Finite Group Databasebeta

← C5×#70059 #14313530

Subgroups — C5×#70059 #14313530

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5825C211yesmaximal, characteristic
2C3×C5×#410C211yesmaximal, characteristic
3C5×#5240C211yesmaximal, characteristic
4C5×#4473C214yesmaximal, complemented
5C5×#2129—31nomaximal, Hall
6C2×C3×C5×#41C2211yescharacteristic
7C22×C5×Dic6C411yescharacteristic, complemented
8C3×C5×#117—21no—
9C5×#944C412yescomplemented
10C5×#936C2212yes—
11C5×#935—21no—
12C5×#935—21no—
13C5×#926—21no—
14C5×#926C411yescharacteristic, complemented
15C5×#924—21no—
16C5×#924—21no—
17C3×C5×#99—21no—
18C3×C5×#99C2212yes—
19C5×#834—21no—
20C5×#834C2212yes—
21#70059C511yesmaximal, characteristic, direct product factor, Hall
22C5×#536—31no—
23C5×#410—31no—
24C5×#410D311yescharacteristic
25C5×#354—34no—
26C3×C5×#41—41no—
27C3×C5×#41—41no—
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×D4D411yescharacteristic
30C2×C3×C5×D4D411yescharacteristic
31C2×C3×C5×D4C2311yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×C4×C5D411yescharacteristic
34C22×C3×C4×C5D411yescharacteristic
35C22×C5×Dic3—21no—
36C2×C5×Dic6—21no—
37C2×C5×Dic6—41no—
38C2×C5×Dic6—41no—
39C2×C5×Dic6C2×C412yes—
40C3×C5×C4⋊C4—41no—
41C3×C5×C22⋊C4—22no—
42C3×C42×C5—41no—
43C5×#217—21no—
44C5×#217—22no—
45C5×#217—22no—
46C5×#217—22no—
47C5×C4⋊Dic3—21no—
48C5×C4⋊Dic3—41no—
49C5×Dic3⋊C4—22no—
50C5×Dic3⋊C4—22no—
51C4×C5×Dic3—41no—
52C4×C5×Dic3C2×C412yes—
53#5825C2×C511yescharacteristic
54C3×#410C2×C511yescharacteristic
55#5240C2×C511yescharacteristic
56#4473C2×C514yescomplemented
57C2×C5×#41C2×D311yescharacteristic
58C22×C5×Q8—31no—
59C5×C4⋊Q8—32no—
60C5×#124—32no—
61C5×#123—61no—
62C5×#123—61no—
63C5×C22⋊Q8—61no—
64C5×C22⋊Q8—61no—
65C5×#117—21no—
66C5×#117—31no—
67C5×#117—61no—
68C5×#99—21no—
69C5×#99—32no—
70C5×#99—61no—
71C5×#99C2×D312yes—
72#2129—31noSylow
73C23×C3×C5C22⋊C411yescharacteristic
74C23×C3×C5C2×D411yescharacteristic
75C23×C3×C5C2×D411yescharacteristic
76C3×C5×Q8—41no—
77C3×C5×D4—41no—
78C3×C5×D4—41no—
79C3×C5×D4—41no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×C5×Dic3—22no—
92C2×C5×Dic3—22no—
93C2×C5×Dic3—42no—
94C2×C5×Dic3—42no—
95C5×Dic6—41no—
96C5×Dic6—41no—
97C5×Dic6—42no—
98C2×C3×#41C22×C511yescharacteristic
99C22×Dic6C4×C511yescharacteristic, complemented
100C3×#117—21no—
101#944C4×C512yescomplemented
102#936C22×C512yes—
103#935—21no—
104#935—21no—
105#926—21no—
106#926C4×C511yescharacteristic, complemented
107#924—21no—
108#924—21no—
109C3×#99—21no—
110C3×#99C22×C512yes—
111#834—21no—
112#834C22×C512yes—
113C5×#41—41no—
114C5×#41—41no—
115C2×C5×Q8—32no—
116C2×C5×Q8—61no—
117C2×C5×Q8—121no—
118C2×C5×Q8—121no—
119C2×C5×Q8C4×D311yescharacteristic
120C2×C5×D4D1211yescharacteristic
121C2×C5×D4#6711yescharacteristic
122C2×C5×D4C22×D311yescharacteristic
123C22×C4×C5—61no—
124C22×C4×C5C4×D311yescharacteristic
125C22×C4×C5D1211yescharacteristic
126C22×C4×C5#6711yescharacteristic
127C5×C4⋊C4—41no—
128C5×C4⋊C4—62no—
129C5×C4⋊C4—62no—
130C5×C4⋊C4—61no—
131C5×C4⋊C4—121no—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—61no—
134C5×C22⋊C4—62no—
135C5×C22⋊C4—62no—
136C5×C22⋊C4—62no—
137C42×C5—32no—
138C42×C5—41no—
139C42×C5—121no—
140#536—31no—
141#410—31no—
142#410C5×D311yescharacteristic
143#354—34no—
144C22×C3×C5—21no—
145C22×C3×C5—21no—
146C22×C3×C5—21no—
147C22×C3×C5—41no—
148C22×C3×C5—41no—
149C22×C3×C5—41no—
150C22×C3×C5C2×C22⋊C411yescharacteristic
151C3×C4×C5—21no—
152C3×C4×C5—21no—
153C3×C4×C5—21no—
154C3×C4×C5—82no—
155C3×C4×C5#9911yescharacteristic
156C3×C4×C5#9911yescharacteristic
157C5×Dic3—42no—
158C5×Dic3—42no—
159C5×Dic3—82no—
160C3×#41—41no—
161C3×#41—41no—
162C2×C3×Q8C2×C4×C511yescharacteristic
163C2×C3×D4C5×D411yescharacteristic
164C2×C3×D4C5×D411yescharacteristic
165C2×C3×D4C23×C511yescharacteristic
166C22×C3×C4C2×C4×C511yescharacteristic
167C22×C3×C4C5×D411yescharacteristic
168C22×C3×C4C5×D411yescharacteristic
169C22×Dic3—21no—
170C2×Dic6—21no—
171C2×Dic6—41no—
172C2×Dic6—41no—
173C2×Dic6C2×C4×C512yes—
174C3×C4⋊C4—41no—
175C3×C22⋊C4—22no—
176C3×C42—41no—
177#217—21no—
178#217—22no—
179#217—22no—
180#217—22no—
181C4⋊Dic3—21no—
182C4⋊Dic3—41no—
183Dic3⋊C4—22no—
184Dic3⋊C4—22no—
185C4×Dic3—41no—
186C4×Dic3C2×C4×C512yes—
187C23×C5(C2×D3)⋊C411yescharacteristic
188C23×C5C2×D1211yescharacteristic
189C23×C5C2×#6711yescharacteristic
190C5×Q8—41no—
191C5×Q8—122no—
192C5×Q8—121no—
193C5×Q8—121no—
194C5×D4—41no—
195C5×D4—41no—
196C5×D4—41no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—42no—
204C2×C4×C5—62no—
205C2×C4×C5—62no—
206C2×C4×C5—122no—
207C2×C4×C5—122no—
208C2×C4×C5(C2×D3)⋊C411yescharacteristic
209C2×C4×C5(C2×D3)⋊C411yescharacteristic
210C2×C4×C5(C2×D3)⋊C411yescharacteristic
211C2×C4×C5C2×C4×D311yescharacteristic
212C2×#41C2×C5×D311yescharacteristic
213C22×Q8—31no—
214C4⋊Q8—32no—
215#124—32no—
216#123—61no—
217#123—61no—
218C22⋊Q8—61no—
219C22⋊Q8—61no—
220#117—21no—
221#117—31no—
222#117—61no—
223#99—21no—
224#99—32no—
225#99—61no—
226#99C2×C5×D312yes—
227C2×C3×C5—21no—
228C2×C3×C5—41no—
229C2×C3×C5—41no—
230C2×C3×C5—41no—
231C2×C3×C5C2×#9911yescharacteristic
232C23×C3C5×C22⋊C411yescharacteristic
233C23×C3C2×C5×D411yescharacteristic
234C23×C3C2×C5×D411yescharacteristic
235C3×Q8—41no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—42no—
246C2×C3×C4C5×C22⋊C411yescharacteristic
247C2×C3×C4C5×C22⋊C411yescharacteristic
248C2×C3×C4C5×C22⋊C411yescharacteristic
249C2×C3×C4C22×C4×C511yescharacteristic
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254Dic6—41no—
255Dic6—41no—
256Dic6—42no—
257C22×C5—21no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5C2×(C2×D3)⋊C411yescharacteristic
264C4×C5—21no—
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—82no—
268C4×C5—122no—
269C4×C5—122no—
270C4×C5—242no—
271C4×C5#80611yescharacteristic
272C4×C5#80611yescharacteristic
273#41—41no—
274#41—41no—
275C2×Q8—32no—
276C2×Q8—61no—
277C2×Q8—121no—
278C2×Q8—121no—
279C2×Q8C4×C5×D311yescharacteristic
280C2×D4C5×D1211yescharacteristic
281C2×D4C5×#6711yescharacteristic
282C2×D4C22×C5×D311yescharacteristic
283C22×C4—61no—
284C22×C4C4×C5×D311yescharacteristic
285C22×C4C5×D1211yescharacteristic
286C22×C4C5×#6711yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—121no—
292C22⋊C4—22no—
293C22⋊C4—61no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C42—32no—
298C42—41no—
299C42—121no—
300C3×C5#212911yescharacteristic, complemented, Hall
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3C2×C5×C22⋊C411yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—82no—
312C3×C4C5×#9911yescharacteristic
313C3×C4C5×#9911yescharacteristic
314Dic3—42no—
315Dic3—42no—
316Dic3—82no—
317C2×C5—21no—
318C2×C5—41no—
319C2×C5—41no—
320C2×C5—41no—
321C2×C5C2×#80611yescharacteristic
322C23C5×(C2×D3)⋊C411yescharacteristic
323C23C2×C5×D1211yescharacteristic
324C23C2×C5×#6711yescharacteristic
325Q8—41no—
326Q8—122no—
327Q8—121no—
328Q8—121no—
329D4—41no—
330D4—41no—
331D4—41no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4C5×(C2×D3)⋊C411yescharacteristic
344C2×C4C5×(C2×D3)⋊C411yescharacteristic
345C2×C4C5×(C2×D3)⋊C411yescharacteristic
346C2×C4C2×C4×C5×D311yescharacteristic
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3C2×C5×#9911yescharacteristic
352C5#7005911yescharacteristic, direct product factor, Sylow
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22C2×C5×(C2×D3)⋊C411yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—122no—
365C4—122no—
366C4—242no—
367C4C5×#80611yescharacteristic
368C4C5×#80611yescharacteristic
369C3C5×#212911yescharacteristic, complemented, Sylow
370C2—21no—
371C2—41no—
372C2—41no—
373C2—41no—
374C2C2×C5×#80611yescharacteristic
subgroupG/H
centerC2×C5C2×#806
commutatorC2×C3×C4C22×C4×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#410C2
socleC2×C3×C5C2×#99
prime radicalC2×C3×C5×#41C22
cosocleC2×C3×D4C23×C5
hypercenterC5×#410D3
hypocenterC3C5×#2129
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ C5×#410

Complements

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

NH ≅ G/Ntype
4. C5×#4473C2semidirect
7. C22×C5×Dic6C4semidirect
9. C5×#944C4semidirect
14. C5×#926C4semidirect
21. #70059C5direct (N × H)
56. #4473C2×C5semidirect
99. C22×Dic6C4×C5semidirect
101. #944C4×C5semidirect
106. #926C4×C5semidirect
300. C3×C5#2129semidirect
352. C5#70059direct (N × H)
369. C3C5×#2129semidirect

Maximal subgroups

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

subgroupindex
C5×#58252
C3×C5×#4102
C5×#52402
C5×#44732
C5×#21293
#700595