. #11716971	C₂	semidirect
11. #11716968	C₂	semidirect
77. C₄×D₂₀	D₅	semidirect
78. #4018	D₅	semidirect
214. C₅²	#540	semidirect
319. C₅	#65869	semidirect
320. C₅	#65896	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×#4018	2
#11717030	2
#11717029	2
#11717027	2
#11717026	2
#11717024	2
#11716988	2
#11716971	2
#11716968	2
#65896	5
#65869	5