Subgroups — #13107834

Order 1600.

1766 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717520	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717485	C₂	1	4	yes	maximal, complemented
5	#11717484	C₂	1	4	yes	maximal, complemented
6	D₅×#713	C₂	1	4	yes	maximal, complemented
7	#11717480	C₂	1	4	yes	maximal, complemented
8	#11717472	C₂	1	6	yes	maximal, complemented
9	D₅×#709	C₂	1	6	yes	maximal, complemented
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	#87904	C₂²	1	4	yes	complemented
12	Q₈×C₅⋊D₅	C₂²	1	4	yes	complemented
13	#87900	C₂²	1	6	yes	complemented
14	C₅×#713	C₂²	1	4	yes	complemented
15	C₅×#713	C₂²	1	4	yes	complemented
16	C₅×#713	C₂²	1	4	yes	complemented
17	C₅×Q₈×D₅	C₂²	1	4	yes	complemented
18	C₅×#709	C₂²	1	6	yes	complemented
19	C₂×C₅×D₂₀	C₂²	1	3	yes	complemented
20	C₂×C₄×C₅×D₅	C₂²	1	3	yes	complemented
21	D₅×C₂²⋊D₅	C₂²	1	12	yes	complemented
22	#87879	C₂²	1	4	yes	complemented
23	#87877	C₂²	1	12	yes	complemented
24	D₅×D₂₀	C₂²	1	6	yes	complemented
25	D₅×D₂₀	C₂²	1	12	yes	complemented
26	C₄×D₅²	C₂²	1	12	yes	complemented
27	#87874	C₂²	1	6	yes	complemented
28	#87873	C₂²	1	12	yes	complemented
29	#87871	C₂²	1	12	yes	complemented
30	#87870	C₂²	1	6	yes	complemented
31	#87870	C₂²	1	12	yes	complemented
32	D₅×Dic₁₀	C₂²	1	6	yes	complemented
33	#66287	—	5	1	no	maximal
34	#66283	—	5	1	no	maximal
35	C₂²×C₅×D₅	C₂³	1	3	yes	—
36	C₂×D₅²	C₂³	1	12	yes	—
37	C₅²×Q₈	C₂³	1	4	yes	—
38	C₂×C₄×C₅²	C₂³	1	3	yes	—
39	C₂²⋊(C₅⋊D₅)	C₂³	1	4	yes	—
40	C₄⋊(C₅⋊D₅)	C₂³	1	6	yes	—
41	C₄×C₅⋊D₅	C₂³	1	12	yes	—
42	#6045	C₂³	1	6	yes	—
43	C₅×C₂²⋊D₅	C₂³	1	4	yes	—
44	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
45	C₅×D₂₀	C₂³	1	12	yes	—
46	C₅×D₂₀	C₂³	1	6	yes	—
47	C₄×C₅×D₅	C₂³	1	6	yes	—
48	C₄×C₅×D₅	C₂³	1	12	yes	—
49	C₄×C₅×D₅	C₂³	1	6	yes	—
50	C₅×Dic₁₀	C₂³	1	6	yes	—
51	#6039	C₂³	1	4	yes	—
52	#6038	C₂³	1	12	yes	—
53	#6038	C₂³	1	4	yes	—
54	#6037	C₂³	1	12	yes	—
55	#6036	C₂³	1	4	yes	—
56	D₅×Dic₅	C₂³	1	12	yes	—
57	D₅×Dic₅	C₂³	1	4	yes	—
58	C₅×2₋¹⁺⁴	—	5	1	no	—
59	C₂×C₅×#41	—	5	1	no	—
60	#4150	—	5	4	no	—
61	#4149	—	5	4	no	—
62	#4149	—	5	4	no	—
63	#4149	—	5	6	no	—
64	D₅×#41	—	5	4	no	—
65	D₅×#41	—	5	4	no	—
66	D₅×#41	—	5	4	no	—
67	D₅×#41	—	5	6	no	—
68	#4147	—	5	1	no	—
69	#4147	—	5	1	no	—
70	#4147	—	5	4	no	—
71	#4147	—	10	2	no	—
72	#4147	D₅	1	1	yes	characteristic, complemented
73	C₂×#713	—	5	4	no	—
74	C₂×#713	D₅	1	1	yes	characteristic, complemented
75	#4144	—	5	6	no	—
76	C₂×#709	—	5	6	no	—
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₂×C₅⋊D₅	C₂⁴	1	4	yes	—
79	C₂×C₅×D₅	—	4	3	no	—
80	C₂×C₅×D₅	C₂⁴	1	6	yes	—
81	C₂×C₅×D₅	C₂⁴	1	4	yes	—
82	D₅²	—	4	12	no	—
83	C₄×C₅²	C₂⁴	1	6	yes	—
84	#1039	C₂⁴	1	4	yes	—
85	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
86	C₅×Dic₅	C₂⁴	1	4	yes	—
87	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
88	C₅×#41	—	5	4	no	—
89	C₅×#41	—	5	4	no	—
90	C₅×#41	—	5	4	no	—
91	C₅×#41	—	5	6	no	—
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	5	4	no	—
94	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
95	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
96	C₂×C₅×D₄	—	5	3	no	—
97	C₂²×C₄×C₅	—	5	3	no	—
98	C₂×C₂²⋊D₅	—	5	12	no	—
99	#713	—	5	4	no	—
100	#713	—	5	4	no	—
101	#713	—	5	12	no	—
102	#713	—	5	6	no	—
103	#713	—	10	8	no	—
104	#713	C₂×D₅	1	4	yes	complemented
105	#713	C₂×D₅	1	4	yes	complemented
106	#713	C₂×D₅	1	4	yes	complemented
107	Q₈×D₅	—	5	4	no	—
108	Q₈×D₅	—	5	4	no	—
109	Q₈×D₅	—	5	6	no	—
110	Q₈×D₅	—	10	8	no	—
111	Q₈×D₅	C₂×D₅	1	4	yes	complemented
112	#711	—	5	4	no	—
113	#711	—	5	12	no	—
114	#711	—	5	6	no	—
115	#711	—	5	12	no	—
116	D₄×D₅	—	5	12	no	—
117	D₄×D₅	—	5	12	no	—
118	D₄×D₅	—	5	12	no	—
119	D₄×D₅	—	5	12	no	—
120	D₄×D₅	—	5	6	no	—
121	#709	—	5	6	no	—
122	#709	—	5	6	no	—
123	#709	—	5	4	no	—
124	#709	—	5	6	no	—
125	#709	—	5	12	no	—
126	#709	—	5	12	no	—
127	#709	—	5	12	no	—
128	#709	—	5	6	no	—
129	#709	—	10	12	no	—
130	#709	C₂×D₅	1	6	yes	complemented
131	C₂×D₂₀	—	5	6	no	—
132	C₂×D₂₀	—	5	12	no	—
133	C₂×D₂₀	C₂×D₅	1	3	yes	complemented
134	C₂×C₄×D₅	—	5	12	no	—
135	C₂×C₄×D₅	—	5	12	no	—
136	C₂×C₄×D₅	C₂×D₅	1	3	yes	complemented
137	C₂×Dic₁₀	—	5	6	no	—
138	#585	—	25	1	no	Sylow
139	C₂×C₅²	—	2	1	no	—
140	C₂×C₅²	C₂⁵	1	1	yes	characteristic
141	C₅⋊D₅	—	2	4	no	—
142	C₅×D₅	—	2	4	no	—
143	C₅×D₅	—	2	6	no	—
144	C₂³×C₅	—	5	3	no	—
145	C₂²×D₅	—	5	12	no	—
146	C₂²×D₅	—	5	12	no	—
147	C₂²×D₅	C₂²×D₅	1	3	yes	—
148	C₅×Q₈	—	2	8	no	—
149	C₅×Q₈	—	5	6	no	—
150	C₅×Q₈	C₂²×D₅	1	4	yes	—
151	C₅×Q₈	C₂²×D₅	1	4	yes	—
152	C₅×D₄	—	5	4	no	—
153	C₅×D₄	—	5	6	no	—
154	C₅×D₄	—	5	12	no	—
155	C₂×C₄×C₅	—	2	6	no	—
156	C₂×C₄×C₅	—	5	1	no	—
157	C₂×C₄×C₅	—	5	6	no	—
158	C₂×C₄×C₅	—	5	6	no	—
159	C₂×C₄×C₅	—	5	12	no	—
160	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
161	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
162	C₂²⋊D₅	—	5	4	no	—
163	C₂²⋊D₅	—	5	4	no	—
164	C₂²⋊D₅	—	5	12	no	—
165	C₂²⋊D₅	—	5	12	no	—
166	C₂²⋊D₅	—	5	4	no	—
167	C₂²⋊D₅	—	5	12	no	—
168	C₂²⋊D₅	—	10	8	no	—
169	C₂²⋊D₅	C₂²×D₅	1	4	yes	—
170	C₂×Dic₅	—	5	12	no	—
171	C₂×Dic₅	—	5	4	no	—
172	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
173	D₂₀	—	5	4	no	—
174	D₂₀	—	5	12	no	—
175	D₂₀	—	5	6	no	—
176	D₂₀	—	5	6	no	—
177	D₂₀	—	10	12	no	—
178	D₂₀	C₂²×D₅	1	12	yes	—
179	D₂₀	C₂²×D₅	1	6	yes	—
180	C₄×D₅	—	5	4	no	—
181	C₄×D₅	—	5	12	no	—
182	C₄×D₅	—	5	12	no	—
183	C₄×D₅	—	5	12	no	—
184	C₄×D₅	—	5	4	no	—
185	C₄×D₅	—	5	4	no	—
186	C₄×D₅	—	10	24	no	—
187	C₄×D₅	C₂²×D₅	1	6	yes	—
188	C₄×D₅	C₂²×D₅	1	12	yes	—
189	C₄×D₅	C₂²×D₅	1	6	yes	—
190	Dic₁₀	—	5	6	no	—
191	Dic₁₀	—	5	6	no	—
192	Dic₁₀	—	5	4	no	—
193	Dic₁₀	—	5	4	no	—
194	Dic₁₀	—	10	12	no	—
195	Dic₁₀	C₂²×D₅	1	6	yes	—
196	2₋¹⁺⁴	—	5	1	no	—
197	2₋¹⁺⁴	—	25	4	no	—
198	2₋¹⁺⁴	—	25	1	no	—
199	H(5,𝔽₂)	—	25	6	no	—
200	H(5,𝔽₂)	—	25	4	no	—
201	C₂×#41	—	5	1	no	—
202	C₂×#41	—	25	6	no	—
203	C₂×#41	—	25	4	no	—
204	C₂×#41	—	25	4	no	—
205	C₅²	#585	1	1	yes	characteristic, complemented, Sylow
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	5	6	no	—
208	C₂²×C₅	—	5	4	no	—
209	C₂²×C₅	—	20	3	no	—
210	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
211	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
212	C₂×D₅	—	4	3	no	—
213	C₂×D₅	—	5	4	no	—
214	C₂×D₅	—	5	4	no	—
215	C₂×D₅	—	10	8	no	—
216	C₂×D₅	—	20	12	no	—
217	C₂×D₅	—	20	12	no	—
218	C₂×D₅	C₂³×D₅	1	6	yes	—
219	C₂×D₅	C₂³×D₅	1	4	yes	—
220	C₄×C₅	—	2	12	no	—
221	C₄×C₅	—	5	1	no	—
222	C₄×C₅	—	5	4	no	—
223	C₄×C₅	—	5	1	no	—
224	C₄×C₅	C₂³×D₅	1	6	yes	—
225	C₄×C₅	C₂³×D₅	1	6	yes	—
226	Dic₅	—	5	4	no	—
227	Dic₅	—	5	4	no	—
228	Dic₅	—	10	8	no	—
229	Dic₅	C₂³×D₅	1	1	yes	characteristic
230	Dic₅	C₂³×D₅	1	4	yes	—
231	Dic₅	C₂³×D₅	1	1	yes	characteristic
232	#41	—	5	4	no	—
233	#41	—	5	4	no	—
234	#41	—	5	4	no	—
235	#41	—	5	6	no	—
236	#41	—	25	12	no	—
237	#41	—	25	4	no	—
238	#41	—	25	6	no	—
239	#41	—	25	12	no	—
240	#41	—	25	6	no	—
241	#41	—	25	12	no	—
242	#41	—	25	4	no	—
243	#41	—	25	6	no	—
244	C₂×Q₈	—	5	4	no	—
245	C₂×Q₈	—	25	6	no	—
246	C₂×Q₈	—	25	4	no	—
247	C₂×Q₈	D₅²	1	1	yes	characteristic, complemented
248	C₂×D₄	—	5	3	no	—
249	C₂×D₄	—	25	6	no	—
250	C₂×D₄	—	25	12	no	—
251	C₂×D₄	—	25	12	no	—
252	C₂×D₄	—	25	12	no	—
253	C₂²×C₄	—	5	3	no	—
254	C₂²×C₄	—	25	12	no	—
255	C₂×C₅	—	2	1	no	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	—	10	4	no	—
260	C₂×C₅	—	10	6	no	—
261	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
262	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
263	D₅	—	2	6	no	—
264	D₅	—	2	4	no	—
265	D₅	—	10	4	no	—
266	D₅	—	10	4	no	—
267	D₅	—	20	8	no	—
268	C₂³	—	5	3	no	—
269	C₂³	—	25	12	no	—
270	Q₈	—	5	6	no	—
271	Q₈	—	25	4	no	—
272	Q₈	—	25	6	no	—
273	Q₈	C₂×D₅²	1	4	yes	—
274	D₄	—	5	4	no	—
275	D₄	—	5	12	no	—
276	D₄	—	5	6	no	—
277	D₄	—	25	4	no	—
278	D₄	—	25	12	no	—
279	D₄	—	25	12	no	—
280	D₄	—	25	4	no	—
281	D₄	—	25	6	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	6	no	—
284	C₂×C₄	—	5	6	no	—
285	C₂×C₄	—	5	12	no	—
286	C₂×C₄	—	25	4	no	—
287	C₂×C₄	—	25	4	no	—
288	C₂×C₄	—	25	12	no	—
289	C₂×C₄	—	25	12	no	—
290	C₂×C₄	C₂×D₅²	1	3	yes	—
291	C₅	—	2	2	no	—
292	C₅	#66283	1	1	yes	characteristic, complemented
293	C₅	#66287	1	1	yes	characteristic, complemented
294	C₂²	—	5	4	no	—
295	C₂²	—	5	6	no	—
296	C₂²	—	20	3	no	—
297	C₂²	—	25	4	no	—
298	C₂²	—	100	12	no	—
299	C₂²	C₂²×D₅²	1	1	yes	characteristic
300	C₄	—	5	1	no	—
301	C₄	—	5	1	no	—
302	C₄	—	5	4	no	—
303	C₄	—	25	4	no	—
304	C₄	C₂²×D₅²	1	6	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	10	6	no	—
307	C₂	—	10	4	no	—
308	C₂	—	50	4	no	—
309	C₂	C₂³×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅²
commutator, socle, cosocle	C₂×C₅²	C₂⁵
Fitting	C₂×C₅²×Q₈	C₂²
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#585
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #11717520	C₂	semidirect
2. C₅×#4147	C₂	semidirect
3. C₂×C₅×#713	C₂	semidirect
4. #11717485	C₂	semidirect
5. #11717484	C₂	semidirect
6. D₅×#713	C₂	semidirect
7. #11717480	C₂	semidirect
8. #11717472	C₂	semidirect
9. D₅×#709	C₂	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
11. #87904	C₂²	semidirect
12. Q₈×C₅⋊D₅	C₂²	semidirect
13. #87900	C₂²	semidirect
14. C₅×#713	C₂²	semidirect
15. C₅×#713	C₂²	semidirect
16. C₅×#713	C₂²	semidirect
17. C₅×Q₈×D₅	C₂²	semidirect
18. C₅×#709	C₂²	semidirect
19. C₂×C₅×D₂₀	C₂²	semidirect
20. C₂×C₄×C₅×D₅	C₂²	semidirect
21. D₅×C₂²⋊D₅	C₂²	semidirect
22. #87879	C₂²	semidirect
23. #87877	C₂²	semidirect
24. D₅×D₂₀	C₂²	semidirect
25. D₅×D₂₀	C₂²	semidirect
26. C₄×D₅²	C₂²	semidirect
27. #87874	C₂²	semidirect
28. #87873	C₂²	semidirect
29. #87871	C₂²	semidirect
30. #87870	C₂²	semidirect
31. #87870	C₂²	semidirect
32. D₅×Dic₁₀	C₂²	semidirect
72. #4147	D₅	semidirect
74. C₂×#713	D₅	semidirect
94. C₂×C₅×Q₈	C₂×D₅	semidirect
95. C₂×C₅×Q₈	C₂×D₅	semidirect
104. #713	C₂×D₅	semidirect
105. #713	C₂×D₅	semidirect
106. #713	C₂×D₅	semidirect
111. Q₈×D₅	C₂×D₅	semidirect
130. #709	C₂×D₅	semidirect
133. C₂×D₂₀	C₂×D₅	semidirect
136. C₂×C₄×D₅	C₂×D₅	semidirect
205. C₅²	#585	semidirect
247. C₂×Q₈	D₅²	semidirect
292. C₅	#66283	semidirect
293. C₅	#66287	semidirect

Maximal subgroups

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

subgroup	index
#11717520	2
C₅×#4147	2
C₂×C₅×#713	2
#11717485	2
#11717484	2
D₅×#713	2
#11717480	2
#11717472	2
D₅×#709	2
#66287	5
#66283	5