Finite Group Databasebeta

← #14417263

Subgroups — #14417263

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756282C211yesmaximal, characteristic
2C5×#5516C211yesmaximal, characteristic, complemented
3C2×#91321C211yesmaximal, characteristic
4C3×#65286C211yesmaximal, characteristic, complemented
5#11748460C211yesmaximal, characteristic, complemented
6C2×#90973C211yesmaximal, characteristic, complemented
7#11748393C211yesmaximal, characteristic
8#11747093C212yesmaximal, complemented
9C4×#6663C212yesmaximal, complemented
10#11747058C212yesmaximal, complemented
11C4×#6660C212yesmaximal, complemented
12#10614278—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17#91781—21no—
18C4⋊C4×D15—21no—
19#91770—22no—
20C4×D60C2212yes—
21C4×Dic30C2212yes—
22C5×#926C2211yescharacteristic
23C5×#887—21no—
24C5×#887—21no—
25C5×#877—22no—
26C4×C5×Dic6C2212yes—
27C4×C5×Dic6C2212yes—
28C2×C3×#681C2211yescharacteristic
29C22×C3×#168C2211yescharacteristic
30#91321C418yescomplemented
31C2×#6663C2212yes—
32C2×#6660C2212yes—
33C3×#3935C2212yes—
34C3×C4×#168C2212yes—
35#90988C2212yes—
36#90985C2212yes—
37#90975C2211yescharacteristic
38#90975C2211yescharacteristic
39#90973C2212yes—
40#90973C2212yes—
41#90969C2211yescharacteristic
42#90969C2211yescharacteristic
43#78571—51nomaximal, Hall
44C5×#519—31no—
45C2×#4088—31no—
46#65857—31no—
47C2×#3967—31no—
48#65319—32no—
49C4×#689—32no—
50#65315—32no—
51C4×#688—32no—
52#65291—31no—
53#65286D311yescharacteristic, complemented
54#65284—31no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×#1220—21no—
57C22×Dic15—21no—
58#6817—22no—
59#6817C2×C414yes—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×Dic30C2311yescharacteristic
64C22×C5×Dic3—21no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6C2×C414yescomplemented
68C2×C5×Dic6C2312yes—
69C3×C5×C4⋊C4D411yescharacteristic
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5C2312yes—
73#6714—22no—
74C4⋊Dic15—21no—
75C4⋊Dic15C2312yes—
76#6712—22no—
77C4×Dic15—22no—
78C5×#217—21no—
79C5×C4⋊Dic3C2311yescharacteristic
80C5×C4⋊Dic3C2311yescharacteristic
81C5×Dic3⋊C4—22no—
82C5×Dic3⋊C4—22no—
83C4×C5×Dic3—22no—
84C4×C5×Dic3—22no—
85C4×C5×Dic3—21no—
86C3×#681C2×C414yescomplemented
87C2×C3×#168—22no—
88C2×C3×#168C2311yescharacteristic
89C2×C3×#168C2311yescharacteristic
90C2×C3×#168C2312yes—
91#6663C2×C418yescomplemented
92#6660C2×C418yescomplemented
93C2×#909—51no—
94#5523—51no—
95#5516D511yescharacteristic, complemented
96C3×#405—51no—
97#5121—51no—
98#5113—51no—
99C2×#816—51no—
100#4716—52no—
101#4710—52no—
102C4×Dic12—52no—
103C4×#204—52no—
104C22×C5×Q8—31no—
105C2×#709—31no—
106C4×C5×Q8—32no—
107C4×C5×Q8—32no—
108C4×C5×Q8—61no—
109C4×C5×Q8—61no—
110C5×#117—31no—
111C5×#117—62no—
112C5×#117C2×D311yescharacteristic
113C2×#689—32no—
114#4088—38no—
115C2×#688—32no—
116C2×C4⋊Dic5—31no—
117C2×#681C2×D311yescharacteristic
118C22×#168C2×D311yescharacteristic
119#4038—61no—
120D5×C4⋊C4—61no—
121#4027—62no—
122C4×D20—32no—
123C4×Dic10—32no—
124#3967—32no—
125#3967—32no—
126#3942—31no—
127#3942—31no—
128#3941—31no—
129#3941—31no—
130#3940—32no—
131#3939—32no—
132#3935C2×D312yes—
133C4×#168C2×D312yes—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—21no—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5#4112yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—24no—
144C2×Dic15—21no—
145C2×Dic15—22no—
146C2×Dic15—81no—
147D60—21no—
148D60C22×C412yes—
149C4×D15—24no—
150C4×D15—82no—
151Dic30—21no—
152Dic30C22×C412yes—
153C2×C5×Dic3—24no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C5×Dic6—24no—
158C5×Dic6—22no—
159C5×Dic6C22×C414yes—
160C3×#168—21no—
161C3×#168—21no—
162C3×#168C22×C414yes—
163C2×#235—51no—
164C22×Dic6C2×D511yescharacteristic
165C2×C3×M16—51no—
166C22×C3×C8—51no—
167C3×#117C2×D511yescharacteristic
168#926C2×D511yescharacteristic
169C2×C4⋊Dic3—51no—
170#909—58no—
171C2×Dic12—52no—
172C2×#204—52no—
173#892—101no—
174D3×C4⋊C4—101no—
175#887—21no—
176#887—21no—
177#67⋊C4—102no—
178#877—22no—
179C4×D12—52no—
180C4×Dic6—52no—
181C4×Dic6C2×D512yes—
182C4×Dic6C2×D512yes—
183C3×#97—52no—
184C3×C4×C8—52no—
185#821—51no—
186#821—51no—
187#818—52no—
188#817—52no—
189#816—52no—
190#816—52no—
191#816—51no—
192#816—51no—
193C2×C5×Q8—34no—
194C2×C5×Q8—32no—
195C2×C5×Q8—61no—
196C2×C5×Q8—62no—
197C22×C4×C5—61no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—61no—
200C22×Dic5—61no—
201#709—34no—
202#709—62no—
203C2×D20—31no—
204C2×C4×D5—61no—
205C2×C4×D5—62no—
206C2×Dic10—31no—
207C5×C4⋊C4—31no—
208C5×C4⋊C4—31no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4—62no—
211C5×C4⋊C4D1211yescharacteristic
212C5×C4⋊C4D1211yescharacteristic
213C5×C22⋊C4—61no—
214C5×C22⋊C4D1212yes—
215C42×C5—61no—
216C42×C5—62no—
217C42×C5—62no—
218C42×C5C22×D312yes—
219#689—38no—
220#688—38no—
221#685—62no—
222C4⋊Dic5—32no—
223C4⋊Dic5—61no—
224#683—62no—
225C4×Dic5—62no—
226#681C4×D314yescomplemented
227C2×#168—22no—
228C2×#168C22×D311yescharacteristic
229C2×#168C22×D311yescharacteristic
230C2×#168C22×D312yes—
231C2×#137—151no—
232#520—151no—
233#519—31no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—151no—
240C2×#103—151no—
241#405—51no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5C4×D412yes—
245C22×C3×C5C22×D411yescharacteristic
246C2×D15—22no—
247C2×D15—41no—
248C2×D15—41no—
249C3×C4×C5—22no—
250C3×C4×C5—22no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254Dic15—22no—
255Dic15—42no—
256C5×Dic3—24no—
257C5×Dic3—41no—
258C5×Dic3—41no—
259C22×C3×C4C22×D511yescharacteristic
260C2×#67—101no—
261C22×Dic3—21no—
262C22×Dic3—101no—
263#235—54no—
264#235—102no—
265C2×D12—51no—
266C2×C4×D3—101no—
267C2×C4×D3—102no—
268C2×Dic6—22no—
269C2×Dic6—21no—
270C2×Dic6—51no—
271C2×Dic6C4×D514yescomplemented
272C2×Dic6C22×D512yes—
273C3×M16—54no—
274C2×C3×C8—52no—
275C2×C3×C8—51no—
276C2×C3×C8—51no—
277C2×C3×C8—102no—
278C3×C4⋊C4C22⋊D511yescharacteristic
279C3×C4⋊C4C22⋊D511yescharacteristic
280C3×C22⋊C4C22⋊D512yes—
281C3×C42C22×D512yes—
282#217—21no—
283(C2×D3)⋊C4—102no—
284C4⋊Dic3—52no—
285C4⋊Dic3—101no—
286C4⋊Dic3C22×D511yescharacteristic
287C4⋊Dic3C22×D511yescharacteristic
288Dic3⋊C4—22no—
289Dic3⋊C4—22no—
290Dic3⋊C4—102no—
291C4×Dic3—22no—
292C4×Dic3—22no—
293C4×Dic3—21no—
294C4×Dic3—102no—
295Dic12—58no—
296#204—58no—
297C23×C5C2×D1211yescharacteristic
298C22×D5—61no—
299C5×Q8—34no—
300C5×Q8—64no—
301C5×Q8—62no—
302C2×C4×C5—64no—
303C2×C4×C5—62no—
304C2×C4×C5—61no—
305C2×C4×C5—61no—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5C2×D1212yes—
308C2×C4×C5C2×D1212yes—
309C2×C4×C5C2×D1211yescharacteristic
310C2×C4×C5#23512yes—
311C2×C4×C5C23×D311yescharacteristic
312C22⋊D5—64no—
313C2×Dic5—61no—
314C2×Dic5—62no—
315C2×Dic5—241no—
316D20—32no—
317D20—61no—
318C4×D5—64no—
319C4×D5—242no—
320Dic10—32no—
321Dic10—61no—
322#168—21no—
323#168—21no—
324#168C2×C4×D314yes—
325C2×#41—151no—
326C22×Q8—31no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—51no—
331C22×C8—51no—
332C4×Q8—32no—
333C4×Q8—32no—
334C4×Q8—61no—
335C4×Q8—61no—
336C4×Q8—152no—
337C4×D4—152no—
338C4×D4—302no—
339#117—31no—
340#117—62no—
341#117—301no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—151no—
344C2×C4⋊C4—301no—
345#107—152no—
346#106—152no—
347#103—151no—
348#103—151no—
349#103—152no—
350#103—152no—
351#102—151no—
352#102—151no—
353#97—52no—
354C4×C8—52no—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359D15—42no—
360C23×C3C2×C22⋊D511yescharacteristic
361C22×D3—101no—
362C2×C3×C4C2×C4×D512yes—
363C2×C3×C4#70912yes—
364C2×C3×C4C2×C22⋊D512yes—
365C2×C3×C4C2×C22⋊D512yes—
366C2×C3×C4C2×C22⋊D511yescharacteristic
367C2×C3×C4C23×D511yescharacteristic
368#67—104no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—101no—
374C2×Dic3—102no—
375C2×Dic3—401no—
376D12—52no—
377D12—101no—
378C4×D3—104no—
379C4×D3—402no—
380Dic6—22no—
381Dic6—24no—
382Dic6—52no—
383Dic6—101no—
384Dic6C2×C4×D514yes—
385C3×C8—54no—
386C3×C8—101no—
387C3×C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C4×D1212yes—
391C22×C5C22×D1211yescharacteristic
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—64no—
398C4×C5—121no—
399C4×C5—121no—
400C4×C5C4×D1212yes—
401C4×C5C22×C4×D311yescharacteristic
402C4×C5C2×#23511yescharacteristic
403Dic5—62no—
404Dic5—122no—
405#41—154no—
406#41—302no—
407C2×Q8—34no—
408C2×Q8—32no—
409C2×Q8—62no—
410C2×Q8—61no—
411C2×Q8—151no—
412C2×D4—151no—
413C2×D4—301no—
414C22×C4—61no—
415C22×C4—301no—
416C22×C4—301no—
417C22×C4—302no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—54no—
422C2×C8—52no—
423C2×C8—51no—
424C2×C8—51no—
425C2×C8—102no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—152no—
431C4⋊C4—301no—
432C4⋊C4—302no—
433C4⋊C4#120311yescharacteristic
434C4⋊C4#120311yescharacteristic
435C22⋊C4—61no—
436C22⋊C4—302no—
437C22⋊C4#120312yes—
438C42—61no—
439C42—62no—
440C42—62no—
441C42—302no—
442C42C2×D3×D512yes—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3C4×C22⋊D512yes—
447C22×C3C22×C22⋊D511yescharacteristic
448C2×D3—102no—
449C2×D3—201no—
450C2×D3—201no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4C4×C22⋊D512yes—
454C3×C4C22×C4×D511yescharacteristic
455C3×C4C2×#70911yescharacteristic
456Dic3—24no—
457Dic3—41no—
458Dic3—41no—
459Dic3—102no—
460Dic3—202no—
461C2×C5—22no—
462C2×C5C2×C4×D1211yescharacteristic
463C2×C5#575111yescharacteristic
464C2×C5#575311yescharacteristic
465D5—122no—
466C23—301no—
467C23C2×#120311yescharacteristic
468Q8—34no—
469Q8—64no—
470Q8—62no—
471Q8—152no—
472Q8—301no—
473D4—152no—
474D4—304no—
475D4—301no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—301no—
481C2×C4—304no—
482C2×C4—302no—
483C2×C4—1202no—
484C2×C4—1201no—
485C2×C4#677112yes—
486C2×C4C4×D3×D512yes—
487C2×C4C2×#120312yes—
488C2×C4C2×#120312yes—
489C2×C4C2×#120311yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—54no—
492C8—101no—
493C8—101no—
494C2×C3—22no—
495C2×C3C2×C4×C22⋊D511yescharacteristic
496C2×C3#6617111yescharacteristic
497C2×C3#6617211yescharacteristic
498D3—202no—
499C5#7857111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—302no—
503C22—601no—
504C22—601no—
505C22C4×#120312yes—
506C22C22×#120311yescharacteristic
507C4—22no—
508C4—22no—
509C4—64no—
510C4—121no—
511C4—121no—
512C4—302no—
513C4—602no—
514C4C4×#120312yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061427811yescharacteristic, complemented, Sylow
518C2—22no—
519C2—602no—
520C2C2×C4×#120311yescharacteristic
521C2#1175530611yescharacteristic
522C2#1175530811yescharacteristic
subgroupG/H
centerC22C22×#1203
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C5×#5516C2semidirect
4. C3×#65286C2semidirect
5. #11748460C2semidirect
6. C2×#90973C2semidirect
8. #11747093C2semidirect
9. C4×#6663C2semidirect
10. #11747058C2semidirect
11. C4×#6660C2semidirect
30. #91321C4semidirect
53. #65286D3semidirect
67. C2×C5×Dic6C2×C4semidirect
86. C3×#681C2×C4semidirect
91. #6663C2×C4semidirect
92. #6660C2×C4semidirect
95. #5516D5semidirect
226. #681C4×D3semidirect
271. C2×Dic6C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#78571semidirect
517. C3#10614278semidirect

Maximal subgroups

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

subgroupindex
#117562822
C5×#55162
C2×#913212
C3×#652862
#117484602
C2×#909732
#117483932
#117470932
C4×#66632
#117470582
C4×#66602
#106142783
#785715