Finite Group Databasebeta

← #14460251

Subgroups — #14460251

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic, complemented
2#11756312C211yesmaximal, characteristic
3#11751195C211yesmaximal, characteristic, complemented
4#11751189C211yesmaximal, characteristic, complemented
5#11751149C212yesmaximal, complemented
6#11751148C212yesmaximal, complemented
7C4×#6718C211yesmaximal, characteristic, complemented
8#11751128C211yesmaximal, characteristic, complemented
9D4×C3⋊#168C211yesmaximal, characteristic, complemented
10#11751104C212yesmaximal, complemented
11#11751103C212yesmaximal, complemented
12#10615404—31nomaximal, Hall
13C2×#6817C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C3×C5×#126—22no—
16C3×C5×#124—21no—
17C3×C5×C22⋊Q8C2212yes—
18C3×C4×C5×Q8—21no—
19C3×C4×C5×Q8C2211yescharacteristic
20C3×C4×C5×D4C2211yescharacteristic
21C3×C5×#117C2212yes—
22C2×#6718C2211yescharacteristic
23C2×#6717C2212yes—
24#91829—21no—
25#91819C2212yes—
26C22×C3⋊#168C2212yes—
27#91782—21no—
28#91773—22no—
29#91763C2211yescharacteristic
30C4×Dic30C2211yescharacteristic
31#91119C2211yescharacteristic
32#91114C2212yes—
33#91094C2212yes—
34#91094C2211yescharacteristic
35#91093C2212yes—
36#91093C2212yes—
37#91093C2212yes—
38#91092C2212yes—
39#91091C2211yescharacteristic
40C4⋊(C3⋊#168)C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#83673—51nomaximal, Hall
43C5×#556D311yescharacteristic, complemented
44#65887—31no—
45#65383—31no—
46#65377—31no—
47#65337—32no—
48#65336—32no—
49C4×#689—31no—
50#65316—31no—
51D4×#168—31no—
52#65292—32no—
53#65291—32no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8C2311yescharacteristic
56C2×C3×C5×D4D411yescharacteristic
57C22×C3×C4×C5C2312yes—
58C2×#1220—22no—
59C2×#1220—22no—
60#6817—24no—
61#6817—42no—
62C2×D60—21no—
63C2×D60—21no—
64C2×D60C2312yes—
65C2×C4×D15—22no—
66C2×Dic30C2311yescharacteristic
67C3×C5×C4⋊C4—22no—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C5×C4⋊C4C2312yes—
73C3×C5×C22⋊C4—22no—
74C3×C5×C22⋊C4—22no—
75C3×C5×C22⋊C4D412yes—
76C3×C42×C5—22no—
77C3×C42×C5—21no—
78C3×C42×C5C2311yescharacteristic
79#6718—22no—
80#6717—42no—
81#6714—22no—
82C4⋊Dic15C2311yescharacteristic
83#6712—21no—
84C4×Dic15—21no—
85C2×C3⋊#168—42no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2312yes—
88C2×C3⋊#168C2311yescharacteristic
89C3×#556D511yescharacteristic, complemented
90#5553—51no—
91#5094—51no—
92#5088—51no—
93#5048—52no—
94#5047—52no—
95C4×#216—51no—
96#5027—51no—
97D4×C3⋊C8—51no—
98#5003—52no—
99#5002—52no—
100C2×#709—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C5×#126—22no—
103C5×#124—21no—
104C5×C22⋊Q8C2×D312yes—
105C4×C5×Q8—21no—
106C4×C5×Q8C2×D311yescharacteristic
107C4×C5×D4C2×D311yescharacteristic
108C5×#117C2×D312yes—
109C2×#689—31no—
110C2×#688—32no—
111#4086—61no—
112#4076—32no—
113C22×#168—32no—
114#4039—61no—
115#4030—62no—
116#4020—31no—
117C4×Dic10—31no—
118#3967—31no—
119C22⋊#168—32no—
120#3942—31no—
121#3942—32no—
122#3941—32no—
123#3941—32no—
124#3941—32no—
125#3940—32no—
126#3939—31no—
127C4⋊#168—31no—
128C4×#168—31no—
129#3304—151noSylow
130C23×C3×C5C2×D412yes—
131C22×D15—22no—
132C3×C5×Q8—21no—
133C3×C5×Q8—21no—
134C3×C5×Q8—42no—
135C3×C5×D4—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2411yescharacteristic
145#1220—44no—
146#1220—82no—
147C2×Dic15—21no—
148C2×Dic15—21no—
149D60—22no—
150D60—22no—
151D60—42no—
152D60—81no—
153C4×D15—44no—
154Dic30—21no—
155Dic30#4112yes—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3⋊#168—22no—
159C2×#235—52no—
160C3×C4⋊Q8C2×D511yescharacteristic
161C3×#126—22no—
162C3×#124—21no—
163C3×C22⋊Q8C2×D512yes—
164C3×C4×Q8—21no—
165C3×C4×Q8C2×D511yescharacteristic
166C3×C4×D4C2×D511yescharacteristic
167C3×#117C2×D512yes—
168C2×#216—51no—
169C2×#215—52no—
170#940—101no—
171#930—52no—
172C22×C3⋊C8—52no—
173#893—101no—
174#884—102no—
175#874—51no—
176C4×Dic6—51no—
177#835—51no—
178#830—52no—
179#810—51no—
180#810—52no—
181#809—52no—
182#809—52no—
183#809—52no—
184#808—52no—
185#807—51no—
186C4⋊(C3⋊C8)—51no—
187C4×C3⋊C8—51no—
188C2×C5×Q8—21no—
189C2×C5×Q8C22×D311yescharacteristic
190C2×C5×D4#6711yescharacteristic
191C22×C4×C5C22×D312yes—
192C2×C22⋊D5—62no—
193C2×C22⋊D5—62no—
194#709—64no—
195#709—122no—
196C2×D20—32no—
197C2×D20—61no—
198C2×D20—61no—
199C2×C4×D5—62no—
200C2×Dic10—31no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4#6711yescharacteristic
205C5×C4⋊C4C22×D311yescharacteristic
206C5×C4⋊C4C22×D312yes—
207C5×C22⋊C4—22no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4#6712yes—
210C42×C5—22no—
211C42×C5—21no—
212C42×C5C22×D311yescharacteristic
213#689—62no—
214#688—122no—
215#685—62no—
216C4⋊Dic5—31no—
217#683—61no—
218C4×Dic5—61no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—32no—
222C2×#168—122no—
223#556D1511yescharacteristic, complemented
224#550—151no—
225#486—151no—
226#482—152no—
227#475—151no—
228#465—152no—
229#460—151no—
230#452—152no—
231C4×Q16—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—22no—
235C22×C3×C5—22no—
236C22×C3×C5—22no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—42no—
239C2×D15—42no—
240C2×D15—42no—
241C3×C4×C5—22no—
242C3×C4×C5—21no—
243C3×C4×C5—41no—
244C3×C4×C5—42no—
245C3×C4×C5—41no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—22no—
250Dic15—41no—
251C2×C3×Q8—21no—
252C2×C3×Q8C22×D511yescharacteristic
253C2×C3×D4C22⋊D511yescharacteristic
254C22×C3×C4C22×D512yes—
255C2×#67—102no—
256C2×#67—102no—
257#235—104no—
258#235—202no—
259C2×D12—52no—
260C2×D12—101no—
261C2×D12—101no—
262C2×C4×D3—102no—
263C2×Dic6—51no—
264C3×C4⋊C4—22no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C22⋊D511yescharacteristic
268C3×C4⋊C4C22×D511yescharacteristic
269C3×C4⋊C4C22×D512yes—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4C22⋊D512yes—
273C3×C42—22no—
274C3×C42—21no—
275C3×C42C22×D511yescharacteristic
276#216—102no—
277#215—202no—
278(C2×D3)⋊C4—102no—
279C4⋊Dic3—51no—
280Dic3⋊C4—101no—
281C4×Dic3—101no—
282C2×C3⋊C8—51no—
283C2×C3⋊C8—51no—
284C2×C3⋊C8—52no—
285C2×C3⋊C8—202no—
286C23×C5C2×#6712yes—
287C22×D5—62no—
288C5×Q8—21no—
289C5×Q8—21no—
290C5×Q8—42no—
291C5×D4—22no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—22no—
297C2×C4×C5C2×#6711yescharacteristic
298C2×C4×C5C2×#6712yes—
299C2×C4×C5C2×#6711yescharacteristic
300C2×C4×C5C23×D311yescharacteristic
301C22⋊D5—124no—
302C22⋊D5—242no—
303C2×Dic5—61no—
304C2×Dic5—61no—
305D20—62no—
306D20—62no—
307D20—122no—
308D20—241no—
309C4×D5—124no—
310Dic10—32no—
311Dic10—61no—
312#168—61no—
313#168—61no—
314#168—62no—
315C2×#41—152no—
316C2×Q16—151no—
317C2×SD16—152no—
318C22×C8—152no—
319C4⋊Q8C2×D1511yescharacteristic
320#127—151no—
321#127—301no—
322#126—22no—
323#124—21no—
324C22⋊Q8C2×D1512yes—
325#121—152no—
326#121—302no—
327C4×Q8—21no—
328C4×Q8—151no—
329C4×Q8C2×D1511yescharacteristic
330C4×D4—301no—
331C4×D4C2×D1511yescharacteristic
332#117C2×D1512yes—
333#107—151no—
334#106—152no—
335C4⋊C8—151no—
336#103—151no—
337#103—151no—
338#103—152no—
339#102—152no—
340#102—152no—
341#102—152no—
342C22⋊C8—152no—
343C4×C8—151no—
344C2×C3×C5—42no—
345C2×C3×C5#54611yescharacteristic
346C2×C3×C5C2×#13511yescharacteristic
347C2×C3×C5#57711yescharacteristic
348D15—82no—
349C23×C3C2×C22⋊D512yes—
350C22×D3—102no—
351C3×Q8—21no—
352C3×Q8—21no—
353C3×Q8—42no—
354C3×D4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4C2×C22⋊D511yescharacteristic
361C2×C3×C4C2×C22⋊D512yes—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C23×D511yescharacteristic
364#67—204no—
365#67—402no—
366C2×Dic3—101no—
367C2×Dic3—101no—
368D12—102no—
369D12—102no—
370D12—202no—
371D12—401no—
372C4×D3—204no—
373Dic6—52no—
374Dic6—101no—
375C3⋊C8—101no—
376C3⋊C8—101no—
377C3⋊C8—102no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5C22×#6711yescharacteristic
382C2×D5—122no—
383C2×D5—122no—
384C2×D5—122no—
385C4×C5—22no—
386C4×C5—21no—
387C4×C5—42no—
388C4×C5—41no—
389C4×C5—41no—
390C4×C5#95012yes—
391C4×C5D3×#4111yescharacteristic
392C4×C5#100911yescharacteristic
393Dic5—62no—
394Dic5—121no—
395#41—304no—
396#41—602no—
397C2×Q8—21no—
398C2×Q8—151no—
399C2×Q8C22×D1511yescharacteristic
400C2×D4—152no—
401C2×D4—302no—
402C2×D4—302no—
403C2×D4—301no—
404C2×D4—301no—
405C2×D4#122011yescharacteristic
406C22×C4—302no—
407C22×C4C22×D1512yes—
408Q16—302no—
409SD16—602no—
410C2×C8—151no—
411C2×C8—151no—
412C2×C8—152no—
413C2×C8—602no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—151no—
418C4⋊C4—301no—
419C4⋊C4#122011yescharacteristic
420C4⋊C4C22×D1512yes—
421C4⋊C4C22×D1511yescharacteristic
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—302no—
425C22⋊C4#122012yes—
426C42—22no—
427C42—21no—
428C42—301no—
429C42C22×D1511yescharacteristic
430C3×C5#330411yescharacteristic, complemented, Hall
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3C22×C22⋊D511yescharacteristic
435C2×D3—202no—
436C2×D3—202no—
437C2×D3—202no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4#409612yes—
444C3×C4D5×#4111yescharacteristic
445C3×C4#414911yescharacteristic
446Dic3—102no—
447Dic3—201no—
448C2×C5—42no—
449C2×C5C2×#95011yescharacteristic
450C2×C5#582911yescharacteristic
451C2×C5#583711yescharacteristic
452D5—242no—
453C23—302no—
454C23C2×#122012yes—
455Q8—21no—
456Q8—21no—
457Q8—42no—
458Q8—152no—
459Q8—301no—
460D4—22no—
461D4—302no—
462D4—302no—
463D4—604no—
464D4—602no—
465D4—1201no—
466D4—1202no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—301no—
473C2×C4—301no—
474C2×C4—604no—
475C2×C4C2×#122012yes—
476C2×C4C2×#122011yescharacteristic
477C2×C4C2×#122011yescharacteristic
478C2×C4C23×D1511yescharacteristic
479C8—301no—
480C8—301no—
481C8—302no—
482C2×C3—42no—
483C2×C3C2×#409611yescharacteristic
484C2×C3#6616311yescharacteristic
485C2×C3#6617111yescharacteristic
486D3—402no—
487C5#8367311yescharacteristic, complemented, Sylow
488C22—22no—
489C22—22no—
490C22—22no—
491C22—602no—
492C22—602no—
493C22—602no—
494C22C22×#122011yescharacteristic
495C4—22no—
496C4—21no—
497C4—42no—
498C4—41no—
499C4—41no—
500C4—302no—
501C4—601no—
502C4#9183912yes—
503C4#41×D1511yescharacteristic
504C4#9210011yescharacteristic
505C3#1061540411yescharacteristic, complemented, Sylow
506C2—42no—
507C2—1202no—
508C2C2×#9183911yescharacteristic
509C2#1175658811yescharacteristic
510C2#1175659611yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#556C2
socleC22×C3×C5C22×D4
prime radical#11756312C2
hypercenter#556D15
hypocenterC3×C5#3304
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×C5×#556C2semidirect
3. #11751195C2semidirect
4. #11751189C2semidirect
5. #11751149C2semidirect
6. #11751148C2semidirect
7. C4×#6718C2semidirect
8. #11751128C2semidirect
9. D4×C3⋊#168C2semidirect
10. #11751104C2semidirect
11. #11751103C2semidirect
43. C5×#556D3semidirect
89. C3×#556D5semidirect
223. #556D15semidirect
430. C3×C5#3304semidirect
487. C5#83673semidirect
505. C3#10615404semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5562
#117563122
#117511952
#117511892
#117511492
#117511482
C4×#67182
#117511282
D4×C3⋊#1682
#117511042
#117511032
#106154043
#836735