Finite Group Databasebeta

← C5×#70056 #14313131

Subgroups — C5×#70056 #14313131

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5828C211yesmaximal, characteristic
2C3×C5×#410C211yesmaximal, characteristic, complemented
3C5×#5237C211yesmaximal, characteristic, complemented
4C5×#4472C212yesmaximal, complemented
5C5×#4471C212yesmaximal, complemented
6C5×#2120—31nomaximal, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C22×C4×C5×D3C411yescharacteristic, complemented
9C3×C5×#117—21no—
10C5×#946—21no—
11C5×#938—21no—
12C5×#937C411yescharacteristic, complemented
13C5×#937C2211yescharacteristic
14C5×#935C411yescharacteristic, complemented
15C5×#935C2211yescharacteristic
16C4×C5×#67—21no—
17C4×C5×#67—21no—
18C5×#926C411yescharacteristic, complemented
19C2×C5×#208—21no—
20C3×C5×#99—21no—
21C3×C5×#99C2212yes—
22C5×#836—21no—
23C5×#833C2212yes—
24#70056C511yesmaximal, characteristic, direct product factor, Hall
25C5×#525—31no—
26C5×#413—31no—
27C5×#410D311yescharacteristic, complemented
28C5×#352—32no—
29C5×AΣL(1,𝔽16)—32no—
30C23×C5×D3C2×C411yescharacteristic
31C3×C5×#41—41no—
32C3×C5×#41—41no—
33C2×C3×C5×Q8D411yescharacteristic
34C2×C3×C5×D4C2×C411yescharacteristic
35C2×C3×C5×D4D411yescharacteristic
36C2×C3×C5×D4C2311yescharacteristic
37C22×C3×C4×C5C2×C411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C22×C3×C4×C5D411yescharacteristic
40C2×C5×#67—21no—
41C2×C5×#67—21no—
42C22×C5×Dic3C2×C411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C4×C5×D3—21no—
45C2×C4×C5×D3—41no—
46C2×C4×C5×D3—41no—
47C3×C5×C4⋊C4—41no—
48C3×C5×C22⋊C4—22no—
49C3×C42×C5—41no—
50C5×#217—21no—
51C5×#217—21no—
52C5×#217C2×C411yescharacteristic
53C5×#217C2×C411yescharacteristic
54C5×(C2×D3)⋊C4—21no—
55C5×(C2×D3)⋊C4—21no—
56C5×C4⋊Dic3—21no—
57C5×Dic3⋊C4—21no—
58C5×Dic3⋊C4—21no—
59C4×C5×Dic3—21no—
60C5×#208—22no—
61C5×#208—41no—
62C2×C5×C3⋊C8—41no—
63#5828C2×C511yescharacteristic
64C3×#410C2×C511yescharacteristic, complemented
65#5237C2×C511yescharacteristic, complemented
66#4472C2×C512yescomplemented
67#4471C2×C512yescomplemented
68C2×C5×#41C2×D311yescharacteristic
69C23×C4×C5—31no—
70C2×C5×M16—61no—
71C5×#123—31no—
72C5×#123—31no—
73C5×C22⋊Q8—61no—
74C5×#121—61no—
75C5×#120—31no—
76C5×#120—31no—
77C4×C5×D4—61no—
78C4×C5×D4—61no—
79C5×#117—21no—
80C5×#117—31no—
81C5×#101—61no—
82C5×#100—32no—
83C5×#99—21no—
84C5×#99C2×D312yes—
85#2120—31noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C22×C411yescharacteristic
88C23×C3×C5C2×D411yescharacteristic
89C22×C5×D3—21no—
90C22×C5×D3—42no—
91C22×C5×D3—41no—
92C3×C5×Q8—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—42no—
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×C4×C5C22⋊C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C5×#67—42no—
108C5×#67—42no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C2×C5×Dic3—21no—
112C2×C5×Dic3—41no—
113C4×C5×D3—42no—
114C4×C5×D3—41no—
115C4×C5×D3—41no—
116C5×C3⋊C8—42no—
117C2×C3×#41C22×C511yescharacteristic
118C22×C4×D3C4×C511yescharacteristic, complemented
119C3×#117—21no—
120#946—21no—
121#938—21no—
122#937C4×C511yescharacteristic, complemented
123#937C22×C511yescharacteristic
124#935C4×C511yescharacteristic, complemented
125#935C22×C511yescharacteristic
126C4×#67—21no—
127C4×#67—21no—
128#926C4×C511yescharacteristic, complemented
129C2×#208—21no—
130C3×#99—21no—
131C3×#99C22×C512yes—
132#836—21no—
133#833C22×C512yes—
134C24×C5—31no—
135C5×#41—41no—
136C5×#41—41no—
137C2×C5×Q8D1211yescharacteristic
138C2×C5×D4—61no—
139C2×C5×D4—61no—
140C2×C5×D4C4×D311yescharacteristic
141C2×C5×D4#6711yescharacteristic
142C2×C5×D4C22×D311yescharacteristic
143C22×C4×C5—31no—
144C22×C4×C5—61no—
145C22×C4×C5—61no—
146C22×C4×C5—121no—
147C22×C4×C5—121no—
148C22×C4×C5C4×D311yescharacteristic
149C22×C4×C5D1211yescharacteristic
150C22×C4×C5#6711yescharacteristic
151C5×M16—62no—
152C5×M16—121no—
153C2×C5×C8—121no—
154C5×C4⋊C4—41no—
155C5×C4⋊C4—61no—
156C5×C4⋊C4—61no—
157C5×C4⋊C4—61no—
158C5×C22⋊C4—22no—
159C5×C22⋊C4—31no—
160C5×C22⋊C4—31no—
161C5×C22⋊C4—61no—
162C5×C22⋊C4—61no—
163C5×C22⋊C4—61no—
164C5×C22⋊C4—61no—
165C42×C5—41no—
166C42×C5—61no—
167#525—31no—
168#413—31no—
169#410C5×D311yescharacteristic, complemented
170#352—32no—
171AΣL(1,𝔽16)—32no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—41no—
176C22×C3×C5—41no—
177C22×C3×C5—41no—
178C22×C3×C5C2×C22⋊C411yescharacteristic
179C2×C5×D3—22no—
180C2×C5×D3—41no—
181C2×C5×D3—42no—
182C2×C5×D3—41no—
183C2×C5×D3—42no—
184C3×C4×C5—21no—
185C3×C4×C5—21no—
186C3×C4×C5—21no—
187C3×C4×C5—82no—
188C3×C4×C5#9911yescharacteristic
189C3×C4×C5#9911yescharacteristic
190C5×Dic3—41no—
191C5×Dic3—41no—
192C5×Dic3—41no—
193C23×D3C2×C4×C511yescharacteristic
194C3×#41—41no—
195C3×#41—41no—
196C2×C3×Q8C5×D411yescharacteristic
197C2×C3×D4C2×C4×C511yescharacteristic
198C2×C3×D4C5×D411yescharacteristic
199C2×C3×D4C23×C511yescharacteristic
200C22×C3×C4C2×C4×C511yescharacteristic
201C22×C3×C4C5×D411yescharacteristic
202C22×C3×C4C5×D411yescharacteristic
203C2×#67—21no—
204C2×#67—21no—
205C22×Dic3C2×C4×C511yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—41no—
209C2×C4×D3—41no—
210C3×C4⋊C4—41no—
211C3×C22⋊C4—22no—
212C3×C42—41no—
213#217—21no—
214#217—21no—
215#217C2×C4×C511yescharacteristic
216#217C2×C4×C511yescharacteristic
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219C4⋊Dic3—21no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222C4×Dic3—21no—
223#208—22no—
224#208—41no—
225C2×C3⋊C8—41no—
226C23×C5—61no—
227C23×C5—121no—
228C23×C5—122no—
229C23×C5(C2×D3)⋊C411yescharacteristic
230C23×C5C2×C4×D311yescharacteristic
231C23×C5C2×#6711yescharacteristic
232C5×Q8—41no—
233C5×D4—41no—
234C5×D4—41no—
235C5×D4—41no—
236C5×D4—122no—
237C5×D4—122no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—42no—
245C2×C4×C5—61no—
246C2×C4×C5—61no—
247C2×C4×C5—61no—
248C2×C4×C5—122no—
249C2×C4×C5—121no—
250C2×C4×C5—121no—
251C2×C4×C5—121no—
252C2×C4×C5(C2×D3)⋊C411yescharacteristic
253C2×C4×C5(C2×D3)⋊C411yescharacteristic
254C2×C4×C5(C2×D3)⋊C411yescharacteristic
255C2×C4×C5C2×D1211yescharacteristic
256C5×C8—122no—
257C2×#41C2×C5×D311yescharacteristic
258C23×C4—31no—
259C2×M16—61no—
260#123—31no—
261#123—31no—
262C22⋊Q8—61no—
263#121—61no—
264#120—31no—
265#120—31no—
266C4×D4—61no—
267C4×D4—61no—
268#117—21no—
269#117—31no—
270#101—61no—
271#100—32no—
272#99—21no—
273#99C2×C5×D312yes—
274C2×C3×C5—21no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5—41no—
278C2×C3×C5C2×#9911yescharacteristic
279C5×D3—42no—
280C23×C3C5×C22⋊C411yescharacteristic
281C23×C3C22×C4×C511yescharacteristic
282C23×C3C2×C5×D411yescharacteristic
283C22×D3—21no—
284C22×D3—42no—
285C22×D3—41no—
286C3×Q8—41no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4—41no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—42no—
297C2×C3×C4C5×C22⋊C411yescharacteristic
298C2×C3×C4C5×C22⋊C411yescharacteristic
299C2×C3×C4C5×C22⋊C411yescharacteristic
300C2×C3×C4C2×C5×D411yescharacteristic
301#67—42no—
302#67—42no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—41no—
307C4×D3—42no—
308C4×D3—41no—
309C4×D3—41no—
310C3⋊C8—42no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5—41no—
317C22×C5—62no—
318C22×C5—121no—
319C22×C5—122no—
320C22×C5—122no—
321C22×C5—121no—
322C22×C5C2×(C2×D3)⋊C411yescharacteristic
323C4×C5—21no—
324C4×C5—21no—
325C4×C5—21no—
326C4×C5—82no—
327C4×C5—121no—
328C4×C5—121no—
329C4×C5—121no—
330C4×C5#80611yescharacteristic
331C4×C5#80611yescharacteristic
332C24—31no—
333#41—41no—
334#41—41no—
335C2×Q8C5×D1211yescharacteristic
336C2×D4—61no—
337C2×D4—61no—
338C2×D4C4×C5×D311yescharacteristic
339C2×D4C5×#6711yescharacteristic
340C2×D4C22×C5×D311yescharacteristic
341C22×C4—31no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4C4×C5×D311yescharacteristic
347C22×C4C5×D1211yescharacteristic
348C22×C4C5×#6711yescharacteristic
349M16—62no—
350M16—121no—
351C2×C8—121no—
352C4⋊C4—41no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C22⋊C4—22no—
357C22⋊C4—31no—
358C22⋊C4—31no—
359C22⋊C4—61no—
360C22⋊C4—61no—
361C22⋊C4—61no—
362C22⋊C4—61no—
363C42—41no—
364C42—61no—
365C3×C5#212011yescharacteristic, complemented, Hall
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3C2×C5×C22⋊C411yescharacteristic
373C2×D3—22no—
374C2×D3—41no—
375C2×D3—42no—
376C2×D3—41no—
377C2×D3—42no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—82no—
382C3×C4C5×#9911yescharacteristic
383C3×C4C5×#9911yescharacteristic
384Dic3—41no—
385Dic3—41no—
386Dic3—41no—
387C2×C5—21no—
388C2×C5—41no—
389C2×C5—41no—
390C2×C5—41no—
391C2×C5—122no—
392C2×C5C2×#80611yescharacteristic
393C23—61no—
394C23—121no—
395C23—122no—
396C23C5×(C2×D3)⋊C411yescharacteristic
397C23C2×C4×C5×D311yescharacteristic
398C23C2×C5×#6711yescharacteristic
399Q8—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—122no—
404D4—122no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4C5×(C2×D3)⋊C411yescharacteristic
420C2×C4C5×(C2×D3)⋊C411yescharacteristic
421C2×C4C5×(C2×D3)⋊C411yescharacteristic
422C2×C4C2×C5×D1211yescharacteristic
423C8—122no—
424C2×C3—21no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3C2×C5×#9911yescharacteristic
429D3—42no—
430C5#7005611yescharacteristic, direct product factor, Sylow
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—62no—
438C22—121no—
439C22—122no—
440C22—122no—
441C22—121no—
442C22C2×C5×(C2×D3)⋊C411yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—82no—
447C4—121no—
448C4—121no—
449C4—121no—
450C4C5×#80611yescharacteristic
451C4C5×#80611yescharacteristic
452C3C5×#212011yescharacteristic, complemented, Sylow
453C2—21no—
454C2—41no—
455C2—41no—
456C2—41no—
457C2—122no—
458C2C2×C5×#80611yescharacteristic
subgroupG/H
centerC4×C5#806
commutatorC23×C3C22×C4×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#410C2
socleC2×C3×C5C2×#99
prime radicalC5×#5828C2
cosocleC2×C3×D4C23×C5
hypercenterC5×#410D3
hypocenterC3C5×#2120
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×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×#4472C2semidirect
5. C5×#4471C2semidirect
8. C22×C4×C5×D3C4semidirect
12. C5×#937C4semidirect
14. C5×#935C4semidirect
18. C5×#926C4semidirect
24. #70056C5direct (N × H)
27. C5×#410D3semidirect
64. C3×#410C2×C5semidirect
65. #5237C2×C5semidirect
66. #4472C2×C5semidirect
67. #4471C2×C5semidirect
118. C22×C4×D3C4×C5semidirect
122. #937C4×C5semidirect
124. #935C4×C5semidirect
128. #926C4×C5semidirect
169. #410C5×D3semidirect
365. C3×C5#2120semidirect
430. C5#70056direct (N × H)
452. C3C5×#2120semidirect

Maximal subgroups

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

subgroupindex
C5×#58282
C3×C5×#4102
C5×#52372
C5×#44722
C5×#44712
C5×#21203
#700565