Subgroups — #10618124

Order 640.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	#66164	C₂	1	2	yes	maximal, complemented
3	#66161	C₂	1	2	yes	maximal, complemented
4	#66066	C₂	1	2	yes	maximal, complemented
5	#66052	C₂	1	2	yes	maximal, complemented
6	#66015	C₂	1	4	yes	maximal, complemented
7	#65951	C₂	1	4	yes	maximal, complemented
8	#65940	C₂	1	4	yes	maximal, complemented
9	#65899	C₂	1	4	yes	maximal, complemented
10	#65897	C₂	1	2	yes	maximal, complemented
11	#65889	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
13	#65804	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₅×#41	C₂²	1	2	yes	—
15	C₂×#711	C₂²	1	4	yes	complemented
16	C₂²×D₂₀	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₅	C₂²	1	2	yes	—
18	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₅×#127	C₂²	1	1	yes	characteristic, complemented
20	C₅×#124	C₂²	1	2	yes	complemented
21	C₅×#121	C₂²	1	2	yes	—
22	C₅×#121	C₂²	1	2	yes	—
23	C₄×C₅×D₄	C₂²	1	2	yes	complemented
24	C₄×C₅×D₄	C₂²	1	2	yes	complemented
25	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
26	#4093	C₂²	1	4	yes	—
27	Q₈×Dic₅	C₂²	1	4	yes	complemented
28	#4086	C₂²	1	4	yes	—
29	#4085	C₂²	1	8	yes	complemented
30	#4084	C₂²	1	4	yes	complemented
31	#4083	C₂²	1	8	yes	—
32	D₄×Dic₅	C₂²	1	8	yes	—
33	D₄×Dic₅	C₂²	1	4	yes	complemented
34	#4076	C₂²	1	4	yes	—
35	#4076	C₂²	1	4	yes	—
36	C₂×#685	C₂²	1	2	yes	—
37	#4072	C₂²	1	2	yes	—
38	#4072	C₂²	1	4	yes	complemented
39	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅	C₂²	1	4	yes	complemented
41	#4070	C₂²	1	2	yes	—
42	#4070	C₂²	1	2	yes	—
43	#4044	C₂²	1	8	yes	complemented
44	#4043	C₂²	1	4	yes	complemented
45	#4038	C₂²	1	4	yes	complemented
46	#4032	C₂²	1	8	yes	complemented
47	#4031	C₂²	1	8	yes	complemented
48	#4029	C₂²	1	8	yes	complemented
49	#4027	C₂²	1	8	yes	complemented
50	C₄×D₂₀	C₂²	1	2	yes	—
51	C₄×D₂₀	C₂²	1	2	yes	—
52	C₄×D₂₀	C₂²	1	2	yes	—
53	C₄×D₂₀	C₂²	1	2	yes	—
54	#4016	C₂²	1	2	yes	complemented
55	#4015	C₂²	1	1	yes	characteristic, complemented
56	#4015	C₂²	1	1	yes	characteristic, complemented
57	C₄×Dic₁₀	C₂²	1	2	yes	complemented
58	C₄×Dic₁₀	C₂²	1	2	yes	complemented
59	#3506	—	5	1	no	maximal, Sylow
60	C₂³×D₅	C₂³	1	2	yes	—
61	C₅×#41	—	2	8	no	—
62	C₂×C₅×Q₈	C₂³	1	2	yes	—
63	C₂×C₅×D₄	C₂³	1	2	yes	—
64	C₂×C₅×D₄	C₂³	1	4	yes	—
65	C₂×C₅×D₄	C₂³	1	2	yes	—
66	C₂×C₅×D₄	C₂³	1	2	yes	—
67	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂³	1	4	yes	—
69	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
72	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
73	C₂²×Dic₅	C₂³	1	2	yes	—
74	C₂²×Dic₅	C₂³	1	8	yes	—
75	#711	—	4	8	no	—
76	C₂×D₂₀	—	2	4	no	—
77	C₂×D₂₀	C₂³	1	2	yes	—
78	C₂×D₂₀	C₂³	1	2	yes	—
79	C₂×C₄×D₅	—	4	4	no	—
80	C₂×C₄×D₅	C₂³	1	4	yes	—
81	C₂×C₄×D₅	C₂³	1	4	yes	—
82	C₂×Dic₁₀	C₂³	1	2	yes	—
83	C₂×Dic₁₀	C₂³	1	2	yes	—
84	C₅×C₄⋊C₄	C₂³	1	2	yes	—
85	C₅×C₄⋊C₄	C₂³	1	2	yes	—
86	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
87	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
88	C₄²×C₅	C₂³	1	1	yes	characteristic
89	C₄²×C₅	C₂³	1	1	yes	characteristic
90	C₄²×C₅	C₂³	1	1	yes	characteristic
91	C₄²×C₅	C₂³	1	1	yes	characteristic
92	C₂²⋊Dic₅	C₂³	1	8	yes	—
93	C₂²⋊Dic₅	C₂³	1	4	yes	—
94	#685	C₂³	1	4	yes	—
95	#685	C₂³	1	4	yes	—
96	#685	C₂³	1	8	yes	—
97	#685	C₂³	1	8	yes	—
98	C₄⋊Dic₅	C₂³	1	2	yes	—
99	C₄⋊Dic₅	C₂³	1	2	yes	—
100	C₄⋊Dic₅	C₂³	1	2	yes	—
101	C₄⋊Dic₅	C₂³	1	2	yes	—
102	C₄⋊Dic₅	C₂³	1	2	yes	—
103	C₄⋊Dic₅	C₂³	1	2	yes	—
104	C₄⋊Dic₅	C₂³	1	8	yes	—
105	#683	C₂³	1	4	yes	—
106	#683	C₂³	1	4	yes	—
107	C₄×Dic₅	C₂³	1	4	yes	—
108	C₄×Dic₅	C₂³	1	8	yes	—
109	#559	—	5	4	no	—
110	#556	—	5	2	no	—
111	#555	—	5	2	no	—
112	#553	—	5	4	no	—
113	#550	—	5	2	no	—
114	#547	—	5	2	no	—
115	#538	—	5	4	no	—
116	#535	—	5	2	no	—
117	#533	—	5	1	no	—
118	#532	D₅	1	1	yes	characteristic, complemented
119	#525	—	5	4	no	—
120	#520	—	5	2	no	—
121	C₂×C₄×D₄	—	5	1	no	—
122	C₂³×C₅	C₂⁴	1	4	yes	—
123	C₂³×C₅	C₂⁴	1	1	yes	characteristic
124	C₂²×D₅	—	2	4	no	—
125	C₂²×D₅	—	4	2	no	—
126	C₂²×D₅	—	4	2	no	—
127	C₂²×D₅	C₂⁴	1	4	yes	—
128	C₅×Q₈	—	2	4	no	—
129	C₅×D₄	—	2	8	no	—
130	C₅×D₄	—	2	4	no	—
131	C₅×D₄	—	2	4	no	—
132	C₅×D₄	—	4	2	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	#41	1	4	yes	complemented
137	C₂×C₄×C₅	C₂⁴	1	4	yes	—
138	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
144	C₂²⋊D₅	—	4	8	no	—
145	C₂²⋊D₅	—	4	8	no	—
146	C₂×Dic₅	—	4	2	no	—
147	C₂×Dic₅	—	4	8	no	—
148	C₂×Dic₅	C₂⁴	1	4	yes	—
149	C₂×Dic₅	C₂⁴	1	4	yes	—
150	C₂×Dic₅	C₂⁴	1	4	yes	—
151	D₂₀	—	2	4	no	—
152	D₂₀	—	2	4	no	—
153	C₄×D₅	—	2	8	no	—
154	C₄×D₅	—	4	4	no	—
155	Dic₁₀	—	2	4	no	—
156	Dic₁₀	—	4	2	no	—
157	C₂×#41	—	5	4	no	—
158	C₂×#41	C₂×D₅	1	2	yes	—
159	C₂²×D₄	—	5	1	no	—
160	C₂³×C₄	—	5	2	no	—
161	C₄⋊Q₈	—	5	1	no	—
162	C₄⋊Q₈	—	5	1	no	—
163	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic, complemented
164	#127	—	5	4	no	—
165	#127	C₂×D₅	1	1	yes	characteristic, complemented
166	#126	—	5	8	no	—
167	#125	—	5	2	no	—
168	#124	—	5	4	no	—
169	#124	—	5	8	no	—
170	#124	C₂×D₅	1	2	yes	complemented
171	#123	—	5	8	no	—
172	#123	—	5	8	no	—
173	C₂²⋊Q₈	—	5	2	no	—
174	C₂²⋊Q₈	—	5	2	no	—
175	C₂²⋊Q₈	—	5	4	no	—
176	#121	—	5	4	no	—
177	#121	—	5	8	no	—
178	#121	—	5	4	no	—
179	#121	—	5	4	no	—
180	#121	C₂×D₅	1	2	yes	—
181	#121	C₂×D₅	1	2	yes	—
182	#120	—	5	8	no	—
183	C₄×Q₈	—	5	4	no	—
184	C₄×Q₈	—	5	2	no	—
185	C₄×Q₈	—	5	2	no	—
186	C₄×D₄	—	5	8	no	—
187	C₄×D₄	—	5	4	no	—
188	C₄×D₄	—	5	8	no	—
189	C₄×D₄	—	5	2	no	—
190	C₄×D₄	—	5	2	no	—
191	C₄×D₄	—	5	2	no	—
192	C₄×D₄	—	5	2	no	—
193	C₄×D₄	C₂×D₅	1	2	yes	complemented
194	C₄×D₄	C₂×D₅	1	2	yes	complemented
195	#117	—	5	2	no	—
196	#117	—	5	4	no	—
197	#117	—	5	4	no	—
198	C₂×C₄⋊C₄	—	5	1	no	—
199	C₂×C₄⋊C₄	—	5	4	no	—
200	C₂×C₂²⋊C₄	—	5	2	no	—
201	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	—	2	4	no	—
207	C₂²×C₅	C₂×#41	1	2	yes	—
208	C₂²×C₅	C₂⁵	1	1	yes	characteristic
209	C₂×D₅	—	2	4	no	—
210	C₂×D₅	—	2	4	no	—
211	C₂×D₅	—	2	4	no	—
212	C₂×D₅	—	4	8	no	—
213	C₄×C₅	—	2	4	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	C₂×#41	1	2	yes	—
218	C₄×C₅	C₂×#41	1	2	yes	—
219	C₄×C₅	H(5,𝔽₂)	1	2	yes	—
220	Dic₅	—	2	4	no	—
221	Dic₅	—	2	4	no	—
222	Dic₅	—	2	4	no	—
223	C₂⁴	—	5	2	no	—
224	#41	—	2	8	no	—
225	#41	—	20	8	no	—
226	C₂×Q₈	—	5	2	no	—
227	C₂×Q₈	—	5	2	no	—
228	C₂×Q₈	C₂²×D₅	1	2	yes	—
229	C₂×D₄	—	5	8	no	—
230	C₂×D₄	—	5	8	no	—
231	C₂×D₄	—	5	2	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	10	4	no	—
234	C₂×D₄	C₂²×D₅	1	4	yes	—
235	C₂×D₄	C₂²×D₅	1	2	yes	—
236	C₂×D₄	C₂²×D₅	1	2	yes	—
237	C₂×D₄	C₂²×D₅	1	2	yes	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	5	8	no	—
240	C₂²×C₄	—	5	4	no	—
241	C₂²×C₄	—	5	4	no	—
242	C₂²×C₄	—	20	4	no	—
243	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
244	C₂²×C₄	C₂²×D₅	1	4	yes	—
245	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
247	C₄⋊C₄	—	5	2	no	—
248	C₄⋊C₄	—	5	2	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	5	8	no	—
251	C₄⋊C₄	—	5	2	no	—
252	C₄⋊C₄	—	5	2	no	—
253	C₄⋊C₄	—	5	4	no	—
254	C₄⋊C₄	—	5	4	no	—
255	C₄⋊C₄	—	5	2	no	—
256	C₄⋊C₄	C₂²×D₅	1	2	yes	—
257	C₄⋊C₄	C₂²×D₅	1	2	yes	—
258	C₂²⋊C₄	—	5	4	no	—
259	C₂²⋊C₄	—	5	8	no	—
260	C₂²⋊C₄	—	5	4	no	—
261	C₂²⋊C₄	—	5	4	no	—
262	C₂²⋊C₄	—	5	8	no	—
263	C₂²⋊C₄	—	5	8	no	—
264	C₂²⋊C₄	C₂²×D₅	1	4	yes	—
265	C₂²⋊C₄	C₂²×D₅	1	4	yes	—
266	C₄²	—	5	8	no	—
267	C₄²	—	5	4	no	—
268	C₄²	C₂²×D₅	1	1	yes	characteristic
269	C₄²	C₂²×D₅	1	1	yes	characteristic
270	C₄²	C₂²×D₅	1	1	yes	characteristic
271	C₄²	C₂²×D₅	1	1	yes	characteristic
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	4	4	no	—
274	C₂×C₅	C₂²×#41	1	1	yes	characteristic
275	C₂×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₅	#585	1	1	yes	characteristic
277	D₅	—	4	4	no	—
278	C₂³	—	5	4	no	—
279	C₂³	—	10	4	no	—
280	C₂³	—	20	2	no	—
281	C₂³	—	20	2	no	—
282	C₂³	C₂³×D₅	1	4	yes	—
283	C₂³	C₂³×D₅	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	10	4	no	—
286	Q₈	—	20	2	no	—
287	D₄	—	2	8	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	D₄	—	10	4	no	—
292	D₄	—	10	4	no	—
293	D₄	—	20	8	no	—
294	D₄	—	20	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	5	4	no	—
299	C₂×C₄	—	5	4	no	—
300	C₂×C₄	—	5	4	no	—
301	C₂×C₄	—	10	8	no	—
302	C₂×C₄	—	20	4	no	—
303	C₂×C₄	—	20	2	no	—
304	C₂×C₄	—	20	8	no	—
305	C₂×C₄	#711	1	4	yes	complemented
306	C₂×C₄	C₂³×D₅	1	4	yes	—
307	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
313	C₅	#3506	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	10	4	no	—
320	C₂²	—	10	4	no	—
321	C₂²	—	10	4	no	—
322	C₂²	—	20	8	no	—
323	C₂²	C₂×#711	1	2	yes	—
324	C₂²	C₂⁴×D₅	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	10	4	no	—
330	C₄	—	10	4	no	—
331	C₄	—	10	4	no	—
332	C₄	C₂×#711	1	2	yes	—
333	C₄	C₂×#711	1	2	yes	—
334	C₄	#4149	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	20	4	no	—
338	C₂	C₂²×#711	1	1	yes	characteristic
339	C₂	C₂×#4149	1	1	yes	characteristic
340	C₂	#66283	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁵
Fitting	C₅×#532	C₂
hypercenter	#532	D₅
hypocenter	C₅	#3506
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₅×#532	C₂	semidirect
2. #66164	C₂	semidirect
3. #66161	C₂	semidirect
4. #66066	C₂	semidirect
5. #66052	C₂	semidirect
6. #66015	C₂	semidirect
7. #65951	C₂	semidirect
8. #65940	C₂	semidirect
9. #65899	C₂	semidirect
10. #65897	C₂	semidirect
11. #65889	C₂	semidirect
12. C₂×C₄×D₂₀	C₂	semidirect
13. #65804	C₂	semidirect
15. C₂×#711	C₂²	semidirect
18. C₅×C₄⋊Q₈	C₂²	semidirect
19. C₅×#127	C₂²	semidirect
20. C₅×#124	C₂²	semidirect
23. C₄×C₅×D₄	C₂²	semidirect
24. C₄×C₅×D₄	C₂²	semidirect
27. Q₈×Dic₅	C₂²	semidirect
29. #4085	C₂²	semidirect
30. #4084	C₂²	semidirect
33. D₄×Dic₅	C₂²	semidirect
38. #4072	C₂²	semidirect
40. C₂×C₄⋊Dic₅	C₂²	semidirect
43. #4044	C₂²	semidirect
44. #4043	C₂²	semidirect
45. #4038	C₂²	semidirect
46. #4032	C₂²	semidirect
47. #4031	C₂²	semidirect
48. #4029	C₂²	semidirect
49. #4027	C₂²	semidirect
54. #4016	C₂²	semidirect
55. #4015	C₂²	semidirect
56. #4015	C₂²	semidirect
57. C₄×Dic₁₀	C₂²	semidirect
58. C₄×Dic₁₀	C₂²	semidirect
118. #532	D₅	semidirect
136. C₂×C₄×C₅	#41	semidirect
163. C₄⋊Q₈	C₂×D₅	semidirect
165. #127	C₂×D₅	semidirect
170. #124	C₂×D₅	semidirect
193. C₄×D₄	C₂×D₅	semidirect
194. C₄×D₄	C₂×D₅	semidirect
305. C₂×C₄	#711	semidirect
313. C₅	#3506	semidirect

Maximal subgroups

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

subgroup	index
C₅×#532	2
#66164	2
#66161	2
#66066	2
#66052	2
#66015	2
#65951	2
#65940	2
#65899	2
#65897	2
#65889	2
C₂×C₄×D₂₀	2
#65804	2
#3506	5