Finite Group Databasebeta

← #14429246

Subgroups — #14429246

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
1C2×C5×#901C211yesmaximal, characteristic, complemented
2#11754603C211yesmaximal, characteristic
3#11754255C211yesmaximal, characteristic, complemented
4C2×#91315C211yesmaximal, characteristic, complemented
5#11751214C211yesmaximal, characteristic, complemented
6C2×C3×#3948C211yesmaximal, characteristic, complemented
7#11748453C211yesmaximal, characteristic, complemented
8#11747949C212yesmaximal, complemented
9#11747948C212yesmaximal, complemented
10#11747947C212yesmaximal, complemented
11#11747946C212yesmaximal, complemented
12#10615539—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C22×C3×Dic10C2211yescharacteristic
16C22×C3×C5×C8C2211yescharacteristic
17#91815C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C5×#901—24no—
20C2×C5×ΓL(1,𝔽25)C2212yes—
21C2×C5×D3×C8C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23#91566—22no—
24#91539—22no—
25#91466—21no—
26#91460C2212yes—
27#91385—21no—
28#91383—21no—
29#91370—21no—
30#91369C2212yes—
31#91369C2212yes—
32#91365C2212yes—
33C2×#6662C2212yes—
34#91315—24no—
35C2×#6658C2212yes—
36#91105C2211yescharacteristic
37#91105C2211yescharacteristic
38#91100C2211yescharacteristic
39#91100C2211yescharacteristic
40C3×#3948C2212yes—
41C3×#3948C2212yes—
42#90990C2212yes—
43#90987C2212yes—
44#78800—51nomaximal, Hall
45C2×C5×#131—31no—
46#66167—31no—
47#66159—31no—
48C2×#4055—31no—
49#65426—32no—
50#65423—32no—
51#65410—31no—
52#65405—32no—
53#65403—32no—
54#65402—31no—
55C2×#3948D311yescharacteristic, complemented
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—21no—
58#6817—22no—
59#6817—22no—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×Dic30C2311yescharacteristic
63C2×C5×#67D412yes—
64C5×#235—22no—
65C5×#235—22no—
66C2×C5×D12D411yescharacteristic
67C2×C4×C5×D3C2312yes—
68C2×C5×Dic6D411yescharacteristic
69C22×C3×Dic5—21no—
70C22×C3×Dic5—21no—
71C2×C3×Dic10—22no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10—21no—
74C2×C3×Dic10C2312yes—
75C2×#1203—22no—
76C2×C3×C5×C8—22no—
77C2×C3×C5×C8C2311yescharacteristic
78C2×C3×C5×C8C2311yescharacteristic
79#6719—21no—
80C4⋊Dic15C2311yescharacteristic
81C4⋊Dic15C2311yescharacteristic
82C4×Dic15—21no—
83C5×#208D414yescomplemented
84C2×C5×C3⋊C8C2312yes—
85C5×ΓL(1,𝔽25)—24no—
86C5×D3×C8—24no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89#6669—22no—
90#6669—22no—
91#6668—22no—
92#6667—22no—
93(C2×D3)⋊Dic5—22no—
94(C2×D3)⋊Dic5—22no—
95(C2×D3)⋊Dic5—22no—
96#6662—24no—
97#6658—24no—
98#5816—51no—
99C2×#942—51no—
100C2×#901D511yescharacteristic, complemented
101#5510—51no—
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×#709—31no—
111C22×Dic10C2×D311yescharacteristic
112C5×#131—64no—
113C2×C5×M16—31no—
114C2×C5×M16—32no—
115C22×C5×C8—32no—
116C22×C5×C8C2×D311yescharacteristic
117#4092—61no—
118#4090—61no—
119#4085—62no—
120#4084—61no—
121#4082—61no—
122#4081—62no—
123#4076—32no—
124#4072—31no—
125C2×C4⋊Dic5C2×D311yescharacteristic
126#4070—32no—
127#4070—32no—
128#4070—32no—
129#4055—64no—
130C2×Dic20—32no—
131C2×#677—32no—
132#3953—31no—
133#3953—31no—
134#3950—32no—
135#3949—32no—
136#3948—31no—
137#3948—31no—
138#3948C2×D312yes—
139#3948C2×D312yes—
140#3063—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×D15—21no—
143C22×C5×D3C2×D412yes—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×C3×C4×C5C2411yescharacteristic
148#1220—42no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152D60—21no—
153D60—21no—
154C4×D15—42no—
155Dic30—21no—
156Dic30—21no—
157C5×#67—24no—
158C2×C5×Dic3C2×D412yes—
159C5×D12—22no—
160C4×C5×D3—22no—
161C4×C5×D3—22no—
162C5×Dic6—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—42no—
166C2×C3×Dic5—81no—
167C3×Dic10—22no—
168C3×Dic10—22no—
169C3×Dic10—42no—
170#1203—82no—
171C3×C5×C8—21no—
172C3×C5×C8—21no—
173C5×C3⋊C8C2×D414yes—
174C22×C3×Q8—51no—
175C2×#235—51no—
176C2×#235C2×D511yescharacteristic
177C22×C3×C8C2×D511yescharacteristic
178C2×C3×C4⋊C4—51no—
179#947—101no—
180#946—52no—
181#946—52no—
182#944—101no—
183C2×#216—52no—
184#942—104no—
185C2×#215—52no—
186#930—102no—
187#929—102no—
188#926—51no—
189C2×#208C2×D511yescharacteristic
190#901—24no—
191C2×ΓL(1,𝔽25)C2×D512yes—
192C2×D3×C8C2×D512yes—
193#897—52no—
194#897—101no—
195#895—52no—
196#895—101no—
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—
207C2×C5×Q8—31no—
208C2×C5×D4—32no—
209C2×C5×D4—31no—
210C22×C4×C5—32no—
211C22×C4×C5C22×D311yescharacteristic
212C2×C22⋊D5—62no—
213C2×C22⋊D5—61no—
214C22×Dic5—21no—
215C22×Dic5—21no—
216#709—62no—
217#709—62no—
218C2×D20—31no—
219C2×C4×D5—61no—
220C2×Dic10—22no—
221C2×Dic10—22no—
222C2×Dic10—21no—
223C2×Dic10—31no—
224C2×Dic10C22×D312yes—
225C5×M16—34no—
226C5×M16—64no—
227C2×C5×C8—22no—
228C2×C5×C8—32no—
229C2×C5×C8—64no—
230C2×C5×C8C22×D311yescharacteristic
231C2×C5×C8C22×D311yescharacteristic
232C22⋊Dic5—62no—
233C22⋊Dic5—62no—
234C22⋊Dic5—62no—
235C22⋊Dic5—61no—
236#685—62no—
237C4⋊Dic5—21no—
238C4⋊Dic5—31no—
239C4⋊Dic5—31no—
240C4⋊Dic5C22×D312yes—
241#683—62no—
242#683—62no—
243#683—62no—
244C4×Dic5—61no—
245Dic20—64no—
246#677—64no—
247C2×#137—151no—
248C2×#131—31no—
249#537—151no—
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×D15—41no—
263C2×D15—41no—
264C2×D15—41no—
265C2×C5×D3—22no—
266C2×C5×D3—22no—
267C2×C5×D3—22no—
268C3×C4×C5#12112yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C2×#4111yescharacteristic
271Dic15—41no—
272Dic15—41no—
273Dic15—41no—
274C3×Dic5—42no—
275C3×Dic5—42no—
276C5×Dic3—22no—
277C2×C3×Q8—52no—
278C2×C3×Q8—101no—
279C2×C3×Q8—102no—
280C2×C3×Q8—102no—
281C22×C3×C4—101no—
282C22×C3×C4—101no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—101no—
285C2×#67D2012yes—
286#235—22no—
287#235—22no—
288#235—102no—
289#235—102no—
290C2×D12—51no—
291C2×D12—102no—
292C2×D12D2011yescharacteristic
293C2×C4×D3—101no—
294C2×C4×D3C22×D512yes—
295C2×Dic6—51no—
296C2×Dic6D2011yescharacteristic
297C2×C3×C8—22no—
298C2×C3×C8C22×D511yescharacteristic
299C2×C3×C8C22×D511yescharacteristic
300C3×C4⋊C4—52no—
301C3×C4⋊C4—101no—
302#217—101no—
303#216—104no—
304#215—104no—
305(C2×D3)⋊C4—102no—
306(C2×D3)⋊C4—102no—
307(C2×D3)⋊C4—102no—
308(C2×D3)⋊C4—102no—
309C4⋊Dic3—51no—
310C4⋊Dic3—51no—
311C4⋊Dic3—102no—
312Dic3⋊C4—102no—
313Dic3⋊C4—102no—
314C4×Dic3—101no—
315#208C22⋊D514yescomplemented
316C2×C3⋊C8C22×D512yes—
317ΓL(1,𝔽25)—24no—
318D3×C8—24no—
319C23×C5—32no—
320C23×C5D3×D411yescharacteristic
321C22×D5—61no—
322C5×Q8—62no—
323C5×D4—64no—
324C5×D4—62no—
325C2×C4×C5—32no—
326C2×C4×C5—62no—
327C2×C4×C5—62no—
328C2×C4×C5#23512yes—
329C2×C4×C5D3×D411yescharacteristic
330C2×C4×C5D3×D412yes—
331C2×C4×C5C23×D311yescharacteristic
332C22⋊D5—122no—
333C22⋊D5—242no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5—42no—
337C2×Dic5—61no—
338C2×Dic5—61no—
339C2×Dic5—61no—
340C2×Dic5—81no—
341D20—61no—
342D20—61no—
343C4×D5—122no—
344Dic10—22no—
345Dic10—22no—
346Dic10—42no—
347Dic10—61no—
348Dic10—61no—
349C5×C8—21no—
350C5×C8—21no—
351C5×C8—34no—
352C2×#41—31no—
353C2×#41—151no—
354C22×Q8—51no—
355#137—304no—
356C2×Q16—152no—
357C2×SD16—152no—
358#131—64no—
359C2×M16—31no—
360C2×M16—32no—
361C22×C8—32no—
362C22×C8D3×D511yescharacteristic
363C4⋊Q8—301no—
364#124—301no—
365#123—302no—
366C22⋊Q8—152no—
367C22⋊Q8—152no—
368C22⋊Q8—152no—
369C22⋊Q8—301no—
370#121—152no—
371#121—301no—
372#121—302no—
373#117—151no—
374C2×C4⋊C4—51no—
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
387D15—81no—
388C5×D3—42no—
389C23×C3C2×D2011yescharacteristic
390C22×D3—101no—
391C22×D3C2×D2012yes—
392C3×Q8—102no—
393C3×Q8—102no—
394C3×Q8—202no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4—202no—
398C2×C3×C4—401no—
399C2×C3×C4C2×D2011yescharacteristic
400C2×C3×C4#70912yes—
401C2×C3×C4C2×C22⋊D512yes—
402C2×C3×C4C23×D511yescharacteristic
403#67—24no—
404#67—202no—
405C2×Dic3—101no—
406C2×Dic3—101no—
407C2×Dic3—101no—
408C2×Dic3C2×D2012yes—
409D12—22no—
410D12—101no—
411D12—101no—
412D12—402no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—202no—
416Dic6—22no—
417Dic6—101no—
418Dic6—101no—
419C3×C8—21no—
420C3×C8—21no—
421C3⋊C8C2×C22⋊D514yes—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5—62no—
425C22×C5—62no—
426C22×C5—62no—
427C22×C5#88412yes—
428C22×C5C2×D3×D411yescharacteristic
429C2×D5—121no—
430C2×D5—121no—
431C2×D5—121no—
432C4×C5—62no—
433C4×C5#88412yes—
434C4×C5C2×#23511yescharacteristic
435C4×C5C2×D3×D411yescharacteristic
436Dic5—42no—
437Dic5—42no—
438Dic5—121no—
439Dic5—121no—
440Dic5—121no—
441#41—62no—
442#41—62no—
443#41—302no—
444#41—302no—
445C2×Q8—31no—
446C2×Q8—52no—
447C2×Q8—101no—
448C2×Q8—102no—
449C2×Q8—102no—
450C2×Q8—151no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—151no—
454C2×D4—302no—
455C2×D4—301no—
456C22×C4—32no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—301no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—304no—
462SD16—304no—
463M16—34no—
464M16—64no—
465C2×C8—22no—
466C2×C8—32no—
467C2×C8—64no—
468C2×C8C2×D3×D511yescharacteristic
469C2×C8C2×D3×D511yescharacteristic
470C4⋊C4—52no—
471C4⋊C4—101no—
472C4⋊C4—151no—
473C4⋊C4—151no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4—302no—
477C22⋊C4—302no—
478C22⋊C4—302no—
479C22⋊C4—302no—
480C22⋊C4—302no—
481C22⋊C4—301no—
482C42—301no—
483C3×C5#306311yescharacteristic, complemented, Hall
484C22×C3—21no—
485C22×C3—21no—
486C22×C3#407612yes—
487C22×C3C22×D2011yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—201no—
492C2×D3—201no—
493C2×D3—201no—
494C3×C4—202no—
495C3×C4—202no—
496C3×C4#407612yes—
497C3×C4C2×#70911yescharacteristic
498C3×C4C22×C22⋊D511yescharacteristic
499Dic3—22no—
500Dic3—201no—
501Dic3—201no—
502Dic3—201no—
503C2×C5—22no—
504C2×C5—122no—
505C2×C5C2×#88411yescharacteristic
506C2×C5#576211yescharacteristic
507C2×C5#576611yescharacteristic
508D5—241no—
509C23—32no—
510C23—301no—
511C23D3×D2011yescharacteristic
512Q8—62no—
513Q8—102no—
514Q8—102no—
515Q8—202no—
516Q8—301no—
517Q8—301no—
518D4—64no—
519D4—62no—
520D4—301no—
521D4—301no—
522D4—602no—
523D4—1202no—
524C2×C4—32no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—202no—
530C2×C4—301no—
531C2×C4—301no—
532C2×C4—301no—
533C2×C4—401no—
534C2×C4—602no—
535C2×C4#677112yes—
536C2×C4D3×D2011yescharacteristic
537C2×C4D3×C22⋊D512yes—
538C2×C4C22×D3×D511yescharacteristic
539C8—21no—
540C8—21no—
541C8—34no—
542C2×C3—22no—
543C2×C3#6608511yescharacteristic
544C2×C3#6608711yescharacteristic
545C2×C3C2×#407611yescharacteristic
546D3—42no—
547D3—401no—
548C5#7880011yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—62no—
552C22—62no—
553C22—62no—
554C22—601no—
555C22—601no—
556C22—601no—
557C22#9145412yes—
558C22C2×D3×D2011yescharacteristic
559C4—62no—
560C4—202no—
561C4—202no—
562C4—601no—
563C4—601no—
564C4—601no—
565C4#9145412yes—
566C4C2×#677111yescharacteristic
567C4C2×D3×C22⋊D511yescharacteristic
568C3#1061553911yescharacteristic, complemented, Sylow
569C2—22no—
570C2—122no—
571C2—1201no—
572C2#1175397011yescharacteristic
573C2#1175399111yescharacteristic
574C2C2×#9145411yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
prime radical#11754603C2
hypercenterC22×C8D3×D5
hypocenterC3×C5#3063
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×C5×#901C2semidirect
3. #11754255C2semidirect
4. C2×#91315C2semidirect
5. #11751214C2semidirect
6. C2×C3×#3948C2semidirect
7. #11748453C2semidirect
8. #11747949C2semidirect
9. #11747948C2semidirect
10. #11747947C2semidirect
11. #11747946C2semidirect
55. C2×#3948D3semidirect
83. C5×#208D4semidirect
100. C2×#901D5semidirect
315. #208C22⋊D5semidirect
483. C3×C5#3063semidirect
548. C5#78800semidirect
568. C3#10615539semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9012
#117546032
#117542552
C2×#913152
#117512142
C2×C3×#39482
#117484532
#117479492
#117479482
#117479472
#117479462
#106155393
#788005