Subgroups — #13105037

Order 1600.

682 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87874	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4018	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717001	C₂	1	2	yes	maximal, complemented
8	#11716999	C₂	1	2	yes	maximal, complemented
9	#11716998	C₂	1	2	yes	maximal, complemented
10	Dic₅×Dic₁₀	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×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×#6038	C₂²	1	2	yes	—
17	C₂×#6036	C₂²	1	2	yes	—
18	C₂×#6036	C₂²	1	2	yes	—
19	C₂×D₅×Dic₅	C₂²	1	2	yes	—
20	#87874	C₄	1	8	yes	complemented
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	1	yes	characteristic
25	C₅×#685	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₅×#683	C₂²	1	1	yes	characteristic
28	C₅×#683	C₂²	1	2	yes	—
29	C₅×#683	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₅	C₂²	1	2	yes	—
31	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
32	#87780	C₂²	1	2	yes	—
33	#87780	C₂²	1	2	yes	—
34	#87779	C₂²	1	2	yes	—
35	Dic₅²	C₂²	1	2	yes	—
36	#65908	—	5	1	no	maximal
37	#65857	—	5	1	no	maximal
38	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
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)	C₂×C₄	1	4	yes	complemented
45	C₄×C₅⋊D₅	—	4	2	no	—
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	C₂×C₅×Dic₅	C₂³	1	2	yes	—
51	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
52	C₅×Dic₁₀	C₂×C₄	1	4	yes	complemented
53	#6038	C₂×C₄	1	8	yes	complemented
54	#6036	—	4	2	no	—
55	#6036	C₂×C₄	1	8	yes	complemented
56	D₅×Dic₅	—	4	2	no	—
57	C₂×#713	—	5	1	no	—
58	C₂×#709	—	5	1	no	—
59	C₄×C₅×Q₈	—	5	1	no	—
60	C₅×#117	—	5	1	no	—
61	Q₈×Dic₅	—	5	1	no	—
62	C₂×C₄⋊Dic₅	—	5	1	no	—
63	#4039	—	5	2	no	—
64	#4039	—	5	1	no	—
65	#4038	—	5	2	no	—
66	D₅×C₄⋊C₄	—	5	1	no	—
67	D₅×C₄⋊C₄	—	5	2	no	—
68	D₅×C₄⋊C₄	—	5	2	no	—
69	#4027	—	5	2	no	—
70	#4027	—	5	2	no	—
71	C₄×D₂₀	—	5	1	no	—
72	C₄×D₂₀	—	5	1	no	—
73	C₄×D₂₀	—	5	2	no	—
74	C₄×D₂₀	—	5	1	no	—
75	C₄×D₂₀	—	10	2	no	—
76	#4018	—	5	2	no	—
77	#4018	D₅	1	1	yes	characteristic, complemented
78	C₄×Dic₁₀	—	5	1	no	—
79	C₄×Dic₁₀	D₅	1	1	yes	characteristic, complemented
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	C₂²×C₄	1	4	yes	—
84	C₂×C₅×D₅	—	2	1	no	—
85	C₂×C₅×D₅	—	2	1	no	—
86	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
87	C₄×C₅²	—	2	1	no	—
88	C₄×C₅²	—	2	1	no	—
89	C₄×C₅²	C₂²×C₄	1	2	yes	—
90	#1039	—	2	2	no	—
91	C₅×Dic₅	—	2	2	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₅×Dic₅	—	2	1	no	—
94	C₅×Dic₅	C₂²×C₄	1	2	yes	—
95	C₅×Dic₅	C₂²×C₄	1	4	yes	—
96	C₂×C₅×Q₈	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	1	no	—
98	C₂×C₂²⋊D₅	—	5	2	no	—
99	C₂²×Dic₅	—	5	2	no	—
100	#713	—	5	8	no	—
101	#709	—	5	8	no	—
102	C₂×D₂₀	—	5	2	no	—
103	C₂×D₂₀	—	5	1	no	—
104	C₂×D₂₀	—	5	1	no	—
105	C₂×D₂₀	—	10	2	no	—
106	C₂×C₄×D₅	—	5	2	no	—
107	C₂×C₄×D₅	—	5	2	no	—
108	C₂×C₄×D₅	—	5	2	no	—
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	—	5	2	no	—
113	C₂×C₄×D₅	—	10	4	no	—
114	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
115	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
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	1	no	—
120	C₅×C₂²⋊C₄	—	5	1	no	—
121	C₅×C₂²⋊C₄	—	5	1	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	—	5	1	no	—
124	C₄²×C₅	—	5	2	no	—
125	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
127	#685	—	5	2	no	—
128	#685	—	5	2	no	—
129	#685	—	5	2	no	—
130	#685	—	5	2	no	—
131	#685	—	10	4	no	—
132	#685	C₂×D₅	1	1	yes	characteristic
133	#685	C₂×D₅	1	1	yes	characteristic
134	C₄⋊Dic₅	—	5	1	no	—
135	C₄⋊Dic₅	—	5	1	no	—
136	C₄⋊Dic₅	—	5	2	no	—
137	C₄⋊Dic₅	—	5	2	no	—
138	C₄⋊Dic₅	—	10	2	no	—
139	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
140	#683	—	5	2	no	—
141	#683	—	5	2	no	—
142	#683	C₂×D₅	1	2	yes	—
143	#683	C₂×D₅	1	1	yes	characteristic
144	#683	C₂×D₅	1	1	yes	characteristic
145	C₄×Dic₅	—	5	2	no	—
146	C₄×Dic₅	—	5	2	no	—
147	C₄×Dic₅	C₂×D₅	1	2	yes	—
148	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
149	#520	—	25	1	no	Sylow
150	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
153	C₅⋊D₅	—	2	4	no	—
154	C₅×D₅	—	2	2	no	—
155	C₂³×C₅	—	5	1	no	—
156	C₂²×D₅	—	5	2	no	—
157	C₂²×D₅	—	5	2	no	—
158	C₂²×D₅	—	10	4	no	—
159	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
160	C₅×Q₈	—	5	4	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	5	1	no	—
166	C₂×C₄×C₅	—	5	4	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	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	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	C₂²⋊D₅	—	5	8	no	—
177	C₂×Dic₅	—	5	2	no	—
178	C₂×Dic₅	—	5	2	no	—
179	C₂×Dic₅	—	10	4	no	—
180	C₂×Dic₅	—	20	2	no	—
181	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₅	C₂²×D₅	1	2	yes	—
184	C₂×Dic₅	C₂²×D₅	1	2	yes	—
185	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
186	D₂₀	—	5	8	no	—
187	D₂₀	—	5	4	no	—
188	D₂₀	—	5	4	no	—
189	D₂₀	—	10	8	no	—
190	C₄×D₅	—	5	8	no	—
191	C₄×D₅	—	5	8	no	—
192	C₄×D₅	—	20	2	no	—
193	C₄×D₅	—	20	2	no	—
194	C₄×D₅	—	20	2	no	—
195	C₄×D₅	—	20	2	no	—
196	C₄×D₅	—	20	2	no	—
197	C₄×D₅	—	20	8	no	—
198	C₄×D₅	C₄×D₅	1	4	yes	complemented
199	Dic₁₀	C₄×D₅	1	4	yes	complemented
200	C₂×#41	—	25	1	no	—
201	C₄×Q₈	—	5	1	no	—
202	C₄×Q₈	—	25	1	no	—
203	C₄×D₄	—	25	2	no	—
204	C₄×D₄	—	25	1	no	—
205	C₄×D₄	—	25	2	no	—
206	C₄×D₄	—	25	1	no	—
207	#117	—	5	1	no	—
208	#117	—	25	2	no	—
209	C₂×C₄⋊C₄	—	25	2	no	—
210	C₂×C₄⋊C₄	—	25	1	no	—
211	C₅²	#520	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	C₂³×D₅	1	1	yes	characteristic
217	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
218	C₂×D₅	—	2	1	no	—
219	C₂×D₅	—	2	1	no	—
220	C₂×D₅	—	5	4	no	—
221	C₂×D₅	—	5	4	no	—
222	C₂×D₅	—	10	8	no	—
223	C₂×D₅	—	10	2	no	—
224	C₂×D₅	—	10	2	no	—
225	C₂×D₅	—	10	2	no	—
226	C₂×D₅	—	10	2	no	—
227	C₂×D₅	—	20	4	no	—
228	C₂×D₅	—	20	4	no	—
229	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	4	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	—	5	2	no	—
238	C₄×C₅	—	5	4	no	—
239	C₄×C₅	—	10	1	no	—
240	C₄×C₅	—	10	2	no	—
241	C₄×C₅	—	10	1	no	—
242	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
243	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
244	Dic₅	—	2	1	no	—
245	Dic₅	—	2	2	no	—
246	Dic₅	—	2	1	no	—
247	Dic₅	—	10	2	no	—
248	Dic₅	—	10	2	no	—
249	Dic₅	—	20	4	no	—
250	Dic₅	C₂×C₄×D₅	1	4	yes	—
251	Dic₅	C₂×C₄×D₅	1	2	yes	—
252	#41	—	25	8	no	—
253	C₂×Q₈	—	5	1	no	—
254	C₂×D₄	—	25	2	no	—
255	C₂×D₄	—	25	1	no	—
256	C₂²×C₄	—	5	1	no	—
257	C₂²×C₄	—	25	2	no	—
258	C₂²×C₄	—	25	2	no	—
259	C₂²×C₄	—	25	2	no	—
260	C₂²×C₄	—	25	2	no	—
261	C₄⋊C₄	—	5	1	no	—
262	C₄⋊C₄	—	5	2	no	—
263	C₄⋊C₄	—	5	1	no	—
264	C₄⋊C₄	—	5	1	no	—
265	C₄⋊C₄	—	25	2	no	—
266	C₄⋊C₄	—	25	2	no	—
267	C₄⋊C₄	—	25	1	no	—
268	C₂²⋊C₄	—	5	1	no	—
269	C₂²⋊C₄	—	5	1	no	—
270	C₂²⋊C₄	—	25	2	no	—
271	C₂²⋊C₄	—	25	2	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	5	2	no	—
274	C₄²	—	25	2	no	—
275	C₄²	D₅²	1	1	yes	characteristic
276	C₂×C₅	—	2	2	no	—
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	—	10	2	no	—
280	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
281	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
282	C₂×C₅	#4147	1	1	yes	characteristic
283	C₂×C₅	#4149	1	1	yes	characteristic
284	C₂×C₅	#4149	1	1	yes	characteristic
285	C₂×C₅	#4150	1	1	yes	characteristic
286	D₅	—	2	2	no	—
287	D₅	—	10	4	no	—
288	D₅	—	10	4	no	—
289	D₅	—	20	8	no	—
290	C₂³	—	5	1	no	—
291	C₂³	—	25	2	no	—
292	Q₈	—	5	4	no	—
293	D₄	—	25	8	no	—
294	D₄	—	25	4	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	4	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	1	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	25	8	no	—
302	C₂×C₄	—	25	2	no	—
303	C₂×C₄	—	100	2	no	—
304	C₂×C₄	—	100	2	no	—
305	C₂×C₄	—	100	2	no	—
306	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
308	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
309	C₅	—	2	2	no	—
310	C₅	#65857	1	1	yes	characteristic, complemented
311	C₅	#65908	1	1	yes	characteristic, complemented
312	C₂²	—	5	2	no	—
313	C₂²	—	10	1	no	—
314	C₂²	—	10	1	no	—
315	C₂²	—	25	4	no	—
316	C₂²	—	50	2	no	—
317	C₂²	—	50	2	no	—
318	C₂²	C₂²×D₅²	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	5	2	no	—
322	C₄	—	5	4	no	—
323	C₄	—	10	1	no	—
324	C₄	—	10	1	no	—
325	C₄	—	10	2	no	—
326	C₄	—	50	2	no	—
327	C₄	C₄×D₅²	1	2	yes	—
328	C₂	—	10	2	no	—
329	C₂	—	50	4	no	—
330	C₂	#11717462	1	1	yes	characteristic
331	C₂	C₂×C₄×D₅²	1	1	yes	characteristic
332	C₂	#11717473	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₄²×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×#87874	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	#520
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. C₅×#4018	C₂	semidirect
4. C₄×C₅×Dic₁₀	C₂	semidirect
5. #11717013	C₂	semidirect
6. D₅×C₄⋊Dic₅	C₂	semidirect
7. #11717001	C₂	semidirect
8. #11716999	C₂	semidirect
9. #11716998	C₂	semidirect
10. Dic₅×Dic₁₀	C₂	semidirect
11. #11716954	C₂	semidirect
20. #87874	C₄	semidirect
44. C₄⋊(C₅⋊D₅)	C₂×C₄	semidirect
51. C₄×C₅×D₅	C₂×C₄	semidirect
52. C₅×Dic₁₀	C₂×C₄	semidirect
53. #6038	C₂×C₄	semidirect
55. #6036	C₂×C₄	semidirect
77. #4018	D₅	semidirect
79. C₄×Dic₁₀	D₅	semidirect
198. C₄×D₅	C₄×D₅	semidirect
199. Dic₁₀	C₄×D₅	semidirect
211. C₅²	#520	semidirect
310. C₅	#65857	semidirect
311. C₅	#65908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87874	2
C₄×C₄⋊(C₅⋊D₅)	2
C₅×#4018	2
C₄×C₅×Dic₁₀	2
#11717013	2
D₅×C₄⋊Dic₅	2
#11717001	2
#11716999	2
#11716998	2
Dic₅×Dic₁₀	2
#11716954	2
#65908	5
#65857	5