Subgroups — #13105054

Order 1600.

548 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	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717029	C₂	1	1	yes	maximal, characteristic
5	#11717025	C₂	1	2	yes	maximal, complemented
6	#11717022	C₂	1	2	yes	maximal, complemented
7	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716979	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716972	C₂	1	2	yes	maximal, complemented
10	#11716968	C₂	1	2	yes	maximal, complemented
11	#11716962	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×#6045	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
16	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₂×#6038	C₂²	1	2	yes	—
18	C₂×#6038	C₂²	1	2	yes	—
19	C₂×#6036	C₂²	1	2	yes	—
20	C₄²×C₅²	C₂²	1	1	yes	characteristic
21	#87808	C₂²	1	2	yes	—
22	#87808	C₂²	1	2	yes	—
23	C₅×#685	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×#683	C₂²	1	1	yes	characteristic
27	C₅×#683	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
30	#87781	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	#87778	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#65919	—	5	1	no	maximal
36	#65893	—	5	1	no	maximal
37	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
38	C₂²×C₅×D₅	C₂³	1	2	yes	—
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	1	yes	characteristic
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	C₅×D₂₀	—	2	2	no	—
51	C₄×C₅×D₅	—	4	2	no	—
52	C₅×Dic₁₀	—	2	2	no	—
53	#6038	—	2	4	no	—
54	#6038	—	4	2	no	—
55	#6036	—	4	2	no	—
56	C₄×C₅×Q₈	—	5	1	no	—
57	C₄×C₅×D₄	—	5	1	no	—
58	#4093	—	5	2	no	—
59	#4086	—	5	1	no	—
60	#4082	—	5	1	no	—
61	#4075	—	5	2	no	—
62	C₄×C₂²⋊D₅	—	5	2	no	—
63	#4040	—	5	2	no	—
64	#4040	—	5	2	no	—
65	#4039	—	5	2	no	—
66	#4039	—	5	1	no	—
67	#4033	—	5	1	no	—
68	#4031	—	5	2	no	—
69	#4029	—	5	2	no	—
70	#4021	—	5	1	no	—
71	#4021	—	5	1	no	—
72	#4021	—	5	1	no	—
73	#4021	—	10	2	no	—
74	#4020	—	5	1	no	—
75	C₄×D₂₀	—	5	1	no	—
76	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀	—	5	1	no	—
78	C₄×Dic₁₀	D₅	1	1	yes	characteristic, complemented
79	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
80	C₂×C₅⋊D₅	—	2	2	no	—
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	2	no	—
85	C₂×C₅×D₅	—	2	2	no	—
86	C₄×C₅²	—	2	1	no	—
87	C₄×C₅²	—	2	1	no	—
88	C₄×C₅²	#41	1	2	yes	—
89	#1039	—	2	2	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	2	no	—
93	C₅×Dic₅	—	2	1	no	—
94	C₅×Dic₅	#41	1	2	yes	—
95	C₂×C₅×Q₈	—	5	1	no	—
96	C₂×C₅×D₄	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	2	no	—
98	C₂×C₂²⋊D₅	—	5	2	no	—
99	C₂×C₂²⋊D₅	—	5	2	no	—
100	C₂×D₂₀	—	5	2	no	—
101	C₂×D₂₀	—	5	1	no	—
102	C₂×D₂₀	—	5	1	no	—
103	C₂×D₂₀	—	5	2	no	—
104	C₂×D₂₀	—	10	2	no	—
105	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×D₅	—	5	2	no	—
107	C₂×C₄×D₅	—	5	2	no	—
108	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
109	C₂×Dic₁₀	—	5	1	no	—
110	C₂×Dic₁₀	—	5	1	no	—
111	C₂×Dic₁₀	—	10	2	no	—
112	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	5	1	no	—
114	C₅×C₄⋊C₄	—	5	1	no	—
115	C₅×C₄⋊C₄	—	5	1	no	—
116	C₅×C₄⋊C₄	—	5	1	no	—
117	C₅×C₂²⋊C₄	—	5	2	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	5	1	no	—
120	C₄²×C₅	—	5	1	no	—
121	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
122	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	5	2	no	—
124	C₂²⋊Dic₅	—	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	—	5	2	no	—
131	#685	—	5	2	no	—
132	#685	—	5	2	no	—
133	#685	—	10	4	no	—
134	#685	—	10	4	no	—
135	#685	C₂×D₅	1	2	yes	—
136	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
137	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
138	#683	—	5	2	no	—
139	#683	—	5	2	no	—
140	#683	C₂×D₅	1	1	yes	characteristic
141	#683	C₂×D₅	1	1	yes	characteristic
142	C₄×Dic₅	—	5	2	no	—
143	C₄×Dic₅	—	5	2	no	—
144	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
145	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
146	#559	—	25	1	no	Sylow
147	C₂×C₅²	C₂×#41	1	1	yes	characteristic
148	C₂×C₅²	C₂×#41	1	1	yes	characteristic
149	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
150	C₅⋊D₅	—	4	2	no	—
151	C₅×D₅	—	4	2	no	—
152	C₂³×C₅	—	5	2	no	—
153	C₂²×D₅	—	5	2	no	—
154	C₂²×D₅	—	5	2	no	—
155	C₂²×D₅	—	10	4	no	—
156	C₂²×D₅	C₂²×D₅	1	2	yes	—
157	C₅×Q₈	—	10	2	no	—
158	C₅×D₄	—	10	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	—	5	1	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	5	1	no	—
165	C₂×C₄×C₅	—	5	1	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	20	2	no	—
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	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
174	C₂²⋊D₅	—	10	4	no	—
175	C₂²⋊D₅	—	20	2	no	—
176	C₂×Dic₅	—	5	2	no	—
177	C₂×Dic₅	—	5	2	no	—
178	C₂×Dic₅	—	10	4	no	—
179	C₂×Dic₅	C₂²×D₅	1	2	yes	—
180	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
184	D₂₀	—	2	2	no	—
185	D₂₀	—	10	4	no	—
186	D₂₀	—	10	2	no	—
187	D₂₀	—	10	2	no	—
188	D₂₀	—	20	4	no	—
189	D₂₀	—	20	2	no	—
190	C₄×D₅	—	4	2	no	—
191	C₄×D₅	—	20	2	no	—
192	C₄×D₅	—	20	2	no	—
193	Dic₁₀	—	2	2	no	—
194	Dic₁₀	—	10	2	no	—
195	Dic₁₀	—	10	2	no	—
196	Dic₁₀	—	20	4	no	—
197	#127	—	25	1	no	—
198	#124	—	25	1	no	—
199	#124	—	25	2	no	—
200	#124	—	25	1	no	—
201	#123	—	25	2	no	—
202	#123	—	25	2	no	—
203	C₄×Q₈	—	5	1	no	—
204	C₄×Q₈	—	25	1	no	—
205	C₄×D₄	—	5	1	no	—
206	C₄×D₄	—	25	1	no	—
207	C₄×D₄	—	25	2	no	—
208	C₅²	#559	1	1	yes	characteristic, complemented, Sylow
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	10	2	no	—
211	C₂²×C₅	—	10	2	no	—
212	C₂²×C₅	—	10	2	no	—
213	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
214	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
215	C₂×D₅	—	2	2	no	—
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	10	2	no	—
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	—	20	4	no	—
225	C₂×D₅	—	20	4	no	—
226	C₂×D₅	—	20	4	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	5	2	no	—
235	C₄×C₅	—	10	1	no	—
236	C₄×C₅	—	10	1	no	—
237	C₄×C₅	—	10	1	no	—
238	C₄×C₅	—	10	2	no	—
239	C₄×C₅	#709	1	2	yes	—
240	C₄×C₅	#713	1	2	yes	—
241	Dic₅	—	2	1	no	—
242	Dic₅	—	2	2	no	—
243	Dic₅	—	2	1	no	—
244	Dic₅	—	2	1	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×Q₈	—	25	1	no	—
251	C₂×D₄	—	5	1	no	—
252	C₂×D₄	—	25	2	no	—
253	C₂×D₄	—	25	2	no	—
254	C₂×D₄	—	25	1	no	—
255	C₂²×C₄	—	5	2	no	—
256	C₂²×C₄	—	25	2	no	—
257	C₄⋊C₄	—	5	1	no	—
258	C₄⋊C₄	—	5	1	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₄⋊C₄	—	5	1	no	—
261	C₄⋊C₄	—	25	2	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	C₂²⋊C₄	—	25	2	no	—
268	C₄²	—	5	1	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	25	2	no	—
271	C₄²	D₅²	1	1	yes	characteristic
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	20	2	no	—
276	C₂×C₅	C₂×#709	1	1	yes	characteristic
277	C₂×C₅	C₂×#709	1	1	yes	characteristic
278	C₂×C₅	#4144	1	1	yes	characteristic
279	C₂×C₅	C₂×#713	1	1	yes	characteristic
280	C₂×C₅	D₅×#41	1	1	yes	characteristic
281	C₂×C₅	#4149	1	1	yes	characteristic
282	D₅	—	4	2	no	—
283	D₅	—	20	2	no	—
284	D₅	—	20	2	no	—
285	D₅	—	40	4	no	—
286	C₂³	—	5	2	no	—
287	C₂³	—	25	2	no	—
288	Q₈	—	10	2	no	—
289	Q₈	—	50	2	no	—
290	D₄	—	10	2	no	—
291	D₄	—	50	4	no	—
292	D₄	—	50	2	no	—
293	D₄	—	100	2	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	—	25	2	no	—
301	C₂×C₄	—	100	2	no	—
302	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
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	#65893	1	1	yes	characteristic, complemented
307	C₅	#65919	1	1	yes	characteristic, complemented
308	C₂²	—	10	2	no	—
309	C₂²	—	10	2	no	—
310	C₂²	—	10	2	no	—
311	C₂²	—	50	2	no	—
312	C₂²	—	50	2	no	—
313	C₂²	—	50	2	no	—
314	C₂²	C₂²×D₅²	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	5	2	no	—
318	C₄	—	10	1	no	—
319	C₄	—	10	2	no	—
320	C₄	—	10	1	no	—
321	C₄	—	10	1	no	—
322	C₄	—	50	2	no	—
323	C₄	#87874	1	2	yes	—
324	C₂	—	20	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	#11717472	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	#11717029	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	#559
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. C₄×C₅×D₂₀	C₂	semidirect
3. C₄×C₅×Dic₁₀	C₂	semidirect
5. #11717025	C₂	semidirect
6. #11717022	C₂	semidirect
7. #11717013	C₂	semidirect
8. #11716979	C₂	semidirect
9. #11716972	C₂	semidirect
10. #11716968	C₂	semidirect
11. #11716962	C₂	semidirect
76. C₄×D₂₀	D₅	semidirect
78. C₄×Dic₁₀	D₅	semidirect
208. C₅²	#559	semidirect
306. C₅	#65893	semidirect
307. C₅	#65919	semidirect

Maximal subgroups

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

subgroup	index
#11717195	2
C₄×C₅×D₂₀	2
C₄×C₅×Dic₁₀	2
#11717029	2
#11717025	2
#11717022	2
#11717013	2
#11716979	2
#11716972	2
#11716968	2
#11716962	2
#65919	5
#65893	5