Subgroups — #13105052

Order 1600.

544 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×C₅×D₅	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
5	#11717022	C₂	1	2	yes	maximal, complemented
6	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716987	C₂	1	2	yes	maximal, complemented
8	#11716975	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716968	C₂	1	2	yes	maximal, complemented
10	#11716962	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716954	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×#6045	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	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×#6036	C₂²	1	2	yes	—
19	C₄²×C₅²	C₂²	1	1	yes	characteristic
20	#87808	C₂²	1	2	yes	—
21	#87808	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	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	2	yes	—
26	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
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	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#66025	—	5	1	no	maximal
36	#65813	—	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	#6045	—	2	2	no	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
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	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	#6038	—	2	4	no	—
54	#6036	—	2	4	no	—
55	C₅×#125	—	5	1	no	—
56	C₂×C₄²×C₅	—	5	1	no	—
57	#4076	—	5	1	no	—
58	#4075	—	5	2	no	—
59	C₄×C₂²⋊D₅	—	5	2	no	—
60	#4070	—	5	1	no	—
61	#4044	—	5	2	no	—
62	#4043	—	5	1	no	—
63	#4041	—	5	1	no	—
64	#4040	—	5	2	no	—
65	#4039	—	5	2	no	—
66	#4039	—	5	1	no	—
67	#4038	—	5	2	no	—
68	#4033	—	5	1	no	—
69	#4022	—	5	2	no	—
70	#4021	—	5	1	no	—
71	#4021	—	5	1	no	—
72	#4021	—	10	2	no	—
73	C₄×D₂₀	—	5	1	no	—
74	#4018	—	5	2	no	—
75	C₄²×D₅	D₅	1	1	yes	characteristic, complemented
76	#4016	D₅	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀	—	5	1	no	—
78	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
79	C₂×C₅⋊D₅	—	2	2	no	—
80	C₂×C₅⋊D₅	—	2	2	no	—
81	C₂×C₅⋊D₅	—	2	2	no	—
82	C₂×C₅×D₅	—	2	1	no	—
83	C₂×C₅×D₅	—	2	1	no	—
84	C₂×C₅×D₅	#41	1	2	yes	—
85	C₄×C₅²	—	2	1	no	—
86	C₄×C₅²	—	2	1	no	—
87	C₄×C₅²	#41	1	2	yes	—
88	#1039	—	2	2	no	—
89	C₅×Dic₅	—	2	2	no	—
90	C₅×Dic₅	—	2	2	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₅×Dic₅	#41	1	2	yes	—
94	C₂²×C₄×C₅	—	5	1	no	—
95	C₂²×C₄×C₅	—	5	1	no	—
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂×C₂²⋊D₅	—	5	2	no	—
98	C₂×D₂₀	—	5	2	no	—
99	C₂×D₂₀	—	5	1	no	—
100	C₂×D₂₀	—	5	1	no	—
101	C₂×D₂₀	—	10	2	no	—
102	C₂×C₄×D₅	—	5	2	no	—
103	C₂×C₄×D₅	—	5	2	no	—
104	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
107	C₂×Dic₁₀	—	5	1	no	—
108	C₂×Dic₁₀	—	5	1	no	—
109	C₂×Dic₁₀	—	10	2	no	—
110	C₅×C₄⋊C₄	—	5	1	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	1	no	—
113	C₅×C₄⋊C₄	—	5	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	5	1	no	—
116	C₄²×C₅	—	5	1	no	—
117	C₄²×C₅	—	5	1	no	—
118	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
120	C₂²⋊Dic₅	—	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	—	5	2	no	—
129	#685	—	5	2	no	—
130	#685	—	10	4	no	—
131	#685	—	10	4	no	—
132	C₄⋊Dic₅	—	5	2	no	—
133	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
134	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
135	#683	—	5	2	no	—
136	#683	—	5	2	no	—
137	#683	—	5	2	no	—
138	#683	C₂×D₅	1	2	yes	—
139	#683	C₂×D₅	1	2	yes	—
140	C₄×Dic₅	—	5	2	no	—
141	C₄×Dic₅	—	5	2	no	—
142	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
143	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
144	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
145	#529	—	25	1	no	Sylow
146	C₂×C₅²	C₂×#41	1	1	yes	characteristic
147	C₂×C₅²	C₂×#41	1	1	yes	characteristic
148	C₂×C₅²	C₂×#41	1	1	yes	characteristic
149	C₅⋊D₅	—	4	2	no	—
150	C₅×D₅	—	2	2	no	—
151	C₂³×C₅	—	5	1	no	—
152	C₂²×D₅	—	5	2	no	—
153	C₂²×D₅	—	5	2	no	—
154	C₂²×D₅	—	10	4	no	—
155	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
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	1	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	10	2	no	—
165	C₂×C₄×C₅	—	10	2	no	—
166	C₂×C₄×C₅	—	10	2	no	—
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	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
173	C₂²⋊D₅	—	10	4	no	—
174	C₂×Dic₅	—	5	2	no	—
175	C₂×Dic₅	—	5	2	no	—
176	C₂×Dic₅	—	10	4	no	—
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	C₂×Dic₅	C₂²×D₅	1	2	yes	—
181	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
182	D₂₀	—	10	4	no	—
183	D₂₀	—	10	2	no	—
184	D₂₀	—	10	2	no	—
185	D₂₀	—	20	4	no	—
186	C₄×D₅	—	2	2	no	—
187	C₄×D₅	—	2	2	no	—
188	C₄×D₅	—	2	2	no	—
189	C₄×D₅	—	10	4	no	—
190	C₄×D₅	—	10	4	no	—
191	Dic₁₀	—	10	2	no	—
192	Dic₁₀	—	10	2	no	—
193	Dic₁₀	—	20	4	no	—
194	#126	—	25	2	no	—
195	#125	—	5	1	no	—
196	#124	—	25	1	no	—
197	#123	—	25	2	no	—
198	C₂²⋊Q₈	—	25	1	no	—
199	#121	—	25	1	no	—
200	C₄×Q₈	—	25	1	no	—
201	C₄×D₄	—	25	1	no	—
202	C₄×D₄	—	25	2	no	—
203	#117	—	25	2	no	—
204	C₂×C₄²	—	5	1	no	—
205	C₅²	#529	1	1	yes	characteristic, complemented, Sylow
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	5	2	no	—
208	C₂²×C₅	—	10	1	no	—
209	C₂²×C₅	—	10	1	no	—
210	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
211	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	—	2	1	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	—	10	2	no	—
220	C₂×D₅	—	20	4	no	—
221	C₂×D₅	—	20	4	no	—
222	C₂×D₅	—	20	4	no	—
223	C₂×D₅	#709	1	2	yes	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	5	2	no	—
232	C₄×C₅	—	10	2	no	—
233	C₄×C₅	—	10	2	no	—
234	C₄×C₅	—	10	1	no	—
235	C₄×C₅	—	10	1	no	—
236	C₄×C₅	#709	1	2	yes	—
237	C₄×C₅	#713	1	2	yes	—
238	Dic₅	—	2	1	no	—
239	Dic₅	—	2	1	no	—
240	Dic₅	—	2	2	no	—
241	Dic₅	—	2	2	no	—
242	Dic₅	—	10	2	no	—
243	Dic₅	—	10	2	no	—
244	Dic₅	—	20	4	no	—
245	Dic₅	#709	1	2	yes	—
246	C₂×Q₈	—	25	1	no	—
247	C₂×D₄	—	25	2	no	—
248	C₂×D₄	—	25	1	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	25	2	no	—
253	C₄⋊C₄	—	5	2	no	—
254	C₄⋊C₄	—	5	1	no	—
255	C₄⋊C₄	—	5	2	no	—
256	C₄⋊C₄	—	5	1	no	—
257	C₄⋊C₄	—	25	2	no	—
258	C₄⋊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	C₂²⋊C₄	—	25	2	no	—
264	C₄²	—	5	1	no	—
265	C₄²	—	5	1	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	25	2	no	—
268	C₄²	D₅²	1	1	yes	characteristic
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	—	2	2	no	—
272	C₂×C₅	—	10	2	no	—
273	C₂×C₅	C₂×#709	1	1	yes	characteristic
274	C₂×C₅	C₂×#709	1	1	yes	characteristic
275	C₂×C₅	C₂×#709	1	1	yes	characteristic
276	C₂×C₅	C₂×#713	1	1	yes	characteristic
277	C₂×C₅	D₅×#41	1	1	yes	characteristic
278	C₂×C₅	D₅×#41	1	1	yes	characteristic
279	D₅	—	2	2	no	—
280	D₅	—	20	2	no	—
281	D₅	—	20	2	no	—
282	D₅	—	40	4	no	—
283	C₂³	—	5	1	no	—
284	C₂³	—	25	2	no	—
285	Q₈	—	50	2	no	—
286	D₄	—	50	4	no	—
287	D₄	—	50	2	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	1	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	10	2	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	25	2	no	—
297	C₂×C₄	—	50	4	no	—
298	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
299	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
300	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
301	C₅	—	2	2	no	—
302	C₅	#65813	1	1	yes	characteristic, complemented
303	C₅	#66025	1	1	yes	characteristic, complemented
304	C₂²	—	5	2	no	—
305	C₂²	—	10	1	no	—
306	C₂²	—	10	1	no	—
307	C₂²	—	50	2	no	—
308	C₂²	—	50	2	no	—
309	C₂²	—	50	2	no	—
310	C₂²	C₂²×D₅²	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	5	2	no	—
314	C₄	—	10	1	no	—
315	C₄	—	10	1	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	2	no	—
318	C₄	—	50	2	no	—
319	C₄	#87874	1	2	yes	—
320	C₂	—	10	2	no	—
321	C₂	—	100	2	no	—
322	C₂	C₂×#87874	1	1	yes	characteristic
323	C₂	D₅×#709	1	1	yes	characteristic
324	C₂	D₅×#709	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	#529
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. #11717195	C₂	semidirect
2. C₄²×C₅×D₅	C₂	semidirect
3. C₅×#4016	C₂	semidirect
5. #11717022	C₂	semidirect
6. #11717013	C₂	semidirect
7. #11716987	C₂	semidirect
8. #11716975	C₂	semidirect
9. #11716968	C₂	semidirect
10. #11716962	C₂	semidirect
11. #11716954	C₂	semidirect
75. C₄²×D₅	D₅	semidirect
76. #4016	D₅	semidirect
205. C₅²	#529	semidirect
302. C₅	#65813	semidirect
303. C₅	#66025	semidirect

Maximal subgroups

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

subgroup	index
#11717195	2
C₄²×C₅×D₅	2
C₅×#4016	2
#11717027	2
#11717022	2
#11717013	2
#11716987	2
#11716975	2
#11716968	2
#11716962	2
#11716954	2
#66025	5
#65813	5