Finite Group Databasebeta

← C5×#70149 #14313717

Subgroups — C5×#70149 #14313717

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5830C211yesmaximal, characteristic
2C3×C5×#413C211yesmaximal, characteristic, complemented
3C5×#5237C211yesmaximal, characteristic, complemented
4C5×#4477C214yesmaximal, complemented
5C5×#2134—31nomaximal, Hall
6C2×C3×C5×#41C2211yescharacteristic
7C22×C5×D12C411yescharacteristic
8C2×C3×C5×M16—21no—
9C5×#947C2212yes—
10C5×#940C412yes—
11C5×#937—21no—
12C5×#937—21no—
13C5×#930—21no—
14C5×#930—21no—
15C5×#926C411yescharacteristic
16C2×C5×#208—21no—
17C3×C5×#101C2212yescomplemented
18C3×C5×#100—21no—
19C5×#836C2212yescomplemented
20C5×#833—21no—
21#70149C511yesmaximal, characteristic, direct product factor, Hall
22C5×#535—31no—
23C5×#413—31no—
24C5×#413D311yescharacteristic, complemented
25C5×#355—34no—
26C23×C5×D3—21no—
27C3×C5×#41—41no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8C2311yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C5×D4D411yescharacteristic
32C2×C3×C5×D4D411yescharacteristic
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C2×C5×#67—22no—
37C2×C5×#67—22no—
38C2×C5×D12—21no—
39C2×C5×D12—41no—
40C2×C5×D12—41no—
41C2×C5×D12C2×C412yes—
42C3×C5×M16—22no—
43C3×C5×M16—41no—
44C2×C3×C5×C8—41no—
45C5×#217—21no—
46C5×(C2×D3)⋊C4—22no—
47C5×(C2×D3)⋊C4—22no—
48C5×C4⋊Dic3—21no—
49C4×C5×Dic3C2×C412yes—
50C5×#208—22no—
51C5×#208—41no—
52C2×C5×C3⋊C8—41no—
53#5830C2×C511yescharacteristic
54C3×#413C2×C511yescharacteristic, complemented
55#5237C2×C511yescharacteristic, complemented
56#4477C2×C514yescomplemented
57C2×C5×#41C2×D311yescharacteristic
58C22×C5×D4—31no—
59C2×C5×M16—21no—
60C2×C5×M16—61no—
61C5×#127—32no—
62C5×#124—32no—
63C5×#121—61no—
64C5×#121—61no—
65C5×#120—61no—
66C5×#120—61no—
67C5×#117—31no—
68C5×#101—32no—
69C5×#101C2×D312yescomplemented
70C5×#100—21no—
71C5×#100—61no—
72#2134—31noSylow
73C23×C3×C5C22⋊C411yescharacteristic
74C23×C3×C5C22⋊C411yescharacteristic
75C23×C3×C5C22⋊C411yescharacteristic
76C22×C5×D3—22no—
77C22×C5×D3—42no—
78C22×C5×D3—82no—
79C3×C5×Q8—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C5×#67—82no—
94C5×#67—82no—
95C2×C5×Dic3—22no—
96C5×D12—42no—
97C5×D12—41no—
98C5×D12—41no—
99C3×C5×C8—42no—
100C5×C3⋊C8—42no—
101C2×C3×#41C22×C511yescharacteristic
102C22×D12C4×C511yescharacteristic
103C2×C3×M16—21no—
104#947C22×C512yes—
105#940C4×C512yes—
106#937—21no—
107#937—21no—
108#930—21no—
109#930—21no—
110#926C4×C511yescharacteristic
111C2×#208—21no—
112C3×#101C22×C512yescomplemented
113C3×#100—21no—
114#836C22×C512yescomplemented
115#833—21no—
116C24×C5—61no—
117C5×#41—41no—
118C5×#41—41no—
119C2×C5×Q8C22×D311yescharacteristic
120C2×C5×D4—32no—
121C2×C5×D4—62no—
122C2×C5×D4—62no—
123C2×C5×D4—61no—
124C2×C5×D4—121no—
125C2×C5×D4—121no—
126C2×C5×D4C4×D311yescharacteristic
127C2×C5×D4D1211yescharacteristic
128C2×C5×D4#6711yescharacteristic
129C22×C4×C5C4×D311yescharacteristic
130C22×C4×C5D1211yescharacteristic
131C22×C4×C5#6711yescharacteristic
132C5×M16—22no—
133C5×M16—41no—
134C5×M16—62no—
135C5×M16—121no—
136C2×C5×C8—41no—
137C2×C5×C8—121no—
138C5×C4⋊C4—61no—
139C5×C22⋊C4—61no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—62no—
142C42×C5—32no—
143#535—31no—
144#413—31no—
145#413C5×D311yescharacteristic, complemented
146#355—34no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—21no—
150C22×C3×C5—41no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5C2×C22⋊C411yescharacteristic
154C2×C5×D3—42no—
155C2×C5×D3—42no—
156C2×C5×D3—42no—
157C2×C5×D3—84no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5#9911yescharacteristic
162C3×C4×C5#9911yescharacteristic
163C5×Dic3—42no—
164C23×D3—21no—
165C3×#41—41no—
166C3×#41—41no—
167C2×C3×Q8C23×C511yescharacteristic
168C2×C3×D4C2×C4×C511yescharacteristic
169C2×C3×D4C5×D411yescharacteristic
170C2×C3×D4C5×D411yescharacteristic
171C22×C3×C4C2×C4×C511yescharacteristic
172C22×C3×C4C5×D411yescharacteristic
173C22×C3×C4C5×D411yescharacteristic
174C2×#67—22no—
175C2×#67—22no—
176C2×D12—21no—
177C2×D12—41no—
178C2×D12—41no—
179C2×D12C2×C4×C512yes—
180C3×M16—22no—
181C3×M16—41no—
182C2×C3×C8—41no—
183#217—21no—
184(C2×D3)⋊C4—22no—
185(C2×D3)⋊C4—22no—
186C4⋊Dic3—21no—
187C4×Dic3C2×C4×C512yes—
188#208—22no—
189#208—41no—
190C2×C3⋊C8—41no—
191C23×C5—62no—
192C23×C5—122no—
193C23×C5—242no—
194C23×C5(C2×D3)⋊C411yescharacteristic
195C23×C5(C2×D3)⋊C411yescharacteristic
196C23×C5(C2×D3)⋊C411yescharacteristic
197C5×Q8—41no—
198C5×D4—41no—
199C5×D4—41no—
200C5×D4—41no—
201C5×D4—122no—
202C5×D4—121no—
203C5×D4—121no—
204C5×D4—242no—
205C5×D4—242no—
206C2×C4×C5—21no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—21no—
212C2×C4×C5—62no—
213C2×C4×C5(C2×D3)⋊C411yescharacteristic
214C2×C4×C5C2×C4×D311yescharacteristic
215C2×C4×C5C2×D1211yescharacteristic
216C2×C4×C5C2×#6711yescharacteristic
217C5×C8—42no—
218C5×C8—122no—
219C2×#41C2×C5×D311yescharacteristic
220C22×D4—31no—
221C2×M16—21no—
222C2×M16—61no—
223#127—32no—
224#124—32no—
225#121—61no—
226#121—61no—
227#120—61no—
228#120—61no—
229#117—31no—
230#101—32no—
231#101C2×C5×D312yescomplemented
232#100—21no—
233#100—61no—
234C2×C3×C5—21no—
235C2×C3×C5—41no—
236C2×C3×C5—41no—
237C2×C3×C5—41no—
238C2×C3×C5C2×#9911yescharacteristic
239C5×D3—82no—
240C23×C3C5×C22⋊C411yescharacteristic
241C23×C3C5×C22⋊C411yescharacteristic
242C23×C3C5×C22⋊C411yescharacteristic
243C22×D3—22no—
244C22×D3—42no—
245C22×D3—82no—
246C3×Q8—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—41no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4C5×C22⋊C411yescharacteristic
257C2×C3×C4C22×C4×C511yescharacteristic
258C2×C3×C4C2×C5×D411yescharacteristic
259C2×C3×C4C2×C5×D411yescharacteristic
260#67—82no—
261#67—82no—
262C2×Dic3—22no—
263D12—42no—
264D12—41no—
265D12—41no—
266C3×C8—42no—
267C3⋊C8—42no—
268C22×C5—21no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—41no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—122no—
275C22×C5—122no—
276C22×C5—122no—
277C22×C5—244no—
278C22×C5C2×(C2×D3)⋊C411yescharacteristic
279C4×C5—21no—
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—122no—
283C4×C5#80611yescharacteristic
284C4×C5#80611yescharacteristic
285C24—61no—
286#41—41no—
287#41—41no—
288C2×Q8C22×C5×D311yescharacteristic
289C2×D4—32no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4C4×C5×D311yescharacteristic
296C2×D4C5×D1211yescharacteristic
297C2×D4C5×#6711yescharacteristic
298C22×C4C4×C5×D311yescharacteristic
299C22×C4C5×D1211yescharacteristic
300C22×C4C5×#6711yescharacteristic
301M16—22no—
302M16—41no—
303M16—62no—
304M16—121no—
305C2×C8—41no—
306C2×C8—121no—
307C4⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—62no—
310C22⋊C4—62no—
311C42—32no—
312C3×C5#213411yescharacteristic, complemented, Hall
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3C2×C5×C22⋊C411yescharacteristic
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C2×D3—84no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4C5×#9911yescharacteristic
328C3×C4C5×#9911yescharacteristic
329Dic3—42no—
330C2×C5—21no—
331C2×C5—41no—
332C2×C5—41no—
333C2×C5—41no—
334C2×C5—242no—
335C2×C5C2×#80611yescharacteristic
336C23—62no—
337C23—122no—
338C23—242no—
339C23C5×(C2×D3)⋊C411yescharacteristic
340C23C5×(C2×D3)⋊C411yescharacteristic
341C23C5×(C2×D3)⋊C411yescharacteristic
342Q8—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—122no—
347D4—121no—
348D4—121no—
349D4—242no—
350D4—242no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—62no—
358C2×C4C5×(C2×D3)⋊C411yescharacteristic
359C2×C4C2×C4×C5×D311yescharacteristic
360C2×C4C2×C5×D1211yescharacteristic
361C2×C4C2×C5×#6711yescharacteristic
362C8—42no—
363C8—122no—
364C2×C3—21no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3C2×C5×#9911yescharacteristic
369D3—82no—
370C5#7014911yescharacteristic, direct product factor, Sylow
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—122no—
378C22—122no—
379C22—122no—
380C22—244no—
381C22C2×C5×(C2×D3)⋊C411yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—122no—
386C4C5×#80611yescharacteristic
387C4C5×#80611yescharacteristic
388C3C5×#213411yescharacteristic, complemented, Sylow
389C2—21no—
390C2—41no—
391C2—41no—
392C2—41no—
393C2—242no—
394C2C2×C5×#80611yescharacteristic
subgroupG/H
centerC2×C5C2×#806
commutatorC2×C3×C4C22×C4×C5
FrattiniC2×Q8C22×C5×D3
FittingC3×C5×#413C2
socleC2×C3×C5C2×#99
prime radicalC5×#5830C2
cosocleC2×C3×Q8C23×C5
hypercenterC5×#413D3
hypocenterC3C5×#2134
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ C5×#413

Complements

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

NH ≅ G/Ntype
2. C3×C5×#413C2semidirect
3. C5×#5237C2semidirect
4. C5×#4477C2semidirect
17. C3×C5×#101C22semidirect
19. C5×#836C22semidirect
21. #70149C5direct (N × H)
24. C5×#413D3semidirect
54. C3×#413C2×C5semidirect
55. #5237C2×C5semidirect
56. #4477C2×C5semidirect
69. C5×#101C2×D3semidirect
112. C3×#101C22×C5semidirect
114. #836C22×C5semidirect
145. #413C5×D3semidirect
231. #101C2×C5×D3semidirect
312. C3×C5#2134semidirect
370. C5#70149direct (N × H)
388. C3C5×#2134semidirect

Maximal subgroups

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

subgroupindex
C5×#58302
C3×C5×#4132
C5×#52372
C5×#44772
C5×#21343
#701495