Subgroups — #10610286

Order 640.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4053	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#65858	C₂	1	1	yes	maximal, characteristic
4	C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	#65399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3952	C₂	1	1	yes	maximal, characteristic, complemented
7	#65288	C₂	1	1	yes	maximal, characteristic, complemented
8	#64995	C₂	1	2	yes	maximal, complemented
9	#64995	C₂	1	2	yes	maximal, complemented
10	#64977	C₂	1	2	yes	maximal, complemented
11	C₈×D₂₀	C₂	1	2	yes	maximal, complemented
12	C₂×#709	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#117	C₂²	1	1	yes	characteristic
17	C₂×#681	C₂²	1	1	yes	characteristic
18	C₂²×#168	C₂²	1	1	yes	characteristic
19	#4053	—	2	4	no	—
20	D₅×M₁₆	—	2	4	no	—
21	C₂×#676	C₂²	1	2	yes	—
22	C₂×#676	C₂²	1	2	yes	complemented
23	C₂×C₈×D₅	C₂²	1	2	yes	—
24	C₂×C₈×D₅	C₂²	1	2	yes	complemented
25	#4043	C₄	1	1	yes	characteristic
26	#4043	C₄	1	1	yes	characteristic
27	#4041	C₄	1	1	yes	characteristic
28	#4041	C₄	1	1	yes	characteristic
29	#4032	C₄	1	2	yes	—
30	#4028	C₄	1	2	yes	—
31	C₄×D₂₀	C₂²	1	2	yes	—
32	C₄×D₂₀	C₂²	1	2	yes	—
33	C₅×#97	C₂²	1	2	yes	complemented
34	C₄×C₅×C₈	C₂²	1	2	yes	complemented
35	#3952	C₂²	1	1	yes	characteristic
36	#3952	C₂²	1	1	yes	characteristic
37	#3952	C₂²	1	1	yes	characteristic
38	#3952	C₂²	1	1	yes	characteristic
39	#3952	C₂²	1	2	yes	complemented
40	#3952	C₂²	1	2	yes	complemented
41	C₄⋊#168	C₂²	1	2	yes	complemented
42	C₄⋊#168	C₂²	1	2	yes	complemented
43	#2933	—	5	1	no	maximal, Sylow
44	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
47	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
48	#709	—	2	2	no	—
49	#709	—	2	2	no	—
50	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
51	C₂×D₂₀	C₂×C₄	1	2	yes	—
52	C₂×D₂₀	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₅	—	2	2	no	—
54	C₂×C₄×D₅	—	2	2	no	—
55	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₅	C₂³	1	2	yes	—
60	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
61	C₅×M₁₆	D₄	1	4	yes	complemented
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂³	1	2	yes	—
66	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₄²×C₅	C₂³	1	2	yes	—
70	#685	C₂×C₄	1	2	yes	—
71	#685	C₂×C₄	1	2	yes	—
72	#685	C₂×C₄	1	2	yes	—
73	#685	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
75	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
76	#681	—	2	2	no	—
77	C₂×#168	—	2	2	no	—
78	C₂×#168	C₂³	1	1	yes	characteristic
79	C₂×#168	C₂³	1	1	yes	characteristic
80	C₂×#168	C₂³	1	2	yes	—
81	#676	—	2	4	no	—
82	#676	—	2	4	no	—
83	C₈×D₅	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	C₂×#131	—	5	1	no	—
86	C₂²×M₁₆	—	5	1	no	—
87	#525	—	5	1	no	—
88	#436	—	5	2	no	—
89	#435	—	5	2	no	—
90	#435	—	5	2	no	—
91	C₈×D₄	—	5	2	no	—
92	#424	—	5	1	no	—
93	#408	—	5	1	no	—
94	C₂×C₂²⋊C₈	—	5	1	no	—
95	#405	D₅	1	1	yes	characteristic, complemented
96	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×D₅	—	2	2	no	—
98	C₂²×D₅	—	2	2	no	—
99	C₂²×D₅	—	4	1	no	—
100	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
101	C₂²×D₅	C₂²×C₄	1	2	yes	—
102	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₅	#41	1	2	yes	—
108	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
109	C₂²⋊D₅	—	2	4	no	—
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	C₂²×C₄	1	2	yes	—
112	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
114	D₂₀	—	2	4	no	—
115	D₂₀	—	2	2	no	—
116	D₂₀	—	4	2	no	—
117	C₄×D₅	—	2	2	no	—
118	C₄×D₅	—	2	2	no	—
119	C₄×D₅	—	2	1	no	—
120	C₄×D₅	—	2	2	no	—
121	C₄×D₅	—	2	1	no	—
122	C₄×D₅	—	2	1	no	—
123	C₄×D₅	—	2	1	no	—
124	C₄×D₅	—	2	2	no	—
125	Dic₁₀	—	2	2	no	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	C₂×D₄	1	4	yes	—
129	#168	—	2	1	no	—
130	#168	—	2	2	no	—
131	#168	—	2	1	no	—
132	C₂×#41	—	5	1	no	—
133	C₂³×C₄	—	5	1	no	—
134	#131	—	10	4	no	—
135	C₂×M₁₆	—	5	1	no	—
136	C₂×M₁₆	—	5	2	no	—
137	C₂×M₁₆	—	5	2	no	—
138	C₂×M₁₆	—	10	4	no	—
139	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
140	C₂²×C₈	—	5	1	no	—
141	C₂²×C₈	—	5	2	no	—
142	C₂²×C₈	—	5	2	no	—
143	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
144	#123	—	5	2	no	—
145	C₂²⋊Q₈	—	5	1	no	—
146	C₂²⋊Q₈	—	5	1	no	—
147	#121	—	5	1	no	—
148	#121	—	5	1	no	—
149	#120	—	5	2	no	—
150	C₄×D₄	—	5	2	no	—
151	C₄×D₄	—	5	2	no	—
152	#117	C₂×D₅	1	1	yes	characteristic
153	C₄⋊C₈	—	5	2	no	—
154	C₄⋊C₈	—	5	2	no	—
155	C₂²⋊C₈	—	5	1	no	—
156	C₂²⋊C₈	—	5	1	no	—
157	C₂²⋊C₈	—	5	1	no	—
158	C₂²⋊C₈	—	5	1	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	—	5	2	no	—
161	#97	C₂×D₅	1	2	yes	complemented
162	C₄×C₈	C₂×D₅	1	2	yes	complemented
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₄×D₄	1	2	yes	—
166	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₂×D₅	—	2	1	no	—
168	C₂×D₅	—	2	1	no	—
169	C₂×D₅	—	2	1	no	—
170	C₂×D₅	—	2	2	no	—
171	C₂×D₅	—	2	2	no	—
172	C₂×D₅	—	2	1	no	—
173	C₂×D₅	—	2	1	no	—
174	C₂×D₅	—	2	2	no	—
175	C₂×D₅	—	4	4	no	—
176	C₂×D₅	#131	1	2	yes	complemented
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	C₄×D₄	1	2	yes	—
180	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₅	C₂×#41	1	1	yes	characteristic
182	Dic₅	—	2	2	no	—
183	Dic₅	—	2	1	no	—
184	Dic₅	—	2	1	no	—
185	C₂⁴	—	5	1	no	—
186	#41	—	10	2	no	—
187	#41	—	10	2	no	—
188	C₂×Q₈	—	5	1	no	—
189	C₂×D₄	—	5	2	no	—
190	C₂×D₄	—	5	2	no	—
191	C₂×D₄	—	5	1	no	—
192	C₂×D₄	—	5	2	no	—
193	C₂²×C₄	—	5	1	no	—
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	2	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	10	2	no	—
200	C₂²×C₄	—	10	2	no	—
201	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
202	M₁₆	—	10	2	no	—
203	M₁₆	—	10	4	no	—
204	M₁₆	—	10	4	no	—
205	M₁₆	D₂₀	1	4	yes	complemented
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	10	2	no	—
211	C₂×C₈	—	10	4	no	—
212	C₂×C₈	—	10	4	no	—
213	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
214	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₅	1	2	yes	—
216	C₄⋊C₄	—	5	2	no	—
217	C₄⋊C₄	—	5	2	no	—
218	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
219	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
220	C₂²⋊C₄	—	5	2	no	—
221	C₂²⋊C₄	—	5	2	no	—
222	C₂²⋊C₄	—	5	2	no	—
223	C₂²⋊C₄	—	5	2	no	—
224	C₂²⋊C₄	C₄×D₅	1	2	yes	—
225	C₄²	C₂²×D₅	1	2	yes	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂×C₅	C₂×#131	1	1	yes	characteristic
229	C₂×C₅	#568	1	1	yes	characteristic
230	D₅	—	2	2	no	—
231	D₅	—	4	2	no	—
232	D₅	—	4	1	no	—
233	C₂³	—	5	1	no	—
234	C₂³	—	5	1	no	—
235	C₂³	—	5	2	no	—
236	C₂³	—	10	2	no	—
237	C₂³	—	10	2	no	—
238	C₂³	—	20	1	no	—
239	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
240	Q₈	—	10	2	no	—
241	D₄	—	10	4	no	—
242	D₄	—	10	4	no	—
243	D₄	—	10	2	no	—
244	D₄	—	20	2	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	10	2	no	—
249	C₂×C₄	—	10	1	no	—
250	C₂×C₄	—	10	2	no	—
251	C₂×C₄	—	10	1	no	—
252	C₂×C₄	—	10	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	10	2	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
258	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
260	C₂×C₄	C₂×D₂₀	1	2	yes	—
261	C₂×C₄	#709	1	2	yes	—
262	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	10	1	no	—
266	C₈	—	10	1	no	—
267	C₈	—	10	2	no	—
268	C₈	C₂×D₂₀	1	4	yes	—
269	C₅	#2933	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	5	2	no	—
273	C₂²	—	10	1	no	—
274	C₂²	—	10	1	no	—
275	C₂²	—	10	1	no	—
276	C₂²	—	10	2	no	—
277	C₂²	—	10	2	no	—
278	C₂²	—	10	1	no	—
279	C₂²	—	10	1	no	—
280	C₂²	—	10	2	no	—
281	C₂²	—	20	4	no	—
282	C₂²	C₄×D₂₀	1	2	yes	—
283	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	10	1	no	—
287	C₄	—	10	1	no	—
288	C₄	—	10	2	no	—
289	C₄	C₄×D₂₀	1	2	yes	—
290	C₄	C₂²×D₂₀	1	1	yes	characteristic
291	C₄	C₂×#709	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	10	2	no	—
294	C₂	—	20	1	no	—
295	C₂	—	20	2	no	—
296	C₂	C₂×C₄×D₂₀	1	1	yes	characteristic
297	C₂	D₅×#131	1	1	yes	characteristic
298	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#405	C₂
prime radical	#65858	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#405	D₅
hypocenter	C₅	#2933
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#4053	C₂	semidirect
2. C₂×D₅×M₁₆	C₂	semidirect
4. C₅×#405	C₂	semidirect
5. #65399	C₂	semidirect
6. C₂×#3952	C₂	semidirect
7. #65288	C₂	semidirect
8. #64995	C₂	semidirect
9. #64995	C₂	semidirect
10. #64977	C₂	semidirect
11. C₈×D₂₀	C₂	semidirect
22. C₂×#676	C₂²	semidirect
24. C₂×C₈×D₅	C₂²	semidirect
33. C₅×#97	C₂²	semidirect
34. C₄×C₅×C₈	C₂²	semidirect
39. #3952	C₂²	semidirect
40. #3952	C₂²	semidirect
41. C₄⋊#168	C₂²	semidirect
42. C₄⋊#168	C₂²	semidirect
61. C₅×M₁₆	D₄	semidirect
95. #405	D₅	semidirect
161. #97	C₂×D₅	semidirect
162. C₄×C₈	C₂×D₅	semidirect
176. C₂×D₅	#131	semidirect
205. M₁₆	D₂₀	semidirect
269. C₅	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4053	2
C₂×D₅×M₁₆	2
#65858	2
C₅×#405	2
#65399	2
C₂×#3952	2
#65288	2
#64995	2
#64977	2
C₈×D₂₀	2
#2933	5