Finite Group Databasebeta

← #14302554

Subgroups — #14302554

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756581C211yesmaximal, characteristic
2#11756281C211yesmaximal, characteristic, complemented
3C3×C5×#410C211yesmaximal, characteristic, complemented
4C2×#91121C211yesmaximal, characteristic, complemented
5#11751106C211yesmaximal, characteristic, complemented
6#11746354C211yesmaximal, characteristic, complemented
7#11746353C211yesmaximal, characteristic, complemented
8#10600848—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13#91840—21no—
14#91839—22no—
15#91838—21no—
16#91831C412yescomplemented
17#91821C412yescomplemented
18C2×C4×Dic15C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20C2×#6710C2211yescharacteristic
21#91782—21no—
22#91776—21no—
23#91770—22no—
24#91761—21no—
25C42×D15—21no—
26C3×C5×#99—22no—
27#91121—21no—
28#91121—21no—
29#91089—22no—
30#70055—51nomaximal, Hall
31#66156—31no—
32#65856—31no—
33C5×#410D311yescharacteristic, complemented
34C2×#3969—31no—
35#65294—31no—
36#64753—31no—
37#64752—31no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C2×C3×C5×Q8C2×C411yescharacteristic, complemented
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4C2×C411yescharacteristic, complemented
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C2×#1220—21no—
48C22×Dic15—21no—
49#6817—21no—
50#6817—21no—
51#6817C2×C412yes—
52#6817D412yes—
53C2×D60D411yescharacteristic
54C2×C4×D15—21no—
55C2×C4×D15—22no—
56C2×Dic30D411yescharacteristic
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4D412yes—
61C3×C42×C5—21no—
62C3×C42×C5—21no—
63#6718—22no—
64#6717—22no—
65#6716—22no—
66#6715—22no—
67#6714—22no—
68#6712—22no—
69C4×Dic15—22no—
70C4×Dic15—21no—
71C4×Dic15D411yescharacteristic
72C4×Dic15D411yescharacteristic
73#6710—21no—
74#6710—21no—
75#6710—21no—
76#6710C2×C412yescomplemented
77C2×C3⋊#168—21no—
78C2×C3⋊#168—21no—
79#5822—51no—
80#5522—51no—
81C3×#410D511yescharacteristic, complemented
82C2×#837—51no—
83#5005—51no—
84#4533—51no—
85#4532—51no—
86C2×C5×#41C2×D311yescharacteristic
87C2×#709—31no—
88C5×#117C2×D311yescharacteristic
89C5×#117C2×D311yescharacteristic
90#4097—61no—
91#4096—62no—
92#4095—61no—
93#4088—32no—
94#4078—32no—
95C2×C4×Dic5—31no—
96C2×#681—31no—
97C2×#681—31no—
98#4039—61no—
99#4033—61no—
100#4027—62no—
101#4018—61no—
102C42×D5—61no—
103C5×#99—22no—
104#3969—61no—
105#3969—61no—
106#3937—62no—
107#1906—151noSylow
108C23×C3×C5—21no—
109C23×C3×C5C2×D411yescharacteristic
110C22×D15—21no—
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×D4—21no—
114C3×C5×D4—21no—
115C3×C5×D4—41no—
116C3×C5×D4—41no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5C22×C411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5#4111yescharacteristic
127C2×C3×C4×C5#4111yescharacteristic
128#1220—22no—
129#1220—22no—
130C2×Dic15—21no—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—42no—
134D60—21no—
135D60C22⋊C412yes—
136C4×D15—22no—
137C4×D15—22no—
138C4×D15—44no—
139Dic30—21no—
140Dic30C22⋊C412yes—
141C3⋊#168—22no—
142C3⋊#168—41no—
143C2×C3×#41C2×D511yescharacteristic
144C2×#235—51no—
145C3×#117C2×D511yescharacteristic
146C3×#117C2×D511yescharacteristic
147#951—101no—
148#950—102no—
149#949—101no—
150#942—52no—
151#932—52no—
152C2×C4×Dic3—51no—
153C2×#208—51no—
154C2×#208—51no—
155#893—101no—
156#887—101no—
157#67⋊C4—102no—
158#872—101no—
159C42×D3—101no—
160C3×#99—22no—
161#837—101no—
162#837—101no—
163#805—102no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C2×C5×Q8C4×D311yescharacteristic, complemented
169C2×C5×D4—21no—
170C2×C5×D4C4×D311yescharacteristic, complemented
171C22×C4×C5—21no—
172C22×C4×C5C22×D311yescharacteristic
173C2×C22⋊D5—61no—
174C22×Dic5—61no—
175#709—32no—
176#709—32no—
177#709—61no—
178#709—61no—
179C2×D20—31no—
180C2×C4×D5—61no—
181C2×C4×D5—62no—
182C2×Dic10—31no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C22⋊C4—21no—
186C5×C22⋊C4#6712yes—
187C42×C5—21no—
188C42×C5—21no—
189#689—62no—
190#688—62no—
191#687—62no—
192#686—62no—
193#685—62no—
194#683—62no—
195C4×Dic5—31no—
196C4×Dic5—31no—
197C4×Dic5—62no—
198C4×Dic5—61no—
199#681—32no—
200#681—61no—
201#681—61no—
202#681—61no—
203C2×#168—61no—
204C2×#168—61no—
205#575—151no—
206C4×#41—151no—
207#421—151no—
208C2×#104—151no—
209#410D1511yescharacteristic, complemented
210#344—151no—
211#339—151no—
212C22×C3×C5—41no—
213C22×C3×C5—41no—
214C22×C3×C5—81no—
215C22×C3×C5C4×D411yescharacteristic
216C22×C3×C5#12011yescharacteristic
217C22×C3×C5#12111yescharacteristic
218C2×D15—22no—
219C2×D15—42no—
220C3×C4×C5—42no—
221C3×C4×C5—41no—
222C3×C4×C5—81no—
223C3×C4×C5C2×C22⋊C411yescharacteristic
224C3×C4×C5C4×D411yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C3×C4×C5#12311yescharacteristic
227Dic15—22no—
228Dic15—22no—
229Dic15—41no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C2×C3×Q8C4×D511yescharacteristic, complemented
235C2×C3×D4—21no—
236C2×C3×D4C4×D511yescharacteristic, complemented
237C22×C3×C4—21no—
238C22×C3×C4C22×D511yescharacteristic
239C2×#67—101no—
240C22×Dic3—101no—
241#235—52no—
242#235—52no—
243#235—101no—
244#235—101no—
245C2×D12—51no—
246C2×C4×D3—101no—
247C2×C4×D3—102no—
248C2×Dic6—51no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4C22⋊D512yes—
253C3×C42—21no—
254C3×C42—21no—
255#216—102no—
256#215—102no—
257#214—102no—
258#213—102no—
259(C2×D3)⋊C4—102no—
260Dic3⋊C4—102no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—102no—
264C4×Dic3—101no—
265#208—52no—
266#208—101no—
267#208—101no—
268#208—101no—
269C2×C3⋊C8—101no—
270C2×C3⋊C8—101no—
271C23×C5—21no—
272C23×C5C2×#6711yescharacteristic
273C22×D5—61no—
274C5×Q8—21no—
275C5×Q8—21no—
276C5×D4—21no—
277C5×D4—21no—
278C5×D4—41no—
279C5×D4—41no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5C2×C4×D311yescharacteristic
286C2×C4×C5#23511yescharacteristic
287C2×C4×C5D3×D411yescharacteristic
288C2×C4×C5D3×D411yescharacteristic
289C2×C4×C5D3×D411yescharacteristic
290C2×C4×C5#23711yescharacteristic
291C22⋊D5—62no—
292C22⋊D5—62no—
293C2×Dic5—61no—
294C2×Dic5—61no—
295C2×Dic5—61no—
296C2×Dic5—122no—
297D20—32no—
298D20—61no—
299C4×D5—62no—
300C4×D5—62no—
301C4×D5—124no—
302Dic10—32no—
303Dic10—61no—
304#168—62no—
305#168—121no—
306C2×#41—151no—
307C2×#41C2×D1511yescharacteristic
308#137—152no—
309#137—301no—
310AGL(1,ℤ/8)—152no—
311AGL(1,ℤ/8)—301no—
312#135—302no—
313C2×M16—151no—
314C2×M16—151no—
315C4×Q8—301no—
316C4×D4—301no—
317C4×D4—302no—
318#117—301no—
319#117C2×D1511yescharacteristic
320#117C2×D1511yescharacteristic
321C2×C42—151no—
322C2×C42—301no—
323#104—302no—
324#104—301no—
325#104—301no—
326#99—22no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—81no—
331C2×C3×C5#38611yescharacteristic
332D15—42no—
333C23×C3—21no—
334C23×C3C2×C22⋊D511yescharacteristic
335C22×D3—101no—
336C3×Q8—21no—
337C3×Q8—21no—
338C3×D4—21no—
339C3×D4—21no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4C2×C4×D511yescharacteristic
348C2×C3×C4#70911yescharacteristic
349C2×C3×C4D4×D511yescharacteristic
350C2×C3×C4D4×D511yescharacteristic
351C2×C3×C4D4×D511yescharacteristic
352C2×C3×C4#71111yescharacteristic
353#67—102no—
354#67—102no—
355C2×Dic3—101no—
356C2×Dic3—101no—
357C2×Dic3—101no—
358C2×Dic3—202no—
359D12—52no—
360D12—101no—
361C4×D3—102no—
362C4×D3—102no—
363C4×D3—204no—
364Dic6—52no—
365Dic6—101no—
366C3⋊C8—102no—
367C3⋊C8—201no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—81no—
371C22×C5C4×#6711yescharacteristic
372C22×C5#93711yescharacteristic
373C22×C5#93811yescharacteristic
374C2×D5—62no—
375C2×D5—122no—
376C4×C5—42no—
377C4×C5—41no—
378C4×C5—81no—
379C4×C5D3×C22⋊C411yescharacteristic
380C4×C5#67⋊C411yescharacteristic
381C4×C5#88311yescharacteristic
382C4×C5#88411yescharacteristic
383Dic5—62no—
384Dic5—62no—
385Dic5—121no—
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—152no—
391#41—152no—
392#41—301no—
393#41—301no—
394C2×Q8—151no—
395C2×Q8C4×D1511yescharacteristic, complemented
396C2×D4—21no—
397C2×D4—151no—
398C2×D4—301no—
399C2×D4C4×D1511yescharacteristic, complemented
400C22×C4—21no—
401C22×C4—302no—
402C22×C4—301no—
403C22×C4—301no—
404C22×C4C22×D1511yescharacteristic
405Q16—302no—
406SD16—302no—
407SD16—302no—
408D8—302no—
409M16—152no—
410M16—301no—
411M16—301no—
412M16—301no—
413C2×C8—301no—
414C2×C8—301no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—302no—
418C22⋊C4—21no—
419C22⋊C4—302no—
420C22⋊C4#122012yes—
421C42—21no—
422C42—21no—
423C42—151no—
424C42—151no—
425C42—302no—
426C42—301no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—81no—
431C22×C3C4×C22⋊D511yescharacteristic
432C22×C3#408311yescharacteristic
433C22×C3#408411yescharacteristic
434C2×D3—102no—
435C2×D3—202no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—81no—
439C3×C4D5×C22⋊C411yescharacteristic
440C3×C4#402711yescharacteristic
441C3×C4#402911yescharacteristic
442C3×C4#403011yescharacteristic
443Dic3—102no—
444Dic3—102no—
445Dic3—201no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—81no—
450C2×C5#495611yescharacteristic
451D5—122no—
452C23—21no—
453C23—301no—
454C23C2×#122011yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—152no—
458Q8—301no—
459D4—21no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—152no—
464D4—302no—
465D4—302no—
466D4—301no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—301no—
473C2×C4—301no—
474C2×C4—302no—
475C2×C4—302no—
476C2×C4—301no—
477C2×C4—604no—
478C2×C4—602no—
479C2×C4C2×C4×D1511yescharacteristic
480C2×C4#681711yescharacteristic
481C2×C4D4×D1511yescharacteristic
482C2×C4D4×D1511yescharacteristic
483C2×C4D4×D1511yescharacteristic
484C2×C4#681911yescharacteristic
485C8—302no—
486C8—601no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—81no—
491C2×C3#6524511yescharacteristic
492D3—202no—
493C5#7005511yescharacteristic, complemented, Sylow
494C22—41no—
495C22—41no—
496C22—81no—
497C22—302no—
498C22—602no—
499C22C4×#122011yescharacteristic
500C22#9182611yescharacteristic
501C22#9182711yescharacteristic
502C4—42no—
503C4—41no—
504C4—81no—
505C4—302no—
506C4—302no—
507C4—601no—
508C4C22⋊C4×D1511yescharacteristic
509C4#9177011yescharacteristic
510C4#9177211yescharacteristic
511C4#9177311yescharacteristic
512C3#1060084811yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—81no—
517C2—602no—
518C2#1175105711yescharacteristic
subgroupG/H
centerC4#91773
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#410C2
socleC2×C3×C5#386
prime radical#11756581C2
cosocleC22×C3×C4×C5C23
hypercenter#410D15
hypocenterC3×C5#1906
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #11756281C2semidirect
3. C3×C5×#410C2semidirect
4. C2×#91121C2semidirect
5. #11751106C2semidirect
6. #11746354C2semidirect
7. #11746353C2semidirect
16. #91831C4semidirect
17. #91821C4semidirect
33. C5×#410D3semidirect
42. C2×C3×C5×Q8C2×C4semidirect
44. C2×C3×C5×D4C2×C4semidirect
76. #6710C2×C4semidirect
81. C3×#410D5semidirect
168. C2×C5×Q8C4×D3semidirect
170. C2×C5×D4C4×D3semidirect
209. #410D15semidirect
234. C2×C3×Q8C4×D5semidirect
236. C2×C3×D4C4×D5semidirect
395. C2×Q8C4×D15semidirect
399. C2×D4C4×D15semidirect
427. C3×C5#1906semidirect
493. C5#70055semidirect
512. C3#10600848semidirect

Maximal subgroups

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

subgroupindex
#117565812
#117562812
C3×C5×#4102
C2×#911212
#117511062
#117463542
#117463532
#106008483
#700555