Finite Group Databasebeta

← #14430764

Subgroups — #14430764

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#544C211yesmaximal, characteristic, complemented
2#11756280C211yesmaximal, characteristic
3#11751188C212yesmaximal, complemented
4#11751185C212yesmaximal, complemented
5#11751167C212yesmaximal, complemented
6#11751167C212yesmaximal, complemented
7#11751102C211yesmaximal, characteristic
8#11751100C211yesmaximal, characteristic, complemented
9#11751099C211yesmaximal, characteristic, complemented
10C2×#91093C211yesmaximal, characteristic
11#11751065C211yesmaximal, characteristic, complemented
12#10615205—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C3×C5×C4⋊Q8—21no—
15C3×C5×C4⋊Q8C2212yescomplemented
16C3×C5×#125—21no—
17C3×C5×#125C2212yescomplemented
18C3×C5×C22⋊Q8—22no—
19C3×C5×#117C2211yescharacteristic
20C3×C5×#117C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22#91815C2211yescharacteristic
23C2×#6710C2211yescharacteristic
24C22×C3⋊#168C2211yescharacteristic
25#91782—21no—
26#91782—21no—
27C22⋊C4×D15—22no—
28C4×D60C2212yes—
29C4×D60C2212yes—
30#91093C2212yes—
31#91093C2212yes—
32#91093C2212yes—
33#91093C2212yes—
34#91092C2211yescharacteristic, complemented
35#91092C2212yescomplemented
36#91092C2211yescharacteristic, complemented
37#91091C2211yescharacteristic, complemented
38#91091C2212yescomplemented
39#91091C2211yescharacteristic, complemented
40C4⋊(C3⋊#168)C2212yescomplemented
41C4⋊(C3⋊#168)C2212yescomplemented
42#83474—51nomaximal, Hall
43C5×#544D311yescharacteristic, complemented
44#65855—31no—
45#65376—32no—
46#65373—32no—
47#65355—32no—
48#65355—32no—
49#65290—31no—
50#65288—31no—
51#65287—31no—
52C2×#3941—31no—
53#65253—31no—
54C23×D15—21no—
55C2×C3×C5×Q8—22no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×D60—21no—
59C2×D60—22no—
60C2×D60Q814yescomplemented
61C2×D60C2312yes—
62C2×C4×D15—22no—
63C2×C4×D15—22no—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4—22no—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4—22no—
68C3×C5×C4⋊C4D411yescharacteristic
69C3×C5×C4⋊C4D411yescharacteristic
70C3×C5×C4⋊C4C2312yes—
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C5×C4⋊C4C2311yescharacteristic
73C3×C5×C22⋊C4—21no—
74C3×C5×C22⋊C4D412yes—
75C3×C42×C5—21no—
76C3×C42×C5C2312yes—
77#6719—21no—
78#6714—22no—
79#6714—22no—
80C4⋊Dic15C2311yescharacteristic
81C4⋊Dic15C2311yescharacteristic
82C4×Dic15—21no—
83#6710—22no—
84C2×C3⋊#168—22no—
85C2×C3⋊#168C2311yescharacteristic
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2312yes—
88C3×#544D511yescharacteristic, complemented
89#5521—51no—
90#5087—52no—
91#5084—52no—
92#5066—52no—
93#5066—52no—
94#5001—51no—
95#4999—51no—
96#4998—51no—
97C2×#809—51no—
98#4964—51no—
99C22×D20—31no—
100C5×C4⋊Q8—21no—
101C5×C4⋊Q8C2×D312yescomplemented
102C5×#125—21no—
103C5×#125C2×D312yescomplemented
104C5×C22⋊Q8—22no—
105C5×#117C2×D311yescharacteristic
106C5×#117C2×D311yescharacteristic
107C2×C5×C4⋊C4C2×D311yescharacteristic
108#4072—31no—
109C2×#681—31no—
110C22×#168—31no—
111#4039—61no—
112#4039—61no—
113D5×C22⋊C4—62no—
114C4×D20—32no—
115C4×D20—32no—
116#3941—32no—
117#3941—32no—
118#3941—32no—
119#3941—32no—
120#3940—31no—
121#3940—31no—
122#3940—32no—
123#3939—31no—
124#3939—31no—
125#3939—32no—
126C4⋊#168—32no—
127C4⋊#168—32no—
128#3081—151noSylow
129C23×C3×C5C2×D411yescharacteristic
130C22×D15—22no—
131C22×D15—24no—
132C22×D15—81no—
133C22×D15—81no—
134C3×C5×Q8—82no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—81no—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×Q812yes—
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5C2411yescharacteristic
145C2×Dic15—21no—
146C2×Dic15—21no—
147D60—24no—
148D60—22no—
149D60C2×Q814yes—
150C4×D15—82no—
151C4×D15—82no—
152C3⋊#168—21no—
153C3⋊#168—21no—
154C3⋊#168—22no—
155C22×D12—51no—
156C3×C4⋊Q8—21no—
157C3×C4⋊Q8C2×D512yescomplemented
158C3×#125—21no—
159C3×#125C2×D512yescomplemented
160C3×C22⋊Q8—22no—
161C3×#117C2×D511yescharacteristic
162C3×#117C2×D511yescharacteristic
163C2×C3×C4⋊C4C2×D511yescharacteristic
164#926—51no—
165C2×#208—51no—
166C22×C3⋊C8—51no—
167#893—101no—
168#893—101no—
169D3×C22⋊C4—102no—
170C4×D12—52no—
171C4×D12—52no—
172#809—52no—
173#809—52no—
174#809—52no—
175#809—52no—
176#808—52no—
177#808—51no—
178#808—51no—
179#807—52no—
180#807—51no—
181#807—51no—
182C4⋊(C3⋊C8)—52no—
183C4⋊(C3⋊C8)—52no—
184C23×D5—61no—
185C2×C5×Q8—22no—
186C22×C4×C5—21no—
187C22×C4×C5C22×D311yescharacteristic
188C2×D20—34no—
189C2×D20—32no—
190C2×D20—62no—
191C2×D20—61no—
192C2×C4×D5—62no—
193C2×C4×D5—62no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—22no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4#6711yescharacteristic
199C5×C4⋊C4#6711yescharacteristic
200C5×C4⋊C4C22×D312yes—
201C5×C4⋊C4C22×D311yescharacteristic
202C5×C4⋊C4C22×D311yescharacteristic
203C5×C22⋊C4—21no—
204C5×C22⋊C4#6712yes—
205C42×C5—21no—
206C42×C5C22×D312yes—
207C22⋊Dic5—61no—
208#685—62no—
209#685—62no—
210C4⋊Dic5—31no—
211C4⋊Dic5—31no—
212C4×Dic5—61no—
213#681—62no—
214C2×#168—31no—
215C2×#168—32no—
216C2×#168—31no—
217C2×#168—62no—
218#544D1511yescharacteristic, complemented
219#518—151no—
220#478—152no—
221#478—152no—
222#476—152no—
223#474—152no—
224#428—151no—
225#427—151no—
226#424—151no—
227#418—151no—
228C2×#102—151no—
229C22×C3×C5—21no—
230C22×C3×C5—21no—
231C22×C3×C5C22⋊Q812yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×D15—24no—
234C2×D15—42no—
235C2×D15—42no—
236C2×D15—84no—
237C3×C4×C5—22no—
238C3×C4×C5—22no—
239C3×C4×C5—42no—
240C3×C4×C5—41no—
241C3×C4×C5—41no—
242C3×C4×C5C22⋊Q812yes—
243C3×C4×C5C22×Q811yescharacteristic
244C3×C4×C5C2×#4111yescharacteristic
245Dic15—41no—
246Dic15—41no—
247C23×D3—101no—
248C2×C3×Q8—22no—
249C22×C3×C4—21no—
250C22×C3×C4C22×D511yescharacteristic
251C2×D12—54no—
252C2×D12—52no—
253C2×D12—102no—
254C2×D12—101no—
255C2×C4×D3—102no—
256C2×C4×D3—102no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—22no—
261C3×C4⋊C4C22⋊D511yescharacteristic
262C3×C4⋊C4C22⋊D511yescharacteristic
263C3×C4⋊C4C22×D512yes—
264C3×C4⋊C4C22×D511yescharacteristic
265C3×C4⋊C4C22×D511yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4C22⋊D512yes—
268C3×C42—21no—
269C3×C42C22×D512yes—
270#217—101no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273C4⋊Dic3—51no—
274C4⋊Dic3—51no—
275C4×Dic3—101no—
276#208—102no—
277C2×C3⋊C8—51no—
278C2×C3⋊C8—52no—
279C2×C3⋊C8—51no—
280C2×C3⋊C8—102no—
281C23×C5C2×#6711yescharacteristic
282C22×D5—62no—
283C22×D5—64no—
284C22×D5—241no—
285C22×D5—241no—
286C5×Q8—82no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—22no—
290C2×C4×C5—81no—
291C2×C4×C5D3×Q812yes—
292C2×C4×C5#23912yes—
293C2×C4×C5C2×#6712yes—
294C2×C4×C5C2×#6712yes—
295C2×C4×C5C2×#6711yescharacteristic
296C2×C4×C5C23×D311yescharacteristic
297C2×Dic5—61no—
298C2×Dic5—61no—
299D20—34no—
300D20—64no—
301D20—62no—
302C4×D5—242no—
303C4×D5—242no—
304#168—61no—
305#168—61no—
306#168—62no—
307C22×D4—151no—
308C2×M16—151no—
309C22×C8—151no—
310C4⋊Q8—21no—
311C4⋊Q8C2×D1512yescomplemented
312#125—21no—
313#125C2×D1512yescomplemented
314C22⋊Q8—22no—
315C4×D4—152no—
316C4×D4—152no—
317C4×D4—301no—
318C4×D4—301no—
319#117—151no—
320#117C2×D1511yescharacteristic
321#117C2×D1511yescharacteristic
322C2×C4⋊C4C2×D1511yescharacteristic
323C2×C22⋊C4—302no—
324#107—151no—
325#107—151no—
326#107—152no—
327#106—151no—
328#106—151no—
329#106—152no—
330C4⋊C8—152no—
331C4⋊C8—152no—
332#102—152no—
333#102—152no—
334#102—152no—
335#102—152no—
336C2×C3×C5—22no—
337C2×C3×C5C2×C22⋊Q811yescharacteristic
338C2×C3×C5#57611yescharacteristic
339C2×C3×C5#57711yescharacteristic
340D15—44no—
341C23×C3C2×C22⋊D511yescharacteristic
342C22×D3—102no—
343C22×D3—104no—
344C22×D3—401no—
345C22×D3—401no—
346C3×Q8—82no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—81no—
351C2×C3×C4Q8×D512yes—
352C2×C3×C4#71312yes—
353C2×C3×C4C2×C22⋊D512yes—
354C2×C3×C4C2×C22⋊D512yes—
355C2×C3×C4C2×C22⋊D511yescharacteristic
356C2×C3×C4C23×D511yescharacteristic
357C2×Dic3—101no—
358C2×Dic3—101no—
359D12—54no—
360D12—102no—
361D12—104no—
362C4×D3—402no—
363C4×D3—402no—
364C3⋊C8—101no—
365C3⋊C8—101no—
366C3⋊C8—102no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5#94612yes—
370C22×C5C22×#6711yescharacteristic
371C2×D5—64no—
372C2×D5—122no—
373C2×D5—122no—
374C2×D5—244no—
375C4×C5—22no—
376C4×C5—22no—
377C4×C5—42no—
378C4×C5—41no—
379C4×C5—41no—
380C4×C5#94612yes—
381C4×C5C2×D3×Q811yescharacteristic
382C4×C5C2×#23911yescharacteristic
383Dic5—121no—
384Dic5—121no—
385C24—301no—
386C2×Q8—22no—
387C2×D4—154no—
388C2×D4—152no—
389C2×D4—302no—
390C2×D4—301no—
391C22×C4—21no—
392C22×C4—302no—
393C22×C4—302no—
394C22×C4C22×D1511yescharacteristic
395M16—302no—
396C2×C8—151no—
397C2×C8—152no—
398C2×C8—151no—
399C2×C8—302no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—151no—
405C4⋊C4—151no—
406C4⋊C4#122011yescharacteristic
407C4⋊C4#122011yescharacteristic
408C4⋊C4C22×D1512yes—
409C4⋊C4C22×D1511yescharacteristic
410C4⋊C4C22×D1511yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—301no—
413C22⋊C4—302no—
414C22⋊C4—302no—
415C22⋊C4#122012yes—
416C42—21no—
417C42—301no—
418C42C22×D1512yes—
419C3×C5#308111yescharacteristic, complemented, Hall
420C22×C3—21no—
421C22×C3—21no—
422C22×C3#409212yes—
423C22×C3C22×C22⋊D511yescharacteristic
424C2×D3—104no—
425C2×D3—202no—
426C2×D3—202no—
427C2×D3—404no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4#409212yes—
434C3×C4C2×Q8×D511yescharacteristic
435C3×C4C2×#71311yescharacteristic
436Dic3—201no—
437Dic3—201no—
438C2×C5—22no—
439C2×C5C2×#94611yescharacteristic
440C2×C5#583511yescharacteristic
441C2×C5#583711yescharacteristic
442D5—124no—
443C23—302no—
444C23—304no—
445C23—1201no—
446C23—1201no—
447C23C2×#122011yescharacteristic
448Q8—82no—
449D4—154no—
450D4—304no—
451D4—302no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—81no—
456C2×C4—301no—
457C2×C4—301no—
458C2×C4—1202no—
459C2×C4—1202no—
460C2×C4Q8×D1512yes—
461C2×C4#682112yes—
462C2×C4C2×#122012yes—
463C2×C4C2×#122012yes—
464C2×C4C2×#122011yescharacteristic
465C2×C4C23×D1511yescharacteristic
466C8—301no—
467C8—301no—
468C8—302no—
469C2×C3—22no—
470C2×C3C2×#409211yescharacteristic
471C2×C3#6616911yescharacteristic
472C2×C3#6617111yescharacteristic
473D3—204no—
474C5#8347411yescharacteristic, complemented, Sylow
475C22—21no—
476C22—21no—
477C22—304no—
478C22—602no—
479C22—602no—
480C22—1204no—
481C22#9183512yes—
482C22C22×#122011yescharacteristic
483C4—22no—
484C4—22no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—601no—
489C4—601no—
490C4#9183512yes—
491C4C2×Q8×D1511yescharacteristic
492C4C2×#682111yescharacteristic
493C3#1061520511yescharacteristic, complemented, Sylow
494C2—22no—
495C2—604no—
496C2C2×#9183511yescharacteristic
497C2#1175659411yescharacteristic
498C2#1175659611yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#544C2
socleC22×C3×C5C22×D4
prime radicalC22×D60C22
hypercenter#544D15
hypocenterC3×C5#3081
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C3×C5×#544C2semidirect
3. #11751188C2semidirect
4. #11751185C2semidirect
5. #11751167C2semidirect
6. #11751167C2semidirect
8. #11751100C2semidirect
9. #11751099C2semidirect
11. #11751065C2semidirect
15. C3×C5×C4⋊Q8C22semidirect
17. C3×C5×#125C22semidirect
34. #91092C22semidirect
35. #91092C22semidirect
36. #91092C22semidirect
37. #91091C22semidirect
38. #91091C22semidirect
39. #91091C22semidirect
40. C4⋊(C3⋊#168)C22semidirect
41. C4⋊(C3⋊#168)C22semidirect
43. C5×#544D3semidirect
60. C2×D60Q8semidirect
88. C3×#544D5semidirect
101. C5×C4⋊Q8C2×D3semidirect
103. C5×#125C2×D3semidirect
157. C3×C4⋊Q8C2×D5semidirect
159. C3×#125C2×D5semidirect
218. #544D15semidirect
311. C4⋊Q8C2×D15semidirect
313. #125C2×D15semidirect
419. C3×C5#3081semidirect
474. C5#83474semidirect
493. C3#10615205semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5442
#117562802
#117511882
#117511852
#117511672
#117511022
#117511002
#117510992
C2×#910932
#117510652
#106152053
#834745