Subgroups — #10611628

Order 640.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66102	C₂	1	1	yes	maximal, characteristic
2	D₅×#131	C₂	1	2	yes	maximal, complemented
3	C₅×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	#65485	C₂	1	1	yes	maximal, characteristic, complemented
5	#65434	C₂	1	2	yes	maximal, complemented
6	#65184	C₂	1	1	yes	maximal, characteristic, complemented
7	#65182	C₂	1	1	yes	maximal, characteristic, complemented
8	#65113	C₂	1	2	yes	maximal, complemented
9	#65110	C₂	1	2	yes	maximal, complemented
10	#65006	C₂	1	1	yes	maximal, characteristic, complemented
11	#64997	C₂	1	1	yes	maximal, characteristic, complemented
12	D₅×#41	C₂²	1	2	yes	—
13	C₅×#135	C₂²	1	1	yes	characteristic
14	C₅×#131	C₂²	1	2	yes	complemented
15	#4096	C₂²	1	2	yes	—
16	#4094	C₂²	1	2	yes	complemented
17	#4064	C₄	1	2	yes	complemented
18	#4061	C₄	1	2	yes	complemented
19	#4060	C₄	1	2	yes	complemented
20	#4057	C₄	1	2	yes	complemented
21	#4053	—	2	2	no	—
22	#4053	C₂²	1	2	yes	complemented
23	D₅×M₁₆	—	2	2	no	—
24	D₅×M₁₆	C₂²	1	2	yes	—
25	#4050	C₂²	1	1	yes	characteristic
26	#4047	—	2	2	no	—
27	#4047	C₂²	1	2	yes	complemented
28	C₂×#676	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₅	—	2	2	no	—
30	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
31	#4018	C₂²	1	1	yes	characteristic
32	C₅×#108	C₂²	1	1	yes	characteristic, complemented
33	C₅×#104	C₂²	1	2	yes	complemented
34	C₅×#97	C₂²	1	1	yes	characteristic, complemented
35	#3969	C₂²	1	2	yes	complemented
36	#3957	C₂²	1	2	yes	complemented
37	#3954	C₂²	1	2	yes	complemented
38	#3951	C₂²	1	1	yes	characteristic, complemented
39	#3947	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₅	C₂²	1	1	yes	characteristic, complemented
41	#3937	C₂²	1	2	yes	complemented
42	C₄×#168	C₂²	1	1	yes	characteristic, complemented
43	#2955	—	5	1	no	maximal, Sylow
44	C₅×#41	C₂³	1	2	yes	—
45	#713	C₂×C₄	1	2	yes	—
46	Q₈×D₅	C₂×C₄	1	2	yes	—
47	#711	—	2	2	no	—
48	#711	C₂×C₄	1	2	yes	—
49	D₄×D₅	—	2	2	no	—
50	D₄×D₅	C₂×C₄	1	2	yes	—
51	#709	—	2	2	no	—
52	#709	C₂³	1	2	yes	—
53	C₂×C₄×D₅	—	2	2	no	—
54	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
55	C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
56	C₅×SD₁₆	C₂×C₄	1	2	yes	—
57	C₅×D₈	C₂×C₄	1	1	yes	characteristic
58	C₅×M₁₆	—	2	2	no	—
59	C₅×M₁₆	C₂³	1	2	yes	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄²×C₅	C₂³	1	1	yes	characteristic
64	#689	C₂×C₄	1	2	yes	—
65	#688	C₂×C₄	1	2	yes	—
66	#687	C₂×C₄	1	2	yes	—
67	#686	C₂×C₄	1	2	yes	—
68	#685	—	2	1	no	—
69	#683	—	2	1	no	—
70	C₄×Dic₅	C₂³	1	1	yes	characteristic
71	#681	—	2	2	no	—
72	#681	C₂³	1	2	yes	—
73	C₂×#168	—	2	2	no	—
74	C₂×#168	C₂³	1	1	yes	characteristic
75	C₂×#168	C₂³	1	1	yes	characteristic
76	Dic₂₀	C₂×C₄	1	1	yes	characteristic
77	D₄₀	C₂×C₄	1	1	yes	characteristic
78	#677	C₂×C₄	1	2	yes	—
79	#676	—	2	2	no	—
80	#676	—	2	2	no	—
81	#676	—	2	2	no	—
82	#676	C₂×C₄	1	2	yes	—
83	#676	C₂×C₄	1	2	yes	—
84	C₈×D₅	—	2	2	no	—
85	C₈×D₅	—	2	2	no	—
86	C₈×D₅	—	2	2	no	—
87	C₈×D₅	D₄	1	2	yes	—
88	C₈×D₅	D₄	1	2	yes	—
89	#575	—	5	1	no	—
90	C₂×#131	—	5	2	no	—
91	#444	—	5	2	no	—
92	#444	—	5	1	no	—
93	#444	D₅	1	1	yes	characteristic, complemented
94	#443	—	5	1	no	—
95	#443	—	5	1	no	—
96	#443	—	5	2	no	—
97	#430	—	5	1	no	—
98	#421	—	5	2	no	—
99	#405	—	5	1	no	—
100	C₂²×D₅	—	2	2	no	—
101	C₂²×D₅	#41	1	1	yes	characteristic
102	C₅×Q₈	C₂²×C₄	1	2	yes	—
103	C₅×D₄	—	2	2	no	—
104	C₅×D₄	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
108	C₂²⋊D₅	—	2	2	no	—
109	C₂²⋊D₅	—	2	2	no	—
110	C₂²⋊D₅	—	2	2	no	—
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	—	2	1	no	—
113	C₂×Dic₅	#41	1	1	yes	characteristic
114	D₂₀	—	2	2	no	—
115	D₂₀	C₂²×C₄	1	2	yes	—
116	C₄×D₅	—	2	2	no	—
117	C₄×D₅	—	2	2	no	—
118	C₄×D₅	—	2	2	no	—
119	C₄×D₅	C₂²×C₄	1	2	yes	—
120	C₄×D₅	C₂×D₄	1	2	yes	—
121	Dic₁₀	—	2	2	no	—
122	Dic₁₀	C₂²×C₄	1	2	yes	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₅×C₈	C₂×D₄	1	2	yes	—
127	#168	—	2	2	no	—
128	#168	C₂²×C₄	1	2	yes	—
129	#168	C₂×D₄	1	1	yes	characteristic
130	#168	C₂×D₄	1	1	yes	characteristic
131	C₂×#41	—	5	2	no	—
132	#137	—	5	2	no	—
133	#137	—	5	2	no	—
134	AGL(1,ℤ/8)	—	5	2	no	—
135	AGL(1,ℤ/8)	—	5	2	no	—
136	#135	—	5	2	no	—
137	#135	—	5	1	no	—
138	#135	C₂×D₅	1	1	yes	characteristic
139	#131	—	5	2	no	—
140	#131	—	5	2	no	—
141	#131	—	5	2	no	—
142	#131	—	10	2	no	—
143	#131	—	10	2	no	—
144	#131	C₂×D₅	1	2	yes	complemented
145	C₂×M₁₆	—	5	1	no	—
146	C₂×M₁₆	—	5	2	no	—
147	C₂×M₁₆	—	10	2	no	—
148	C₂²×C₈	—	5	1	no	—
149	C₂²×C₈	—	10	2	no	—
150	#117	—	5	1	no	—
151	#108	—	5	2	no	—
152	#108	—	5	1	no	—
153	#108	C₂×D₅	1	1	yes	characteristic, complemented
154	#104	—	5	2	no	—
155	#104	—	5	2	no	—
156	#104	—	5	2	no	—
157	#104	C₂×D₅	1	2	yes	complemented
158	#97	—	5	1	no	—
159	#97	C₂×D₅	1	1	yes	characteristic, complemented
160	C₄×C₈	—	5	1	no	—
161	C₄×C₈	—	5	1	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×D₅	—	2	2	no	—
165	C₂×D₅	—	4	1	no	—
166	C₂×D₅	—	4	4	no	—
167	C₂×D₅	C₄×D₄	1	2	yes	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
171	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	Dic₅	—	2	2	no	—
173	Dic₅	—	4	1	no	—
174	Dic₅	C₄×D₄	1	2	yes	—
175	#41	—	5	2	no	—
176	#41	—	5	2	no	—
177	#41	—	5	2	no	—
178	#41	—	10	2	no	—
179	#41	—	10	2	no	—
180	#41	C₂²×D₅	1	2	yes	—
181	C₂×Q₈	—	5	2	no	—
182	C₂×D₄	—	5	2	no	—
183	C₂×D₄	—	10	2	no	—
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	—	10	2	no	—
186	Q₁₆	—	5	2	no	—
187	Q₁₆	—	5	1	no	—
188	Q₁₆	C₄×D₅	1	1	yes	characteristic
189	SD₁₆	—	5	2	no	—
190	SD₁₆	—	5	2	no	—
191	SD₁₆	—	5	2	no	—
192	SD₁₆	C₄×D₅	1	2	yes	—
193	D₈	—	5	2	no	—
194	D₈	—	5	1	no	—
195	D₈	C₄×D₅	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	5	2	no	—
198	M₁₆	—	5	2	no	—
199	M₁₆	—	5	2	no	—
200	M₁₆	—	10	2	no	—
201	M₁₆	—	10	2	no	—
202	M₁₆	—	10	2	no	—
203	M₁₆	—	10	2	no	—
204	M₁₆	C₂²×D₅	1	2	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	5	1	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	10	2	no	—
211	C₂×C₈	—	10	2	no	—
212	C₂×C₈	—	10	2	no	—
213	C₂×C₈	—	10	2	no	—
214	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
216	C₄⋊C₄	—	10	1	no	—
217	C₂²⋊C₄	—	10	1	no	—
218	C₄²	—	5	1	no	—
219	C₄²	C₂²×D₅	1	1	yes	characteristic
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	D₅	—	2	2	no	—
224	D₅	—	4	2	no	—
225	C₂³	—	5	1	no	—
226	C₂³	—	10	2	no	—
227	Q₈	—	5	2	no	—
228	Q₈	—	10	2	no	—
229	Q₈	C₂×C₄×D₅	1	2	yes	—
230	D₄	—	2	2	no	—
231	D₄	—	5	2	no	—
232	D₄	—	10	2	no	—
233	D₄	—	10	2	no	—
234	D₄	—	10	2	no	—
235	D₄	—	10	2	no	—
236	D₄	C₂×C₄×D₅	1	2	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	5	2	no	—
242	C₂×C₄	—	10	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	10	2	no	—
246	C₂×C₄	—	10	1	no	—
247	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	5	2	no	—
250	C₈	—	5	1	no	—
251	C₈	—	5	1	no	—
252	C₈	—	10	2	no	—
253	C₈	C₂×C₄×D₅	1	1	yes	characteristic
254	C₈	C₂×C₄×D₅	1	1	yes	characteristic
255	C₈	D₄×D₅	1	2	yes	—
256	C₅	#2955	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	5	2	no	—
259	C₂²	—	10	2	no	—
260	C₂²	—	20	1	no	—
261	C₂²	—	20	4	no	—
262	C₂²	D₅×#41	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	5	2	no	—
266	C₄	—	10	2	no	—
267	C₄	—	20	1	no	—
268	C₄	C₂²×C₄×D₅	1	1	yes	characteristic
269	C₄	C₂×D₄×D₅	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	10	2	no	—
273	C₂	—	20	2	no	—
274	C₂	C₄×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#444	C₂
socle	C₂×C₅	C₂×C₄×D₄
prime radical	#66102	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#444	D₅
hypocenter	C₅	#2955
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. D₅×#131	C₂	semidirect
3. C₅×#444	C₂	semidirect
4. #65485	C₂	semidirect
5. #65434	C₂	semidirect
6. #65184	C₂	semidirect
7. #65182	C₂	semidirect
8. #65113	C₂	semidirect
9. #65110	C₂	semidirect
10. #65006	C₂	semidirect
11. #64997	C₂	semidirect
14. C₅×#131	C₂²	semidirect
16. #4094	C₂²	semidirect
17. #4064	C₄	semidirect
18. #4061	C₄	semidirect
19. #4060	C₄	semidirect
20. #4057	C₄	semidirect
22. #4053	C₂²	semidirect
27. #4047	C₂²	semidirect
32. C₅×#108	C₂²	semidirect
33. C₅×#104	C₂²	semidirect
34. C₅×#97	C₂²	semidirect
35. #3969	C₂²	semidirect
36. #3957	C₂²	semidirect
37. #3954	C₂²	semidirect
38. #3951	C₂²	semidirect
39. #3947	C₂²	semidirect
40. C₈×Dic₅	C₂²	semidirect
41. #3937	C₂²	semidirect
42. C₄×#168	C₂²	semidirect
93. #444	D₅	semidirect
144. #131	C₂×D₅	semidirect
153. #108	C₂×D₅	semidirect
157. #104	C₂×D₅	semidirect
159. #97	C₂×D₅	semidirect
256. C₅	#2955	semidirect

Maximal subgroups

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

subgroup	index
#66102	2
D₅×#131	2
C₅×#444	2
#65485	2
#65434	2
#65184	2
#65182	2
#65113	2
#65110	2
#65006	2
#64997	2
#2955	5