Finite Group Databasebeta

← C5×#70057 #14313257

Subgroups — C5×#70057 #14313257

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5830C211yesmaximal, characteristic
2C3×C5×#410C211yesmaximal, characteristic, complemented
3C5×#5237C211yesmaximal, characteristic, complemented
4C5×#4471C214yesmaximal, complemented
5C5×#2122—31nomaximal, Hall
6C2×C3×C5×#41C2211yescharacteristic
7C22×C5×D12C411yescharacteristic, complemented
8C3×C5×#117—21no—
9C5×#947—21no—
10C5×#940—21no—
11C5×#937C412yescomplemented
12C5×#937C2212yes—
13C5×#930—21no—
14C5×#930—21no—
15C5×#926C411yescharacteristic, complemented
16C2×C5×#208—21no—
17C3×C5×#99—21no—
18C3×C5×#99C2212yes—
19C5×#836—21no—
20C5×#833C2212yes—
21#70057C511yesmaximal, characteristic, direct product factor, Hall
22C5×#535—31no—
23C5×#413—31no—
24C5×#410D311yescharacteristic, complemented
25C5×AΣL(1,𝔽16)—34no—
26C23×C5×D3C2×C412yes—
27C3×C5×#41—41no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8D411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C5×D4D411yescharacteristic
32C2×C3×C5×D4C2311yescharacteristic
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—21no—
40C2×C5×D12—41no—
41C2×C5×D12—41no—
42C3×C5×C4⋊C4—41no—
43C3×C5×C22⋊C4—22no—
44C3×C42×C5—41no—
45C5×#217C2×C412yes—
46C5×(C2×D3)⋊C4—22no—
47C5×(C2×D3)⋊C4—22no—
48C5×C4⋊Dic3—21no—
49C4×C5×Dic3—21no—
50C5×#208—22no—
51C5×#208—41no—
52C2×C5×C3⋊C8—41no—
53#5830C2×C511yescharacteristic
54C3×#410C2×C511yescharacteristic, complemented
55#5237C2×C511yescharacteristic, complemented
56#4471C2×C514yescomplemented
57C2×C5×#41C2×D311yescharacteristic
58C22×C5×D4—31no—
59C2×C5×M16—61no—
60C5×#127—61no—
61C5×#124—61no—
62C5×#121—61no—
63C5×#121—61no—
64C5×#120—32no—
65C5×#120—32no—
66C5×#117—21no—
67C5×#117—31no—
68C5×#101—61no—
69C5×#100—32no—
70C5×#99—21no—
71C5×#99C2×D312yes—
72#2122—31noSylow
73C23×C3×C5C22⋊C411yescharacteristic
74C23×C3×C5C22×C411yescharacteristic
75C23×C3×C5C2×D411yescharacteristic
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×C5—42no—
90C2×C3×C4×C5C22⋊C411yescharacteristic
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94C5×#67—82no—
95C5×#67—82no—
96C2×C5×Dic3—22no—
97C5×D12—42no—
98C5×D12—41no—
99C5×D12—41no—
100C5×C3⋊C8—42no—
101C2×C3×#41C22×C511yescharacteristic
102C22×D12C4×C511yescharacteristic, complemented
103C3×#117—21no—
104#947—21no—
105#940—21no—
106#937C4×C512yescomplemented
107#937C22×C512yes—
108#930—21no—
109#930—21no—
110#926C4×C511yescharacteristic, complemented
111C2×#208—21no—
112C3×#99—21no—
113C3×#99C22×C512yes—
114#836—21no—
115#833C22×C512yes—
116C24×C5—32no—
117C5×#41—41no—
118C5×#41—41no—
119C2×C5×Q8D1211yescharacteristic
120C2×C5×D4—62no—
121C2×C5×D4—62no—
122C2×C5×D4—61no—
123C2×C5×D4—61no—
124C2×C5×D4—121no—
125C2×C5×D4—121no—
126C2×C5×D4C4×D311yescharacteristic
127C2×C5×D4#6711yescharacteristic
128C2×C5×D4C22×D311yescharacteristic
129C22×C4×C5C4×D311yescharacteristic
130C22×C4×C5D1211yescharacteristic
131C22×C4×C5#6711yescharacteristic
132C5×M16—62no—
133C5×M16—121no—
134C2×C5×C8—121no—
135C5×C4⋊C4—41no—
136C5×C4⋊C4—61no—
137C5×C22⋊C4—22no—
138C5×C22⋊C4—32no—
139C5×C22⋊C4—62no—
140C5×C22⋊C4—62no—
141C42×C5—41no—
142C42×C5—61no—
143#535—31no—
144#413—31no—
145#410C5×D311yescharacteristic, complemented
146AΣL(1,𝔽16)—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—82no—
162C3×C4×C5#9911yescharacteristic
163C3×C4×C5#9911yescharacteristic
164C5×Dic3—42no—
165C23×D3C2×C4×C512yes—
166C3×#41—41no—
167C3×#41—41no—
168C2×C3×Q8C5×D411yescharacteristic
169C2×C3×D4C2×C4×C511yescharacteristic
170C2×C3×D4C5×D411yescharacteristic
171C2×C3×D4C23×C511yescharacteristic
172C22×C3×C4C2×C4×C511yescharacteristic
173C22×C3×C4C5×D411yescharacteristic
174C22×C3×C4C5×D411yescharacteristic
175C2×#67—22no—
176C2×#67—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—41no—
180C2×D12—41no—
181C3×C4⋊C4—41no—
182C3×C22⋊C4—22no—
183C3×C42—41no—
184#217C2×C4×C512yes—
185(C2×D3)⋊C4—22no—
186(C2×D3)⋊C4—22no—
187C4⋊Dic3—21no—
188C4×Dic3—21no—
189#208—22no—
190#208—41no—
191C2×C3⋊C8—41no—
192C23×C5—62no—
193C23×C5—122no—
194C23×C5—242no—
195C23×C5(C2×D3)⋊C411yescharacteristic
196C23×C5C2×C4×D311yescharacteristic
197C23×C5C2×#6711yescharacteristic
198C5×Q8—41no—
199C5×D4—41no—
200C5×D4—41no—
201C5×D4—41no—
202C5×D4—122no—
203C5×D4—121no—
204C5×D4—121no—
205C5×D4—242no—
206C5×D4—242no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—42no—
214C2×C4×C5—62no—
215C2×C4×C5(C2×D3)⋊C411yescharacteristic
216C2×C4×C5(C2×D3)⋊C411yescharacteristic
217C2×C4×C5(C2×D3)⋊C411yescharacteristic
218C2×C4×C5C2×D1211yescharacteristic
219C5×C8—122no—
220C2×#41C2×C5×D311yescharacteristic
221C22×D4—31no—
222C2×M16—61no—
223#127—61no—
224#124—61no—
225#121—61no—
226#121—61no—
227#120—32no—
228#120—32no—
229#117—21no—
230#117—31no—
231#101—61no—
232#100—32no—
233#99—21no—
234#99C2×C5×D312yes—
235C2×C3×C5—21no—
236C2×C3×C5—41no—
237C2×C3×C5—41no—
238C2×C3×C5—41no—
239C2×C3×C5C2×#9911yescharacteristic
240C5×D3—82no—
241C23×C3C5×C22⋊C411yescharacteristic
242C23×C3C22×C4×C511yescharacteristic
243C23×C3C2×C5×D411yescharacteristic
244C22×D3—22no—
245C22×D3—42no—
246C22×D3—82no—
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—42no—
258C2×C3×C4C5×C22⋊C411yescharacteristic
259C2×C3×C4C5×C22⋊C411yescharacteristic
260C2×C3×C4C5×C22⋊C411yescharacteristic
261C2×C3×C4C2×C5×D411yescharacteristic
262#67—82no—
263#67—82no—
264C2×Dic3—22no—
265D12—42no—
266D12—41no—
267D12—41no—
268C3⋊C8—42no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5—122no—
276C22×C5—122no—
277C22×C5—122no—
278C22×C5—244no—
279C22×C5C2×(C2×D3)⋊C411yescharacteristic
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—21no—
283C4×C5—82no—
284C4×C5—122no—
285C4×C5#80611yescharacteristic
286C4×C5#80611yescharacteristic
287C24—32no—
288#41—41no—
289#41—41no—
290C2×Q8C5×D1211yescharacteristic
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4C4×C5×D311yescharacteristic
298C2×D4C5×#6711yescharacteristic
299C2×D4C22×C5×D311yescharacteristic
300C22×C4C4×C5×D311yescharacteristic
301C22×C4C5×D1211yescharacteristic
302C22×C4C5×#6711yescharacteristic
303M16—62no—
304M16—121no—
305C2×C8—121no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C22⋊C4—22no—
309C22⋊C4—32no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C42—41no—
313C42—61no—
314C3×C5#212211yescharacteristic, complemented, Hall
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3C2×C5×C22⋊C411yescharacteristic
322C2×D3—42no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—84no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—82no—
330C3×C4C5×#9911yescharacteristic
331C3×C4C5×#9911yescharacteristic
332Dic3—42no—
333C2×C5—21no—
334C2×C5—41no—
335C2×C5—41no—
336C2×C5—41no—
337C2×C5—242no—
338C2×C5C2×#80611yescharacteristic
339C23—62no—
340C23—122no—
341C23—242no—
342C23C5×(C2×D3)⋊C411yescharacteristic
343C23C2×C4×C5×D311yescharacteristic
344C23C2×C5×#6711yescharacteristic
345Q8—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—122no—
350D4—121no—
351D4—121no—
352D4—242no—
353D4—242no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—62no—
362C2×C4C5×(C2×D3)⋊C411yescharacteristic
363C2×C4C5×(C2×D3)⋊C411yescharacteristic
364C2×C4C5×(C2×D3)⋊C411yescharacteristic
365C2×C4C2×C5×D1211yescharacteristic
366C8—122no—
367C2×C3—21no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3C2×C5×#9911yescharacteristic
372D3—82no—
373C5#7005711yescharacteristic, direct product factor, Sylow
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—122no—
381C22—122no—
382C22—122no—
383C22—244no—
384C22C2×C5×(C2×D3)⋊C411yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—122no—
390C4C5×#80611yescharacteristic
391C4C5×#80611yescharacteristic
392C3C5×#212211yescharacteristic, complemented, Sylow
393C2—21no—
394C2—41no—
395C2—41no—
396C2—41no—
397C2—242no—
398C2C2×C5×#80611yescharacteristic
subgroupG/H
centerC2×C5C2×#806
commutatorC23×C3C22×C4×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#410C2
socleC2×C3×C5C2×#99
prime radicalC5×#5830C2
cosocleC2×C3×D4C23×C5
hypercenterC5×#410D3
hypocenterC3C5×#2122
LCSG ▷ C23×C3 ▷ 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
2. C3×C5×#410C2semidirect
3. C5×#5237C2semidirect
4. C5×#4471C2semidirect
7. C22×C5×D12C4semidirect
11. C5×#937C4semidirect
15. C5×#926C4semidirect
21. #70057C5direct (N × H)
24. C5×#410D3semidirect
54. C3×#410C2×C5semidirect
55. #5237C2×C5semidirect
56. #4471C2×C5semidirect
102. C22×D12C4×C5semidirect
106. #937C4×C5semidirect
110. #926C4×C5semidirect
145. #410C5×D3semidirect
314. C3×C5#2122semidirect
373. C5#70057direct (N × H)
392. C3C5×#2122semidirect

Maximal subgroups

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

subgroupindex
C5×#58302
C3×C5×#4102
C5×#52372
C5×#44712
C5×#21223
#700575