Subgroups — #13106002

Order 1600.

504 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#4070	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	#11716996	C₂	1	2	yes	maximal, complemented
8	#11716989	C₂	1	2	yes	maximal, complemented
9	#11716988	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716974	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₀	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	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	C₄×#1039	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#685	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×#683	C₂²	1	2	yes	—
26	C₅×#683	C₂²	1	1	yes	characteristic
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	#65973	—	5	1	no	maximal
36	#65875	—	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	C₅×C₂²⋊D₅	—	2	2	no	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
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	C₄×C₅×D₅	—	2	2	no	—
51	C₅×Dic₁₀	—	2	2	no	—
52	D₅×Dic₅	—	2	4	no	—
53	C₅×C₂²⋊Q₈	—	5	1	no	—
54	C₄×C₅×D₄	—	5	1	no	—
55	#4092	—	5	1	no	—
56	Q₈×Dic₅	—	5	1	no	—
57	#4081	—	5	2	no	—
58	C₄×C₂²⋊D₅	D₅	1	1	yes	characteristic
59	#4072	—	5	1	no	—
60	#4072	—	5	1	no	—
61	#4072	—	10	2	no	—
62	C₂×C₄⋊Dic₅	—	5	1	no	—
63	#4070	—	5	1	no	—
64	#4070	D₅	1	1	yes	characteristic, complemented
65	#4038	—	5	2	no	—
66	D₅×C₄⋊C₄	—	5	1	no	—
67	#4036	—	5	1	no	—
68	#4035	—	5	2	no	—
69	#4035	—	5	2	no	—
70	#4029	—	5	2	no	—
71	#4025	—	5	2	no	—
72	#4025	—	5	2	no	—
73	#4023	—	5	2	no	—
74	#4016	—	5	1	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	2	no	—
89	C₅×Dic₅	—	2	1	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₂×C₅×Q₈	—	5	1	no	—
93	C₂×C₅×D₄	—	5	1	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	5	1	no	—
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
98	C₂×C₂²⋊D₅	C₂×D₅	1	1	yes	characteristic
99	C₂²×Dic₅	—	5	2	no	—
100	C₂×C₄×D₅	—	5	2	no	—
101	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
102	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	5	1	no	—
104	C₅×C₄⋊C₄	—	5	2	no	—
105	C₅×C₄⋊C₄	—	5	1	no	—
106	C₅×C₂²⋊C₄	—	5	1	no	—
107	C₅×C₂²⋊C₄	—	5	2	no	—
108	C₅×C₂²⋊C₄	—	5	1	no	—
109	C₄²×C₅	—	5	1	no	—
110	C₂²⋊Dic₅	—	5	2	no	—
111	C₂²⋊Dic₅	—	5	2	no	—
112	C₂²⋊Dic₅	—	5	2	no	—
113	C₂²⋊Dic₅	—	10	4	no	—
114	C₂²⋊Dic₅	C₂×D₅	1	1	yes	characteristic
115	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
116	#685	—	5	2	no	—
117	#685	C₂×D₅	1	1	yes	characteristic
118	C₄⋊Dic₅	—	5	1	no	—
119	C₄⋊Dic₅	—	5	1	no	—
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	—	10	2	no	—
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	C₂×D₅	1	2	yes	—
136	#683	C₂×D₅	1	1	yes	characteristic
137	C₄×Dic₅	—	5	2	no	—
138	C₄×Dic₅	—	5	2	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	#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×Q₈	—	10	2	no	—
155	C₅×D₄	—	10	2	no	—
156	C₂×C₄×C₅	—	2	8	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	5	1	no	—
162	C₂×C₄×C₅	—	5	2	no	—
163	C₂×C₄×C₅	—	5	1	no	—
164	C₂×C₄×C₅	—	5	1	no	—
165	C₂×C₄×C₅	—	5	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	C₂²⋊D₅	—	2	2	no	—
172	C₂×Dic₅	—	5	2	no	—
173	C₂×Dic₅	—	5	2	no	—
174	C₂×Dic₅	—	5	2	no	—
175	C₂×Dic₅	—	5	2	no	—
176	C₂×Dic₅	—	10	4	no	—
177	C₂×Dic₅	—	10	4	no	—
178	C₂×Dic₅	—	10	4	no	—
179	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
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	C₂×Dic₅	C₂²×D₅	1	2	yes	—
184	C₄×D₅	—	2	2	no	—
185	C₄×D₅	—	10	4	no	—
186	Dic₁₀	—	2	2	no	—
187	#126	—	25	2	no	—
188	#125	—	25	1	no	—
189	#125	—	25	2	no	—
190	#123	—	25	2	no	—
191	C₂²⋊Q₈	—	5	1	no	—
192	C₂²⋊Q₈	—	25	1	no	—
193	C₄×Q₈	—	25	1	no	—
194	C₄×D₄	—	5	1	no	—
195	#117	—	25	2	no	—
196	#117	—	25	1	no	—
197	C₂×C₄⋊C₄	—	25	1	no	—
198	C₅²	#552	1	1	yes	characteristic, complemented, Sylow
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	—	2	1	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	2	no	—
207	C₂²×C₅	—	4	2	no	—
208	C₂²×C₅	—	4	2	no	—
209	C₂²×C₅	—	5	2	no	—
210	C₂²×C₅	—	10	1	no	—
211	C₂²×C₅	—	10	1	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	#709	1	2	yes	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	4	no	—
221	C₄×C₅	—	10	2	no	—
222	C₄×C₅	—	10	1	no	—
223	C₄×C₅	—	10	2	no	—
224	C₄×C₅	—	10	1	no	—
225	C₄×C₅	—	10	1	no	—
226	C₄×C₅	#711	1	2	yes	—
227	C₄×C₅	#711	1	2	yes	—
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	2	no	—
230	Dic₅	—	2	1	no	—
231	Dic₅	—	2	2	no	—
232	Dic₅	—	2	1	no	—
233	Dic₅	—	10	2	no	—
234	Dic₅	—	10	2	no	—
235	Dic₅	—	10	2	no	—
236	Dic₅	—	10	2	no	—
237	Dic₅	—	20	4	no	—
238	Dic₅	—	20	4	no	—
239	C₂×Q₈	—	5	1	no	—
240	C₂×D₄	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	25	2	no	—
243	C₂²×C₄	D₅²	1	1	yes	characteristic
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	—	25	2	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	1	no	—
253	C₄⋊C₄	—	25	1	no	—
254	C₂²⋊C₄	—	5	1	no	—
255	C₂²⋊C₄	—	5	2	no	—
256	C₂²⋊C₄	—	5	1	no	—
257	C₂²⋊C₄	—	25	2	no	—
258	C₂²⋊C₄	—	25	2	no	—
259	C₄²	—	5	1	no	—
260	C₄²	—	25	2	no	—
261	C₄²	—	25	2	no	—
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	4	4	no	—
266	C₂×C₅	—	4	1	no	—
267	C₂×C₅	—	4	1	no	—
268	C₂×C₅	—	10	2	no	—
269	C₂×C₅	C₂×#709	1	1	yes	characteristic
270	C₂×C₅	C₂×#711	1	1	yes	characteristic
271	C₂×C₅	C₂×#711	1	1	yes	characteristic
272	C₂×C₅	#4147	1	1	yes	characteristic
273	C₂×C₅	D₅×#41	1	1	yes	characteristic
274	C₂×C₅	#4150	1	1	yes	characteristic
275	D₅	—	2	2	no	—
276	C₂³	—	5	1	no	—
277	C₂³	C₂×D₅²	1	1	yes	characteristic
278	Q₈	—	10	2	no	—
279	D₄	—	10	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	5	1	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	—	10	2	no	—
287	C₂×C₄	—	25	2	no	—
288	C₂×C₄	—	25	2	no	—
289	C₂×C₄	—	50	4	no	—
290	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
291	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
292	C₅	—	2	2	no	—
293	C₅	#65875	1	1	yes	characteristic, complemented
294	C₅	#65973	1	1	yes	characteristic, complemented
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	5	2	no	—
299	C₂²	—	10	1	no	—
300	C₂²	—	10	1	no	—
301	C₂²	C₂²×D₅²	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	10	1	no	—
304	C₄	—	10	2	no	—
305	C₄	—	10	1	no	—
306	C₄	—	10	1	no	—
307	C₄	—	10	2	no	—
308	C₄	—	50	2	no	—
309	C₄	—	50	2	no	—
310	C₄	#87880	1	2	yes	—
311	C₂	—	4	1	no	—
312	C₂	—	10	2	no	—
313	C₂	#11717462	1	1	yes	characteristic
314	C₂	D₅×#709	1	1	yes	characteristic
315	C₂	C₂×#87880	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. #11717246	C₂	semidirect
3. C₅×#4070	C₂	semidirect
5. #11717074	C₂	semidirect
6. D₅×C₄⋊Dic₅	C₂	semidirect
7. #11716996	C₂	semidirect
8. #11716989	C₂	semidirect
9. #11716988	C₂	semidirect
10. #11716974	C₂	semidirect
11. Dic₅×Dic₁₀	C₂	semidirect
64. #4070	D₅	semidirect
198. C₅²	#552	semidirect
293. C₅	#65875	semidirect
294. C₅	#65973	semidirect

Maximal subgroups

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

subgroup	index
#11717246	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4070	2
#11717087	2
#11717074	2
D₅×C₄⋊Dic₅	2
#11716996	2
#11716989	2
#11716988	2
#11716974	2
Dic₅×Dic₁₀	2
#65973	5
#65875	5