Finite Group Databasebeta

← #14417261

Subgroups — #14417261

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#11756275C211yesmaximal, characteristic, complemented
2C5×#5523C211yesmaximal, characteristic
3C2×#91322C211yesmaximal, characteristic
4C3×#65286C211yesmaximal, characteristic, complemented
5#11748459C211yesmaximal, characteristic, complemented
6C2×#90975C211yesmaximal, characteristic, complemented
7#11748388C211yesmaximal, characteristic
8#11747093C212yesmaximal, complemented
9C4×#6663C212yesmaximal, complemented
10#11747056C212yesmaximal, complemented
11C4×#6659C212yesmaximal, complemented
12#10614302—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16#91815C2211yescharacteristic
17#91776—21no—
18#91776—21no—
19#91766—22no—
20C4×Dic30C2212yes—
21C4×Dic30C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C5×#892—21no—
24C5×D3×C4⋊C4—21no—
25C5×#67⋊C4—22no—
26C4×C5×D12C2212yes—
27C4×C5×Dic6C2212yes—
28C2×C3×#681C2211yescharacteristic
29C22×C3×#168C2211yescharacteristic
30#91322C418yescomplemented
31C2×#6663C2212yes—
32C2×#6659C2212yes—
33C3×#3935C2212yes—
34C3×C4×#168C2212yes—
35#90988C2212yes—
36#90985C2212yes—
37#90975C2212yes—
38#90975C2212yes—
39#90973C2211yescharacteristic
40#90973C2211yescharacteristic
41#90967C2211yescharacteristic
42#90967C2211yescharacteristic
43#78571—51nomaximal, Hall
44C5×#520—31no—
45C2×#4097—31no—
46#65850—31no—
47#65522—31no—
48#65319—32no—
49C4×#689—32no—
50#65307—32no—
51C4×#687—32no—
52#65286D311yescharacteristic, complemented
53C2×#3942—31no—
54#65253—31no—
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15—21no—
57C2×Dic30—21no—
58C2×Dic30—22no—
59C2×Dic30C2×C414yescomplemented
60C2×Dic30C2312yes—
61C2×C5×#67—21no—
62C22×C5×Dic3—21no—
63C5×#235—22no—
64C5×#235C2×C414yes—
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
67C2×C4×C5×D3—22no—
68C2×C5×Dic6C2311yescharacteristic
69C3×C5×C4⋊C4D411yescharacteristic
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5C2312yes—
73#6719—21no—
74C4⋊Dic15C2311yescharacteristic
75C4⋊Dic15C2311yescharacteristic
76#6712—22no—
77#6712—22no—
78C4×Dic15—22no—
79C4×Dic15—22no—
80C4×Dic15—21no—
81C5×(C2×D3)⋊C4—22no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×Dic3⋊C4—22no—
85C4×C5×Dic3—22no—
86C3×#681C2×C414yescomplemented
87C2×C3×#168—22no—
88C2×C3×#168C2311yescharacteristic
89C2×C3×#168C2311yescharacteristic
90C2×C3×#168C2312yes—
91#6663C2×C418yescomplemented
92#6659C2×C418yescomplemented
93C2×#909—51no—
94#5523D511yescharacteristic
95#5516—51no—
96C3×#405—51no—
97#5121—51no—
98#5113—51no—
99C2×#816—51no—
100#4716—52no—
101#4710—52no—
102C4×Dic12—52no—
103C4×#204—52no—
104C2×C5×#41—31no—
105C22×Dic10—31no—
106C4×C5×Q8—32no—
107C4×C5×D4—32no—
108C4×C5×D4—62no—
109C5×#117—61no—
110C5×#117C2×D311yescharacteristic
111C2×C5×C4⋊C4—31no—
112C2×C5×C4⋊C4—61no—
113#4097—38no—
114C2×#689—32no—
115C2×#687—32no—
116#4072—31no—
117C2×#681C2×D311yescharacteristic
118C22×#168C2×D311yescharacteristic
119#4033—61no—
120#4033—61no—
121#4023—62no—
122C4×Dic10—32no—
123C4×Dic10—32no—
124#3967—31no—
125#3967—31no—
126#3964—31no—
127#3964—31no—
128#3942—32no—
129#3942—32no—
130#3940—32no—
131#3939—32no—
132#3935C2×D312yes—
133C4×#168C2×D312yes—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—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
143C2×Dic15—24no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×Dic15—21no—
147Dic30—24no—
148Dic30—22no—
149Dic30C22×C414yes—
150C5×#67—24no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—81no—
154C5×D12—21no—
155C5×D12C22×C412yes—
156C4×C5×D3—24no—
157C4×C5×D3—82no—
158C5×Dic6—21no—
159C5×Dic6C22×C412yes—
160C3×#168—21no—
161C3×#168—21no—
162C3×#168C22×C414yes—
163C2×#235C2×D511yescharacteristic
164C22×Dic6—51no—
165C2×C3×M16—51no—
166C22×C3×C8—51no—
167C3×#117C2×D511yescharacteristic
168#926—51no—
169C2×C4⋊Dic3C2×D511yescharacteristic
170#909—58no—
171C2×Dic12—52no—
172C2×#204—52no—
173#892—21no—
174D3×C4⋊C4—21no—
175#887—101no—
176#887—101no—
177#67⋊C4—22no—
178#877—102no—
179C4×D12C2×D512yes—
180C4×Dic6—52no—
181C4×Dic6—52no—
182C4×Dic6C2×D512yes—
183C3×#97—52no—
184C3×C4×C8—52no—
185#821—51no—
186#821—51no—
187#818—52no—
188#817—52no—
189#816—51no—
190#816—51no—
191#816—52no—
192#816—52no—
193C5×#41—34no—
194C5×#41—62no—
195C2×C5×Q8—31no—
196C2×C5×D4—31no—
197C2×C5×D4—61no—
198C22×C4×C5—61no—
199C22×C4×C5—61no—
200C22×C4×C5—62no—
201C22×C4×C5C22×D311yescharacteristic
202C22×Dic5—61no—
203C2×Dic10—34no—
204C2×Dic10—32no—
205C2×Dic10—61no—
206C2×Dic10—62no—
207C5×C4⋊C4—32no—
208C5×C4⋊C4—61no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4D1211yescharacteristic
211C5×C4⋊C4D1211yescharacteristic
212C5×C22⋊C4—62no—
213C5×C22⋊C4D1212yes—
214C42×C5—62no—
215C42×C5C22×D312yes—
216C22⋊Dic5—61no—
217#689—38no—
218#687—38no—
219C4⋊Dic5—31no—
220C4⋊Dic5—31no—
221#683—62no—
222#683—62no—
223C4×Dic5—62no—
224C4×Dic5—62no—
225C4×Dic5—61no—
226#681C4×D314yescomplemented
227C2×#168—22no—
228C2×#168C22×D311yescharacteristic
229C2×#168C22×D311yescharacteristic
230C2×#168C22×D312yes—
231C2×#137—151no—
232#520—31no—
233#519—151no—
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×C5×D3—22no—
247C2×C5×D3—41no—
248C2×C5×D3—41no—
249C3×C4×C5—22no—
250C3×C4×C5—22no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254Dic15—24no—
255Dic15—41no—
256Dic15—41no—
257C5×Dic3—22no—
258C5×Dic3—42no—
259C22×C3×C4C22×D511yescharacteristic
260C2×#67—21no—
261C22×Dic3—21no—
262C22×Dic3—101no—
263#235—22no—
264#235C4×D514yes—
265C2×D12C22×D511yescharacteristic
266C2×C4×D3—21no—
267C2×C4×D3—22no—
268C2×Dic6—54no—
269C2×Dic6—52no—
270C2×Dic6—102no—
271C2×Dic6—101no—
272C2×Dic6C22×D511yescharacteristic
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—101no—
283(C2×D3)⋊C4—22no—
284C4⋊Dic3—21no—
285C4⋊Dic3—51no—
286C4⋊Dic3—51no—
287C4⋊Dic3C22×D512yes—
288Dic3⋊C4—22no—
289Dic3⋊C4—102no—
290Dic3⋊C4—102no—
291C4×Dic3—22no—
292C4×Dic3—102no—
293C4×Dic3—102no—
294C4×Dic3—101no—
295Dic12—58no—
296#204—58no—
297C23×C5—61no—
298C23×C5C2×D1211yescharacteristic
299C5×Q8—32no—
300C5×Q8—61no—
301C5×D4—32no—
302C5×D4—64no—
303C5×D4—61no—
304C2×C4×C5—61no—
305C2×C4×C5—64no—
306C2×C4×C5—62no—
307C2×C4×C5—242no—
308C2×C4×C5—241no—
309C2×C4×C5C2×C4×D312yes—
310C2×C4×C5C2×D1212yes—
311C2×C4×C5C2×D1212yes—
312C2×C4×C5C2×D1211yescharacteristic
313C2×C4×C5#23512yes—
314C2×C4×C5C23×D311yescharacteristic
315C2×Dic5—64no—
316C2×Dic5—62no—
317C2×Dic5—61no—
318C2×Dic5—61no—
319Dic10—34no—
320Dic10—64no—
321Dic10—62no—
322#168—21no—
323#168—21no—
324#168C2×C4×D314yes—
325C2×#41—31no—
326C22×Q8—151no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—51no—
331C22×C8—51no—
332C4×Q8—32no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—32no—
338C4×D4—62no—
339#117—61no—
340#117—151no—
341#117—302no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—31no—
344C2×C4⋊C4—61no—
345#107—152no—
346#106—152no—
347#103—152no—
348#103—152no—
349#103—151no—
350#103—151no—
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
359C5×D3—42no—
360C23×C3C2×C22⋊D511yescharacteristic
361C22×D3—21no—
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—24no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—81no—
372C2×Dic3—102no—
373C2×Dic3—104no—
374C2×Dic3—101no—
375C2×Dic3—101no—
376D12—21no—
377D12C2×C4×D512yes—
378C4×D3—24no—
379C4×D3—82no—
380Dic6—21no—
381Dic6—54no—
382Dic6—102no—
383Dic6—104no—
384Dic6C2×C4×D512yes—
385C3×C8—54no—
386C3×C8—101no—
387C3×C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5—62no—
391C22×C5—121no—
392C22×C5—121no—
393C22×C5C4×D1212yes—
394C22×C5C22×D1211yescharacteristic
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—62no—
398C4×C5—122no—
399C4×C5C4×D1212yes—
400C4×C5C22×C4×D311yescharacteristic
401C4×C5C2×#23511yescharacteristic
402Dic5—64no—
403Dic5—121no—
404Dic5—121no—
405#41—34no—
406#41—62no—
407C2×Q8—31no—
408C2×Q8—154no—
409C2×Q8—152no—
410C2×Q8—302no—
411C2×Q8—301no—
412C2×D4—31no—
413C2×D4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—54no—
422C2×C8—52no—
423C2×C8—51no—
424C2×C8—51no—
425C2×C8—102no—
426C4⋊C4—32no—
427C4⋊C4—61no—
428C4⋊C4—62no—
429C4⋊C4—151no—
430C4⋊C4—151no—
431C4⋊C4—302no—
432C4⋊C4—302no—
433C4⋊C4#120311yescharacteristic
434C4⋊C4#120311yescharacteristic
435C22⋊C4—62no—
436C22⋊C4—301no—
437C22⋊C4#120312yes—
438C42—62no—
439C42—301no—
440C42—302no—
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—22no—
449C2×D3—41no—
450C2×D3—41no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4C4×C22⋊D512yes—
454C3×C4C22×C4×D511yescharacteristic
455C3×C4C2×#70911yescharacteristic
456Dic3—22no—
457Dic3—42no—
458Dic3—104no—
459Dic3—201no—
460Dic3—201no—
461C2×C5—22no—
462C2×C5—122no—
463C2×C5C2×C4×D1211yescharacteristic
464C2×C5#575111yescharacteristic
465C2×C5#575311yescharacteristic
466C23—61no—
467C23C2×#120311yescharacteristic
468Q8—32no—
469Q8—61no—
470Q8—154no—
471Q8—304no—
472Q8—302no—
473D4—32no—
474D4—64no—
475D4—61no—
476C2×C4—61no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—242no—
480C2×C4—241no—
481C2×C4—304no—
482C2×C4—302no—
483C2×C4—301no—
484C2×C4—301no—
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#6617011yescharacteristic
497C2×C3#6617211yescharacteristic
498D3—42no—
499C5#7857111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—62no—
503C22—121no—
504C22—121no—
505C22C4×#120312yes—
506C22C22×#120311yescharacteristic
507C4—22no—
508C4—22no—
509C4—62no—
510C4—122no—
511C4—304no—
512C4—601no—
513C4—601no—
514C4C4×#120312yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061430211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—122no—
520C2C2×C4×#120311yescharacteristic
521C2#1175530611yescharacteristic
522C2#1175530711yescharacteristic
subgroupG/H
centerC22C22×#1203
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#235C22
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
1. #11756275C2semidirect
4. C3×#65286C2semidirect
5. #11748459C2semidirect
6. C2×#90975C2semidirect
8. #11747093C2semidirect
9. C4×#6663C2semidirect
10. #11747056C2semidirect
11. C4×#6659C2semidirect
30. #91322C4semidirect
52. #65286D3semidirect
59. C2×Dic30C2×C4semidirect
86. C3×#681C2×C4semidirect
91. #6663C2×C4semidirect
92. #6659C2×C4semidirect
226. #681C4×D3semidirect
443. C3×C5#2947semidirect
499. C5#78571semidirect
517. C3#10614302semidirect

Maximal subgroups

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

subgroupindex
#117562752
C5×#55232
C2×#913222
C3×#652862
#117484592
C2×#909752
#117483882
#117470932
C4×#66632
#117470562
C4×#66592
#106143023
#785715