Subgroups — #13101397

Order 1600.

560 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87778	C₂	1	2	yes	maximal
5	#11716620	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×#1039	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	—	2	2	no	—
12	C₂×C₄×C₅×D₅	—	2	2	no	—
13	C₂×C₄×C₅×D₅	C₄	1	4	yes	complemented
14	C₄⋊#1039	—	2	2	no	—
15	C₅×C₄⋊Dic₅	—	2	2	no	—
16	#87778	—	2	4	no	—
17	#65501	—	5	1	no	maximal
18	#65276	—	5	1	no	maximal
19	C₂³×C₅²	C₂³	1	1	yes	characteristic
20	C₂²×C₅×D₅	—	2	1	no	—
21	C₂²×C₅×D₅	—	2	1	no	—
22	C₂²×C₅×D₅	—	2	1	no	—
23	C₂²×C₅×D₅	—	2	1	no	—
24	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
25	C₂²×C₅×D₅	D₄	1	2	yes	—
26	C₂²×C₅×D₅	Q₈	1	2	yes	—
27	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₅²	D₄	1	2	yes	—
29	C₂×C₄×C₅²	D₄	1	2	yes	—
30	C₂×#1039	—	2	2	no	—
31	C₂×#1039	—	2	2	no	—
32	C₂×#1039	—	2	2	no	—
33	C₂×C₅×Dic₅	—	2	1	no	—
34	C₂×C₅×Dic₅	—	2	1	no	—
35	C₂×C₅×Dic₅	—	2	2	no	—
36	C₂×C₅×Dic₅	—	2	2	no	—
37	C₂×C₅×Dic₅	—	2	2	no	—
38	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
39	C₄×C₅×D₅	—	2	8	no	—
40	C₂³×C₄×C₅	—	5	1	no	—
41	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
42	C₂×C₅×C₄⋊C₄	—	5	1	no	—
43	C₂×C₂²⋊Dic₅	—	5	2	no	—
44	C₂×#685	—	5	2	no	—
45	C₂×C₄⋊Dic₅	—	5	1	no	—
46	C₂×C₄⋊Dic₅	—	5	1	no	—
47	C₂×C₄⋊Dic₅	—	10	2	no	—
48	C₂×C₄⋊Dic₅	D₅	1	1	yes	characteristic, complemented
49	#3963	—	5	2	no	—
50	#3963	—	5	2	no	—
51	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₅²	#41	1	1	yes	characteristic
57	C₂²×C₅²	#41	1	1	yes	characteristic
58	C₂×C₅×D₅	—	2	2	no	—
59	C₂×C₅×D₅	—	2	2	no	—
60	C₂×C₅×D₅	—	2	2	no	—
61	C₂×C₅×D₅	—	2	2	no	—
62	C₂×C₅×D₅	—	2	2	no	—
63	C₂×C₅×D₅	—	2	2	no	—
64	C₂×C₅×D₅	C₄⋊C₄	1	4	yes	complemented
65	C₄×C₅²	C₂²⋊C₄	1	4	yes	complemented
66	#1039	—	4	2	no	—
67	C₅×Dic₅	—	2	2	no	—
68	C₅×Dic₅	—	4	2	no	—
69	C₂⁴×C₅	—	5	1	no	—
70	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	5	1	no	—
73	C₂²×C₄×C₅	—	5	4	no	—
74	C₂²×C₄×C₅	—	5	2	no	—
75	C₂²×C₄×C₅	—	10	2	no	—
76	C₂²×C₄×C₅	—	10	2	no	—
77	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
79	C₂²×Dic₅	—	5	2	no	—
80	C₂²×Dic₅	—	5	2	no	—
81	C₂²×Dic₅	—	10	4	no	—
82	C₂²×Dic₅	C₂×D₅	1	2	yes	—
83	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
84	C₂×C₄×D₅	—	2	2	no	—
85	C₂×C₄×D₅	—	2	2	no	—
86	C₂×C₄×D₅	Dic₅	1	4	yes	complemented
87	C₅×C₄⋊C₄	—	10	2	no	—
88	C₂²⋊Dic₅	—	10	4	no	—
89	#685	—	10	4	no	—
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	—	10	8	no	—
92	C₄⋊Dic₅	—	10	2	no	—
93	C₄⋊Dic₅	—	10	2	no	—
94	#380	—	25	1	no	Sylow
95	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₅²	#117	1	1	yes	characteristic
98	C₂×C₅²	#121	1	1	yes	characteristic
99	C₂×C₅²	#121	1	1	yes	characteristic
100	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
102	C₅×D₅	—	2	4	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	5	2	no	—
105	C₂³×C₅	—	5	2	no	—
106	C₂³×C₅	—	5	2	no	—
107	C₂³×C₅	—	10	1	no	—
108	C₂³×C₅	—	10	1	no	—
109	C₂³×C₅	—	10	1	no	—
110	C₂³×C₅	—	10	1	no	—
111	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
112	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	—	2	1	no	—
115	C₂²×D₅	—	2	1	no	—
116	C₂²×D₅	—	2	1	no	—
117	C₂²×D₅	Dic₁₀	1	2	yes	—
118	C₂²×D₅	D₂₀	1	2	yes	—
119	C₂²×D₅	C₂×Dic₅	1	2	yes	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	5	2	no	—
124	C₂×C₄×C₅	—	10	1	no	—
125	C₂×C₄×C₅	—	10	1	no	—
126	C₂×C₄×C₅	—	10	8	no	—
127	C₂×C₄×C₅	—	10	2	no	—
128	C₂×C₄×C₅	—	10	2	no	—
129	C₂×C₄×C₅	—	10	2	no	—
130	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
131	C₂×C₄×C₅	D₂₀	1	2	yes	—
132	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
133	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
134	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
135	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
136	C₂×Dic₅	—	2	1	no	—
137	C₂×Dic₅	—	2	1	no	—
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	2	no	—
141	C₂×Dic₅	—	10	8	no	—
142	C₂×Dic₅	—	10	2	no	—
143	C₂×Dic₅	—	10	2	no	—
144	C₂×Dic₅	—	10	2	no	—
145	C₂×Dic₅	—	10	2	no	—
146	C₂×Dic₅	—	10	2	no	—
147	C₂×Dic₅	—	10	2	no	—
148	C₂×Dic₅	—	20	4	no	—
149	C₂×Dic₅	—	20	4	no	—
150	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
151	C₄×D₅	—	2	8	no	—
152	C₂³×C₄	—	5	1	no	—
153	C₂×C₄⋊C₄	—	5	1	no	—
154	C₂×C₄⋊C₄	—	25	1	no	—
155	C₂×C₂²⋊C₄	—	25	2	no	—
156	#95	—	25	2	no	—
157	C₅²	#380	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	5	4	no	—
166	C₂²×C₅	—	10	2	no	—
167	C₂²×C₅	—	10	2	no	—
168	C₂²×C₅	—	10	2	no	—
169	C₂²×C₅	—	10	2	no	—
170	C₂²×C₅	—	10	2	no	—
171	C₂²×C₅	—	10	2	no	—
172	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
173	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
175	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
176	C₂²×C₅	#709	1	1	yes	characteristic
177	C₂²×C₅	#709	1	1	yes	characteristic
178	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
179	C₂²×C₅	#711	1	1	yes	characteristic
180	C₂²×C₅	Q₈×D₅	1	1	yes	characteristic
181	C₂²×C₅	#713	1	1	yes	characteristic
182	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
183	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×D₅	—	2	2	no	—
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	—	2	2	no	—
189	C₂×D₅	—	2	2	no	—
190	C₂×D₅	—	2	2	no	—
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	C₄⋊Dic₅	1	4	yes	complemented
193	C₄×C₅	—	2	8	no	—
194	C₄×C₅	—	10	2	no	—
195	C₄×C₅	—	20	2	no	—
196	C₄×C₅	#685	1	4	yes	complemented
197	C₄×C₅	C₂²⋊Dic₅	1	4	yes	complemented
198	Dic₅	—	2	2	no	—
199	Dic₅	—	4	2	no	—
200	Dic₅	—	20	8	no	—
201	Dic₅	—	20	2	no	—
202	Dic₅	—	20	2	no	—
203	C₂⁴	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	4	no	—
206	C₂²×C₄	—	5	2	no	—
207	C₂²×C₄	—	10	2	no	—
208	C₂²×C₄	—	10	2	no	—
209	C₂²×C₄	—	25	2	no	—
210	C₂²×C₄	D₅²	1	1	yes	characteristic
211	C₄⋊C₄	—	10	2	no	—
212	C₄⋊C₄	—	50	2	no	—
213	C₂²⋊C₄	—	50	4	no	—
214	C₂×C₅	—	2	2	no	—
215	C₂×C₅	—	2	2	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	2	2	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	—	10	4	no	—
222	C₂×C₅	D₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₅	#4038	1	1	yes	characteristic
224	C₂×C₅	#4041	1	1	yes	characteristic
225	C₂×C₅	#4043	1	1	yes	characteristic
226	C₂×C₅	#4070	1	1	yes	characteristic
227	C₂×C₅	#4070	1	1	yes	characteristic
228	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
229	C₂×C₅	#4072	1	1	yes	characteristic
230	C₂×C₅	C₂×#685	1	1	yes	characteristic
231	C₂×C₅	#4076	1	1	yes	characteristic
232	C₂×C₅	#4076	1	1	yes	characteristic
233	C₂×C₅	#4084	1	1	yes	characteristic
234	C₂×C₅	#4092	1	1	yes	characteristic
235	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
236	D₅	—	2	4	no	—
237	C₂³	—	5	2	no	—
238	C₂³	—	5	2	no	—
239	C₂³	—	5	2	no	—
240	C₂³	—	10	1	no	—
241	C₂³	—	10	1	no	—
242	C₂³	—	10	1	no	—
243	C₂³	—	10	1	no	—
244	C₂³	C₂×D₅²	1	1	yes	characteristic
245	C₂×C₄	—	5	2	no	—
246	C₂×C₄	—	10	1	no	—
247	C₂×C₄	—	10	1	no	—
248	C₂×C₄	—	10	8	no	—
249	C₂×C₄	—	10	2	no	—
250	C₂×C₄	—	10	2	no	—
251	C₂×C₄	—	10	2	no	—
252	C₂×C₄	—	50	2	no	—
253	C₂×C₄	—	50	2	no	—
254	C₂×C₄	—	50	2	no	—
255	C₂×C₄	D₅×Dic₅	1	2	yes	—
256	C₂×C₄	#6037	1	2	yes	—
257	C₂×C₄	#6038	1	2	yes	—
258	C₅	—	2	2	no	—
259	C₅	#65276	1	1	yes	characteristic, complemented
260	C₅	#65501	1	1	yes	characteristic, complemented
261	C₂²	—	5	4	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	10	2	no	—
264	C₂²	—	10	2	no	—
265	C₂²	—	10	2	no	—
266	C₂²	—	10	2	no	—
267	C₂²	—	10	2	no	—
268	C₂²	D₅×Dic₁₀	1	1	yes	characteristic
269	C₂²	#87870	1	1	yes	characteristic
270	C₂²	#87874	1	1	yes	characteristic
271	C₂²	D₅×D₂₀	1	1	yes	characteristic
272	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
273	C₂²	C₂×#6037	1	1	yes	characteristic
274	C₂²	C₂×#6038	1	1	yes	characteristic
275	C₄	—	10	2	no	—
276	C₄	—	20	2	no	—
277	C₄	—	100	2	no	—
278	C₄	#87778	1	4	yes	complemented
279	C₂	—	10	4	no	—
280	C₂	#11716959	1	1	yes	characteristic
281	C₂	#11716974	1	1	yes	characteristic
282	C₂	#11716975	1	1	yes	characteristic
283	C₂	D₅×C₄⋊Dic₅	1	1	yes	characteristic
284	C₂	#11717026	1	1	yes	characteristic
285	C₂	#11717027	1	1	yes	characteristic
286	C₂	C₂×#87778	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₅²
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂²×C₄×C₅²	C₂²
socle, cosocle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#380
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#1039	C₂	semidirect
3. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
5. #11716620	C₂	semidirect
13. C₂×C₄×C₅×D₅	C₄	semidirect
48. C₂×C₄⋊Dic₅	D₅	semidirect
64. C₂×C₅×D₅	C₄⋊C₄	semidirect
65. C₄×C₅²	C₂²⋊C₄	semidirect
86. C₂×C₄×D₅	Dic₅	semidirect
157. C₅²	#380	semidirect
192. C₂×D₅	C₄⋊Dic₅	semidirect
196. C₄×C₅	#685	semidirect
197. C₄×C₅	C₂²⋊Dic₅	semidirect
259. C₅	#65276	semidirect
260. C₅	#65501	semidirect
278. C₄	#87778	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×D₅	2
C₂×C₄⋊#1039	2
C₂×C₅×C₄⋊Dic₅	2
C₂×#87778	2
#11716620	2
#65501	5
#65276	5