Subgroups — #10618245

Order 640.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#536	C₂	1	1	yes	maximal, characteristic, complemented
2	#66164	C₂	1	1	yes	maximal, characteristic
3	#41×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4093	C₂	1	1	yes	maximal, characteristic, complemented
5	#66146	C₂	1	1	yes	maximal, characteristic, complemented
6	#66066	C₂	1	4	yes	maximal, complemented
7	#66015	C₂	1	4	yes	maximal, complemented
8	#65992	C₂	1	8	yes	maximal, complemented
9	#65977	C₂	1	4	yes	maximal, complemented
10	#65958	C₂	1	4	yes	maximal, complemented
11	#65855	C₂	1	1	yes	maximal, characteristic, complemented
12	#65851	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×D₂₀	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₅×#124	C₂²	1	2	yes	complemented
18	C₅×#123	C₂²	1	4	yes	—
19	C₅×C₂²⋊Q₈	C₂²	1	4	yes	complemented
20	C₅×#117	C₂²	1	1	yes	characteristic
21	#4093	C₂²	1	2	yes	—
22	#4093	C₂²	1	4	yes	—
23	#4093	C₂²	1	2	yes	—
24	#4093	C₂²	1	2	yes	—
25	Q₈×Dic₅	C₂²	1	2	yes	complemented
26	Q₈×Dic₅	C₂²	1	4	yes	complemented
27	Q₈×Dic₅	C₂²	1	4	yes	complemented
28	#4086	C₂²	1	2	yes	—
29	#4083	C₂²	1	4	yes	—
30	D₄×Dic₅	C₂²	1	4	yes	—
31	D₄×Dic₅	C₂²	1	2	yes	complemented
32	#4076	C₂²	1	4	yes	—
33	C₂×#685	C₂²	1	4	yes	—
34	#4072	C₂²	1	1	yes	characteristic
35	#4072	C₂²	1	1	yes	characteristic
36	#4072	C₂²	1	4	yes	—
37	#4072	C₂²	1	2	yes	complemented
38	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₅	C₂²	1	2	yes	complemented
40	#4044	C₂²	1	8	yes	complemented
41	#4044	C₂²	1	8	yes	complemented
42	#4043	C₂²	1	8	yes	complemented
43	#4039	C₂²	1	4	yes	—
44	#4038	C₂²	1	8	yes	complemented
45	#4036	C₂²	1	2	yes	complemented
46	#4034	C₂²	1	2	yes	complemented
47	#4032	C₂²	1	8	yes	complemented
48	#4031	C₂²	1	8	yes	complemented
49	#4029	C₂²	1	8	yes	complemented
50	#4027	C₂²	1	8	yes	complemented
51	D₅×C₂²⋊C₄	C₂²	1	4	yes	—
52	#4024	C₂²	1	4	yes	—
53	C₄×D₂₀	C₂²	1	4	yes	—
54	C₄×Dic₁₀	C₂²	1	4	yes	complemented
55	#3510	—	5	1	no	maximal, Sylow
56	C₂³×D₅	C₂³	1	2	yes	—
57	C₅×#41	—	2	4	no	—
58	C₂×C₅×Q₈	—	2	4	no	—
59	C₂×C₅×Q₈	C₂³	1	2	yes	—
60	C₂×C₅×Q₈	C₂³	1	2	yes	—
61	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×D₄	C₂³	1	2	yes	—
63	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
68	C₂²×Dic₅	C₂³	1	2	yes	—
69	C₂²×Dic₅	C₂³	1	4	yes	—
70	C₂×D₂₀	—	2	4	no	—
71	C₂×D₂₀	C₂³	1	2	yes	—
72	C₂×D₂₀	C₂³	1	2	yes	—
73	C₂×C₄×D₅	C₂³	1	8	yes	—
74	C₂×Dic₁₀	C₂³	1	4	yes	—
75	C₅×C₄⋊C₄	C₂³	1	2	yes	—
76	C₅×C₄⋊C₄	C₂³	1	8	yes	—
77	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
78	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
79	C₄²×C₅	C₂³	1	2	yes	—
80	C₂²⋊Dic₅	C₂³	1	2	yes	—
81	C₂²⋊Dic₅	C₂³	1	2	yes	—
82	C₂²⋊Dic₅	C₂³	1	4	yes	—
83	#685	C₂³	1	8	yes	—
84	#685	C₂³	1	8	yes	—
85	#685	C₂³	1	8	yes	—
86	#685	C₂³	1	8	yes	—
87	C₄⋊Dic₅	C₂³	1	4	yes	—
88	C₄⋊Dic₅	C₂³	1	2	yes	—
89	C₄⋊Dic₅	C₂³	1	8	yes	—
90	C₄⋊Dic₅	C₂³	1	2	yes	—
91	C₄⋊Dic₅	C₂³	1	4	yes	—
92	#683	C₂³	1	4	yes	—
93	#683	C₂³	1	4	yes	—
94	C₄×Dic₅	C₂³	1	2	yes	—
95	C₄×Dic₅	C₂³	1	2	yes	—
96	C₄×Dic₅	C₂³	1	8	yes	—
97	C₄×Dic₅	C₂³	1	1	yes	characteristic
98	C₄×Dic₅	C₂³	1	1	yes	characteristic
99	C₄×Dic₅	C₂³	1	4	yes	—
100	C₄×Dic₅	C₂³	1	2	yes	—
101	#559	—	5	4	no	—
102	#556	—	5	4	no	—
103	#551	—	5	8	no	—
104	#542	—	5	4	no	—
105	#536	D₅	1	1	yes	characteristic, complemented
106	#535	—	5	1	no	—
107	#533	—	5	1	no	—
108	#529	—	5	4	no	—
109	C₂×#124	—	5	1	no	—
110	#519	—	5	1	no	—
111	#518	—	5	1	no	—
112	C₄×#41	—	5	1	no	—
113	C₂³×C₅	C₂⁴	1	2	yes	—
114	C₂³×C₅	C₂⁴	1	1	yes	characteristic
115	C₂²×D₅	—	2	4	no	—
116	C₂²×D₅	—	4	4	no	—
117	C₂²×D₅	C₂⁴	1	4	yes	—
118	C₅×Q₈	—	2	4	no	—
119	C₅×Q₈	—	2	4	no	—
120	C₅×Q₈	—	2	2	no	—
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	#41	1	4	yes	—
126	C₂×C₄×C₅	C₂⁴	1	4	yes	—
127	C₂×C₄×C₅	C₂⁴	1	2	yes	—
128	C₂×C₄×C₅	C₂⁴	1	4	yes	—
129	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₅	—	4	8	no	—
132	C₂×Dic₅	—	4	2	no	—
133	C₂×Dic₅	—	4	4	no	—
134	C₂×Dic₅	C₂⁴	1	2	yes	—
135	C₂×Dic₅	C₂⁴	1	4	yes	—
136	C₂×Dic₅	C₂⁴	1	4	yes	—
137	C₂×Dic₅	C₂⁴	1	2	yes	—
138	D₂₀	—	2	4	no	—
139	D₂₀	—	2	4	no	—
140	C₄×D₅	—	4	8	no	—
141	Dic₁₀	—	4	4	no	—
142	C₂×#41	C₂×D₅	1	1	yes	characteristic
143	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
144	C₂²×D₄	—	5	1	no	—
145	C₄⋊Q₈	—	5	2	no	—
146	C₄⋊Q₈	C₂×D₅	1	2	yes	complemented
147	#127	—	5	2	no	—
148	#126	—	5	8	no	—
149	#126	—	5	8	no	—
150	#125	—	5	2	no	—
151	#124	—	5	2	no	—
152	#124	—	5	2	no	—
153	#124	—	5	4	no	—
154	#124	—	5	2	no	—
155	#124	—	5	8	no	—
156	#124	C₂×D₅	1	2	yes	complemented
157	#123	—	5	8	no	—
158	#123	—	5	8	no	—
159	#123	C₂×D₅	1	4	yes	—
160	C₂²⋊Q₈	—	5	8	no	—
161	C₂²⋊Q₈	—	5	4	no	—
162	C₂²⋊Q₈	C₂×D₅	1	4	yes	complemented
163	#121	—	5	4	no	—
164	#120	—	5	4	no	—
165	C₄×Q₈	—	5	2	no	—
166	C₄×Q₈	—	5	4	no	—
167	C₄×Q₈	—	5	4	no	—
168	C₄×Q₈	—	5	4	no	—
169	C₄×D₄	—	5	4	no	—
170	C₄×D₄	—	5	2	no	—
171	C₄×D₄	—	5	4	no	—
172	C₄×D₄	—	5	8	no	—
173	C₄×D₄	—	5	4	no	—
174	#117	—	5	1	no	—
175	#117	—	5	2	no	—
176	#117	—	5	1	no	—
177	#117	—	5	4	no	—
178	#117	—	5	8	no	—
179	#117	C₂×D₅	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	5	4	no	—
181	C₂×C₂²⋊C₄	—	5	4	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₂×C₄²	—	5	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	C₂×#41	1	2	yes	—
188	C₂²×C₅	C₂⁵	1	1	yes	characteristic
189	C₂×D₅	—	2	4	no	—
190	C₂×D₅	—	2	8	no	—
191	C₂×D₅	—	8	4	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	4	no	—
194	C₄×C₅	—	2	4	no	—
195	C₄×C₅	C₂×#41	1	2	yes	—
196	C₄×C₅	C₂×#41	1	2	yes	—
197	Dic₅	—	2	2	no	—
198	Dic₅	—	2	4	no	—
199	Dic₅	—	2	4	no	—
200	Dic₅	—	2	2	no	—
201	C₂⁴	—	5	2	no	—
202	#41	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	5	4	no	—
205	C₂×Q₈	C₂²×D₅	1	2	yes	—
206	C₂×Q₈	C₂²×D₅	1	2	yes	—
207	C₂×Q₈	C₂²×D₅	1	1	yes	characteristic
208	C₂×D₄	—	5	8	no	—
209	C₂×D₄	—	5	2	no	—
210	C₂×D₄	—	5	2	no	—
211	C₂×D₄	—	10	4	no	—
212	C₂×D₄	C₂²×D₅	1	2	yes	—
213	C₂×D₄	C₂²×D₅	1	1	yes	characteristic
214	C₂²×C₄	—	5	2	no	—
215	C₂²×C₄	—	5	4	no	—
216	C₂²×C₄	—	5	8	no	—
217	C₂²×C₄	C₂²×D₅	1	2	yes	—
218	C₂²×C₄	C₂²×D₅	1	2	yes	—
219	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
220	C₄⋊C₄	—	5	4	no	—
221	C₄⋊C₄	—	5	2	no	—
222	C₄⋊C₄	—	5	4	no	—
223	C₄⋊C₄	—	5	2	no	—
224	C₄⋊C₄	—	5	8	no	—
225	C₄⋊C₄	—	5	4	no	—
226	C₄⋊C₄	—	5	4	no	—
227	C₄⋊C₄	C₂²×D₅	1	8	yes	—
228	C₄⋊C₄	C₂²×D₅	1	2	yes	—
229	C₂²⋊C₄	—	5	2	no	—
230	C₂²⋊C₄	—	5	2	no	—
231	C₂²⋊C₄	—	5	4	no	—
232	C₂²⋊C₄	—	5	8	no	—
233	C₂²⋊C₄	—	5	8	no	—
234	C₂²⋊C₄	—	5	8	no	—
235	C₂²⋊C₄	—	5	8	no	—
236	C₂²⋊C₄	C₂²×D₅	1	2	yes	—
237	C₂²⋊C₄	C₂²×D₅	1	8	yes	—
238	C₄²	—	5	1	no	—
239	C₄²	—	5	2	no	—
240	C₄²	—	5	4	no	—
241	C₄²	—	5	1	no	—
242	C₄²	—	5	2	no	—
243	C₄²	—	5	8	no	—
244	C₄²	—	5	2	no	—
245	C₄²	C₂²×D₅	1	2	yes	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	4	2	no	—
248	C₂×C₅	C₂²×#41	1	1	yes	characteristic
249	C₂×C₅	#585	1	2	yes	—
250	D₅	—	4	4	no	—
251	C₂³	—	5	4	no	—
252	C₂³	—	10	4	no	—
253	C₂³	—	20	4	no	—
254	C₂³	C₂³×D₅	1	2	yes	—
255	C₂³	C₂³×D₅	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	2	no	—
259	Q₈	—	20	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	2	no	—
262	D₄	—	10	4	no	—
263	D₄	—	10	4	no	—
264	D₄	—	20	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	4	no	—
269	C₂×C₄	—	5	4	no	—
270	C₂×C₄	—	5	2	no	—
271	C₂×C₄	—	20	8	no	—
272	C₂×C₄	—	20	4	no	—
273	C₂×C₄	—	20	2	no	—
274	C₂×C₄	#711	1	4	yes	—
275	C₂×C₄	C₂³×D₅	1	4	yes	—
276	C₂×C₄	C₂³×D₅	1	2	yes	—
277	C₂×C₄	C₂³×D₅	1	4	yes	—
278	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
280	C₅	#3510	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	10	4	no	—
285	C₂²	—	10	8	no	—
286	C₂²	—	40	4	no	—
287	C₂²	C₂×#711	1	2	yes	—
288	C₂²	C₂⁴×D₅	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	10	4	no	—
293	C₄	—	10	2	no	—
294	C₄	—	10	2	no	—
295	C₄	—	10	4	no	—
296	C₄	C₂×#711	1	2	yes	—
297	C₄	C₂×#711	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	20	4	no	—
301	C₂	C₂²×#711	1	1	yes	characteristic
302	C₂	#66287	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁵
Fitting	C₅×#536	C₂
prime radical	#66164	C₂
hypercenter	#536	D₅
hypocenter	C₅	#3510
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #536

Complements

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

N	H ≅ G/N	type
1. C₅×#536	C₂	semidirect
3. #41×Dic₅	C₂	semidirect
4. C₂×#4093	C₂	semidirect
5. #66146	C₂	semidirect
6. #66066	C₂	semidirect
7. #66015	C₂	semidirect
8. #65992	C₂	semidirect
9. #65977	C₂	semidirect
10. #65958	C₂	semidirect
11. #65855	C₂	semidirect
12. #65851	C₂	semidirect
16. C₅×C₄⋊Q₈	C₂²	semidirect
17. C₅×#124	C₂²	semidirect
19. C₅×C₂²⋊Q₈	C₂²	semidirect
25. Q₈×Dic₅	C₂²	semidirect
26. Q₈×Dic₅	C₂²	semidirect
27. Q₈×Dic₅	C₂²	semidirect
31. D₄×Dic₅	C₂²	semidirect
37. #4072	C₂²	semidirect
39. C₂×C₄×Dic₅	C₂²	semidirect
40. #4044	C₂²	semidirect
41. #4044	C₂²	semidirect
42. #4043	C₂²	semidirect
44. #4038	C₂²	semidirect
45. #4036	C₂²	semidirect
46. #4034	C₂²	semidirect
47. #4032	C₂²	semidirect
48. #4031	C₂²	semidirect
49. #4029	C₂²	semidirect
50. #4027	C₂²	semidirect
54. C₄×Dic₁₀	C₂²	semidirect
105. #536	D₅	semidirect
146. C₄⋊Q₈	C₂×D₅	semidirect
156. #124	C₂×D₅	semidirect
162. C₂²⋊Q₈	C₂×D₅	semidirect
280. C₅	#3510	semidirect

Maximal subgroups

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

subgroup	index
C₅×#536	2
#66164	2
#41×Dic₅	2
C₂×#4093	2
#66146	2
#66066	2
#66015	2
#65992	2
#65977	2
#65958	2
#65855	2
#65851	2
#3510	5