Subgroups — #13106016

Order 1600.

562 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717250	C₂	1	1	yes	maximal, characteristic
2	C₅×#4070	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717084	C₂	1	2	yes	maximal, complemented
5	#11717070	C₂	1	2	yes	maximal, complemented
6	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716987	C₂	1	2	yes	maximal, complemented
9	#11716982	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716954	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×#6036	C₂²	1	2	yes	—
19	#87808	C₂²	1	2	yes	—
20	C₄⋊#1039	C₂²	1	1	yes	characteristic
21	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#683	C₂²	1	2	yes	—
24	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	#87781	C₂²	1	2	yes	—
29	#87780	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	Dic₅²	C₂²	1	2	yes	—
35	#65958	—	5	1	no	maximal
36	#65813	—	5	1	no	maximal
37	C₂³×C₅²	C₂³	1	1	yes	characteristic
38	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	—	2	2	no	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
43	C₂×#1039	C₂³	1	2	yes	—
44	C₄⋊(C₅⋊D₅)	—	2	2	no	—
45	C₂×C₅×Dic₅	—	2	2	no	—
46	C₂×C₅×Dic₅	—	2	2	no	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
52	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
53	C₅×Dic₁₀	—	2	2	no	—
54	#6036	—	2	4	no	—
55	C₅×C₂²⋊Q₈	—	5	1	no	—
56	C₂×C₄²×C₅	—	5	1	no	—
57	#4093	—	5	1	no	—
58	Q₈×Dic₅	—	5	1	no	—
59	#4076	—	5	1	no	—
60	#4076	—	5	1	no	—
61	#4076	—	10	2	no	—
62	#4075	—	5	2	no	—
63	C₄×C₂²⋊D₅	—	5	2	no	—
64	#4070	D₅	1	1	yes	characteristic, complemented
65	C₂×C₄×Dic₅	D₅	1	1	yes	characteristic, complemented
66	#4044	—	5	2	no	—
67	#4039	—	5	1	no	—
68	#4038	—	5	2	no	—
69	#4036	—	5	1	no	—
70	#4032	—	5	2	no	—
71	#4027	—	5	2	no	—
72	#4022	—	5	2	no	—
73	#4021	—	5	1	no	—
74	C₄×D₂₀	—	5	1	no	—
75	#4018	—	5	2	no	—
76	#4016	—	5	1	no	—
77	C₄×Dic₁₀	—	5	1	no	—
78	C₂²×C₅²	—	2	1	no	—
79	C₂²×C₅²	—	2	1	no	—
80	C₂²×C₅²	#41	1	2	yes	—
81	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
82	C₂×C₅⋊D₅	—	2	2	no	—
83	C₂×C₅⋊D₅	—	2	2	no	—
84	C₂×C₅⋊D₅	—	2	2	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	2	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₅×Dic₅	#41	1	2	yes	—
94	C₂×C₅×Q₈	—	5	1	no	—
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	1	no	—
98	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	—	5	2	no	—
101	C₂×C₂²⋊D₅	—	5	2	no	—
102	C₂×C₂²⋊D₅	—	10	4	no	—
103	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
104	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
105	C₂×D₂₀	—	5	1	no	—
106	C₂×D₂₀	—	5	1	no	—
107	C₂×D₂₀	—	10	2	no	—
108	C₂×C₄×D₅	—	5	2	no	—
109	C₂×C₄×D₅	—	5	2	no	—
110	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	5	1	no	—
112	C₅×C₄⋊C₄	—	5	2	no	—
113	C₅×C₂²⋊C₄	—	5	2	no	—
114	C₄²×C₅	—	5	1	no	—
115	C₄²×C₅	—	5	1	no	—
116	C₄²×C₅	—	5	1	no	—
117	C₄²×C₅	—	5	1	no	—
118	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
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	—	10	4	no	—
128	C₄⋊Dic₅	—	5	1	no	—
129	C₄⋊Dic₅	—	5	1	no	—
130	C₄⋊Dic₅	—	5	2	no	—
131	C₄⋊Dic₅	—	5	2	no	—
132	C₄⋊Dic₅	—	10	2	no	—
133	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
134	#683	—	5	2	no	—
135	#683	—	5	2	no	—
136	#683	—	5	2	no	—
137	#683	—	5	2	no	—
138	#683	C₂×D₅	1	2	yes	—
139	C₄×Dic₅	—	5	2	no	—
140	C₄×Dic₅	—	5	2	no	—
141	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
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	—	2	2	no	—
147	C₂×C₅²	C₂×#41	1	1	yes	characteristic
148	C₂×C₅²	C₂×#41	1	1	yes	characteristic
149	C₂×C₅²	C₂×#41	1	1	yes	characteristic
150	C₅⋊D₅	—	4	2	no	—
151	C₂³×C₅	—	2	2	no	—
152	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
154	C₂²×D₅	—	5	2	no	—
155	C₂²×D₅	—	5	2	no	—
156	C₂²×D₅	—	10	4	no	—
157	C₅×Q₈	—	10	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	—	5	1	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	5	1	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	5	1	no	—
168	C₂×C₄×C₅	—	5	2	no	—
169	C₂×C₄×C₅	—	10	2	no	—
170	C₂×C₄×C₅	—	10	2	no	—
171	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
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	C₂²⋊D₅	—	10	4	no	—
176	C₂²⋊D₅	—	10	4	no	—
177	C₂²⋊D₅	—	20	8	no	—
178	C₂×Dic₅	—	2	2	no	—
179	C₂×Dic₅	—	2	2	no	—
180	C₂×Dic₅	—	5	2	no	—
181	C₂×Dic₅	—	5	2	no	—
182	C₂×Dic₅	—	10	4	no	—
183	C₂×Dic₅	C₂²×D₅	1	2	yes	—
184	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
186	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₅	C₂²×D₅	1	2	yes	—
188	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
189	D₂₀	—	10	2	no	—
190	D₂₀	—	10	2	no	—
191	D₂₀	—	20	4	no	—
192	C₄×D₅	—	10	4	no	—
193	C₄×D₅	—	10	4	no	—
194	Dic₁₀	—	2	2	no	—
195	#126	—	25	2	no	—
196	#125	—	25	1	no	—
197	#124	—	25	1	no	—
198	#123	—	25	2	no	—
199	C₂²⋊Q₈	—	5	1	no	—
200	#121	—	25	1	no	—
201	C₄×Q₈	—	25	1	no	—
202	C₄×D₄	—	25	1	no	—
203	C₄×D₄	—	25	2	no	—
204	#117	—	25	2	no	—
205	C₂×C₄²	—	5	1	no	—
206	C₅²	#529	1	1	yes	characteristic, complemented, Sylow
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	4	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	#709	1	2	yes	—
216	C₂²×C₅	#713	1	2	yes	—
217	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
218	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
219	C₂×D₅	—	10	2	no	—
220	C₂×D₅	—	10	2	no	—
221	C₂×D₅	—	10	2	no	—
222	C₂×D₅	—	10	2	no	—
223	C₂×D₅	—	10	2	no	—
224	C₂×D₅	—	10	2	no	—
225	C₂×D₅	—	20	4	no	—
226	C₂×D₅	—	20	4	no	—
227	C₂×D₅	—	20	4	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	—	5	2	no	—
233	C₄×C₅	—	10	2	no	—
234	C₄×C₅	—	10	2	no	—
235	C₄×C₅	—	10	1	no	—
236	C₄×C₅	—	10	1	no	—
237	C₄×C₅	—	10	1	no	—
238	C₄×C₅	#709	1	2	yes	—
239	C₄×C₅	#711	1	2	yes	—
240	Dic₅	—	2	1	no	—
241	Dic₅	—	2	1	no	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	2	2	no	—
244	Dic₅	—	2	2	no	—
245	Dic₅	—	10	2	no	—
246	Dic₅	—	10	2	no	—
247	Dic₅	—	20	4	no	—
248	Dic₅	#709	1	2	yes	—
249	C₂×Q₈	—	5	1	no	—
250	C₂×D₄	—	25	2	no	—
251	C₂×D₄	—	25	1	no	—
252	C₂²×C₄	—	5	1	no	—
253	C₂²×C₄	—	5	1	no	—
254	C₂²×C₄	—	25	2	no	—
255	C₂²×C₄	D₅²	1	1	yes	characteristic
256	C₄⋊C₄	—	5	2	no	—
257	C₄⋊C₄	—	5	1	no	—
258	C₄⋊C₄	—	25	2	no	—
259	C₄⋊C₄	—	25	2	no	—
260	C₄⋊C₄	—	25	2	no	—
261	C₄⋊C₄	—	25	1	no	—
262	C₂²⋊C₄	—	5	2	no	—
263	C₂²⋊C₄	—	25	2	no	—
264	C₂²⋊C₄	—	25	2	no	—
265	C₂²⋊C₄	—	25	2	no	—
266	C₂²⋊C₄	—	25	2	no	—
267	C₄²	—	5	1	no	—
268	C₄²	—	5	1	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	5	1	no	—
271	C₄²	—	25	2	no	—
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	2	2	no	—
277	C₂×C₅	—	4	4	no	—
278	C₂×C₅	C₂×#709	1	1	yes	characteristic
279	C₂×C₅	C₂×#709	1	1	yes	characteristic
280	C₂×C₅	C₂×#709	1	1	yes	characteristic
281	C₂×C₅	C₂×#711	1	1	yes	characteristic
282	C₂×C₅	C₂×#713	1	1	yes	characteristic
283	C₂×C₅	D₅×#41	1	1	yes	characteristic
284	D₅	—	20	2	no	—
285	D₅	—	20	2	no	—
286	D₅	—	40	4	no	—
287	C₂³	—	25	2	no	—
288	C₂³	C₂×D₅²	1	1	yes	characteristic
289	Q₈	—	10	2	no	—
290	D₄	—	50	4	no	—
291	D₄	—	50	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	25	2	no	—
302	C₂×C₄	—	50	4	no	—
303	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
304	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
305	C₅	—	2	2	no	—
306	C₅	#65813	1	1	yes	characteristic, complemented
307	C₅	#65958	1	1	yes	characteristic, complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	50	2	no	—
311	C₂²	—	50	2	no	—
312	C₂²	—	50	2	no	—
313	C₂²	#87874	1	2	yes	—
314	C₂²	C₂²×D₅²	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	5	2	no	—
317	C₄	—	10	1	no	—
318	C₄	—	10	1	no	—
319	C₄	—	10	1	no	—
320	C₄	—	10	2	no	—
321	C₄	—	10	2	no	—
322	C₄	—	50	2	no	—
323	C₄	#87879	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	100	2	no	—
326	C₂	C₂×#87874	1	1	yes	characteristic
327	C₂	D₅×#709	1	1	yes	characteristic
328	C₂	C₂×#87879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂²×C₄×C₅²	C₂²
prime radical	#11717250	C₂
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#529
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₅×#4070	C₂	semidirect
3. C₂×C₄×C₅×Dic₅	C₂	semidirect
4. #11717084	C₂	semidirect
5. #11717070	C₂	semidirect
6. #11717013	C₂	semidirect
7. #11716988	C₂	semidirect
8. #11716987	C₂	semidirect
9. #11716982	C₂	semidirect
10. Dic₅×Dic₁₀	C₂	semidirect
11. #11716954	C₂	semidirect
64. #4070	D₅	semidirect
65. C₂×C₄×Dic₅	D₅	semidirect
206. C₅²	#529	semidirect
306. C₅	#65813	semidirect
307. C₅	#65958	semidirect

Maximal subgroups

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

subgroup	index
#11717250	2
C₅×#4070	2
C₂×C₄×C₅×Dic₅	2
#11717084	2
#11717070	2
#11717013	2
#11716988	2
#11716987	2
#11716982	2
Dic₅×Dic₁₀	2
#11716954	2
#65958	5
#65813	5