Subgroups — #13101352

Order 1600.

472 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
3	C₂×#87778	C₂	1	1	yes	maximal, characteristic
4	C₂×#87778	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3963	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×#1039	C₂²	1	1	yes	characteristic
11	C₂²×#1039	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₄⋊#1039	—	2	2	no	—
16	C₅×#685	C₄	1	4	yes	complemented
17	#87778	—	2	2	no	—
18	#87778	—	2	2	no	—
19	Dic₅²	—	2	2	no	—
20	#65240	—	5	1	no	maximal
21	#64959	—	5	1	no	maximal
22	C₂³×C₅²	C₂³	1	1	yes	characteristic
23	C₂²×C₅×D₅	—	2	1	no	—
24	C₂²×C₅×D₅	—	2	1	no	—
25	C₂²×C₅×D₅	—	2	1	no	—
26	C₂²×C₅×D₅	—	2	1	no	—
27	C₂²×C₅×D₅	—	2	1	no	—
28	C₂²×C₅×D₅	—	2	1	no	—
29	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₅²	—	2	1	no	—
31	C₂×C₄×C₅²	—	2	1	no	—
32	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
33	C₂×#1039	—	2	1	no	—
34	C₂×#1039	—	2	1	no	—
35	C₂×#1039	—	2	1	no	—
36	C₂×#1039	—	2	1	no	—
37	C₂×#1039	—	2	1	no	—
38	C₂×#1039	D₄	1	2	yes	—
39	C₂×C₅×Dic₅	—	2	1	no	—
40	C₂×C₅×Dic₅	—	2	1	no	—
41	C₂×C₅×Dic₅	—	2	1	no	—
42	C₂×C₅×Dic₅	—	2	1	no	—
43	C₂×C₅×Dic₅	—	2	1	no	—
44	C₂×C₅×Dic₅	—	2	1	no	—
45	C₂×C₅×Dic₅	—	2	1	no	—
46	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
47	C₂×C₅×Dic₅	D₄	1	2	yes	—
48	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
49	C₂×C₂²⋊Dic₅	—	5	1	no	—
50	C₂×C₂²⋊Dic₅	—	5	1	no	—
51	C₂×#685	—	5	1	no	—
52	C₂×#685	—	5	1	no	—
53	C₂×#685	D₅	1	1	yes	characteristic
54	C₂×C₄⋊Dic₅	—	5	1	no	—
55	C₂×C₄⋊Dic₅	—	5	1	no	—
56	C₂×C₄⋊Dic₅	—	10	2	no	—
57	C₂×C₄×Dic₅	—	5	1	no	—
58	C₂×C₄×Dic₅	—	5	1	no	—
59	C₅×#95	—	5	1	no	—
60	#3963	—	5	1	no	—
61	#3963	—	5	1	no	—
62	#3963	D₅	1	1	yes	characteristic, complemented
63	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₅²	#41	1	1	yes	characteristic
67	C₂²×C₅²	#41	1	1	yes	characteristic
68	C₂²×C₅²	#41	1	1	yes	characteristic
69	C₂²×C₅²	#41	1	1	yes	characteristic
70	C₂×C₅×D₅	—	2	2	no	—
71	C₂×C₅×D₅	—	2	2	no	—
72	C₂×C₅×D₅	—	2	2	no	—
73	C₂×C₅×D₅	—	4	1	no	—
74	C₂×C₅×D₅	—	4	1	no	—
75	C₂×C₅×D₅	—	4	1	no	—
76	C₂×C₅×D₅	—	4	1	no	—
77	C₄×C₅²	—	2	2	no	—
78	#1039	—	2	2	no	—
79	#1039	—	4	1	no	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	—	2	2	no	—
82	C₅×Dic₅	—	4	1	no	—
83	C₂⁴×C₅	—	5	1	no	—
84	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	5	1	no	—
87	C₂²×C₄×C₅	—	5	1	no	—
88	C₂²×C₄×C₅	—	5	1	no	—
89	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
91	C₂²×Dic₅	—	5	1	no	—
92	C₂²×Dic₅	—	5	1	no	—
93	C₂²×Dic₅	—	5	1	no	—
94	C₂²×Dic₅	—	5	1	no	—
95	C₂²×Dic₅	—	10	2	no	—
96	C₂²×Dic₅	—	10	2	no	—
97	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
98	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
99	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
100	C₅×C₂²⋊C₄	—	5	4	no	—
101	C₂²⋊Dic₅	—	10	2	no	—
102	C₂²⋊Dic₅	—	10	2	no	—
103	#685	—	10	2	no	—
104	#685	—	10	2	no	—
105	#685	Dic₅	1	4	yes	complemented
106	C₄⋊Dic₅	—	10	2	no	—
107	C₄⋊Dic₅	—	10	2	no	—
108	C₄⋊Dic₅	—	10	8	no	—
109	C₄×Dic₅	—	10	2	no	—
110	C₄×Dic₅	—	10	2	no	—
111	#388	—	25	1	no	Sylow
112	C₂×C₅²	#117	1	1	yes	characteristic
113	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
114	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
115	C₂×C₅²	#121	1	1	yes	characteristic
116	C₂×C₅²	#123	1	1	yes	characteristic
117	C₂×C₅²	#124	1	1	yes	characteristic
118	C₂×C₅²	#126	1	1	yes	characteristic
119	C₅×D₅	—	4	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	5	2	no	—
122	C₂³×C₅	—	10	1	no	—
123	C₂³×C₅	—	10	1	no	—
124	C₂³×C₅	—	10	1	no	—
125	C₂³×C₅	—	10	1	no	—
126	C₂³×C₅	—	10	1	no	—
127	C₂³×C₅	—	10	1	no	—
128	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂²×D₅	—	2	1	no	—
131	C₂²×D₅	—	2	1	no	—
132	C₂²×D₅	—	2	1	no	—
133	C₂²×D₅	—	2	1	no	—
134	C₂²×D₅	—	2	1	no	—
135	C₂²×D₅	—	2	1	no	—
136	C₂²×D₅	C₂×Dic₅	1	2	yes	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	5	2	no	—
145	C₂×C₄×C₅	—	5	2	no	—
146	C₂×C₄×C₅	—	10	1	no	—
147	C₂×C₄×C₅	—	10	1	no	—
148	C₂×C₄×C₅	—	10	1	no	—
149	C₂×C₄×C₅	—	10	1	no	—
150	C₂×C₄×C₅	—	10	1	no	—
151	C₂×C₄×C₅	—	10	1	no	—
152	C₂×C₄×C₅	—	10	1	no	—
153	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
154	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
155	C₂×Dic₅	—	2	1	no	—
156	C₂×Dic₅	—	2	1	no	—
157	C₂×Dic₅	—	2	1	no	—
158	C₂×Dic₅	—	2	1	no	—
159	C₂×Dic₅	—	2	1	no	—
160	C₂×Dic₅	—	2	1	no	—
161	C₂×Dic₅	—	2	1	no	—
162	C₂×Dic₅	—	5	2	no	—
163	C₂×Dic₅	—	5	2	no	—
164	C₂×Dic₅	—	10	4	no	—
165	C₂×Dic₅	—	10	1	no	—
166	C₂×Dic₅	—	10	1	no	—
167	C₂×Dic₅	—	10	1	no	—
168	C₂×Dic₅	—	10	1	no	—
169	C₂×Dic₅	—	10	4	no	—
170	C₂×Dic₅	—	10	1	no	—
171	C₂×Dic₅	—	10	1	no	—
172	C₂×Dic₅	—	10	1	no	—
173	C₂×Dic₅	—	10	1	no	—
174	C₂×Dic₅	—	10	1	no	—
175	C₂×Dic₅	—	10	1	no	—
176	C₂×Dic₅	—	20	2	no	—
177	C₂×Dic₅	—	20	2	no	—
178	C₂×Dic₅	—	20	2	no	—
179	C₂×Dic₅	—	20	2	no	—
180	C₂×Dic₅	D₂₀	1	2	yes	—
181	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
182	C₂×C₄⋊C₄	—	25	1	no	—
183	C₂×C₂²⋊C₄	—	5	1	no	—
184	C₂×C₂²⋊C₄	—	25	1	no	—
185	C₂×C₂²⋊C₄	—	25	1	no	—
186	C₂×C₄²	—	25	1	no	—
187	#95	—	5	1	no	—
188	#95	—	25	1	no	—
189	C₅²	#388	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	10	2	no	—
198	C₂²×C₅	—	10	2	no	—
199	C₂²×C₅	—	10	2	no	—
200	C₂²×C₅	—	20	1	no	—
201	C₂²×C₅	—	20	1	no	—
202	C₂²×C₅	—	20	1	no	—
203	C₂²×C₅	—	20	1	no	—
204	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
205	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
206	C₂²×C₅	#709	1	1	yes	characteristic
207	C₂²×C₅	#709	1	1	yes	characteristic
208	C₂²×C₅	#709	1	1	yes	characteristic
209	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
210	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
211	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
212	C₂²×C₅	#711	1	1	yes	characteristic
213	C₂²×C₅	#711	1	1	yes	characteristic
214	C₂²×C₅	#711	1	1	yes	characteristic
215	C₂²×C₅	#711	1	1	yes	characteristic
216	C₂²×C₅	#713	1	1	yes	characteristic
217	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	4	1	no	—
222	C₂×D₅	—	4	1	no	—
223	C₂×D₅	—	4	1	no	—
224	C₂×D₅	—	4	1	no	—
225	C₄×C₅	—	2	8	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	10	2	no	—
229	C₄×C₅	—	10	2	no	—
230	C₄×C₅	—	20	1	no	—
231	Dic₅	—	2	2	no	—
232	Dic₅	—	2	2	no	—
233	Dic₅	—	4	1	no	—
234	Dic₅	—	10	2	no	—
235	Dic₅	—	10	2	no	—
236	Dic₅	—	20	4	no	—
237	Dic₅	—	20	4	no	—
238	Dic₅	—	20	1	no	—
239	Dic₅	—	20	1	no	—
240	C₂⁴	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	25	1	no	—
245	C₂²×C₄	—	25	1	no	—
246	C₂²×C₄	D₅²	1	1	yes	characteristic
247	C₄⋊C₄	—	50	2	no	—
248	C₂²⋊C₄	—	5	4	no	—
249	C₂²⋊C₄	—	50	2	no	—
250	C₂²⋊C₄	—	50	2	no	—
251	C₄²	—	50	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	20	2	no	—
260	C₂×C₅	C₄×D₂₀	1	1	yes	characteristic
261	C₂×C₅	#4025	1	1	yes	characteristic
262	C₂×C₅	#4027	1	1	yes	characteristic
263	C₂×C₅	#4029	1	1	yes	characteristic
264	C₂×C₅	#4030	1	1	yes	characteristic
265	C₂×C₅	#4031	1	1	yes	characteristic
266	C₂×C₅	#4031	1	1	yes	characteristic
267	C₂×C₅	#4032	1	1	yes	characteristic
268	C₂×C₅	#4038	1	1	yes	characteristic
269	C₂×C₅	#4041	1	1	yes	characteristic
270	C₂×C₅	#4044	1	1	yes	characteristic
271	C₂×C₅	#4072	1	1	yes	characteristic
272	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
273	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
274	D₅	—	4	2	no	—
275	C₂³	—	5	2	no	—
276	C₂³	—	10	1	no	—
277	C₂³	—	10	1	no	—
278	C₂³	—	10	1	no	—
279	C₂³	—	10	1	no	—
280	C₂³	—	10	1	no	—
281	C₂³	—	10	1	no	—
282	C₂³	C₂×D₅²	1	1	yes	characteristic
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	5	2	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	10	1	no	—
288	C₂×C₄	—	10	1	no	—
289	C₂×C₄	—	10	1	no	—
290	C₂×C₄	—	10	1	no	—
291	C₂×C₄	—	10	1	no	—
292	C₂×C₄	—	10	1	no	—
293	C₂×C₄	—	10	1	no	—
294	C₂×C₄	—	25	2	no	—
295	C₂×C₄	—	50	1	no	—
296	C₂×C₄	—	50	1	no	—
297	C₂×C₄	—	50	1	no	—
298	C₂×C₄	—	50	1	no	—
299	C₂×C₄	—	50	1	no	—
300	C₂×C₄	D₅×Dic₅	1	2	yes	—
301	C₅	—	2	2	no	—
302	C₅	#64959	1	1	yes	characteristic, complemented
303	C₅	#65240	1	1	yes	characteristic, complemented
304	C₂²	—	10	2	no	—
305	C₂²	—	10	2	no	—
306	C₂²	—	10	2	no	—
307	C₂²	—	20	1	no	—
308	C₂²	—	20	1	no	—
309	C₂²	—	20	1	no	—
310	C₂²	—	20	1	no	—
311	C₂²	#87870	1	1	yes	characteristic
312	C₂²	#87870	1	1	yes	characteristic
313	C₂²	#87874	1	1	yes	characteristic
314	C₂²	D₅×D₂₀	1	1	yes	characteristic
315	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
316	C₂²	#87880	1	1	yes	characteristic
317	C₂²	#87886	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	10	2	no	—
320	C₄	—	10	2	no	—
321	C₄	—	20	1	no	—
322	C₄	—	50	2	no	—
323	C₄	—	100	1	no	—
324	C₂	—	20	2	no	—
325	C₂	#11716959	1	1	yes	characteristic
326	C₂	#11716977	1	1	yes	characteristic
327	C₂	#11717003	1	1	yes	characteristic
328	C₂	Dic₅×D₂₀	1	1	yes	characteristic
329	C₂	#11717028	1	1	yes	characteristic
330	C₂	#11717031	1	1	yes	characteristic
331	C₂	#11717091	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	#388
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
1. C₂×C₄⋊#1039	C₂	semidirect
5. C₂×Dic₅²	C₂	semidirect
6. C₅×#3963	C₂	semidirect
7. #11716620	C₂	semidirect
16. C₅×#685	C₄	semidirect
62. #3963	D₅	semidirect
105. #685	Dic₅	semidirect
189. C₅²	#388	semidirect
302. C₅	#64959	semidirect
303. C₅	#65240	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#1039	2
C₂×C₅×#685	2
C₂×#87778	2
C₂×Dic₅²	2
C₅×#3963	2
#11716620	2
#65240	5
#64959	5