Subgroups — #13105045

Order 1600.

538 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×#6045	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717026	C₂	1	1	yes	maximal, characteristic
5	#11717023	C₂	1	2	yes	maximal, complemented
6	C₄×#6037	C₂	1	2	yes	maximal, complemented
7	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716979	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716978	C₂	1	2	yes	maximal, complemented
10	#11716975	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716973	C₂	1	2	yes	maximal, complemented
12	C₂×#6045	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×#6037	C₂²	1	2	yes	—
19	C₄²×C₅²	C₂²	1	1	yes	characteristic
20	C₄⋊#1039	C₂²	1	1	yes	characteristic
21	#87806	C₂²	1	2	yes	—
22	C₄×#1039	C₂²	1	2	yes	—
23	C₅×#685	C₂²	1	2	yes	—
24	C₅×C₄⋊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	#87778	C₂²	1	2	yes	—
32	#87778	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	#65891	—	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	C₂×#1039	C₂³	1	2	yes	—
44	#6045	—	2	2	no	—
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₅×D₂₀	—	2	2	no	—
50	C₄×C₅×D₅	—	2	2	no	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	C₄×C₅×D₅	—	2	4	no	—
54	#6037	—	2	4	no	—
55	C₄×C₅×D₄	—	5	1	no	—
56	C₂×C₄²×C₅	—	5	1	no	—
57	#4084	—	5	1	no	—
58	#4082	—	5	1	no	—
59	#4076	—	5	1	no	—
60	#4075	—	5	2	no	—
61	C₄×C₂²⋊D₅	—	5	2	no	—
62	C₄×C₂²⋊D₅	—	5	2	no	—
63	#4072	—	5	2	no	—
64	#4070	—	5	1	no	—
65	#4043	—	5	1	no	—
66	#4036	—	5	1	no	—
67	#4029	—	5	2	no	—
68	#4025	—	5	2	no	—
69	#4022	—	5	2	no	—
70	#4021	—	5	1	no	—
71	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
72	#4018	—	5	2	no	—
73	C₄²×D₅	D₅	1	1	yes	characteristic, complemented
74	#4016	—	5	1	no	—
75	C₄×Dic₁₀	—	5	1	no	—
76	C₄×Dic₁₀	—	5	1	no	—
77	C₄×Dic₁₀	—	10	2	no	—
78	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
79	C₂×C₅×D₅	—	2	1	no	—
80	C₂×C₅×D₅	—	2	1	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	#41	1	2	yes	—
85	C₄×C₅²	—	2	1	no	—
86	C₄×C₅²	#41	1	2	yes	—
87	C₄×C₅²	#41	1	2	yes	—
88	#1039	—	2	2	no	—
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	C₂×C₅×D₄	—	5	1	no	—
95	C₂²×C₄×C₅	—	5	1	no	—
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	1	no	—
98	C₂²×C₄×C₅	—	5	2	no	—
99	C₂×C₂²⋊D₅	—	5	2	no	—
100	C₂×C₂²⋊D₅	—	5	2	no	—
101	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
103	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
106	C₂×Dic₁₀	—	5	1	no	—
107	C₂×Dic₁₀	—	5	1	no	—
108	C₂×Dic₁₀	—	10	2	no	—
109	C₅×C₄⋊C₄	—	5	1	no	—
110	C₅×C₂²⋊C₄	—	5	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	5	1	no	—
113	C₄²×C₅	—	5	1	no	—
114	C₄²×C₅	—	5	1	no	—
115	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
117	C₂²⋊Dic₅	—	5	2	no	—
118	C₂²⋊Dic₅	—	5	2	no	—
119	C₂²⋊Dic₅	—	5	2	no	—
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	C₂×D₅	1	2	yes	—
126	C₄⋊Dic₅	—	5	1	no	—
127	C₄⋊Dic₅	—	5	1	no	—
128	C₄⋊Dic₅	—	5	2	no	—
129	C₄⋊Dic₅	—	5	2	no	—
130	C₄⋊Dic₅	—	10	2	no	—
131	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
132	#683	—	5	2	no	—
133	#683	—	5	2	no	—
134	#683	—	5	2	no	—
135	#683	—	5	2	no	—
136	#683	—	5	2	no	—
137	#683	—	5	2	no	—
138	#683	—	10	4	no	—
139	C₄×Dic₅	—	5	2	no	—
140	C₄×Dic₅	—	5	2	no	—
141	C₄×Dic₅	—	10	4	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	—	2	2	no	—
150	C₅×D₅	—	4	2	no	—
151	C₂³×C₅	—	5	1	no	—
152	C₂³×C₅	—	5	2	no	—
153	C₂²×D₅	C₂²×D₅	1	2	yes	—
154	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
155	C₅×D₄	—	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	1	no	—
161	C₂×C₄×C₅	—	5	1	no	—
162	C₂×C₄×C₅	—	5	2	no	—
163	C₂×C₄×C₅	—	10	2	no	—
164	C₂×C₄×C₅	—	10	4	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	C₂²⋊D₅	—	10	4	no	—
175	C₂×Dic₅	—	5	2	no	—
176	C₂×Dic₅	—	5	2	no	—
177	C₂×Dic₅	—	5	2	no	—
178	C₂×Dic₅	—	5	2	no	—
179	C₂×Dic₅	—	10	4	no	—
180	C₂×Dic₅	—	10	4	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	1	yes	characteristic
184	C₂×Dic₅	C₂²×D₅	1	2	yes	—
185	D₂₀	—	2	2	no	—
186	C₄×D₅	—	2	2	no	—
187	C₄×D₅	—	2	2	no	—
188	C₄×D₅	—	2	2	no	—
189	C₄×D₅	—	2	4	no	—
190	Dic₁₀	—	10	8	no	—
191	Dic₁₀	—	10	2	no	—
192	Dic₁₀	—	10	2	no	—
193	#126	—	25	2	no	—
194	#125	—	25	1	no	—
195	#124	—	25	1	no	—
196	#123	—	25	2	no	—
197	C₂²⋊Q₈	—	25	1	no	—
198	#121	—	25	1	no	—
199	C₄×Q₈	—	25	1	no	—
200	C₄×D₄	—	5	1	no	—
201	C₄×D₄	—	25	2	no	—
202	#117	—	25	2	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₅²	#529	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	5	2	no	—
207	C₂²×C₅	—	10	1	no	—
208	C₂²×C₅	—	10	1	no	—
209	C₂²×C₅	—	10	2	no	—
210	C₂²×C₅	—	10	2	no	—
211	C₂²×C₅	—	10	2	no	—
212	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
213	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	#709	1	2	yes	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	4	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	10	2	no	—
226	C₄×C₅	—	10	1	no	—
227	C₄×C₅	—	10	1	no	—
228	C₄×C₅	—	10	1	no	—
229	C₄×C₅	#709	1	2	yes	—
230	C₄×C₅	#709	1	2	yes	—
231	C₄×C₅	#709	1	2	yes	—
232	C₄×C₅	#711	1	2	yes	—
233	Dic₅	—	2	1	no	—
234	Dic₅	—	2	1	no	—
235	Dic₅	—	2	1	no	—
236	Dic₅	—	2	2	no	—
237	Dic₅	—	10	8	no	—
238	Dic₅	—	10	2	no	—
239	Dic₅	—	10	2	no	—
240	Dic₅	—	10	2	no	—
241	Dic₅	—	10	2	no	—
242	Dic₅	—	20	4	no	—
243	C₂×Q₈	—	25	1	no	—
244	C₂×D₄	—	5	1	no	—
245	C₂×D₄	—	25	2	no	—
246	C₂²×C₄	—	5	1	no	—
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	5	1	no	—
249	C₂²×C₄	—	5	2	no	—
250	C₄⋊C₄	—	5	1	no	—
251	C₄⋊C₄	—	25	2	no	—
252	C₄⋊C₄	—	25	2	no	—
253	C₄⋊C₄	—	25	2	no	—
254	C₄⋊C₄	—	25	1	no	—
255	C₄⋊C₄	—	25	2	no	—
256	C₂²⋊C₄	—	5	2	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	C₄²	—	5	1	no	—
262	C₄²	—	5	1	no	—
263	C₄²	—	5	1	no	—
264	C₄²	—	25	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	—	10	2	no	—
270	C₂×C₅	—	20	2	no	—
271	C₂×C₅	C₂×#709	1	1	yes	characteristic
272	C₂×C₅	C₂×#709	1	1	yes	characteristic
273	C₂×C₅	C₂×#709	1	1	yes	characteristic
274	C₂×C₅	C₂×#709	1	1	yes	characteristic
275	C₂×C₅	C₂×#711	1	1	yes	characteristic
276	C₂×C₅	D₅×#41	1	1	yes	characteristic
277	D₅	—	2	2	no	—
278	D₅	—	4	2	no	—
279	C₂³	—	5	1	no	—
280	C₂³	—	5	2	no	—
281	Q₈	—	50	2	no	—
282	D₄	—	10	2	no	—
283	D₄	—	50	4	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	10	2	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	10	4	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	25	2	no	—
293	C₂×C₄	—	25	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	#65813	1	1	yes	characteristic, complemented
299	C₅	#65891	1	1	yes	characteristic, complemented
300	C₂²	—	5	2	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	—	10	2	no	—
304	C₂²	—	10	2	no	—
305	C₂²	—	10	2	no	—
306	C₂²	C₂²×D₅²	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	10	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	1	no	—
312	C₄	—	50	2	no	—
313	C₄	—	50	2	no	—
314	C₄	#87870	1	2	yes	—
315	C₄	#87873	1	2	yes	—
316	C₂	—	10	2	no	—
317	C₂	—	20	2	no	—
318	C₂	C₂×#87870	1	1	yes	characteristic
319	C₂	C₂×#87873	1	1	yes	characteristic
320	C₂	D₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Frattini	C₂²	C₂²×D₅²
Fitting	C₄²×C₅²	C₂²
prime radical	#11717026	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	#529
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×#6045	C₂	semidirect
2. C₄×C₅×D₂₀	C₂	semidirect
3. C₄²×C₅×D₅	C₂	semidirect
5. #11717023	C₂	semidirect
6. C₄×#6037	C₂	semidirect
7. #11716988	C₂	semidirect
8. #11716979	C₂	semidirect
9. #11716978	C₂	semidirect
10. #11716975	C₂	semidirect
11. #11716973	C₂	semidirect
71. C₄×D₂₀	D₅	semidirect
73. C₄²×D₅	D₅	semidirect
204. C₅²	#529	semidirect
298. C₅	#65813	semidirect
299. C₅	#65891	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6045	2
C₄×C₅×D₂₀	2
C₄²×C₅×D₅	2
#11717026	2
#11717023	2
C₄×#6037	2
#11716988	2
#11716979	2
#11716978	2
#11716975	2
#11716973	2
#65891	5
#65813	5