Finite Group Databasebeta

← C5×#64297 #13089310

Subgroups — C5×#64297 #13089310

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3828C211yesmaximal, characteristic, complemented
2C3×C5×#235C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic6C211yesmaximal, characteristic, complemented
4C5×#3804C214yesmaximal, complemented
5C5×#3799C212yesmaximal, complemented
6C5×#3797C212yesmaximal, complemented
7C5×#3796C212yesmaximal, complemented
8C5×D3×Dic6C212yesmaximal, complemented
9C5×#1010—31nomaximal
10C5×#1007—31nomaximal
11C2×C32×C4×C5C2211yescharacteristic
12C5×#638C2212yes—
13C5×C4⋊(C3⋊D3)C2211yescharacteristic
14C4×C5×C3⋊D3C2212yes—
15C5×#634C2211yescharacteristic
16C3×C5×#67C2212yes—
17C2×C3×C5×Dic3C2212yes—
18C3×C5×D12C2211yescharacteristic
19C3×C4×C5×D3C2212yes—
20C3×C5×Dic6C2212yes—
21C3×C5×Dic6C2212yes—
22C3×C5×Dic6C2211yescharacteristic
23C5×#627C2214yes—
24C5×#626C2214yes—
25C5×#624C2214yes—
26C5×D3×Dic3C2214yes—
27#64297C511yesmaximal, characteristic, direct product factor, Hall
28C3×C5×#41—31no—
29C2×C3×C5×Q8—31no—
30C5×#239—32no—
31C5×#239—32no—
32C5×D3×Q8—32no—
33C5×D3×Q8—32no—
34C5×D3×Q8—32no—
35C5×#237—34no—
36C5×#237—32no—
37C5×#235—31no—
38C5×#235—31no—
39C5×#235—34no—
40C5×#235—32no—
41C5×#235—61no—
42C5×#235D311yescharacteristic, complemented
43C2×C5×Dic6—32no—
44C2×C5×Dic6D311yescharacteristic, complemented
45C22×C32×C5C2311yescharacteristic
46C2×C5×C3⋊D3C2312yes—
47C2×C3×C5×D3C2312yes—
48C32×C4×C5C2311yescharacteristic
49C32×C4×C5C2311yescharacteristic
50C5×C3⋊Dic3C2312yes—
51C3×C5×Dic3C2314yes—
52C3×C5×Dic3C2312yes—
53C5×2−1+4—91noHall
54#3828C2×C511yescharacteristic, complemented
55C3×#235C2×C511yescharacteristic, complemented
56C2×C3×Dic6C2×C511yescharacteristic, complemented
57#3804C2×C514yescomplemented
58#3799C2×C512yescomplemented
59#3797C2×C512yescomplemented
60#3796C2×C512yescomplemented
61D3×Dic6C2×C512yescomplemented
62C3×C5×Q8—31no—
63C3×C5×Q8—32no—
64C3×C5×Q8—32no—
65C3×C5×D4—32no—
66C3×C5×D4—31no—
67C2×C3×C4×C5—21no—
68C2×C3×C4×C5—32no—
69C2×C3×C4×C5—32no—
70C2×C3×C4×C5C2×D311yescharacteristic
71C2×C3×C4×C5C2×D311yescharacteristic
72C5×#67—32no—
73C5×#67—32no—
74C5×#67—34no—
75C5×#67—62no—
76C5×#67C2×D312yes—
77C2×C5×Dic3—34no—
78C2×C5×Dic3C2×D312yes—
79C5×D12—34no—
80C5×D12—31no—
81C5×D12—31no—
82C5×D12—61no—
83C5×D12C2×D311yescharacteristic
84C4×C5×D3—34no—
85C4×C5×D3—32no—
86C4×C5×D3—32no—
87C4×C5×D3—34no—
88C4×C5×D3—34no—
89C4×C5×D3—62no—
90C4×C5×D3C2×D312yes—
91C5×Dic6—31no—
92C5×Dic6—31no—
93C5×Dic6—34no—
94C5×Dic6—34no—
95C5×Dic6—61no—
96C5×Dic6C2×D312yes—
97C5×Dic6C2×D312yes—
98C5×Dic6C2×D311yescharacteristic
99#1010—31no—
100#1007—31no—
101C2×C32×C5—21no—
102C2×C32×C5C2411yescharacteristic
103C5×C3⋊D3—22no—
104C3×C5×D3—22no—
105C5×#41—31no—
106C5×#41—94no—
107C5×#41—92no—
108C5×#41—92no—
109C5×#41—91no—
110C2×C5×Q8—31no—
111C2×C5×Q8—92no—
112C2×C5×Q8—92no—
113C2×C32×C4C22×C511yescharacteristic
114#638C22×C512yes—
115C4⋊(C3⋊D3)C22×C511yescharacteristic
116C4×C3⋊D3C22×C512yes—
117#634C22×C511yescharacteristic
118C3×#67C22×C512yes—
119C2×C3×Dic3C22×C512yes—
120C3×D12C22×C511yescharacteristic
121C3×C4×D3C22×C512yes—
122C3×Dic6C22×C512yes—
123C3×Dic6C22×C512yes—
124C3×Dic6C22×C511yescharacteristic
125#627C22×C514yes—
126#626C22×C514yes—
127#624C22×C514yes—
128D3×Dic3C22×C514yes—
129C22×C3×C5—21no—
130C22×C3×C5—32no—
131C22×C3×C5C22×D311yescharacteristic
132C22×C3×C5C22×D311yescharacteristic
133C2×C5×D3—32no—
134C2×C5×D3—32no—
135C2×C5×D3—62no—
136C2×C5×D3C22×D312yes—
137C3×C4×C5—21no—
138C3×C4×C5—21no—
139C3×C4×C5—32no—
140C3×C4×C5—34no—
141C3×C4×C5C22×D311yescharacteristic
142C3×C4×C5C22×D311yescharacteristic
143C3×C4×C5C22×D311yescharacteristic
144C3×C4×C5C22×D311yescharacteristic
145C5×Dic3—32no—
146C5×Dic3—32no—
147C5×Dic3—62no—
148C5×Dic3C22×D312yes—
149C5×Dic3C22×D314yes—
150C3×#41—31no—
151C2×C3×Q8—31no—
152#239—32no—
153#239—32no—
154D3×Q8—32no—
155D3×Q8—32no—
156D3×Q8—32no—
157#237—34no—
158#237—32no—
159#235—31no—
160#235—31no—
161#235—34no—
162#235—32no—
163#235—61no—
164#235C5×D311yescharacteristic, complemented
165C2×Dic6—32no—
166C2×Dic6C5×D311yescharacteristic, complemented
167C32×C52−1+411yescharacteristic, complemented, Hall
168C5×Q8—32no—
169C5×Q8—32no—
170C5×Q8—31no—
171C5×Q8—94no—
172C5×Q8—91no—
173C5×D4—32no—
174C5×D4—31no—
175C5×D4—94no—
176C5×D4—92no—
177C5×D4—91no—
178C2×C4×C5—32no—
179C2×C4×C5—32no—
180C2×C4×C5—94no—
181C2×C4×C5—94no—
182C2×C4×C5—92no—
183C2×C4×C5D3211yescharacteristic
184C22×C32C23×C511yescharacteristic
185C2×C3⋊D3C23×C512yes—
186C2×C3×D3C23×C512yes—
187C32×C4C23×C511yescharacteristic
188C32×C4C23×C511yescharacteristic
189C3⋊Dic3C23×C512yes—
190C3×Dic3C23×C514yes—
191C3×Dic3C23×C512yes—
1922−1+4—91noSylow
193C2×C3×C5—21no—
194C2×C3×C5—21no—
195C2×C3×C5—21no—
196C2×C3×C5—41no—
197C2×C3×C5—62no—
198C2×C3×C5C23×D311yescharacteristic
199C2×C3×C5C23×D311yescharacteristic
200C5×D3—22no—
201C5×D3—62no—
202C5×D3—62no—
203C5×D3—122no—
204C3×Q8—31no—
205C3×Q8—32no—
206C3×Q8—32no—
207C3×D4—32no—
208C3×D4—31no—
209C2×C3×C4—21no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4C2×C5×D311yescharacteristic
213C2×C3×C4C2×C5×D311yescharacteristic
214#67—32no—
215#67—32no—
216#67—34no—
217#67—62no—
218#67C2×C5×D312yes—
219C2×Dic3—34no—
220C2×Dic3C2×C5×D312yes—
221D12—34no—
222D12—31no—
223D12—31no—
224D12—61no—
225D12C2×C5×D311yescharacteristic
226C4×D3—34no—
227C4×D3—32no—
228C4×D3—32no—
229C4×D3—34no—
230C4×D3—34no—
231C4×D3—62no—
232C4×D3C2×C5×D312yes—
233Dic6—31no—
234Dic6—31no—
235Dic6—34no—
236Dic6—34no—
237Dic6—61no—
238Dic6C2×C5×D312yes—
239Dic6C2×C5×D312yes—
240Dic6C2×C5×D311yescharacteristic
241C22×C5—32no—
242C22×C5—92no—
243C22×C5C2×D3211yescharacteristic
244C4×C5—32no—
245C4×C5—34no—
246C4×C5—92no—
247C4×C5C2×D3211yescharacteristic
248C4×C5C2×D3211yescharacteristic
249C2×C32—21no—
250C2×C32C24×C511yescharacteristic
251C3⋊D3—22no—
252C3×D3—22no—
253#41—31no—
254#41—94no—
255#41—92no—
256#41—92no—
257#41—91no—
258C2×Q8—31no—
259C2×Q8—92no—
260C2×Q8—92no—
261C3×C5—21no—
262C3×C5#100711yescharacteristic, complemented
263C3×C5#101011yescharacteristic, complemented
264C22×C3—21no—
265C22×C3—32no—
266C22×C3C22×C5×D311yescharacteristic
267C22×C3C22×C5×D311yescharacteristic
268C2×D3—32no—
269C2×D3—32no—
270C2×D3—62no—
271C2×D3C22×C5×D312yes—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—32no—
275C3×C4—34no—
276C3×C4C22×C5×D311yescharacteristic
277C3×C4C22×C5×D311yescharacteristic
278C3×C4C22×C5×D311yescharacteristic
279C3×C4C22×C5×D311yescharacteristic
280Dic3—32no—
281Dic3—32no—
282Dic3—62no—
283Dic3C22×C5×D312yes—
284Dic3C22×C5×D314yes—
285C2×C5—21no—
286C2×C5—62no—
287C2×C5—182no—
288C2×C5C22×D3211yescharacteristic
289C32C5×2−1+411yescharacteristic, complemented, Sylow
290Q8—32no—
291Q8—32no—
292Q8—31no—
293Q8—94no—
294Q8—91no—
295D4—32no—
296D4—31no—
297D4—94no—
298D4—92no—
299D4—91no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—94no—
303C2×C4—94no—
304C2×C4—92no—
305C2×C4C5×D3211yescharacteristic
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—41no—
310C2×C3—62no—
311C2×C3C23×C5×D311yescharacteristic
312C2×C3C23×C5×D311yescharacteristic
313D3—22no—
314D3—62no—
315D3—62no—
316D3—122no—
317C5#6429711yescharacteristic, direct product factor, Sylow
318C22—32no—
319C22—92no—
320C22C2×C5×D3211yescharacteristic
321C4—32no—
322C4—34no—
323C4—92no—
324C4C2×C5×D3211yescharacteristic
325C4C2×C5×D3211yescharacteristic
326C3—21no—
327C3C5×#100711yescharacteristic, complemented
328C3C5×#101011yescharacteristic, complemented
329C2—21no—
330C2—62no—
331C2—182no—
332C2C22×C5×D3211yescharacteristic
subgroupG/H
centerC2×C5C22×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC2×C32×C4×C5C22
socleC2×C32×C5C24
hypercenterC2×C4×C5D32
hypocenterC32C5×2−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C5×#3828C2semidirect
2. C3×C5×#235C2semidirect
3. C2×C3×C5×Dic6C2semidirect
4. C5×#3804C2semidirect
5. C5×#3799C2semidirect
6. C5×#3797C2semidirect
7. C5×#3796C2semidirect
8. C5×D3×Dic6C2semidirect
27. #64297C5direct (N × H)
42. C5×#235D3semidirect
44. C2×C5×Dic6D3semidirect
54. #3828C2×C5semidirect
55. C3×#235C2×C5semidirect
56. C2×C3×Dic6C2×C5semidirect
57. #3804C2×C5semidirect
58. #3799C2×C5semidirect
59. #3797C2×C5semidirect
60. #3796C2×C5semidirect
61. D3×Dic6C2×C5semidirect
164. #235C5×D3semidirect
166. C2×Dic6C5×D3semidirect
167. C32×C52−1+4semidirect
262. C3×C5#1007semidirect
263. C3×C5#1010semidirect
289. C32C5×2−1+4semidirect
317. C5#64297direct (N × H)
327. C3C5×#1007semidirect
328. C3C5×#1010semidirect

Maximal subgroups

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

subgroupindex
C5×#38282
C3×C5×#2352
C2×C3×C5×Dic62
C5×#38042
C5×#37992
C5×#37972
C5×#37962
C5×D3×Dic62
C5×#10103
C5×#10073
#642975