Subgroups — #10611713

Order 640.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65919	C₂	1	1	yes	maximal, characteristic
2	#65901	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#65137	C₂	1	1	yes	maximal, characteristic
5	#65136	C₂	1	1	yes	maximal, characteristic, complemented
6	#65134	C₂	1	1	yes	maximal, characteristic, complemented
7	#65134	C₂	1	1	yes	maximal, characteristic, complemented
8	#65131	C₂	1	1	yes	maximal, characteristic, complemented
9	#65118	C₂	1	1	yes	maximal, characteristic, complemented
10	#65005	C₂	1	1	yes	maximal, characteristic, complemented
11	#65003	C₂	1	1	yes	maximal, characteristic, complemented
12	#65002	C₂	1	1	yes	maximal, characteristic
13	#64999	C₂	1	1	yes	maximal, characteristic, complemented
14	#64986	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#677	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
17	#4093	—	2	1	no	—
18	#4090	—	2	1	no	—
19	C₂×Dic₂₀	C₂²	1	1	yes	characteristic
20	C₂×#677	C₂²	1	1	yes	characteristic
21	C₂×#677	C₂²	1	1	yes	characteristic
22	#4040	—	2	1	no	—
23	#4040	—	2	1	no	—
24	#4039	—	2	1	no	—
25	#4035	—	2	1	no	—
26	#4035	—	2	1	no	—
27	#4033	—	2	1	no	—
28	#4021	C₂²	1	1	yes	characteristic
29	#4021	C₂²	1	1	yes	characteristic
30	#4020	C₂²	1	1	yes	characteristic
31	C₄×D₂₀	C₂²	1	1	yes	characteristic
32	C₄×D₂₀	C₂²	1	1	yes	characteristic
33	#4016	C₂²	1	1	yes	characteristic
34	#4016	C₂²	1	1	yes	characteristic
35	#4015	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
38	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
39	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
40	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₅×#97	C₂²	1	1	yes	characteristic
44	C₅×#97	C₂²	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
46	#3953	C₂²	1	1	yes	characteristic
47	#3953	C₂²	1	1	yes	characteristic
48	#3953	C₂²	1	1	yes	characteristic
49	#3953	C₂²	1	1	yes	characteristic
50	#3953	C₂²	1	1	yes	characteristic
51	#3953	C₂²	1	1	yes	characteristic
52	#3950	C₂²	1	1	yes	characteristic
53	#3949	C₂²	1	1	yes	characteristic
54	#3949	C₂²	1	1	yes	characteristic
55	#3948	C₂²	1	1	yes	characteristic
56	#3948	C₂²	1	1	yes	characteristic
57	#3948	C₂²	1	1	yes	characteristic
58	#3948	C₂²	1	1	yes	characteristic
59	#3371	—	5	1	no	maximal, Sylow
60	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
61	C₂×D₂₀	—	2	1	no	—
62	C₂×D₂₀	—	2	1	no	—
63	C₂×D₂₀	C₂³	1	1	yes	characteristic
64	C₂×D₂₀	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×Dic₁₀	—	2	1	no	—
68	C₂×Dic₁₀	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₀	C₂³	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
76	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
77	C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₄²×C₅	C₂³	1	1	yes	characteristic
80	#685	—	2	1	no	—
81	#685	—	2	1	no	—
82	#685	—	2	1	no	—
83	#685	—	2	1	no	—
84	#685	—	2	1	no	—
85	#685	—	2	1	no	—
86	C₄⋊Dic₅	—	2	1	no	—
87	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
91	#683	—	2	1	no	—
92	#683	—	2	1	no	—
93	#683	—	2	1	no	—
94	#683	—	2	1	no	—
95	#683	—	2	1	no	—
96	#683	—	2	1	no	—
97	#683	—	2	1	no	—
98	#683	—	2	1	no	—
99	C₄×Dic₅	—	2	1	no	—
100	C₄×Dic₅	—	2	1	no	—
101	C₄×Dic₅	—	2	1	no	—
102	C₄×Dic₅	—	2	1	no	—
103	Dic₂₀	—	2	2	no	—
104	#677	—	2	2	no	—
105	#677	—	2	2	no	—
106	#559	—	5	1	no	—
107	#557	—	5	1	no	—
108	#490	—	5	1	no	—
109	#489	—	5	1	no	—
110	#488	—	5	1	no	—
111	#479	—	5	1	no	—
112	#478	—	5	1	no	—
113	#476	—	5	1	no	—
114	#464	—	5	1	no	—
115	#464	—	5	1	no	—
116	#460	—	5	1	no	—
117	#446	D₅	1	1	yes	characteristic, complemented
118	#441	—	5	1	no	—
119	#440	—	5	1	no	—
120	C₄×SD₁₆	—	5	1	no	—
121	C₂²×D₅	—	2	1	no	—
122	C₂²×D₅	—	2	1	no	—
123	C₅×Q₈	—	4	1	no	—
124	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	1	no	—
133	C₂×Dic₅	—	2	1	no	—
134	C₂×Dic₅	—	2	1	no	—
135	C₂×Dic₅	—	2	1	no	—
136	C₂×Dic₅	—	2	1	no	—
137	D₂₀	—	2	1	no	—
138	D₂₀	—	2	1	no	—
139	D₂₀	—	2	1	no	—
140	D₂₀	—	2	1	no	—
141	D₂₀	—	4	2	no	—
142	D₂₀	—	8	1	no	—
143	C₄×D₅	—	8	1	no	—
144	C₄×D₅	—	8	1	no	—
145	Dic₁₀	—	2	1	no	—
146	Dic₁₀	—	2	1	no	—
147	Dic₁₀	—	2	1	no	—
148	Dic₁₀	—	4	2	no	—
149	Dic₁₀	#41	1	2	yes	—
150	C₅×C₈	—	2	1	no	—
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	2	1	no	—
154	C₂×Q₁₆	—	5	1	no	—
155	C₂×SD₁₆	—	5	1	no	—
156	C₂×SD₁₆	—	5	1	no	—
157	C₄⋊Q₈	—	5	1	no	—
158	C₄⋊Q₈	—	10	1	no	—
159	#127	—	5	1	no	—
160	#125	—	5	1	no	—
161	#125	—	5	1	no	—
162	#125	—	10	1	no	—
163	#125	—	10	1	no	—
164	#124	—	5	1	no	—
165	#124	—	5	1	no	—
166	#124	—	10	1	no	—
167	#123	—	10	1	no	—
168	#123	—	10	1	no	—
169	C₄×Q₈	—	5	1	no	—
170	C₄×Q₈	—	5	1	no	—
171	C₄×Q₈	—	5	1	no	—
172	C₄×Q₈	—	5	1	no	—
173	C₄×Q₈	—	10	1	no	—
174	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
175	C₄×D₄	—	5	1	no	—
176	C₄×D₄	—	5	1	no	—
177	C₄×D₄	—	10	1	no	—
178	#107	—	5	1	no	—
179	#106	—	5	1	no	—
180	#106	—	5	1	no	—
181	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
182	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
183	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
184	#103	—	5	1	no	—
185	#103	—	5	1	no	—
186	#103	—	5	1	no	—
187	#103	—	5	1	no	—
188	#102	—	5	1	no	—
189	#102	—	5	1	no	—
190	#102	—	5	1	no	—
191	#102	—	5	1	no	—
192	#102	—	5	1	no	—
193	#102	—	5	1	no	—
194	#97	C₂×D₅	1	1	yes	characteristic
195	#97	C₂×D₅	1	1	yes	characteristic
196	C₄×C₈	C₂×D₅	1	1	yes	characteristic
197	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₂×D₅	—	4	1	no	—
199	C₂×D₅	—	4	1	no	—
200	C₂×D₅	—	4	1	no	—
201	C₂×D₅	—	4	1	no	—
202	C₂×D₅	—	4	1	no	—
203	C₂×D₅	—	4	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	C₂×#41	1	1	yes	characteristic
211	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
212	Dic₅	—	2	2	no	—
213	Dic₅	—	4	1	no	—
214	Dic₅	—	4	1	no	—
215	Dic₅	—	4	1	no	—
216	Dic₅	—	4	1	no	—
217	Dic₅	—	4	1	no	—
218	C₂×Q₈	—	5	1	no	—
219	C₂×Q₈	—	5	1	no	—
220	C₂×Q₈	—	10	1	no	—
221	C₂×Q₈	D₂₀	1	1	yes	characteristic
222	C₂×D₄	—	5	1	no	—
223	C₂×D₄	—	5	1	no	—
224	C₂×D₄	—	10	1	no	—
225	C₂×D₄	—	10	1	no	—
226	C₂²×C₄	—	10	1	no	—
227	C₂²×C₄	—	10	1	no	—
228	Q₁₆	—	10	2	no	—
229	SD₁₆	—	10	2	no	—
230	SD₁₆	—	10	2	no	—
231	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	5	1	no	—
236	C₄⋊C₄	—	5	1	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	1	no	—
239	C₄⋊C₄	—	10	1	no	—
240	C₄⋊C₄	—	10	1	no	—
241	C₄⋊C₄	—	10	1	no	—
242	C₄⋊C₄	—	10	1	no	—
243	C₄⋊C₄	—	10	1	no	—
244	C₄⋊C₄	—	10	1	no	—
245	C₄⋊C₄	—	10	1	no	—
246	C₄⋊C₄	—	10	1	no	—
247	C₄⋊C₄	—	10	1	no	—
248	C₄⋊C₄	D₂₀	1	1	yes	characteristic
249	C₄⋊C₄	D₂₀	1	1	yes	characteristic
250	C₄⋊C₄	D₂₀	1	1	yes	characteristic
251	C₂²⋊C₄	—	10	1	no	—
252	C₂²⋊C₄	—	10	1	no	—
253	C₂²⋊C₄	—	10	1	no	—
254	C₂²⋊C₄	—	10	1	no	—
255	C₂²⋊C₄	—	10	1	no	—
256	C₂²⋊C₄	—	10	1	no	—
257	C₄²	—	10	1	no	—
258	C₄²	—	10	1	no	—
259	C₄²	—	10	1	no	—
260	C₄²	—	10	1	no	—
261	C₄²	C₂²×D₅	1	1	yes	characteristic
262	C₄²	C₂²×D₅	1	1	yes	characteristic
263	C₄²	C₂²×D₅	1	1	yes	characteristic
264	C₂×C₅	#548	1	1	yes	characteristic
265	C₂×C₅	#575	1	1	yes	characteristic
266	C₂×C₅	#577	1	1	yes	characteristic
267	D₅	—	8	1	no	—
268	D₅	—	8	1	no	—
269	C₂³	—	10	1	no	—
270	C₂³	—	10	1	no	—
271	Q₈	—	4	1	no	—
272	Q₈	—	5	2	no	—
273	Q₈	—	10	1	no	—
274	Q₈	—	10	1	no	—
275	Q₈	—	10	1	no	—
276	Q₈	—	20	2	no	—
277	D₄	—	10	1	no	—
278	D₄	—	10	1	no	—
279	D₄	—	10	1	no	—
280	D₄	—	10	1	no	—
281	D₄	—	20	2	no	—
282	D₄	—	40	1	no	—
283	C₂×C₄	—	10	1	no	—
284	C₂×C₄	—	10	1	no	—
285	C₂×C₄	—	10	1	no	—
286	C₂×C₄	—	10	1	no	—
287	C₂×C₄	—	10	1	no	—
288	C₂×C₄	—	10	1	no	—
289	C₂×C₄	—	40	1	no	—
290	C₂×C₄	—	40	1	no	—
291	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
292	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
293	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
294	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
295	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
296	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₅	#3371	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	20	1	no	—
304	C₂²	—	20	1	no	—
305	C₂²	—	20	1	no	—
306	C₂²	—	20	1	no	—
307	C₂²	—	20	1	no	—
308	C₂²	—	20	1	no	—
309	C₂²	C₂²×D₂₀	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	10	2	no	—
317	C₄	—	20	1	no	—
318	C₄	—	20	1	no	—
319	C₄	—	20	1	no	—
320	C₄	—	20	1	no	—
321	C₄	—	20	1	no	—
322	C₄	#4147	1	1	yes	characteristic
323	C₄	D₅×#41	1	1	yes	characteristic
324	C₂	—	40	1	no	—
325	C₂	—	40	1	no	—
326	C₂	#65910	1	1	yes	characteristic
327	C₂	#66082	1	1	yes	characteristic
328	C₂	#66085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#446	C₂
socle	C₂²×C₅	C₂²×D₄
prime radical	#4020	C₂²
hypercenter	#446	D₅
hypocenter	C₅	#3371
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #65901	C₂	semidirect
3. C₅×#446	C₂	semidirect
5. #65136	C₂	semidirect
6. #65134	C₂	semidirect
7. #65134	C₂	semidirect
8. #65131	C₂	semidirect
9. #65118	C₂	semidirect
10. #65005	C₂	semidirect
11. #65003	C₂	semidirect
13. #64999	C₂	semidirect
14. #64986	C₂	semidirect
15. C₄×#677	C₂	semidirect
117. #446	D₅	semidirect
302. C₅	#3371	semidirect

Maximal subgroups

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

subgroup	index
#65919	2
#65901	2
C₅×#446	2
#65137	2
#65136	2
#65134	2
#65131	2
#65118	2
#65005	2
#65003	2
#65002	2
#64999	2
#64986	2
C₄×#677	2
#3371	5