Subgroups — #10607968

Order 640.

468 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4044	C₂	1	2	yes	maximal
2	C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
3	#65276	C₂	1	1	yes	maximal, characteristic
4	#65276	C₂	1	1	yes	maximal, characteristic
5	#65274	C₂	1	1	yes	maximal, characteristic, complemented
6	#65267	C₂	1	2	yes	maximal, complemented
7	#65265	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#685	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
10	#64964	C₂	1	2	yes	maximal
11	#64949	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂×#685	C₂²	1	2	yes	—
18	C₂×#685	C₂²	1	2	yes	—
19	C₂×#685	C₂²	1	1	yes	characteristic
20	C₂×#685	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₂×#683	C₂²	1	2	yes	—
25	C₂×#683	C₂²	1	2	yes	—
26	C₂×C₄×Dic₅	C₂²	1	2	yes	—
27	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	#4044	—	2	8	no	—
29	C₅×#95	C₂²	1	2	yes	—
30	#3963	C₂²	1	1	yes	characteristic
31	#3963	C₂²	1	2	yes	—
32	#3963	C₂²	1	2	yes	—
33	#3963	C₂²	1	2	yes	—
34	#3963	C₂²	1	2	yes	—
35	#3963	C₂²	1	1	yes	characteristic
36	#2496	—	5	1	no	maximal, Sylow
37	C₂³×D₅	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂³	1	2	yes	—
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂²×Dic₅	C₂³	1	1	yes	characteristic
44	C₂²×Dic₅	C₂³	1	2	yes	—
45	C₂²×Dic₅	C₂³	1	2	yes	—
46	C₂²×Dic₅	C₂³	1	1	yes	characteristic
47	C₂²×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₅	—	2	2	no	—
49	C₂×C₄×D₅	—	2	2	no	—
50	C₂×C₄×D₅	—	2	2	no	—
51	C₅×C₄⋊C₄	—	2	4	no	—
52	C₅×C₄⋊C₄	—	2	2	no	—
53	C₅×C₄⋊C₄	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	#685	—	2	4	no	—
56	#685	—	2	4	no	—
57	#685	—	2	2	no	—
58	#685	D₄	1	4	yes	—
59	C₄⋊Dic₅	—	2	2	no	—
60	C₄⋊Dic₅	—	2	2	no	—
61	C₄⋊Dic₅	—	2	4	no	—
62	#683	—	2	4	no	—
63	#683	—	2	4	no	—
64	C₄×Dic₅	—	2	4	no	—
65	C₄×Dic₅	—	2	2	no	—
66	C₂×#126	—	5	2	no	—
67	#400	—	5	2	no	—
68	#397	—	5	2	no	—
69	#389	—	5	1	no	—
70	#389	D₅	1	1	yes	characteristic, complemented
71	#387	—	5	2	no	—
72	#384	—	5	1	no	—
73	#380	—	5	1	no	—
74	#380	—	5	1	no	—
75	C₄×C₄⋊C₄	—	5	1	no	—
76	C₄×C₂²⋊C₄	—	5	1	no	—
77	C₂³×C₅	C₂⁴	1	1	yes	characteristic
78	C₂²×D₅	—	2	1	no	—
79	C₂²×D₅	—	2	1	no	—
80	C₂²×D₅	—	2	1	no	—
81	C₂²×D₅	—	2	1	no	—
82	C₂²×D₅	—	2	1	no	—
83	C₂²×D₅	—	2	1	no	—
84	C₂²×D₅	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	1	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅	#41	1	2	yes	—
98	C₂×C₄×C₅	#41	1	2	yes	—
99	C₂×C₄×C₅	#41	1	2	yes	—
100	C₂×Dic₅	—	2	1	no	—
101	C₂×Dic₅	—	2	1	no	—
102	C₂×Dic₅	—	2	1	no	—
103	C₂×Dic₅	—	2	1	no	—
104	C₂×Dic₅	—	2	2	no	—
105	C₂×Dic₅	—	2	2	no	—
106	C₂×Dic₅	—	2	2	no	—
107	C₂×Dic₅	—	2	2	no	—
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	—	2	1	no	—
110	C₂×Dic₅	—	2	1	no	—
111	C₂×Dic₅	—	2	1	no	—
112	C₂×Dic₅	—	2	2	no	—
113	C₂×Dic₅	C₂×D₄	1	2	yes	—
114	C₂×Dic₅	#41	1	2	yes	—
115	C₄×D₅	—	4	4	no	—
116	C₂³×C₄	—	5	1	no	—
117	#126	—	10	8	no	—
118	C₂×C₄⋊C₄	—	5	2	no	—
119	C₂×C₄⋊C₄	—	5	2	no	—
120	C₂×C₄⋊C₄	—	5	1	no	—
121	C₂×C₄⋊C₄	—	5	1	no	—
122	C₂×C₄⋊C₄	—	5	2	no	—
123	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
124	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	5	1	no	—
127	C₂×C₂²⋊C₄	—	5	2	no	—
128	C₂×C₂²⋊C₄	—	5	2	no	—
129	C₂×C₂²⋊C₄	—	5	1	no	—
130	C₂×C₄²	—	5	2	no	—
131	C₂×C₄²	—	5	1	no	—
132	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
133	#95	—	5	1	no	—
134	#95	—	5	1	no	—
135	#95	—	5	2	no	—
136	#95	—	5	2	no	—
137	#95	—	5	2	no	—
138	#95	—	5	2	no	—
139	#95	C₂×D₅	1	2	yes	—
140	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₅	C₂×#41	1	1	yes	characteristic
142	C₂²×C₅	C₂×#41	1	1	yes	characteristic
143	C₂²×C₅	C₂×#41	1	1	yes	characteristic
144	C₂²×C₅	C₂×#41	1	1	yes	characteristic
145	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×D₅	—	2	2	no	—
148	C₂×D₅	—	2	2	no	—
149	C₂×D₅	—	2	2	no	—
150	C₂×D₅	—	4	1	no	—
151	C₂×D₅	—	4	1	no	—
152	C₂×D₅	—	4	1	no	—
153	C₂×D₅	—	4	1	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	4	2	no	—
156	C₄×C₅	—	4	2	no	—
157	C₄×C₅	—	4	1	no	—
158	C₄×C₅	#126	1	4	yes	complemented
159	Dic₅	—	2	2	no	—
160	Dic₅	—	2	2	no	—
161	Dic₅	—	4	1	no	—
162	Dic₅	—	4	2	no	—
163	Dic₅	—	4	2	no	—
164	C₂⁴	—	5	1	no	—
165	C₂²×C₄	—	5	1	no	—
166	C₂²×C₄	—	5	1	no	—
167	C₂²×C₄	—	5	1	no	—
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	—	5	2	no	—
170	C₂²×C₄	—	10	2	no	—
171	C₂²×C₄	—	10	2	no	—
172	C₂²×C₄	—	10	2	no	—
173	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₅	1	2	yes	—
175	C₂²×C₄	C₂²×D₅	1	2	yes	—
176	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	10	4	no	—
182	C₄⋊C₄	—	10	4	no	—
183	C₄⋊C₄	—	10	4	no	—
184	C₄⋊C₄	—	10	2	no	—
185	C₄⋊C₄	—	10	2	no	—
186	C₂²⋊C₄	—	5	4	no	—
187	C₂²⋊C₄	—	10	2	no	—
188	C₂²⋊C₄	—	10	4	no	—
189	C₂²⋊C₄	—	10	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	10	4	no	—
192	C₄²	—	10	2	no	—
193	C₂×C₅	C₂×#126	1	1	yes	characteristic
194	C₂×C₅	#532	1	1	yes	characteristic
195	C₂×C₅	#541	1	1	yes	characteristic
196	C₂×C₅	#547	1	1	yes	characteristic
197	C₂×C₅	#547	1	1	yes	characteristic
198	C₂×C₅	#552	1	1	yes	characteristic
199	C₂×C₅	#556	1	1	yes	characteristic
200	D₅	—	4	2	no	—
201	C₂³	—	5	2	no	—
202	C₂³	—	10	1	no	—
203	C₂³	—	10	1	no	—
204	C₂³	—	10	1	no	—
205	C₂³	—	10	1	no	—
206	C₂³	—	10	1	no	—
207	C₂³	—	10	1	no	—
208	C₂³	C₂³×D₅	1	1	yes	characteristic
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	5	2	no	—
222	C₂×C₄	—	10	1	no	—
223	C₂×C₄	—	10	1	no	—
224	C₂×C₄	—	10	1	no	—
225	C₂×C₄	—	10	1	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	1	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	—	10	2	no	—
230	C₂×C₄	—	10	1	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	1	no	—
233	C₂×C₄	—	10	2	no	—
234	C₂×C₄	—	10	2	no	—
235	C₂×C₄	—	20	4	no	—
236	C₂×C₄	#709	1	2	yes	—
237	C₂×C₄	D₄×D₅	1	2	yes	—
238	C₂×C₄	#711	1	2	yes	—
239	C₂×C₄	#713	1	2	yes	—
240	C₅	#2496	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	10	2	no	—
242	C₂²	—	10	2	no	—
243	C₂²	—	10	2	no	—
244	C₂²	—	20	1	no	—
245	C₂²	—	20	1	no	—
246	C₂²	—	20	1	no	—
247	C₂²	—	20	1	no	—
248	C₂²	C₂×#709	1	1	yes	characteristic
249	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
250	C₂²	C₂×#711	1	1	yes	characteristic
251	C₂²	C₂×#713	1	1	yes	characteristic
252	C₂²	#4147	1	1	yes	characteristic
253	C₂²	D₅×#41	1	1	yes	characteristic
254	C₂²	#4150	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	10	2	no	—
260	C₄	—	10	2	no	—
261	C₄	—	20	2	no	—
262	C₄	—	20	1	no	—
263	C₄	—	20	2	no	—
264	C₄	#4044	1	4	yes	complemented
265	C₂	—	20	2	no	—
266	C₂	C₂×#4044	1	1	yes	characteristic
267	C₂	#65882	1	1	yes	characteristic
268	C₂	#65917	1	1	yes	characteristic
269	C₂	#65945	1	1	yes	characteristic
270	C₂	#65975	1	1	yes	characteristic
271	C₂	#66028	1	1	yes	characteristic
272	C₂	#66060	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#389	C₂
prime radical	C₂³×D₅	C₂³
hypercenter	#389	D₅
hypocenter	C₅	#2496
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. C₅×#389	C₂	semidirect
5. #65274	C₂	semidirect
6. #65267	C₂	semidirect
7. #65265	C₂	semidirect
9. C₄×C₄⋊Dic₅	C₂	semidirect
11. #64949	C₂	semidirect
70. #389	D₅	semidirect
158. C₄×C₅	#126	semidirect
240. C₅	#2496	semidirect
264. C₄	#4044	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4044	2
C₅×#389	2
#65276	2
#65274	2
#65267	2
#65265	2
C₄×#685	2
C₄×C₄⋊Dic₅	2
#64964	2
#64949	2
#2496	5