Finite Group Databasebeta

← C5×#63952 #13087238

Subgroups — C5×#63952 #13087238

Order 1440.

430 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#638C211yesmaximal, characteristic
2C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
3C2×C5×#624C211yesmaximal, characteristic, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5C5×#3724C211yesmaximal, characteristic, complemented
6C5×#3722C211yesmaximal, characteristic, complemented
7C5×Dic32C211yesmaximal, characteristic, complemented
8C4×C5×#67—31nomaximal
9C5×#67⋊C4—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C5×C3⋊D3C2211yescharacteristic
12C5×#638C414yescomplemented
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×C5×Dic3—22no—
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19C5×#624—22no—
20#63952C511yesmaximal, characteristic, direct product factor, Hall
21C22×C3×C4×C5—31no—
22C2×C5×#67—31no—
23C2×C5×#67—31no—
24C2×C5×#67—61no—
25C22×C5×Dic3D311yescharacteristic, complemented
26C2×C4×C5×D3—31no—
27C2×C4×C5×D3—31no—
28C3×C5×C22⋊C4—31no—
29C5×#217D311yescharacteristic, complemented
30C5×(C2×D3)⋊C4—31no—
31C5×(C2×D3)⋊C4—31no—
32C5×Dic3⋊C4—31no—
33C5×Dic3⋊C4—31no—
34C4×C5×Dic3—31no—
35C4×C5×Dic3—31no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2×C412yescomplemented
39C22×C32×C5C2311yescharacteristic
40C2×C5×C3⋊D3—21no—
41C2×C5×C3⋊D3—21no—
42C2×C5×C3⋊D3C2×C412yescomplemented
43C5×C3⋊Dic3C2×C412yescomplemented
44C3×C5×Dic3—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3—21no—
47C3×C5×Dic3D412yescomplemented
48C4×C5×D4—91noHall
49C2×#638C2×C511yescharacteristic
50C22×C3×Dic3C2×C511yescharacteristic, complemented
51C2×#624C2×C511yescharacteristic, complemented
52C3×#217C2×C511yescharacteristic, complemented
53#3724C2×C511yescharacteristic, complemented
54#3722C2×C511yescharacteristic, complemented
55Dic32C2×C511yescharacteristic, complemented
56C23×C3×C5—21no—
57C23×C3×C5C2×D311yescharacteristic
58C23×C3×C5C2×D311yescharacteristic
59C22×C5×D3—31no—
60C22×C5×D3—31no—
61C22×C5×D3—61no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5—31no—
64C2×C3×C4×C5—31no—
65C2×C3×C4×C5—31no—
66C2×C3×C4×C5—62no—
67C5×#67—34no—
68C5×#67—34no—
69C5×#67—64no—
70C2×C5×Dic3—22no—
71C2×C5×Dic3—31no—
72C2×C5×Dic3—31no—
73C2×C5×Dic3—61no—
74C2×C5×Dic3C2×D311yescharacteristic
75C2×C5×Dic3C2×D311yescharacteristic
76C2×C5×Dic3C2×D311yescharacteristic
77C2×C5×Dic3C2×D311yescharacteristic
78C4×C5×D3—62no—
79C4×C5×D3—62no—
80C4×#67—31no—
81#67⋊C4—31no—
82C2×C32×C5—22no—
83C2×C32×C5C22×C411yescharacteristic
84C2×C32×C5C2×D411yescharacteristic
85C2×C32×C5#4111yescharacteristic
86C5×C3⋊D3—22no—
87C2×C5×D4—91no—
88C22×C4×C5—31no—
89C22×C4×C5—91no—
90C5×C4⋊C4—91no—
91C5×C22⋊C4—31no—
92C5×C22⋊C4—91no—
93C42×C5—91no—
94C23×C32C22×C511yescharacteristic
95C22×C3⋊D3C22×C511yescharacteristic
96#638C4×C514yescomplemented
97C2×C3⋊Dic3C22×C511yescharacteristic
98C2×C3×Dic3—22no—
99C2×C3×Dic3C22×C511yescharacteristic
100C2×C3×Dic3C22×C511yescharacteristic
101C2×C3×Dic3C22×C511yescharacteristic
102C2×C3×Dic3C22×C511yescharacteristic
103#624—22no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—22no—
109C22×C3×C5—21no—
110C22×C3×C5—41no—
111C22×C3×C5—41no—
112C22×C3×C5C4×D312yescomplemented
113C22×C3×C5C4×D312yescomplemented
114C22×C3×C5C22×D311yescharacteristic
115C22×C3×C5C22×D311yescharacteristic
116C2×C5×D3—32no—
117C2×C5×D3—32no—
118C2×C5×D3—62no—
119C2×C5×D3—61no—
120C2×C5×D3—61no—
121C2×C5×D3—61no—
122C2×C5×D3—61no—
123C2×C5×D3—121no—
124C2×C5×D3—121no—
125C3×C4×C5—32no—
126C3×C4×C5—61no—
127C3×C4×C5—61no—
128C3×C4×C5—61no—
129C5×Dic3—21no—
130C5×Dic3—21no—
131C5×Dic3—21no—
132C5×Dic3—32no—
133C5×Dic3—32no—
134C5×Dic3—62no—
135C5×Dic3#6712yescomplemented
136C22×C3×C4—31no—
137C2×#67—31no—
138C2×#67—31no—
139C2×#67—61no—
140C22×Dic3C5×D311yescharacteristic, complemented
141C2×C4×D3—31no—
142C2×C4×D3—31no—
143C3×C22⋊C4—31no—
144#217C5×D311yescharacteristic, complemented
145(C2×D3)⋊C4—31no—
146(C2×D3)⋊C4—31no—
147Dic3⋊C4—31no—
148Dic3⋊C4—31no—
149C4×Dic3—31no—
150C4×Dic3—31no—
151C32×C5C4×D411yescharacteristic, complemented, Hall
152C23×C5—91no—
153C23×C5D3211yescharacteristic
154C5×D4—94no—
155C2×C4×C5—31no—
156C2×C4×C5—31no—
157C2×C4×C5—31no—
158C2×C4×C5—31no—
159C2×C4×C5—62no—
160C2×C4×C5—91no—
161C2×C4×C5—182no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32C2×C4×C512yescomplemented
165C22×C32C23×C511yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3C2×C4×C512yescomplemented
169C3⋊Dic3C2×C4×C512yescomplemented
170C3×Dic3—21no—
171C3×Dic3—21no—
172C3×Dic3—21no—
173C3×Dic3C5×D412yescomplemented
174C4×D4—91noSylow
175C2×C3×C5—22no—
176C2×C3×C5—22no—
177C2×C3×C5—21no—
178C2×C3×C5—21no—
179C2×C3×C5—21no—
180C2×C3×C5—42no—
181C2×C3×C5C2×C4×D311yescharacteristic
182C2×C3×C5C2×C4×D311yescharacteristic
183C2×C3×C5#23511yescharacteristic
184C2×C3×C5D3×D411yescharacteristic
185C2×C3×C5#23711yescharacteristic
186C2×C3×C5C2×#6711yescharacteristic
187C5×D3—62no—
188C5×D3—62no—
189C5×D3—122no—
190C23×C3—21no—
191C23×C3C2×C5×D311yescharacteristic
192C23×C3C2×C5×D311yescharacteristic
193C22×D3—31no—
194C22×D3—31no—
195C22×D3—61no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—62no—
201#67—34no—
202#67—34no—
203#67—64no—
204C2×Dic3—22no—
205C2×Dic3—31no—
206C2×Dic3—31no—
207C2×Dic3—61no—
208C2×Dic3C2×C5×D311yescharacteristic
209C2×Dic3C2×C5×D311yescharacteristic
210C2×Dic3C2×C5×D311yescharacteristic
211C2×Dic3C2×C5×D311yescharacteristic
212C4×D3—62no—
213C4×D3—62no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—92no—
217C22×C5—181no—
218C22×C5—181no—
219C22×C5#62412yescomplemented
220C22×C5C2×D3211yescharacteristic
221C4×C5—32no—
222C4×C5—61no—
223C4×C5—61no—
224C4×C5—61no—
225C4×C5—92no—
226C2×C32—22no—
227C2×C32C22×C4×C511yescharacteristic
228C2×C32C2×C5×D411yescharacteristic
229C2×C32C5×#4111yescharacteristic
230C3⋊D3—22no—
231C2×D4—91no—
232C22×C4—31no—
233C22×C4—91no—
234C4⋊C4—91no—
235C22⋊C4—31no—
236C22⋊C4—91no—
237C42—91no—
238C3×C5—21no—
239C3×C5#67⋊C411yescharacteristic, complemented
240C3×C5C4×#6711yescharacteristic, complemented
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3C4×C5×D312yescomplemented
250C22×C3C4×C5×D312yescomplemented
251C22×C3C22×C5×D311yescharacteristic
252C22×C3C22×C5×D311yescharacteristic
253C2×D3—32no—
254C2×D3—32no—
255C2×D3—62no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—121no—
261C2×D3—121no—
262C3×C4—32no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266Dic3—21no—
267Dic3—21no—
268Dic3—21no—
269Dic3—32no—
270Dic3—32no—
271Dic3—62no—
272Dic3C5×#6712yescomplemented
273C2×C5—22no—
274C2×C5—182no—
275C2×C5#380411yescharacteristic
276C2×C5C2×#62411yescharacteristic
277C2×C5D3×#6711yescharacteristic
278C32C4×C5×D411yescharacteristic, complemented, Sylow
279C23—91no—
280C23C5×D3211yescharacteristic
281D4—94no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—62no—
287C2×C4—91no—
288C2×C4—182no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—42no—
295C2×C3C2×C4×C5×D311yescharacteristic
296C2×C3C2×C4×C5×D311yescharacteristic
297C2×C3C5×#23511yescharacteristic
298C2×C3C5×D3×D411yescharacteristic
299C2×C3C5×#23711yescharacteristic
300C2×C3C2×C5×#6711yescharacteristic
301D3—62no—
302D3—62no—
303D3—122no—
304C5#6395211yescharacteristic, direct product factor, Sylow
305C22—21no—
306C22—21no—
307C22—92no—
308C22—181no—
309C22—181no—
310C22C5×#62412yescomplemented
311C22C2×C5×D3211yescharacteristic
312C4—32no—
313C4—61no—
314C4—61no—
315C4—61no—
316C4—92no—
317C3—21no—
318C3C5×#67⋊C411yescharacteristic, complemented
319C3C4×C5×#6711yescharacteristic, complemented
320C2—22no—
321C2—182no—
322C2C5×#380411yescharacteristic
323C2C2×C5×#62411yescharacteristic
324C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutatorC2×C32C22×C4×C5
FrattiniC22C2×C5×D32
FittingC23×C32×C5C22
socleC22×C32×C5C23
prime radicalC2×C5×#638C2
cosocleC22×C32C23×C5
hypercenterC23×C5D32
hypocenterC32C4×C5×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×Dic3C2semidirect
3. C2×C5×#624C2semidirect
4. C3×C5×#217C2semidirect
5. C5×#3724C2semidirect
6. C5×#3722C2semidirect
7. C5×Dic32C2semidirect
12. C5×#638C4semidirect
20. #63952C5direct (N × H)
25. C22×C5×Dic3D3semidirect
29. C5×#217D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C5×C3⋊D3C2×C4semidirect
43. C5×C3⋊Dic3C2×C4semidirect
47. C3×C5×Dic3D4semidirect
50. C22×C3×Dic3C2×C5semidirect
51. C2×#624C2×C5semidirect
52. C3×#217C2×C5semidirect
53. #3724C2×C5semidirect
54. #3722C2×C5semidirect
55. Dic32C2×C5semidirect
96. #638C4×C5semidirect
112. C22×C3×C5C4×D3semidirect
113. C22×C3×C5C4×D3semidirect
135. C5×Dic3#67semidirect
140. C22×Dic3C5×D3semidirect
144. #217C5×D3semidirect
151. C32×C5C4×D4semidirect
164. C22×C32C2×C4×C5semidirect
168. C2×C3⋊D3C2×C4×C5semidirect
169. C3⋊Dic3C2×C4×C5semidirect
173. C3×Dic3C5×D4semidirect
219. C22×C5#624semidirect
239. C3×C5#67⋊C4semidirect
240. C3×C5C4×#67semidirect
249. C22×C3C4×C5×D3semidirect
250. C22×C3C4×C5×D3semidirect
272. Dic3C5×#67semidirect
278. C32C4×C5×D4semidirect
304. C5#63952direct (N × H)
310. C22C5×#624semidirect
318. C3C5×#67⋊C4semidirect
319. C3C4×C5×#67semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C22×C3×C5×Dic32
C2×C5×#6242
C3×C5×#2172
C5×#37242
C5×#37222
C5×Dic322
C4×C5×#673
C5×#67⋊C43
#639525