Finite Group Databasebeta

← C5×#71886 #14288326

Subgroups — C5×#71886 #14288326

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#539C211yesmaximal, characteristic
2C5×#5043C211yesmaximal, characteristic, complemented
3C5×#5033C211yesmaximal, characteristic
4C5×#4981C211yesmaximal, characteristic, complemented
5C5×#4538C211yesmaximal, characteristic, complemented
6C5×#4537C211yesmaximal, characteristic
7C5×#4528C211yesmaximal, characteristic, complemented
8C5×#1609—31nomaximal, Hall
9C3×C5×#126—21no—
10C3×C5×#125—21no—
11C3×C5×#123—21no—
12C3×C5×#123—21no—
13C3×C5×C22⋊Q8—21no—
14C3×C5×C22⋊Q8C2211yescharacteristic
15C3×C5×#121C2211yescharacteristic
16C3×C4×C5×D4—21no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×Dic3⋊C4—21no—
20C5×#835—21no—
21C5×#832—21no—
22C5×C3⋊#95—21no—
23C5×C3⋊#95C2211yescharacteristic
24C5×#830C2211yescharacteristic
25C5×#830C2211yescharacteristic
26C5×#808—21no—
27C5×#807—21no—
28#71886C511yesmaximal, characteristic, direct product factor, Hall
29C5×#539D311yescharacteristic
30C5×#483—31no—
31C5×#480—31no—
32C5×#398—31no—
33C5×#328—31no—
34C5×#327—31no—
35C5×#324—31no—
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4D411yescharacteristic
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5C2311yescharacteristic
43C22×C5×Dic3—21no—
44C22×C5×Dic3—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4D411yescharacteristic
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59C5×C4⋊Dic3—21no—
60C5×C4⋊Dic3—21no—
61C5×Dic3⋊C4—42no—
62C2×C5×C3⋊C8—21no—
63C2×C5×C3⋊C8—21no—
64C3×#539C2×C511yescharacteristic
65#5043C2×C511yescharacteristic, complemented
66#5033C2×C511yescharacteristic
67#4981C2×C511yescharacteristic, complemented
68#4538C2×C511yescharacteristic, complemented
69#4537C2×C511yescharacteristic
70#4528C2×C511yescharacteristic, complemented
71C5×#126—21no—
72C5×#125—21no—
73C5×#123—21no—
74C5×#123—21no—
75C5×C22⋊Q8—21no—
76C5×C22⋊Q8C2×D311yescharacteristic
77C5×#121C2×D311yescharacteristic
78C4×C5×D4—21no—
79C2×C5×C4⋊C4—31no—
80C2×C5×C4⋊C4—61no—
81C2×C5×C4⋊C4C2×D311yescharacteristic
82C5×#107—61no—
83C5×#106—61no—
84C5×#103—61no—
85C5×#102—61no—
86C5×C22⋊C8—31no—
87C5×C22⋊C8—31no—
88C5×#95—31no—
89C5×#95—61no—
90#1609—31noSylow
91C23×C3×C5—21no—
92C23×C3×C5C2×D411yescharacteristic
93C3×C5×Q8—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C3×C5×D4—41no—
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—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—81no—
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C5×Dic3—22no—
110C2×C5×Dic3—41no—
111C2×C5×Dic3—41no—
112C2×C5×Dic3—41no—
113C2×C5×Dic3—41no—
114C2×C5×Dic3—41no—
115C5×C3⋊C8—41no—
116C5×C3⋊C8—41no—
117C3×#126—21no—
118C3×#125—21no—
119C3×#123—21no—
120C3×#123—21no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8C22×C511yescharacteristic
123C3×#121C22×C511yescharacteristic
124C3×C4×D4—21no—
125C2×C3×C4⋊C4C22×C511yescharacteristic
126C2×C4⋊Dic3C22×C511yescharacteristic
127C2×Dic3⋊C4—21no—
128#835—21no—
129#832—21no—
130C3⋊#95—21no—
131C3⋊#95C22×C511yescharacteristic
132#830C22×C511yescharacteristic
133#830C22×C511yescharacteristic
134#808—21no—
135#807—21no—
136C2×C5×Q8#6711yescharacteristic
137C2×C5×D4—21no—
138C2×C5×D4#6711yescharacteristic
139C22×C4×C5—21no—
140C22×C4×C5—61no—
141C22×C4×C5—61no—
142C22×C4×C5#6711yescharacteristic
143C22×C4×C5#6711yescharacteristic
144C22×C4×C5C22×D311yescharacteristic
145C2×C5×C8—61no—
146C2×C5×C8—61no—
147C5×C4⋊C4—21no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—21no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4—61no—
154C5×C4⋊C4—61no—
155C5×C4⋊C4—122no—
156C5×C4⋊C4#6711yescharacteristic
157C5×C4⋊C4#6711yescharacteristic
158C5×C22⋊C4—21no—
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C42×C5—21no—
164#539C5×D311yescharacteristic
165#483—31no—
166#480—31no—
167#398—31no—
168#328—31no—
169#327—31no—
170#324—31no—
171C22×C3×C5—21no—
172C22×C3×C5—21no—
173C22×C3×C5—41no—
174C22×C3×C5—41no—
175C22×C3×C5—41no—
176C22×C3×C5#12011yescharacteristic
177C22×C3×C5#13511yescharacteristic
178C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
179C3×C4×C5—21no—
180C3×C4×C5—21no—
181C3×C4×C5—41no—
182C3×C4×C5—41no—
183C3×C4×C5—41no—
184C3×C4×C5—41no—
185C3×C4×C5—41no—
186C5×Dic3—42no—
187C5×Dic3—81no—
188C2×C3×Q8C5×D411yescharacteristic
189C2×C3×D4—21no—
190C2×C3×D4C5×D411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4C5×D411yescharacteristic
193C22×C3×C4C5×D411yescharacteristic
194C22×C3×C4C23×C511yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C5×D411yescharacteristic
204C3×C4⋊C4C5×D411yescharacteristic
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C42—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213Dic3⋊C4—42no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C23×C5—21no—
217C23×C5C2×#6711yescharacteristic
218C5×Q8—41no—
219C5×D4—41no—
220C5×D4—41no—
221C5×D4—41no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5—41no—
230C2×C4×C5—41no—
231C2×C4×C5—62no—
232C2×C4×C5—81no—
233C2×C4×C5—121no—
234C2×C4×C5—121no—
235C2×C4×C5—121no—
236C2×C4×C5—121no—
237C2×C4×C5—121no—
238C2×C4×C5C2×#6711yescharacteristic
239C2×C4×C5C2×#6711yescharacteristic
240C5×C8—121no—
241C5×C8—121no—
242#126—21no—
243#125—21no—
244#123—21no—
245#123—21no—
246C22⋊Q8—21no—
247C22⋊Q8C2×C5×D311yescharacteristic
248#121C2×C5×D311yescharacteristic
249C4×D4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4C2×C5×D311yescharacteristic
253#107—61no—
254#106—61no—
255#103—61no—
256#102—61no—
257C22⋊C8—31no—
258C22⋊C8—31no—
259#95—31no—
260#95—61no—
261C2×C3×C5—21no—
262C2×C3×C5—21no—
263C2×C3×C5—81no—
264C2×C3×C5#44911yescharacteristic
265C2×C3×C5#45611yescharacteristic
266C2×C3×C5#45811yescharacteristic
267C23×C3—21no—
268C23×C3C2×C5×D411yescharacteristic
269C3×Q8—41no—
270C3×D4—41no—
271C3×D4—41no—
272C3×D4—41no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—81no—
283C2×C3×C4C2×C5×D411yescharacteristic
284C2×C3×C4C2×C5×D411yescharacteristic
285C2×Dic3—22no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C3⋊C8—41no—
292C3⋊C8—41no—
293C22×C5—21no—
294C22×C5—21no—
295C22×C5—41no—
296C22×C5—41no—
297C22×C5—41no—
298C22×C5#94911yescharacteristic
299C22×C5#95011yescharacteristic
300C22×C5#95311yescharacteristic
301C4×C5—21no—
302C4×C5—21no—
303C4×C5—41no—
304C4×C5—41no—
305C4×C5—41no—
306C4×C5—41no—
307C4×C5—41no—
308C4×C5—122no—
309C4×C5—241no—
310C2×Q8C5×#6711yescharacteristic
311C2×D4—21no—
312C2×D4C5×#6711yescharacteristic
313C22×C4—21no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4C5×#6711yescharacteristic
317C22×C4C5×#6711yescharacteristic
318C22×C4C22×C5×D311yescharacteristic
319C2×C8—61no—
320C2×C8—61no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—122no—
330C4⋊C4C5×#6711yescharacteristic
331C4⋊C4C5×#6711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C3×C5#160911yescharacteristic, complemented, Hall
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3C5×#12011yescharacteristic
345C22×C3C5×#13511yescharacteristic
346C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—41no—
354Dic3—42no—
355Dic3—81no—
356C2×C5—21no—
357C2×C5—21no—
358C2×C5—81no—
359C2×C5#523811yescharacteristic
360C2×C5#524311yescharacteristic
361C2×C5#524611yescharacteristic
362C23—21no—
363C23C2×C5×#6711yescharacteristic
364Q8—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—62no—
378C2×C4—81no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4C2×C5×#6711yescharacteristic
385C2×C4C2×C5×#6711yescharacteristic
386C8—121no—
387C8—121no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—81no—
391C2×C3C5×#44911yescharacteristic
392C2×C3C5×#45611yescharacteristic
393C2×C3C5×#45811yescharacteristic
394C5#7188611yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22C5×#94911yescharacteristic
401C22C5×#95011yescharacteristic
402C22C5×#95311yescharacteristic
403C4—21no—
404C4—21no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—122no—
411C4—241no—
412C3C5×#160911yescharacteristic, complemented, Sylow
413C2—21no—
414C2—21no—
415C2—81no—
416C2C5×#523811yescharacteristic
417C2C5×#524311yescharacteristic
418C2C5×#524611yescharacteristic
subgroupG/H
centerC22×C5#953
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#539C2
socleC22×C3×C5#120
prime radicalC3×C5×#121C22
hypercenterC5×#539D3
hypocenterC3C5×#1609
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#539

Complements

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

NH ≅ G/Ntype
2. C5×#5043C2semidirect
4. C5×#4981C2semidirect
5. C5×#4538C2semidirect
7. C5×#4528C2semidirect
28. #71886C5direct (N × H)
65. #5043C2×C5semidirect
67. #4981C2×C5semidirect
68. #4538C2×C5semidirect
70. #4528C2×C5semidirect
338. C3×C5#1609semidirect
394. C5#71886direct (N × H)
412. C3C5×#1609semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5392
C5×#50432
C5×#50332
C5×#49812
C5×#45382
C5×#45372
C5×#45282
C5×#16093
#718865