Subgroups — #13105998

Order 1600.

502 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717244	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717087	C₂	1	2	yes	maximal
5	#11717074	C₂	1	2	yes	maximal, complemented
6	D₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716995	C₂	1	2	yes	maximal, complemented
8	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716986	C₂	1	2	yes	maximal, complemented
10	#11716975	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716962	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×#6045	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×D₅×Dic₅	C₂²	1	2	yes	—
17	C₂²⋊#1039	C₂²	1	2	yes	—
18	C₄⋊#1039	C₂²	1	1	yes	characteristic
19	#87806	C₂²	1	2	yes	—
20	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₅×#685	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×#683	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₅	C₂²	1	2	yes	—
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	#87781	C₂²	1	2	yes	—
29	#87781	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87780	C₂²	1	2	yes	—
32	#87780	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#65875	—	5	1	no	maximal
36	#65863	—	5	1	no	maximal
37	C₂³×C₅²	C₂³	1	1	yes	characteristic
38	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅²	—	2	2	no	—
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	C₅×C₂²⋊D₅	—	2	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	—	2	2	no	—
52	D₅×Dic₅	—	2	4	no	—
53	C₄×C₅×D₄	—	5	1	no	—
54	C₅×#117	—	5	1	no	—
55	#4081	—	5	2	no	—
56	C₄×C₂²⋊D₅	D₅	1	1	yes	characteristic
57	#4072	D₅	1	1	yes	characteristic, complemented
58	C₂×C₄⋊Dic₅	—	5	1	no	—
59	#4070	—	5	1	no	—
60	#4070	—	5	1	no	—
61	#4070	—	5	1	no	—
62	#4070	—	10	2	no	—
63	#4043	—	5	1	no	—
64	D₅×C₄⋊C₄	—	5	1	no	—
65	#4036	—	5	1	no	—
66	#4035	—	5	2	no	—
67	#4033	—	5	1	no	—
68	#4029	—	5	2	no	—
69	#4025	—	5	2	no	—
70	#4025	—	5	2	no	—
71	#4023	—	5	2	no	—
72	#4018	—	5	2	no	—
73	#4016	—	5	1	no	—
74	#4016	—	5	2	no	—
75	C₄×Dic₁₀	—	5	1	no	—
76	C₂²×C₅²	—	2	1	no	—
77	C₂²×C₅²	—	2	1	no	—
78	C₂²×C₅²	—	2	1	no	—
79	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×D₅	—	2	1	no	—
81	C₂×C₅×D₅	—	2	1	no	—
82	C₂×C₅×D₅	#41	1	2	yes	—
83	C₄×C₅²	—	2	1	no	—
84	C₄×C₅²	#41	1	2	yes	—
85	#1039	—	2	2	no	—
86	#1039	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	1	no	—
89	C₅×Dic₅	—	2	1	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	2	no	—
92	C₂×C₅×D₄	—	5	1	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	5	1	no	—
95	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
97	C₂×C₂²⋊D₅	C₂×D₅	1	1	yes	characteristic
98	C₂²×Dic₅	—	5	2	no	—
99	C₂×C₄×D₅	—	5	2	no	—
100	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
101	C₂×Dic₁₀	—	5	1	no	—
102	C₂×Dic₁₀	—	5	1	no	—
103	C₂×Dic₁₀	—	10	2	no	—
104	C₅×C₄⋊C₄	—	5	1	no	—
105	C₅×C₄⋊C₄	—	5	1	no	—
106	C₅×C₄⋊C₄	—	5	1	no	—
107	C₅×C₂²⋊C₄	—	5	2	no	—
108	C₅×C₂²⋊C₄	—	5	1	no	—
109	C₅×C₂²⋊C₄	—	5	1	no	—
110	C₄²×C₅	—	5	2	no	—
111	C₄²×C₅	—	5	1	no	—
112	C₂²⋊Dic₅	—	5	2	no	—
113	C₂²⋊Dic₅	—	5	2	no	—
114	C₂²⋊Dic₅	—	5	2	no	—
115	C₂²⋊Dic₅	—	10	4	no	—
116	C₂²⋊Dic₅	C₂×D₅	1	1	yes	characteristic
117	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
118	#685	—	5	2	no	—
119	#685	C₂×D₅	1	1	yes	characteristic
120	C₄⋊Dic₅	—	5	1	no	—
121	C₄⋊Dic₅	—	5	1	no	—
122	C₄⋊Dic₅	—	5	2	no	—
123	C₄⋊Dic₅	—	5	2	no	—
124	C₄⋊Dic₅	—	5	2	no	—
125	C₄⋊Dic₅	—	10	2	no	—
126	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
127	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
128	#683	—	5	2	no	—
129	#683	—	5	2	no	—
130	#683	—	5	2	no	—
131	#683	—	5	2	no	—
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	—	10	4	no	—
138	#683	C₂×D₅	1	1	yes	characteristic
139	C₄×Dic₅	—	5	2	no	—
140	C₄×Dic₅	—	5	2	no	—
141	C₄×Dic₅	C₂×D₅	1	2	yes	—
142	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
143	#552	—	25	1	no	Sylow
144	C₂×C₅²	—	4	1	no	—
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₂×C₅²	C₂×#41	1	1	yes	characteristic
147	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
148	C₅×D₅	—	2	2	no	—
149	C₂³×C₅	—	2	2	no	—
150	C₂³×C₅	—	5	1	no	—
151	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
152	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
154	C₅×D₄	—	10	2	no	—
155	C₂×C₄×C₅	—	2	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	—	4	4	no	—
160	C₂×C₄×C₅	—	5	1	no	—
161	C₂×C₄×C₅	—	5	2	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	1	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	10	2	no	—
166	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
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	C₂²⋊D₅	—	2	2	no	—
171	C₂×Dic₅	—	5	2	no	—
172	C₂×Dic₅	—	5	2	no	—
173	C₂×Dic₅	—	5	2	no	—
174	C₂×Dic₅	—	5	2	no	—
175	C₂×Dic₅	—	10	4	no	—
176	C₂×Dic₅	—	10	4	no	—
177	C₂×Dic₅	—	10	4	no	—
178	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
179	C₂×Dic₅	C₂²×D₅	1	2	yes	—
180	C₂×Dic₅	C₂²×D₅	1	2	yes	—
181	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	10	4	no	—
185	Dic₁₀	—	10	2	no	—
186	Dic₁₀	—	10	2	no	—
187	Dic₁₀	—	20	4	no	—
188	#126	—	25	2	no	—
189	#125	—	25	2	no	—
190	#125	—	25	1	no	—
191	#123	—	25	2	no	—
192	C₂²⋊Q₈	—	25	1	no	—
193	C₂²⋊Q₈	—	25	1	no	—
194	C₄×Q₈	—	25	1	no	—
195	C₄×D₄	—	5	1	no	—
196	#117	—	5	1	no	—
197	#117	—	25	2	no	—
198	C₂×C₄⋊C₄	—	25	1	no	—
199	C₅²	#552	1	1	yes	characteristic, complemented, Sylow
200	C₂²×C₅	—	2	4	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	4	2	no	—
209	C₂²×C₅	—	4	2	no	—
210	C₂²×C₅	—	5	2	no	—
211	C₂²×C₅	—	10	1	no	—
212	C₂²×C₅	—	10	1	no	—
213	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
214	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₂×D₅	#709	1	2	yes	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	4	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	—	10	2	no	—
223	C₄×C₅	—	10	1	no	—
224	C₄×C₅	—	10	1	no	—
225	C₄×C₅	—	10	1	no	—
226	C₄×C₅	—	10	2	no	—
227	C₄×C₅	#709	1	2	yes	—
228	C₄×C₅	#711	1	2	yes	—
229	Dic₅	—	2	1	no	—
230	Dic₅	—	2	1	no	—
231	Dic₅	—	2	2	no	—
232	Dic₅	—	2	2	no	—
233	Dic₅	—	2	1	no	—
234	Dic₅	—	10	2	no	—
235	Dic₅	—	10	2	no	—
236	Dic₅	—	10	2	no	—
237	Dic₅	—	10	2	no	—
238	Dic₅	—	20	4	no	—
239	Dic₅	—	20	4	no	—
240	C₂×Q₈	—	25	1	no	—
241	C₂×D₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	25	2	no	—
244	C₂²×C₄	D₅²	1	1	yes	characteristic
245	C₄⋊C₄	—	5	1	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	—	5	1	no	—
248	C₄⋊C₄	—	25	2	no	—
249	C₄⋊C₄	—	25	2	no	—
250	C₄⋊C₄	—	25	2	no	—
251	C₄⋊C₄	—	25	2	no	—
252	C₄⋊C₄	—	25	2	no	—
253	C₄⋊C₄	—	25	1	no	—
254	C₄⋊C₄	—	25	2	no	—
255	C₂²⋊C₄	—	5	2	no	—
256	C₂²⋊C₄	—	5	1	no	—
257	C₂²⋊C₄	—	5	1	no	—
258	C₂²⋊C₄	—	25	2	no	—
259	C₂²⋊C₄	—	25	2	no	—
260	C₄²	—	5	2	no	—
261	C₄²	—	5	1	no	—
262	C₄²	—	25	2	no	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	4	4	no	—
267	C₂×C₅	—	4	1	no	—
268	C₂×C₅	—	4	1	no	—
269	C₂×C₅	—	10	2	no	—
270	C₂×C₅	C₂×#709	1	1	yes	characteristic
271	C₂×C₅	C₂×#709	1	1	yes	characteristic
272	C₂×C₅	C₂×#711	1	1	yes	characteristic
273	C₂×C₅	D₅×#41	1	1	yes	characteristic
274	C₂×C₅	#4150	1	1	yes	characteristic
275	C₂×C₅	#4150	1	1	yes	characteristic
276	D₅	—	2	2	no	—
277	C₂³	—	5	1	no	—
278	C₂³	C₂×D₅²	1	1	yes	characteristic
279	Q₈	—	50	2	no	—
280	D₄	—	10	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	5	2	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	10	2	no	—
288	C₂×C₄	—	25	2	no	—
289	C₂×C₄	—	25	2	no	—
290	C₂×C₄	—	50	4	no	—
291	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
292	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
293	C₅	—	2	2	no	—
294	C₅	#65863	1	1	yes	characteristic, complemented
295	C₅	#65875	1	1	yes	characteristic, complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	5	2	no	—
300	C₂²	—	10	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	C₂²×D₅²	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	10	1	no	—
305	C₄	—	10	2	no	—
306	C₄	—	10	2	no	—
307	C₄	—	10	1	no	—
308	C₄	—	10	1	no	—
309	C₄	—	50	2	no	—
310	C₄	—	50	2	no	—
311	C₄	#87879	1	2	yes	—
312	C₂	—	4	1	no	—
313	C₂	—	10	2	no	—
314	C₂	#11717463	1	1	yes	characteristic
315	C₂	D₅×#709	1	1	yes	characteristic
316	C₂	C₂×#87879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂²×C₄×C₅²	C₂²
prime radical	C₂×C₅×C₂²⋊D₅	C₂²
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#552
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #11717244	C₂	semidirect
3. C₅×#4072	C₂	semidirect
5. #11717074	C₂	semidirect
6. D₅×C₄⋊Dic₅	C₂	semidirect
7. #11716995	C₂	semidirect
8. #11716988	C₂	semidirect
9. #11716986	C₂	semidirect
10. #11716975	C₂	semidirect
11. #11716962	C₂	semidirect
57. #4072	D₅	semidirect
199. C₅²	#552	semidirect
294. C₅	#65863	semidirect
295. C₅	#65875	semidirect

Maximal subgroups

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

subgroup	index
#11717244	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4072	2
#11717087	2
#11717074	2
D₅×C₄⋊Dic₅	2
#11716995	2
#11716988	2
#11716986	2
#11716975	2
#11716962	2
#65875	5
#65863	5