Finite Group Databasebeta

← C5×#68763 #14308905

Subgroups — C5×#68763 #14308905

Order 1920.

564 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5519C211yesmaximal, characteristic, complemented
2C5×#5131C211yesmaximal, characteristic
3C5×#5001C211yesmaximal, characteristic
4C5×#4939C211yesmaximal, characteristic
5C3×C5×#339C211yesmaximal, characteristic, complemented
6C5×#4554C211yesmaximal, characteristic, complemented
7C5×#4523C211yesmaximal, characteristic, complemented
8C5×#2027—31nomaximal, Hall
9C22×C5×D12C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C3×C42×C5C2211yescharacteristic
13C2×C5×(C2×D3)⋊C4—22no—
14C5×#926C2211yescharacteristic
15C2×C5×C4⋊Dic3C2211yescharacteristic
16C2×C5×#208C2211yescharacteristic
17C5×#890—21no—
18C5×#888—21no—
19C5×#878—22no—
20C5×#870—21no—
21C5×#869—21no—
22C5×#821—21no—
23C5×#821—21no—
24C5×#809—21no—
25C5×#809—21no—
26#68763C511yesmaximal, characteristic, direct product factor, Hall
27C5×#544—31no—
28C5×#418—31no—
29C5×#418—31no—
30C5×#390—31no—
31C5×#343—31no—
32C5×#339—31no—
33C5×#339D311yescharacteristic, complemented
34C23×C5×D3—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×C5×Dic3D412yes—
38C2×C5×D12—21no—
39C2×C5×D12—21no—
40C2×C5×D12—21no—
41C2×C5×D12—21no—
42C2×C5×D12—21no—
43C2×C5×D12Q812yescomplemented
44C2×C5×Dic6—22no—
45C3×C5×M16—21no—
46C3×C5×M16—21no—
47C2×C3×C5×C8—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52C3×C42×C5—21no—
53C3×C42×C5—21no—
54C5×#217—21no—
55C5×(C2×D3)⋊C4—44no—
56C5×C4⋊Dic3—22no—
57C5×C4⋊Dic3—21no—
58C5×C4⋊Dic3—21no—
59C5×C4⋊Dic3D411yescharacteristic
60C5×C4⋊Dic3D411yescharacteristic
61C5×Dic3⋊C4—22no—
62C5×Dic3⋊C4—22no—
63C4×C5×Dic3—21no—
64C5×#208—21no—
65C5×#208—21no—
66C2×C5×C3⋊C8—21no—
67#5519C2×C511yescharacteristic, complemented
68#5131C2×C511yescharacteristic
69#5001C2×C511yescharacteristic
70#4939C2×C511yescharacteristic
71C3×#339C2×C511yescharacteristic, complemented
72#4554C2×C511yescharacteristic, complemented
73#4523C2×C511yescharacteristic, complemented
74C22×C5×D4—31no—
75C2×C5×M16—31no—
76C2×C5×M16C2×D311yescharacteristic
77C5×C4⋊Q8—61no—
78C5×C4⋊Q8—61no—
79C5×#125—61no—
80C5×#125—61no—
81C5×C22⋊Q8—62no—
82C5×#117—31no—
83C5×#117C2×D311yescharacteristic
84C2×C5×C4⋊C4—31no—
85C2×C5×C22⋊C4—62no—
86C2×C42×C5C2×D311yescharacteristic
87C5×#102—61no—
88C5×#102—61no—
89C5×#102—61no—
90C5×#102—61no—
91#2027—31noSylow
92C23×C3×C5C2×D411yescharacteristic
93C22×C5×D3—22no—
94C22×C5×D3—42no—
95C22×C5×D3—42no—
96C22×C5×D3—41no—
97C22×C5×D3—41no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×Q811yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C2×C5×Dic3—22no—
112C2×C5×Dic3—42no—
113C5×D12—22no—
114C5×D12—22no—
115C5×D12—41no—
116C5×D12—41no—
117C5×Dic6—82no—
118C3×C5×C8—41no—
119C5×C3⋊C8—41no—
120C22×D12C22×C511yescharacteristic
121C2×C3×M16C22×C511yescharacteristic
122C3×#117C22×C511yescharacteristic
123C2×C3×C42C22×C511yescharacteristic
124C2×(C2×D3)⋊C4—22no—
125#926C22×C511yescharacteristic
126C2×C4⋊Dic3C22×C511yescharacteristic
127C2×#208C22×C511yescharacteristic
128#890—21no—
129#888—21no—
130#878—22no—
131#870—21no—
132#869—21no—
133#821—21no—
134#821—21no—
135#809—21no—
136#809—21no—
137C24×C5—61no—
138C2×C5×Q8—62no—
139C2×C5×D4—32no—
140C2×C5×D4—61no—
141C2×C5×D4—61no—
142C2×C5×D4—61no—
143C2×C5×D4—61no—
144C2×C5×D4—61no—
145C22×C4×C5—21no—
146C22×C4×C5—32no—
147C22×C4×C5C22×D311yescharacteristic
148C5×M16—21no—
149C5×M16—21no—
150C5×M16—61no—
151C5×M16—61no—
152C2×C5×C8—21no—
153C2×C5×C8—61no—
154C5×C4⋊C4—31no—
155C5×C4⋊C4—31no—
156C5×C4⋊C4—62no—
157C5×C4⋊C4—62no—
158C5×C4⋊C4—62no—
159C5×C4⋊C4—61no—
160C5×C4⋊C4—61no—
161C5×C4⋊C4D1211yescharacteristic
162C5×C4⋊C4D1211yescharacteristic
163C5×C22⋊C4—21no—
164C5×C22⋊C4—61no—
165C5×C22⋊C4—124no—
166C42×C5—21no—
167C42×C5—21no—
168C42×C5—21no—
169C42×C5—61no—
170#544—31no—
171#418—31no—
172#418—31no—
173#390—31no—
174#343—31no—
175#339—31no—
176#339C5×D311yescharacteristic, complemented
177C22×C3×C5—21no—
178C22×C3×C5—21no—
179C22×C3×C5#12011yescharacteristic
180C22×C3×C5C22⋊Q811yescharacteristic
181C22×C3×C5#12411yescharacteristic
182C2×C5×D3—44no—
183C2×C5×D3—42no—
184C2×C5×D3—82no—
185C2×C5×D3—81no—
186C2×C5×D3—81no—
187C3×C4×C5—42no—
188C3×C4×C5—41no—
189C3×C4×C5—41no—
190C3×C4×C5C22⋊Q811yescharacteristic
191C3×C4×C5C22⋊Q811yescharacteristic
192C3×C4×C5#12411yescharacteristic
193C3×C4×C5#12411yescharacteristic
194C5×Dic3—41no—
195C5×Dic3—42no—
196C5×Dic3—41no—
197C23×D3—21no—
198C22×C3×C4—21no—
199C22×C3×C4C23×C511yescharacteristic
200C22×Dic3C5×D412yes—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C5×Q812yescomplemented
207C2×Dic6—22no—
208C3×M16—21no—
209C3×M16—21no—
210C2×C3×C8—21no—
211C3×C4⋊C4C5×D411yescharacteristic
212C3×C4⋊C4C5×D411yescharacteristic
213C3×C22⋊C4—21no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—21no—
217#217—21no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3C5×D411yescharacteristic
223C4⋊Dic3C5×D411yescharacteristic
224Dic3⋊C4—22no—
225Dic3⋊C4—22no—
226C4×Dic3—21no—
227#208—21no—
228#208—21no—
229C2×C3⋊C8—21no—
230C23×C5—62no—
231C23×C5—122no—
232C23×C5—122no—
233C23×C5—121no—
234C23×C5—121no—
235C23×C5D3×D411yescharacteristic
236C5×Q8—242no—
237C5×D4—62no—
238C5×D4—62no—
239C5×D4—121no—
240C5×D4—121no—
241C2×C4×C5—22no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—61no—
247C2×C4×C5—61no—
248C2×C4×C5—62no—
249C2×C4×C5—122no—
250C2×C4×C5C2×D1211yescharacteristic
251C2×C4×C5#23511yescharacteristic
252C2×C4×C5#23511yescharacteristic
253C2×C4×C5D3×D411yescharacteristic
254C2×C4×C5#23711yescharacteristic
255C2×C4×C5D3×Q811yescharacteristic
256C5×C8—41no—
257C5×C8—121no—
258C22×D4—31no—
259C2×M16—31no—
260C2×M16C2×C5×D311yescharacteristic
261C4⋊Q8—61no—
262C4⋊Q8—61no—
263#125—61no—
264#125—61no—
265C22⋊Q8—62no—
266#117—31no—
267#117C2×C5×D311yescharacteristic
268C2×C4⋊C4—31no—
269C2×C22⋊C4—62no—
270C2×C42C2×C5×D311yescharacteristic
271#102—61no—
272#102—61no—
273#102—61no—
274#102—61no—
275C2×C3×C5—21no—
276C2×C3×C5—21no—
277C2×C3×C5#39311yescharacteristic
278C2×C3×C5#45311yescharacteristic
279C2×C3×C5#45411yescharacteristic
280C5×D3—82no—
281C23×C3C2×C5×D411yescharacteristic
282C22×D3—22no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—41no—
286C22×D3—41no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4C2×C5×D411yescharacteristic
293C2×C3×C4C2×C5×D411yescharacteristic
294C2×C3×C4C2×C5×Q811yescharacteristic
295C2×C3×C4C5×#4111yescharacteristic
296C2×C3×C4C5×#4111yescharacteristic
297C2×C3×C4C5×#4111yescharacteristic
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302D12—22no—
303D12—22no—
304D12—41no—
305D12—41no—
306Dic6—82no—
307C3×C8—41no—
308C3⋊C8—41no—
309C22×C5—21no—
310C22×C5—21no—
311C22×C5—124no—
312C22×C5—122no—
313C22×C5—242no—
314C22×C5—241no—
315C22×C5—241no—
316C22×C5#88211yescharacteristic
317C22×C5#88511yescharacteristic
318C22×C5#89611yescharacteristic
319C4×C5—42no—
320C4×C5—41no—
321C4×C5—41no—
322C4×C5—122no—
323C4×C5—121no—
324C4×C5—121no—
325C4×C5#87511yescharacteristic
326C4×C5#88511yescharacteristic
327C4×C5#89611yescharacteristic
328C4×C5#89711yescharacteristic
329C24—61no—
330C2×Q8—62no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C22×C4—21no—
338C22×C4—32no—
339C22×C4C22×C5×D311yescharacteristic
340M16—21no—
341M16—21no—
342M16—61no—
343M16—61no—
344C2×C8—21no—
345C2×C8—61no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4C5×D1211yescharacteristic
354C4⋊C4C5×D1211yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—61no—
357C22⋊C4—124no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—61no—
362C3×C5#202711yescharacteristic, complemented, Hall
363C22×C3—21no—
364C22×C3—21no—
365C22×C3C5×#12011yescharacteristic
366C22×C3C5×C22⋊Q811yescharacteristic
367C22×C3C5×#12411yescharacteristic
368C2×D3—44no—
369C2×D3—42no—
370C2×D3—82no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4C5×C22⋊Q811yescharacteristic
377C3×C4C5×C22⋊Q811yescharacteristic
378C3×C4C5×#12411yescharacteristic
379C3×C4C5×#12411yescharacteristic
380Dic3—41no—
381Dic3—42no—
382Dic3—41no—
383C2×C5—21no—
384C2×C5—21no—
385C2×C5—242no—
386C2×C5#467311yescharacteristic
387C2×C5#482211yescharacteristic
388C2×C5#482711yescharacteristic
389C23—62no—
390C23—122no—
391C23—122no—
392C23—121no—
393C23—121no—
394C23C5×D3×D411yescharacteristic
395Q8—242no—
396D4—62no—
397D4—62no—
398D4—121no—
399D4—121no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—62no—
408C2×C4—122no—
409C2×C4C2×C5×D1211yescharacteristic
410C2×C4C5×#23511yescharacteristic
411C2×C4C5×#23511yescharacteristic
412C2×C4C5×D3×D411yescharacteristic
413C2×C4C5×#23711yescharacteristic
414C2×C4C5×D3×Q811yescharacteristic
415C8—41no—
416C8—121no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3C5×#39311yescharacteristic
420C2×C3C5×#45311yescharacteristic
421C2×C3C5×#45411yescharacteristic
422D3—82no—
423C5#6876311yescharacteristic, direct product factor, Sylow
424C22—21no—
425C22—21no—
426C22—124no—
427C22—122no—
428C22—242no—
429C22—241no—
430C22—241no—
431C22C5×#88211yescharacteristic
432C22C5×#88511yescharacteristic
433C22C5×#89611yescharacteristic
434C4—42no—
435C4—41no—
436C4—41no—
437C4—122no—
438C4—121no—
439C4—121no—
440C4C5×#87511yescharacteristic
441C4C5×#88511yescharacteristic
442C4C5×#89611yescharacteristic
443C4C5×#89711yescharacteristic
444C3C5×#202711yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—242no—
448C2C5×#467311yescharacteristic
449C2C5×#482211yescharacteristic
450C2C5×#482711yescharacteristic
subgroupG/H
centerC22×C5#882
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#339C2
socleC22×C3×C5#120
prime radicalC22×C5×D12C22
hypercenterC5×#339D3
hypocenterC3C5×#2027
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#339

Complements

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

NH ≅ G/Ntype
1. C5×#5519C2semidirect
5. C3×C5×#339C2semidirect
6. C5×#4554C2semidirect
7. C5×#4523C2semidirect
26. #68763C5direct (N × H)
33. C5×#339D3semidirect
43. C2×C5×D12Q8semidirect
67. #5519C2×C5semidirect
71. C3×#339C2×C5semidirect
72. #4554C2×C5semidirect
73. #4523C2×C5semidirect
176. #339C5×D3semidirect
206. C2×D12C5×Q8semidirect
362. C3×C5#2027semidirect
423. C5#68763direct (N × H)
444. C3C5×#2027semidirect

Maximal subgroups

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

subgroupindex
C5×#55192
C5×#51312
C5×#50012
C5×#49392
C3×C5×#3392
C5×#45542
C5×#45232
C5×#20273
#687635