Finite Group Databasebeta

← #14460278

Subgroups — #14460278

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
1#11756477C211yesmaximal, characteristic
2#11756343C211yesmaximal, characteristic, complemented
3C3×C5×#460C211yesmaximal, characteristic, complemented
4#11751274C212yesmaximal, complemented
5#11751272C212yesmaximal, complemented
6#11751171C211yesmaximal, characteristic, complemented
7C4×#6718C211yesmaximal, characteristic, complemented
8#11750952C211yesmaximal, characteristic, complemented
9#11750935C211yesmaximal, characteristic, complemented
10#11750880C212yesmaximal, complemented
11#11750873C212yesmaximal, complemented
12#10612127—31nomaximal, Hall
13C2×#6819C2212yes—
14C2×C3×C5×SD16C2212yes—
15C3×C5×#127C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17#91835C2212yes—
18C2×#6718C2211yescharacteristic
19#91829—21no—
20#91828—22no—
21#91827C2212yes—
22D4×Dic15—21no—
23C2×C8×D15C2212yes—
24#91787—22no—
25#91781C2212yes—
26#91764—21no—
27C4×D60C2211yescharacteristic
28#91758C2211yescharacteristic
29C4×Dic30—21no—
30C4×Dic30C2211yescharacteristic
31C3×C5×C4⋊C8C2211yescharacteristic
32C3×C5×#102C2212yes—
33#91119C2211yescharacteristic
34#91116C2212yes—
35#91116C2212yes—
36#91104C2212yes—
37#91101C2212yes—
38#91100C2212yes—
39#91094C2211yescharacteristic
40#91091C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#80396—51nomaximal, Hall
43#66052—31no—
44#65918—31no—
45C5×#460D311yescharacteristic, complemented
46#65462—32no—
47#65460—32no—
48#65359—31no—
49C4×#689—31no—
50#65140—31no—
51#65123—31no—
52#65068—32no—
53#65061—32no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4C2312yes—
58C2×#1220—22no—
59C2×#1220—22no—
60C22×Dic15—22no—
61#6819—24no—
62#6819—42no—
63C2×D60D411yescharacteristic
64C2×C4×D15C2312yes—
65C2×Dic30—21no—
66C2×Dic30C2311yescharacteristic
67C3×C5×SD16—42no—
68C2×C3×C5×C8C2312yes—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2311yescharacteristic
71C3×C42×C5—21no—
72C3×C42×C5C2311yescharacteristic
73#6719—22no—
74#6718—22no—
75#6714—22no—
76#6714—22no—
77#6714D412yes—
78C4⋊Dic15—21no—
79C4⋊Dic15D411yescharacteristic
80C4⋊Dic15C2311yescharacteristic
81C4⋊Dic15C2312yes—
82#6712—21no—
83#6712—22no—
84C4×Dic15—21no—
85C4×Dic15—22no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2311yescharacteristic
88C8×D15—42no—
89#5718—51no—
90#5584—51no—
91C3×#460D511yescharacteristic, complemented
92#5173—52no—
93#5171—52no—
94#5070—51no—
95C4×#216—51no—
96#4851—51no—
97#4834—51no—
98#4779—52no—
99#4772—52no—
100C2×#711—32no—
101C2×C5×SD16C2×D312yes—
102C5×#127C2×D311yescharacteristic
103C4×C5×Q8C2×D311yescharacteristic
104#4092—32no—
105C2×#689—31no—
106#4086—61no—
107#4085—62no—
108#4084—32no—
109D4×Dic5—61no—
110C2×C8×D5—32no—
111#4044—62no—
112#4038—32no—
113#4021—61no—
114C4×D20—31no—
115#4015—31no—
116C4×Dic10—31no—
117C4×Dic10—61no—
118C5×C4⋊C8C2×D311yescharacteristic
119C5×#102C2×D312yes—
120#3967—31no—
121#3964—32no—
122#3964—32no—
123#3952—32no—
124#3949—32no—
125#3948—32no—
126#3942—31no—
127#3939—31no—
128C4×#168—31no—
129#3304—151noSylow
130C23×C3×C5—22no—
131C22×D15C2×D412yes—
132C3×C5×Q8—21no—
133C3×C5×Q8—21no—
134C3×C5×D4—22no—
135C3×C5×D4—22no—
136C3×C5×D4—42no—
137C3×C5×D4—81no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2411yescharacteristic
143#1220—44no—
144#1220—82no—
145C2×Dic15—21no—
146C2×Dic15—22no—
147C2×Dic15—21no—
148C2×Dic15—44no—
149C2×Dic15C2×D412yes—
150D60—22no—
151C4×D15—22no—
152C4×D15—22no—
153Dic30—21no—
154Dic30—42no—
155Dic30#4112yes—
156C3×C5×C8—22no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C2×#237—52no—
160C2×C3×SD16C2×D512yes—
161C3×#127C2×D511yescharacteristic
162C3×C4×Q8C2×D511yescharacteristic
163#946—52no—
164C2×#216—51no—
165#940—101no—
166#939—102no—
167#938—52no—
168D4×Dic3—101no—
169C2×D3×C8—52no—
170#898—102no—
171#892—52no—
172#875—101no—
173C4×D12—51no—
174#869—51no—
175C4×Dic6—51no—
176C4×Dic6—101no—
177C3×C4⋊C8C2×D511yescharacteristic
178C3×#102C2×D512yes—
179#835—51no—
180#832—52no—
181#832—52no—
182(C2×D3)⋊C8—52no—
183#817—52no—
184#816—52no—
185#810—51no—
186#807—51no—
187C4×C3⋊C8—51no—
188C2×C5×Q8C22×D311yescharacteristic
189C2×C5×D4—21no—
190C2×C5×D4—21no—
191C2×C5×D4C22×D312yes—
192C2×C22⋊D5—62no—
193C2×C22⋊D5—62no—
194C22×Dic5—62no—
195#711—64no—
196#711—122no—
197C2×D20—31no—
198C2×C4×D5—32no—
199C2×Dic10—31no—
200C2×Dic10—61no—
201C5×SD16—42no—
202C2×C5×C8C22×D312yes—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C22×D311yescharacteristic
205C42×C5—21no—
206C42×C5C22×D311yescharacteristic
207C22⋊Dic5—62no—
208#689—62no—
209#685—32no—
210#685—62no—
211#685—62no—
212C4⋊Dic5—31no—
213C4⋊Dic5—31no—
214C4⋊Dic5—32no—
215C4⋊Dic5—61no—
216#683—62no—
217#683—61no—
218C4×Dic5—62no—
219C4×Dic5—61no—
220C2×#168—31no—
221C2×#168—31no—
222C8×D5—122no—
223#556—151no—
224#550—151no—
225#486—151no—
226#482—152no—
227#475—151no—
228#465—152no—
229#460D1511yescharacteristic, complemented
230#452—152no—
231C4×Q16—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—42no—
235C22×C3×C5—42no—
236C22×C3×C5—42no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—22no—
239C2×D15—22no—
240C2×D15—22no—
241C3×C4×C5—21no—
242C3×C4×C5—41no—
243C3×C4×C5—41no—
244C3×C4×C5#13512yes—
245C3×C4×C5C2×#4111yescharacteristic
246C3×C4×C5H(5,𝔽2)11yescharacteristic
247Dic15—22no—
248Dic15—22no—
249Dic15—42no—
250Dic15—41no—
251C2×C3×Q8C22×D511yescharacteristic
252C2×C3×D4—21no—
253C2×C3×D4—21no—
254C2×C3×D4C22×D512yes—
255C2×#67—102no—
256C2×#67—102no—
257C22×Dic3—102no—
258#237—104no—
259#237—202no—
260C2×D12—51no—
261C2×C4×D3—52no—
262C2×Dic6—51no—
263C2×Dic6—101no—
264C3×SD16—42no—
265C2×C3×C8C22×D512yes—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C22×D511yescharacteristic
268C3×C42—21no—
269C3×C42C22×D511yescharacteristic
270#217—102no—
271#216—102no—
272(C2×D3)⋊C4—52no—
273(C2×D3)⋊C4—102no—
274(C2×D3)⋊C4—102no—
275C4⋊Dic3—51no—
276C4⋊Dic3—51no—
277C4⋊Dic3—52no—
278C4⋊Dic3—101no—
279Dic3⋊C4—102no—
280Dic3⋊C4—101no—
281C4×Dic3—102no—
282C4×Dic3—101no—
283C2×C3⋊C8—51no—
284C2×C3⋊C8—51no—
285D3×C8—202no—
286C23×C5—22no—
287C22×D5—32no—
288C5×Q8—21no—
289C5×Q8—21no—
290C5×D4—22no—
291C5×D4—22no—
292C5×D4—42no—
293C5×D4—81no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5D3×D411yescharacteristic
297C2×C4×C5D3×D411yescharacteristic
298C2×C4×C5C23×D311yescharacteristic
299C22⋊D5—124no—
300C22⋊D5—242no—
301C2×Dic5—32no—
302C2×Dic5—61no—
303C2×Dic5—62no—
304C2×Dic5—61no—
305C2×Dic5—124no—
306D20—62no—
307C4×D5—62no—
308C4×D5—62no—
309Dic10—32no—
310Dic10—61no—
311Dic10—122no—
312C5×C8—22no—
313#168—61no—
314#168—61no—
315C2×#41—152no—
316C2×Q16—151no—
317C2×SD16C2×D1512yes—
318C22×C8—152no—
319C4⋊Q8—151no—
320#127—301no—
321#127C2×D1511yescharacteristic
322#126—302no—
323#124—301no—
324C22⋊Q8—152no—
325#121—152no—
326#121—302no—
327C4×Q8—151no—
328C4×Q8—301no—
329C4×Q8C2×D1511yescharacteristic
330C4×D4—151no—
331C4×D4—301no—
332#117—152no—
333#107—151no—
334#106—152no—
335C4⋊C8C2×D1511yescharacteristic
336#103—151no—
337#103—152no—
338#103—151no—
339#102—152no—
340#102—152no—
341#102C2×D1512yes—
342C22⋊C8—152no—
343C4×C8—151no—
344C2×C3×C5—82no—
345C2×C3×C5#54611yescharacteristic
346C2×C3×C5C2×#13511yescharacteristic
347C2×C3×C5#57711yescharacteristic
348D15—42no—
349C23×C3—22no—
350C22×D3—52no—
351C3×Q8—21no—
352C3×Q8—21no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—42no—
356C3×D4—81no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4D4×D511yescharacteristic
360C2×C3×C4D4×D511yescharacteristic
361C2×C3×C4C23×D511yescharacteristic
362#67—204no—
363#67—402no—
364C2×Dic3—52no—
365C2×Dic3—101no—
366C2×Dic3—102no—
367C2×Dic3—101no—
368C2×Dic3—204no—
369D12—102no—
370C4×D3—102no—
371C4×D3—102no—
372Dic6—52no—
373Dic6—101no—
374Dic6—202no—
375C3×C8—22no—
376C3⋊C8—101no—
377C3⋊C8—101no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5C2×D3×D411yescharacteristic
382C2×D5—62no—
383C2×D5—62no—
384C2×D5—62no—
385C4×C5—21no—
386C4×C5—41no—
387C4×C5—41no—
388C4×C5#91612yes—
389C4×C5#100411yescharacteristic
390C4×C5D3×#4111yescharacteristic
391Dic5—62no—
392Dic5—62no—
393Dic5—122no—
394Dic5—121no—
395#41—304no—
396#41—602no—
397C2×Q8—151no—
398C2×Q8—301no—
399C2×Q8C22×D1511yescharacteristic
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—151no—
403C2×D4—302no—
404C2×D4—302no—
405C2×D4C22×D1512yes—
406C22×C4—152no—
407C22×C4—302no—
408Q16—302no—
409SD16—42no—
410C2×C8—151no—
411C2×C8—151no—
412C2×C8—602no—
413C2×C8C22×D1512yes—
414C4⋊C4—21no—
415C4⋊C4—151no—
416C4⋊C4—152no—
417C4⋊C4—151no—
418C4⋊C4—302no—
419C4⋊C4—301no—
420C4⋊C4—301no—
421C4⋊C4C22×D1511yescharacteristic
422C22⋊C4—152no—
423C22⋊C4—302no—
424C22⋊C4—302no—
425C22⋊C4—302no—
426C42—21no—
427C42—302no—
428C42—301no—
429C42C22×D1511yescharacteristic
430C3×C5#330411yescharacteristic, complemented, Hall
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3C2×D4×D511yescharacteristic
435C2×D3—102no—
436C2×D3—102no—
437C2×D3—102no—
438C3×C4—21no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4#406212yes—
442C3×C4#414411yescharacteristic
443C3×C4D5×#4111yescharacteristic
444Dic3—102no—
445Dic3—102no—
446Dic3—202no—
447Dic3—201no—
448C2×C5—82no—
449C2×C5#561211yescharacteristic
450C2×C5C2×#91611yescharacteristic
451C2×C5#577511yescharacteristic
452D5—122no—
453C23—22no—
454C23—152no—
455Q8—21no—
456Q8—21no—
457Q8—152no—
458Q8—301no—
459Q8—602no—
460D4—22no—
461D4—22no—
462D4—42no—
463D4—81no—
464D4—302no—
465D4—604no—
466D4—1202no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—152no—
470C2×C4—301no—
471C2×C4—302no—
472C2×C4—302no—
473C2×C4—302no—
474C2×C4—301no—
475C2×C4—604no—
476C2×C4D4×D1511yescharacteristic
477C2×C4D4×D1511yescharacteristic
478C2×C4C23×D1511yescharacteristic
479C8—22no—
480C8—301no—
481C8—301no—
482C2×C3—82no—
483C2×C3#6594611yescharacteristic
484C2×C3C2×#406211yescharacteristic
485C2×C3#6610911yescharacteristic
486D3—202no—
487C5#8039611yescharacteristic, complemented, Sylow
488C22—42no—
489C22—42no—
490C22—42no—
491C22—302no—
492C22—302no—
493C22—302no—
494C22C2×D4×D1511yescharacteristic
495C4—21no—
496C4—41no—
497C4—41no—
498C4—302no—
499C4—302no—
500C4—602no—
501C4—601no—
502C4#9180512yes—
503C4#9209511yescharacteristic
504C4#41×D1511yescharacteristic
505C3#1061212711yescharacteristic, complemented, Sylow
506C2—82no—
507C2—602no—
508C2#1175637111yescharacteristic
509C2C2×#9180511yescharacteristic
510C2#1175653411yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#460C2
socleC22×C3×C5C22×D4
prime radical#11756477C2
hypercenter#460D15
hypocenterC3×C5#3304
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #11756343C2semidirect
3. C3×C5×#460C2semidirect
4. #11751274C2semidirect
5. #11751272C2semidirect
6. #11751171C2semidirect
7. C4×#6718C2semidirect
8. #11750952C2semidirect
9. #11750935C2semidirect
10. #11750880C2semidirect
11. #11750873C2semidirect
45. C5×#460D3semidirect
91. C3×#460D5semidirect
229. #460D15semidirect
430. C3×C5#3304semidirect
487. C5#80396semidirect
505. C3#10612127semidirect

Maximal subgroups

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

subgroupindex
#117564772
#117563432
C3×C5×#4602
#117512742
#117512722
#117511712
C4×#67182
#117509522
#117509352
#117508802
#117508732
#106121273
#803965