Finite Group Databasebeta

← #14429253

Subgroups — #14429253

Order 1920.

944 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5510C211yesmaximal, characteristic
2#11754255C211yesmaximal, characteristic, complemented
3C2×#91309C211yesmaximal, characteristic, complemented
4C2×#91286C211yesmaximal, characteristic, complemented
5#11751065C211yesmaximal, characteristic, complemented
6C2×C3×#3942C211yesmaximal, characteristic, complemented
7#11748388C211yesmaximal, characteristic, complemented
8#11747355C212yesmaximal, complemented
9#11747354C212yesmaximal, complemented
10#11747353C212yesmaximal, complemented
11#11747352C212yesmaximal, complemented
12#10615154—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C22×C3×Dic10C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17#91815C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C5×#930—22no—
20C5×#897—21no—
21C5×#897C2212yes—
22C5×#895—21no—
23C5×#895C2212yes—
24C22×C3×#168C2211yescharacteristic
25#91539—22no—
26#91385—21no—
27#91370—21no—
28#91369C2212yes—
29#91369C2212yes—
30C2×#6661C2212yes—
31#91309—24no—
32C2×#6656C2212yes—
33C2×#6650C2212yes—
34#91286—24no—
35C2×D3×#168C2212yes—
36#91092C2212yes—
37#91091C2212yes—
38C3×#3942C2212yes—
39C3×#3942C2212yes—
40#90973C2211yescharacteristic
41#90973C2211yescharacteristic
42#90967C2211yescharacteristic
43#90967C2211yescharacteristic
44#78800—51nomaximal, Hall
45C5×#537—31no—
46#66159—31no—
47C2×#4097—31no—
48C2×#4094—31no—
49#65522—31no—
50#65341—32no—
51#65340—32no—
52#65331—32no—
53#65330—32no—
54C2×#3942D311yescharacteristic, complemented
55#65253—31no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×C5×#67—21no—
59C2×C5×#67D412yes—
60C5×#235—22no—
61C5×#235—22no—
62C5×#235—22no—
63C5×#235—22no—
64C2×C5×D12—22no—
65C2×C5×D12D411yescharacteristic
66C2×C5×D12C2311yescharacteristic
67C2×C4×C5×D3—21no—
68C2×C4×C5×D3C2312yes—
69C2×C5×Dic6D411yescharacteristic
70C2×C5×Dic6C2311yescharacteristic
71C22×C3×Dic5—21no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10—22no—
74C2×C3×Dic10—21no—
75C2×C3×Dic10C2312yes—
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78#6719—21no—
79C4⋊Dic15C2311yescharacteristic
80C4⋊Dic15C2311yescharacteristic
81C4×Dic15—21no—
82#6710D414yescomplemented
83C2×C3⋊#168C2312yes—
84C5×(C2×D3)⋊C4—22no—
85C5×(C2×D3)⋊C4—22no—
86C5×C4⋊Dic3—22no—
87C2×C3×#168—22no—
88C2×C3×#168C2311yescharacteristic
89C2×C3×#168C2311yescharacteristic
90#6669—22no—
91#6669—22no—
92(C2×D3)⋊Dic5—22no—
93(C2×D3)⋊Dic5—22no—
94#6661—24no—
95#6656—24no—
96#6650—24no—
97D3×#168—24no—
98#5816—51no—
99C2×#942—51no—
100C2×#901—51no—
101#5510D511yescharacteristic
102C2×C3×#103—51no—
103#5113—51no—
104#4964—51no—
105#4816—52no—
106#4813—52no—
107#4812—52no—
108#4807—52no—
109C2×C5×#41—31no—
110C2×C5×#41—31no—
111C22×Dic10C2×D311yescharacteristic
112C5×C22⋊Q8—32no—
113C5×C22⋊Q8—61no—
114C5×#121—32no—
115C5×#121—61no—
116C5×#121—62no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118#4097—64no—
119#4094—64no—
120#4090—61no—
121C2×#689—32no—
122#4082—61no—
123#4081—62no—
124C2×#687—32no—
125#4072—31no—
126#4070—32no—
127#4070—32no—
128C2×#681—31no—
129C2×#681—32no—
130C22×#168—32no—
131C22×#168C2×D311yescharacteristic
132#3967—31no—
133#3967—31no—
134#3964—31no—
135#3964—31no—
136#3942C2×D312yes—
137#3942C2×D312yes—
138#3940—32no—
139#3939—32no—
140#3063—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×C5×D3—21no—
143C22×C5×D3C2×D412yes—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—81no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×C3×C4×C5C2411yescharacteristic
150C2×Dic15—21no—
151C2×Dic15—21no—
152C5×#67—24no—
153C5×#67—42no—
154C2×C5×Dic3—21no—
155C2×C5×Dic3C2×D412yes—
156C5×D12—22no—
157C5×D12—21no—
158C5×D12—21no—
159C5×D12—82no—
160C4×C5×D3—22no—
161C4×C5×D3—22no—
162C4×C5×D3—42no—
163C5×Dic6—22no—
164C5×Dic6—21no—
165C5×Dic6—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—42no—
168C3×Dic10—22no—
169C3×Dic10—22no—
170C3×Dic10—42no—
171C3⋊#168C2×D414yes—
172C3×#168—21no—
173C3×#168—21no—
174C22×C3×Q8—51no—
175C2×#235C2×D511yescharacteristic
176C2×#235C2×D511yescharacteristic
177C22×C3×C8—51no—
178C2×C3×C4⋊C4C2×D511yescharacteristic
179#947—101no—
180#946—52no—
181#946—52no—
182#944—101no—
183C2×#216—52no—
184#942—104no—
185C2×#215—52no—
186#930—22no—
187#929—102no—
188#926—51no—
189C2×#208—51no—
190#901—104no—
191C2×ΓL(1,𝔽25)—52no—
192C2×D3×C8—52no—
193#897—21no—
194#897C2×D512yes—
195#895—21no—
196#895C2×D512yes—
197C3×#103—52no—
198C3×#103—52no—
199#821—51no—
200#821—51no—
201#816—51no—
202#816—51no—
203#808—52no—
204#807—52no—
205C5×#41—62no—
206C5×#41—62no—
207C5×#41—62no—
208C5×#41—62no—
209C2×C5×Q8—31no—
210C2×C5×Q8—31no—
211C2×C5×D4—32no—
212C2×C5×D4—31no—
213C2×C5×D4—31no—
214C2×C5×D4—62no—
215C2×C5×D4—61no—
216C22×C4×C5—21no—
217C22×C4×C5—32no—
218C22×C4×C5—61no—
219C22×C4×C5C22×D311yescharacteristic
220C22×Dic5—21no—
221C2×Dic10—22no—
222C2×Dic10—22no—
223C2×Dic10—21no—
224C2×Dic10C22×D312yes—
225C5×C4⋊C4—21no—
226C5×C4⋊C4—62no—
227C5×C4⋊C4C22×D312yes—
228C5×C22⋊C4—62no—
229C5×C22⋊C4—62no—
230C22⋊Dic5—62no—
231C22⋊Dic5—62no—
232C22⋊Dic5—61no—
233#689—64no—
234#687—64no—
235C4⋊Dic5—31no—
236C4⋊Dic5—31no—
237#683—62no—
238#683—62no—
239C4×Dic5—61no—
240#681—34no—
241#681—64no—
242C2×#168—22no—
243C2×#168—32no—
244C2×#168—64no—
245C2×#168C22×D311yescharacteristic
246C2×#168C22×D311yescharacteristic
247C2×#137—151no—
248C2×#131—151no—
249#537—31no—
250#536—151no—
251#470—152no—
252#468—152no—
253#467—152no—
254#465—152no—
255#428—151no—
256#416—151no—
257C2×#103—51no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5#12112yes—
261C22×C3×C5C22×D411yescharacteristic
262C2×C5×D3—22no—
263C2×C5×D3—22no—
264C2×C5×D3—22no—
265C2×C5×D3—41no—
266C2×C5×D3—41no—
267C2×C5×D3—41no—
268C3×C4×C5—42no—
269C3×C4×C5#12112yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C2×#4111yescharacteristic
272Dic15—41no—
273Dic15—41no—
274C3×Dic5—42no—
275C5×Dic3—22no—
276C5×Dic3—41no—
277C2×C3×Q8—52no—
278C2×C3×Q8—101no—
279C2×C3×Q8—102no—
280C2×C3×Q8—102no—
281C22×C3×C4—21no—
282C22×C3×C4—101no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—21no—
285C2×#67C22⋊D512yes—
286#235—22no—
287#235—22no—
288#235—22no—
289#235—22no—
290C2×D12—22no—
291C2×D12C22⋊D511yescharacteristic
292C2×D12C22×D511yescharacteristic
293C2×C4×D3—21no—
294C2×C4×D3C22×D512yes—
295C2×Dic6C22⋊D511yescharacteristic
296C2×Dic6C22×D511yescharacteristic
297C2×C3×C8—51no—
298C2×C3×C8—51no—
299C2×C3×C8—102no—
300C3×C4⋊C4—21no—
301C3×C4⋊C4C22×D512yes—
302#217—101no—
303#216—104no—
304#215—104no—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—22no—
307(C2×D3)⋊C4—102no—
308(C2×D3)⋊C4—102no—
309C4⋊Dic3—22no—
310C4⋊Dic3—51no—
311C4⋊Dic3—51no—
312Dic3⋊C4—102no—
313Dic3⋊C4—102no—
314C4×Dic3—101no—
315#208—54no—
316C2×C3⋊C8—52no—
317ΓL(1,𝔽25)—104no—
318D3×C8—104no—
319C23×C5—32no—
320C23×C5—61no—
321C23×C5D3×D411yescharacteristic
322C5×Q8—62no—
323C5×Q8—61no—
324C5×Q8—61no—
325C5×D4—64no—
326C5×D4—62no—
327C5×D4—61no—
328C5×D4—61no—
329C5×D4—122no—
330C5×D4—242no—
331C2×C4×C5—22no—
332C2×C4×C5—32no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—61no—
336C2×C4×C5—81no—
337C2×C4×C5—122no—
338C2×C4×C5#23512yes—
339C2×C4×C5D3×D412yes—
340C2×C4×C5D3×D411yescharacteristic
341C2×C4×C5C23×D311yescharacteristic
342C2×Dic5—22no—
343C2×Dic5—42no—
344C2×Dic5—61no—
345C2×Dic5—61no—
346Dic10—22no—
347Dic10—22no—
348Dic10—42no—
349#168—21no—
350#168—21no—
351#168—34no—
352C2×#41—31no—
353C2×#41—31no—
354C22×Q8—51no—
355#137—304no—
356C2×Q16—152no—
357C2×SD16—152no—
358#131—304no—
359C2×M16—152no—
360C2×M16—151no—
361C22×C8—51no—
362C22×C8—152no—
363C4⋊Q8—301no—
364#124—301no—
365#123—302no—
366C22⋊Q8—32no—
367C22⋊Q8—61no—
368C22⋊Q8—152no—
369C22⋊Q8—152no—
370#121—32no—
371#121—61no—
372#121—62no—
373#117—151no—
374C2×C4⋊C4D3×D511yescharacteristic
375#107—152no—
376#106—152no—
377#103—52no—
378#103—52no—
379#103—151no—
380#103—151no—
381#102—151no—
382#102—151no—
383C2×C3×C5—22no—
384C2×C3×C5C2×#12111yescharacteristic
385C2×C3×C5#57711yescharacteristic
386C2×C3×C5#57811yescharacteristic
387C5×D3—42no—
388C5×D3—81no—
389C23×C3C2×C22⋊D511yescharacteristic
390C22×D3—21no—
391C22×D3C2×C22⋊D512yes—
392C3×Q8—102no—
393C3×Q8—102no—
394C3×Q8—202no—
395C2×C3×C4—22no—
396C2×C3×C4—81no—
397C2×C3×C4—102no—
398C2×C3×C4—202no—
399C2×C3×C4D4×D512yes—
400C2×C3×C4#71112yes—
401C2×C3×C4C2×C22⋊D511yescharacteristic
402C2×C3×C4C23×D511yescharacteristic
403#67—24no—
404#67—42no—
405C2×Dic3—21no—
406C2×Dic3—101no—
407C2×Dic3—101no—
408C2×Dic3C2×C22⋊D512yes—
409D12—21no—
410D12—21no—
411D12—22no—
412D12—82no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—42no—
416Dic6—22no—
417Dic6—21no—
418Dic6—21no—
419C3×C8—101no—
420C3×C8—101no—
421C3⋊C8—54no—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5—62no—
425C22×C5—62no—
426C22×C5—62no—
427C22×C5—121no—
428C22×C5—121no—
429C22×C5—121no—
430C22×C5#88412yes—
431C22×C5C2×D3×D411yescharacteristic
432C4×C5—42no—
433C4×C5—62no—
434C4×C5—121no—
435C4×C5#88412yes—
436C4×C5C2×#23511yescharacteristic
437C4×C5C2×D3×D411yescharacteristic
438Dic5—42no—
439Dic5—121no—
440Dic5—121no—
441#41—62no—
442#41—62no—
443#41—62no—
444#41—62no—
445C2×Q8—31no—
446C2×Q8—31no—
447C2×Q8—52no—
448C2×Q8—101no—
449C2×Q8—102no—
450C2×Q8—102no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—31no—
454C2×D4—62no—
455C2×D4—61no—
456C22×C4—21no—
457C22×C4—32no—
458C22×C4—61no—
459C22×C4—101no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—304no—
462SD16—304no—
463M16—154no—
464M16—304no—
465C2×C8—51no—
466C2×C8—51no—
467C2×C8—102no—
468C2×C8—152no—
469C2×C8—304no—
470C4⋊C4—21no—
471C4⋊C4—62no—
472C4⋊C4—151no—
473C4⋊C4—151no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4C2×D3×D512yes—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—302no—
480C22⋊C4—302no—
481C22⋊C4—301no—
482C42—301no—
483C3×C5#306311yescharacteristic, complemented, Hall
484C22×C3—21no—
485C22×C3—21no—
486C22×C3#408512yes—
487C22×C3C22×C22⋊D511yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C3×C4—42no—
495C3×C4—202no—
496C3×C4#408512yes—
497C3×C4C2×D4×D511yescharacteristic
498C3×C4C2×#71111yescharacteristic
499Dic3—22no—
500Dic3—41no—
501Dic3—201no—
502Dic3—201no—
503C2×C5—22no—
504C2×C5—122no—
505C2×C5—241no—
506C2×C5C2×#88411yescharacteristic
507C2×C5#576211yescharacteristic
508C2×C5#576611yescharacteristic
509C23—32no—
510C23—61no—
511C23D3×C22⋊D511yescharacteristic
512Q8—62no—
513Q8—61no—
514Q8—61no—
515Q8—102no—
516Q8—102no—
517Q8—202no—
518D4—64no—
519D4—62no—
520D4—61no—
521D4—61no—
522D4—122no—
523D4—242no—
524C2×C4—22no—
525C2×C4—32no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—81no—
530C2×C4—102no—
531C2×C4—122no—
532C2×C4—202no—
533C2×C4—301no—
534C2×C4—301no—
535C2×C4#677212yes—
536C2×C4#677712yes—
537C2×C4D3×C22⋊D511yescharacteristic
538C2×C4C22×D3×D511yescharacteristic
539C8—101no—
540C8—101no—
541C8—154no—
542C2×C3—22no—
543C2×C3C2×#408511yescharacteristic
544C2×C3#6617011yescharacteristic
545C2×C3#6617211yescharacteristic
546D3—42no—
547D3—81no—
548C5#7880011yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—62no—
552C22—62no—
553C22—62no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22#9145312yes—
558C22C2×D3×C22⋊D511yescharacteristic
559C4—42no—
560C4—62no—
561C4—121no—
562C4—202no—
563C4—601no—
564C4—601no—
565C4#9145312yes—
566C4C2×#677211yescharacteristic
567C4C2×#677711yescharacteristic
568C3#1061515411yescharacteristic, complemented, Sylow
569C2—22no—
570C2—122no—
571C2—241no—
572C2C2×#9145311yescharacteristic
573C2#1175471111yescharacteristic
574C2#1175510511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC5×#5510C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#3063
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11754255C2semidirect
3. C2×#91309C2semidirect
4. C2×#91286C2semidirect
5. #11751065C2semidirect
6. C2×C3×#3942C2semidirect
7. #11748388C2semidirect
8. #11747355C2semidirect
9. #11747354C2semidirect
10. #11747353C2semidirect
11. #11747352C2semidirect
54. C2×#3942D3semidirect
82. #6710D4semidirect
483. C3×C5#3063semidirect
548. C5#78800semidirect
568. C3#10615154semidirect

Maximal subgroups

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

subgroupindex
C5×#55102
#117542552
C2×#913092
C2×#912862
#117510652
C2×C3×#39422
#117483882
#117473552
#117473542
#117473532
#117473522
#106151543
#788005