Finite Group Databasebeta

← #14460260

Subgroups — #14460260

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#11756491C211yesmaximal, characteristic, complemented
2#11756336C211yesmaximal, characteristic
3C3×C5×#475C211yesmaximal, characteristic, complemented
4#11751189C211yesmaximal, characteristic, complemented
5C4×#6718C211yesmaximal, characteristic, complemented
6#11750967C212yesmaximal, complemented
7#11750965C212yesmaximal, complemented
8#11750949C211yesmaximal, characteristic, complemented
9#11750935C211yesmaximal, characteristic, complemented
10#11750909C212yesmaximal, complemented
11#11750905C212yesmaximal, complemented
12#10612754—31nomaximal, Hall
13C2×#6821C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C3×C4×C5×Q8C2211yescharacteristic
16#91836—21no—
17Q8×Dic15—21no—
18C2×#6718C2211yescharacteristic
19C2×#6706C2212yes—
20C2×C8×D15C2212yes—
21#91787—22no—
22#91786C2212yes—
23#91784—22no—
24#91784C2212yes—
25#91781C2212yes—
26#91763—21no—
27#91763C2211yescharacteristic
28C4×D60—21no—
29C4×D60C2211yescharacteristic
30C4×Dic30C2211yescharacteristic
31C3×C5×#106C2212yes—
32C3×C5×C4⋊C8C2211yescharacteristic
33C3×C5×#103C2212yes—
34#91119C2211yescharacteristic
35#91105C2212yes—
36#91104C2212yes—
37#91094C2211yescharacteristic
38#91093C2212yes—
39#91093C2212yes—
40#91091C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#81023—51nomaximal, Hall
43#66066—31no—
44#65911—31no—
45C5×#475D311yescharacteristic, complemented
46#65377—31no—
47C4×#689—31no—
48#65155—32no—
49#65153—32no—
50#65137—31no—
51#65123—31no—
52#65099—32no—
53#65093—32no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8C2311yescharacteristic
56#6821—24no—
57#6821—42no—
58C2×D60—22no—
59C2×D60—21no—
60C2×D60—22no—
61C2×D60—21no—
62C2×D60D411yescharacteristic
63C2×D60C2312yes—
64C2×C4×D15—22no—
65C2×C4×D15C2312yes—
66C2×Dic30C2311yescharacteristic
67C2×C3×C5×C8C2312yes—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2312yes—
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C42×C5—21no—
73C3×C42×C5C2311yescharacteristic
74#6718—22no—
75#6714—22no—
76#6714—22no—
77#6714—22no—
78#6714D412yes—
79C4⋊Dic15—22no—
80C4⋊Dic15D411yescharacteristic
81C4⋊Dic15C2311yescharacteristic
82#6712—21no—
83C4×Dic15—21no—
84C4×Dic15—22no—
85C2×C3⋊#168C2311yescharacteristic
86C2×C3⋊#168C2311yescharacteristic
87#6706—42no—
88C8×D15—42no—
89#5732—51no—
90#5577—51no—
91C3×#475D511yescharacteristic, complemented
92#5088—51no—
93C4×#216—51no—
94#4866—52no—
95#4864—52no—
96#4848—51no—
97#4834—51no—
98#4808—52no—
99#4804—52no—
100C2×#713—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C4×C5×Q8C2×D311yescharacteristic
103#4093—61no—
104Q8×Dic5—61no—
105C2×#689—31no—
106C2×#677—32no—
107C2×C8×D5—32no—
108#4044—62no—
109#4043—32no—
110#4041—32no—
111#4041—62no—
112#4038—32no—
113#4020—31no—
114#4020—61no—
115C4×D20—31no—
116C4×D20—61no—
117C4×Dic10—31no—
118C5×#106C2×D312yes—
119C5×C4⋊C8C2×D311yescharacteristic
120C5×#103C2×D312yes—
121#3967—31no—
122#3953—32no—
123#3952—32no—
124#3942—31no—
125#3941—32no—
126#3941—32no—
127#3939—31no—
128C4×#168—31no—
129#3304—151noSylow
130C22×D15—22no—
131C22×D15C2×D412yes—
132C3×C5×Q8—21no—
133C3×C5×Q8—42no—
134C3×C5×Q8#4112yes—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2411yescharacteristic
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15C2×D412yes—
144D60—22no—
145D60—22no—
146D60—22no—
147D60—44no—
148D60—42no—
149D60—81no—
150D60—82no—
151C4×D15—22no—
152C4×D15—22no—
153C4×D15—44no—
154Dic30—21no—
155Dic30—21no—
156C3×C5×C8—22no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C2×#239—52no—
160C3×C4⋊Q8C2×D511yescharacteristic
161C3×C4×Q8C2×D511yescharacteristic
162#947—101no—
163Q8×Dic3—101no—
164C2×#216—51no—
165C2×#204—52no—
166C2×D3×C8—52no—
167#898—102no—
168#897—52no—
169#895—52no—
170#895—102no—
171#892—52no—
172#874—51no—
173#874—101no—
174C4×D12—51no—
175C4×D12—101no—
176C4×Dic6—51no—
177C3×#106C2×D512yes—
178C3×C4⋊C8C2×D511yescharacteristic
179C3×#103C2×D512yes—
180#835—51no—
181#821—52no—
182(C2×D3)⋊C8—52no—
183#810—51no—
184#809—52no—
185#809—52no—
186#807—51no—
187C4×C3⋊C8—51no—
188C2×C5×Q8—21no—
189C2×C5×Q8C22×D311yescharacteristic
190#713—64no—
191#713—122no—
192C2×D20—31no—
193C2×D20—32no—
194C2×D20—62no—
195C2×D20—61no—
196C2×D20—62no—
197C2×D20—61no—
198C2×C4×D5—32no—
199C2×C4×D5—62no—
200C2×Dic10—31no—
201C2×C5×C8C22×D312yes—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C22×D311yescharacteristic
205C5×C4⋊C4C22×D312yes—
206C42×C5—21no—
207C42×C5C22×D311yescharacteristic
208#689—62no—
209#685—32no—
210#685—62no—
211#685—62no—
212#685—62no—
213C4⋊Dic5—31no—
214C4⋊Dic5—31no—
215C4⋊Dic5—62no—
216#683—61no—
217C4×Dic5—62no—
218C4×Dic5—61no—
219C2×#168—31no—
220C2×#168—31no—
221#677—122no—
222C8×D5—122no—
223#556—151no—
224#550—151no—
225#486—151no—
226#482—152no—
227#475D1511yescharacteristic, complemented
228#465—152no—
229#460—151no—
230#452—152no—
231C4×Q16—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5C22×D411yescharacteristic
235C2×D15—22no—
236C2×D15—22no—
237C2×D15—22no—
238C2×D15—42no—
239C2×D15—42no—
240C2×D15—42no—
241C3×C4×C5—22no—
242C3×C4×C5—21no—
243C3×C4×C5—42no—
244C3×C4×C5—41no—
245C3×C4×C5#13512yes—
246C3×C4×C5C2×#4111yescharacteristic
247C3×C4×C5H(5,𝔽2)11yescharacteristic
248Dic15—22no—
249Dic15—41no—
250Dic15—41no—
251C2×C3×Q8—21no—
252C2×C3×Q8C22×D511yescharacteristic
253#239—104no—
254#239—202no—
255C2×D12—51no—
256C2×D12—52no—
257C2×D12—102no—
258C2×D12—101no—
259C2×D12—102no—
260C2×D12—101no—
261C2×C4×D3—52no—
262C2×C4×D3—102no—
263C2×Dic6—51no—
264C2×C3×C8C22×D512yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C22×D511yescharacteristic
268C3×C4⋊C4C22×D512yes—
269C3×C42—21no—
270C3×C42C22×D511yescharacteristic
271#216—102no—
272(C2×D3)⋊C4—52no—
273(C2×D3)⋊C4—102no—
274(C2×D3)⋊C4—102no—
275(C2×D3)⋊C4—102no—
276C4⋊Dic3—51no—
277C4⋊Dic3—51no—
278C4⋊Dic3—102no—
279Dic3⋊C4—101no—
280C4×Dic3—102no—
281C4×Dic3—101no—
282C2×C3⋊C8—51no—
283C2×C3⋊C8—51no—
284#204—202no—
285D3×C8—202no—
286C22×D5—32no—
287C22×D5—62no—
288C5×Q8—21no—
289C5×Q8—42no—
290C5×Q8#23912yes—
291C2×C4×C5—21no—
292C2×C4×C5—22no—
293C2×C4×C5—21no—
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5D3×D411yescharacteristic
296C2×C4×C5C23×D311yescharacteristic
297C2×Dic5—32no—
298C2×Dic5—61no—
299C2×Dic5—61no—
300D20—62no—
301D20—62no—
302D20—62no—
303D20—124no—
304D20—122no—
305D20—241no—
306D20—242no—
307C4×D5—62no—
308C4×D5—62no—
309C4×D5—124no—
310Dic10—61no—
311Dic10—61no—
312C5×C8—22no—
313#168—61no—
314#168—61no—
315C2×#41—152no—
316C2×Q16—151no—
317C2×SD16—152no—
318C22×C8—152no—
319C4⋊Q8C2×D1511yescharacteristic
320#127—151no—
321#127—301no—
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#106C2×D1512yes—
335C4⋊C8C2×D1511yescharacteristic
336#103—151no—
337#103—151no—
338#103C2×D1512yes—
339#102—152no—
340#102—152no—
341#102—152no—
342C22⋊C8—152no—
343C4×C8—151no—
344C2×C3×C5#54611yescharacteristic
345C2×C3×C5C2×#13511yescharacteristic
346C2×C3×C5#57711yescharacteristic
347D15—42no—
348D15—82no—
349C22×D3—52no—
350C22×D3—102no—
351C3×Q8—21no—
352C3×Q8—42no—
353C3×Q8#71312yes—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4D4×D511yescharacteristic
358C2×C3×C4D4×D511yescharacteristic
359C2×C3×C4C23×D511yescharacteristic
360C2×Dic3—52no—
361C2×Dic3—101no—
362C2×Dic3—101no—
363D12—102no—
364D12—102no—
365D12—102no—
366D12—204no—
367D12—202no—
368D12—401no—
369D12—402no—
370C4×D3—102no—
371C4×D3—102no—
372C4×D3—204no—
373Dic6—101no—
374Dic6—101no—
375C3×C8—22no—
376C3⋊C8—101no—
377C3⋊C8—101no—
378C22×C5C2×D3×D411yescharacteristic
379C2×D5—62no—
380C2×D5—62no—
381C2×D5—62no—
382C2×D5—122no—
383C2×D5—122no—
384C2×D5—122no—
385C4×C5—21no—
386C4×C5—22no—
387C4×C5—42no—
388C4×C5—41no—
389C4×C5#91612yes—
390C4×C5C2×#23911yescharacteristic
391C4×C5#100911yescharacteristic
392Dic5—62no—
393Dic5—121no—
394Dic5—121no—
395#41—304no—
396#41—602no—
397C2×Q8—21no—
398C2×Q8—151no—
399C2×Q8C22×D1511yescharacteristic
400C2×D4—151no—
401C2×D4—152no—
402C2×D4—302no—
403C2×D4—302no—
404C2×D4—301no—
405C2×D4—301no—
406C22×C4—152no—
407C22×C4—302no—
408Q16—302no—
409SD16—602no—
410C2×C8—151no—
411C2×C8—151no—
412C2×C8—602no—
413C2×C8C22×D1512yes—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—151no—
417C4⋊C4—151no—
418C4⋊C4—302no—
419C4⋊C4—301no—
420C4⋊C4C22×D1512yes—
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×C3C2×D4×D511yescharacteristic
432C2×D3—102no—
433C2×D3—102no—
434C2×D3—102no—
435C2×D3—202no—
436C2×D3—202no—
437C2×D3—202no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4#406212yes—
443C3×C4C2×#71311yescharacteristic
444C3×C4#414911yescharacteristic
445Dic3—102no—
446Dic3—201no—
447Dic3—201no—
448C2×C5#563011yescharacteristic
449C2×C5C2×#91611yescharacteristic
450C2×C5#577811yescharacteristic
451D5—122no—
452D5—242no—
453C23—152no—
454C23—302no—
455Q8—21no—
456Q8—42no—
457Q8—301no—
458Q8—301no—
459Q8#682112yes—
460D4—302no—
461D4—302no—
462D4—302no—
463D4—604no—
464D4—602no—
465D4—1201no—
466D4—1202no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—152no—
471C2×C4—302no—
472C2×C4—302no—
473C2×C4—301no—
474C2×C4—301no—
475C2×C4—604no—
476C2×C4D4×D1511yescharacteristic
477C2×C4D4×D1511yescharacteristic
478C2×C4C23×D1511yescharacteristic
479C8—22no—
480C8—301no—
481C8—301no—
482C2×C3#6596411yescharacteristic
483C2×C3C2×#406211yescharacteristic
484C2×C3#6611211yescharacteristic
485D3—202no—
486D3—402no—
487C5#8102311yescharacteristic, complemented, Sylow
488C22—302no—
489C22—302no—
490C22—302no—
491C22—602no—
492C22—602no—
493C22—602no—
494C22C2×D4×D1511yescharacteristic
495C4—22no—
496C4—21no—
497C4—42no—
498C4—41no—
499C4—302no—
500C4—601no—
501C4—601no—
502C4#9180512yes—
503C4C2×#682111yescharacteristic
504C4#9210011yescharacteristic
505C3#1061275411yescharacteristic, complemented, Sylow
506C2—602no—
507C2—1202no—
508C2#1175638911yescharacteristic
509C2C2×#9180511yescharacteristic
510C2#1175653711yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#475C2
socleC22×C3×C5C22×D4
prime radical#11756336C2
hypercenter#475D15
hypocenterC3×C5#3304
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #11756491C2semidirect
3. C3×C5×#475C2semidirect
4. #11751189C2semidirect
5. C4×#6718C2semidirect
6. #11750967C2semidirect
7. #11750965C2semidirect
8. #11750949C2semidirect
9. #11750935C2semidirect
10. #11750909C2semidirect
11. #11750905C2semidirect
45. C5×#475D3semidirect
91. C3×#475D5semidirect
227. #475D15semidirect
430. C3×C5#3304semidirect
487. C5#81023semidirect
505. C3#10612754semidirect

Maximal subgroups

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

subgroupindex
#117564912
#117563362
C3×C5×#4752
#117511892
C4×#67182
#117509672
#117509652
#117509492
#117509352
#117509092
#117509052
#106127543
#810235