Subgroups — #13105070

Order 1600.

526 conjugacy classes of subgroups.

The subgroup lattice has 322 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	C₅×#4018	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717027	C₂	1	1	yes	maximal, characteristic
6	#11716987	C₂	1	2	yes	maximal, complemented
7	#11716984	C₂	1	2	yes	maximal, complemented
8	#11716982	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716974	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716971	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×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₂×#6039	C₂²	1	2	yes	—
17	C₂×#6038	C₂²	1	2	yes	—
18	C₄²×C₅²	C₂²	1	1	yes	characteristic
19	#87808	C₂²	1	2	yes	—
20	C₄⋊#1039	C₂²	1	1	yes	characteristic
21	C₅×#685	C₂²	1	1	yes	characteristic
22	C₅×#685	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₅×#683	C₂²	1	2	yes	—
25	C₅×#683	C₂²	1	1	yes	characteristic
26	C₅×#683	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	2	yes	—
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	#87781	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	#87779	C₂²	1	2	yes	—
33	#87779	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	#65916	—	5	1	no	maximal
36	#65868	—	5	1	no	maximal
37	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
38	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅²	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×#1039	C₂³	1	2	yes	—
43	C₄⋊(C₅⋊D₅)	—	2	2	no	—
44	C₄×C₅⋊D₅	—	4	2	no	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
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	C₄×C₅×D₅	—	2	2	no	—
51	C₅×Dic₁₀	—	2	2	no	—
52	#6039	—	4	2	no	—
53	#6038	—	4	2	no	—
54	C₄×C₅×Q₈	—	5	1	no	—
55	C₅×#117	—	5	1	no	—
56	#4093	—	5	1	no	—
57	#4092	—	5	1	no	—
58	#4076	—	5	1	no	—
59	#4070	—	5	1	no	—
60	#4044	—	5	2	no	—
61	#4042	—	5	2	no	—
62	#4042	—	5	2	no	—
63	#4041	—	5	1	no	—
64	#4040	—	5	2	no	—
65	#4034	—	5	1	no	—
66	#4031	—	5	2	no	—
67	#4029	—	5	2	no	—
68	#4022	—	5	2	no	—
69	#4021	—	5	1	no	—
70	#4021	—	5	2	no	—
71	C₄×D₂₀	—	5	1	no	—
72	C₄×D₂₀	—	5	1	no	—
73	C₄×D₂₀	—	10	2	no	—
74	#4018	D₅	1	1	yes	characteristic, complemented
75	#4015	—	5	1	no	—
76	C₄×Dic₁₀	D₅	1	1	yes	characteristic, complemented
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₂×C₅⋊D₅	—	2	2	no	—
79	C₂×C₅⋊D₅	—	2	2	no	—
80	C₂×C₅⋊D₅	—	2	2	no	—
81	C₂×C₅×D₅	—	2	1	no	—
82	C₂×C₅×D₅	—	2	1	no	—
83	C₂×C₅×D₅	—	2	1	no	—
84	C₄×C₅²	—	2	1	no	—
85	C₄×C₅²	—	2	1	no	—
86	C₄×C₅²	#41	1	2	yes	—
87	#1039	—	2	2	no	—
88	C₅×Dic₅	—	2	2	no	—
89	C₅×Dic₅	—	2	1	no	—
90	C₅×Dic₅	—	2	2	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₂×C₅×Q₈	—	5	1	no	—
94	C₂²×C₄×C₅	—	5	1	no	—
95	C₂×C₂²⋊D₅	—	5	2	no	—
96	C₂×D₂₀	—	5	1	no	—
97	C₂×D₂₀	—	5	1	no	—
98	C₂×D₂₀	—	5	2	no	—
99	C₂×D₂₀	—	10	2	no	—
100	C₂×C₄×D₅	—	5	2	no	—
101	C₂×C₄×D₅	—	5	2	no	—
102	C₂×C₄×D₅	—	10	4	no	—
103	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
104	C₂×Dic₁₀	—	5	2	no	—
105	C₂×Dic₁₀	—	5	2	no	—
106	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
107	C₅×C₄⋊C₄	—	5	1	no	—
108	C₅×C₄⋊C₄	—	5	1	no	—
109	C₅×C₄⋊C₄	—	5	1	no	—
110	C₅×C₄⋊C₄	—	5	2	no	—
111	C₅×C₂²⋊C₄	—	5	1	no	—
112	C₅×C₂²⋊C₄	—	5	1	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	5	1	no	—
115	C₄²×C₅	—	5	2	no	—
116	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
118	C₂²⋊Dic₅	—	5	2	no	—
119	#685	—	5	2	no	—
120	#685	—	5	2	no	—
121	#685	—	5	2	no	—
122	#685	—	5	2	no	—
123	#685	—	5	2	no	—
124	#685	—	5	2	no	—
125	#685	—	5	2	no	—
126	#685	—	5	2	no	—
127	#685	—	5	2	no	—
128	#685	—	10	4	no	—
129	#685	C₂×D₅	1	1	yes	characteristic
130	#685	C₂×D₅	1	1	yes	characteristic
131	C₄⋊Dic₅	—	5	1	no	—
132	C₄⋊Dic₅	—	5	1	no	—
133	C₄⋊Dic₅	—	5	2	no	—
134	C₄⋊Dic₅	—	10	2	no	—
135	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
136	#683	—	5	2	no	—
137	#683	—	5	2	no	—
138	#683	—	5	2	no	—
139	#683	C₂×D₅	1	2	yes	—
140	#683	C₂×D₅	1	1	yes	characteristic
141	#683	C₂×D₅	1	1	yes	characteristic
142	C₄×Dic₅	C₂×D₅	1	2	yes	—
143	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
144	#542	—	25	1	no	Sylow
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
148	C₅⋊D₅	—	4	2	no	—
149	C₅×D₅	—	4	1	no	—
150	C₂³×C₅	—	5	1	no	—
151	C₂²×D₅	—	5	2	no	—
152	C₂²×D₅	—	5	2	no	—
153	C₂²×D₅	—	10	4	no	—
154	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
155	C₅×Q₈	—	10	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	5	1	no	—
160	C₂×C₄×C₅	—	5	2	no	—
161	C₂×C₄×C₅	—	5	2	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	1	no	—
164	C₂×C₄×C₅	—	10	2	no	—
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂²⋊D₅	—	20	2	no	—
172	C₂×Dic₅	—	5	2	no	—
173	C₂×Dic₅	—	5	2	no	—
174	C₂×Dic₅	—	10	4	no	—
175	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
176	C₂×Dic₅	C₂²×D₅	1	2	yes	—
177	C₂×Dic₅	C₂²×D₅	1	2	yes	—
178	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
179	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
180	D₂₀	—	10	8	no	—
181	D₂₀	—	10	2	no	—
182	D₂₀	—	10	2	no	—
183	D₂₀	—	20	2	no	—
184	C₄×D₅	—	2	2	no	—
185	C₄×D₅	—	20	2	no	—
186	C₄×D₅	—	20	2	no	—
187	C₄×D₅	—	20	8	no	—
188	Dic₁₀	—	2	2	no	—
189	Dic₁₀	—	20	2	no	—
190	Dic₁₀	—	20	2	no	—
191	C₄⋊Q₈	—	25	1	no	—
192	#126	—	25	2	no	—
193	#124	—	25	2	no	—
194	#124	—	25	1	no	—
195	#123	—	25	2	no	—
196	C₂²⋊Q₈	—	25	2	no	—
197	C₂²⋊Q₈	—	25	1	no	—
198	#121	—	25	1	no	—
199	C₄×Q₈	—	5	1	no	—
200	C₄×D₄	—	25	1	no	—
201	#117	—	5	1	no	—
202	C₅²	#542	1	1	yes	characteristic, complemented, Sylow
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	10	1	no	—
205	C₂²×C₅	—	10	1	no	—
206	C₂²×C₅	—	10	1	no	—
207	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
208	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
209	C₂×D₅	—	2	1	no	—
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	10	8	no	—
213	C₂×D₅	—	10	2	no	—
214	C₂×D₅	—	10	2	no	—
215	C₂×D₅	—	10	2	no	—
216	C₂×D₅	—	10	2	no	—
217	C₂×D₅	—	10	2	no	—
218	C₂×D₅	—	10	2	no	—
219	C₂×D₅	—	20	4	no	—
220	C₂×D₅	—	20	4	no	—
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	4	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	10	1	no	—
229	C₄×C₅	—	10	2	no	—
230	C₄×C₅	—	10	1	no	—
231	C₄×C₅	—	10	1	no	—
232	C₄×C₅	—	10	2	no	—
233	C₄×C₅	#709	1	2	yes	—
234	C₄×C₅	#709	1	2	yes	—
235	Dic₅	—	2	2	no	—
236	Dic₅	—	2	1	no	—
237	Dic₅	—	2	1	no	—
238	Dic₅	—	2	2	no	—
239	Dic₅	—	2	1	no	—
240	Dic₅	—	10	2	no	—
241	Dic₅	—	10	2	no	—
242	Dic₅	—	20	4	no	—
243	C₂×Q₈	—	5	1	no	—
244	C₂×Q₈	—	25	2	no	—
245	C₂×D₄	—	25	2	no	—
246	C₂×D₄	—	25	1	no	—
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	25	2	no	—
249	C₄⋊C₄	—	5	1	no	—
250	C₄⋊C₄	—	5	1	no	—
251	C₄⋊C₄	—	5	1	no	—
252	C₄⋊C₄	—	5	2	no	—
253	C₄⋊C₄	—	25	2	no	—
254	C₄⋊C₄	—	25	2	no	—
255	C₄⋊C₄	—	25	1	no	—
256	C₂²⋊C₄	—	5	1	no	—
257	C₂²⋊C₄	—	5	1	no	—
258	C₂²⋊C₄	—	25	2	no	—
259	C₂²⋊C₄	—	25	2	no	—
260	C₂²⋊C₄	—	25	2	no	—
261	C₂²⋊C₄	—	25	2	no	—
262	C₂²⋊C₄	—	25	2	no	—
263	C₄²	—	5	1	no	—
264	C₄²	—	5	2	no	—
265	C₄²	D₅²	1	1	yes	characteristic
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	—	2	2	no	—
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	20	1	no	—
270	C₂×C₅	C₂×#709	1	1	yes	characteristic
271	C₂×C₅	C₂×#709	1	1	yes	characteristic
272	C₂×C₅	#4147	1	1	yes	characteristic
273	C₂×C₅	#4149	1	1	yes	characteristic
274	C₂×C₅	#4149	1	1	yes	characteristic
275	C₂×C₅	#4150	1	1	yes	characteristic
276	D₅	—	4	1	no	—
277	D₅	—	20	8	no	—
278	D₅	—	20	2	no	—
279	D₅	—	20	2	no	—
280	C₂³	—	5	1	no	—
281	C₂³	—	25	2	no	—
282	Q₈	—	10	2	no	—
283	Q₈	—	100	2	no	—
284	D₄	—	50	2	no	—
285	D₄	—	100	2	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	1	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	25	2	no	—
293	C₂×C₄	—	100	2	no	—
294	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
295	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
296	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
297	C₅	—	2	2	no	—
298	C₅	#65868	1	1	yes	characteristic, complemented
299	C₅	#65916	1	1	yes	characteristic, complemented
300	C₂²	—	10	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	—	50	2	no	—
304	C₂²	—	50	2	no	—
305	C₂²	—	50	2	no	—
306	C₂²	C₂²×D₅²	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	10	1	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	2	no	—
312	C₄	—	10	2	no	—
313	C₄	—	10	1	no	—
314	C₄	—	50	2	no	—
315	C₄	#87873	1	2	yes	—
316	C₂	—	20	1	no	—
317	C₂	—	100	2	no	—
318	C₂	#11717462	1	1	yes	characteristic
319	C₂	C₂×#87873	1	1	yes	characteristic
320	C₂	#11717473	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	#11717027	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	#542
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. C₅×#4018	C₂	semidirect
3. C₄×C₅×Dic₁₀	C₂	semidirect
4. #11717036	C₂	semidirect
6. #11716987	C₂	semidirect
7. #11716984	C₂	semidirect
8. #11716982	C₂	semidirect
9. #11716974	C₂	semidirect
10. #11716971	C₂	semidirect
11. #11716970	C₂	semidirect
74. #4018	D₅	semidirect
76. C₄×Dic₁₀	D₅	semidirect
202. C₅²	#542	semidirect
298. C₅	#65868	semidirect
299. C₅	#65916	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
C₅×#4018	2
C₄×C₅×Dic₁₀	2
#11717036	2
#11717027	2
#11716987	2
#11716984	2
#11716982	2
#11716974	2
#11716971	2
#11716970	2
#65916	5
#65868	5