Finite Group Databasebeta

← #14429251

Subgroups — #14429251

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
1#11756269C211yesmaximal, characteristic
2#11754278C211yesmaximal, characteristic, complemented
3C2×#91315C211yesmaximal, characteristic, complemented
4C2×#91290C211yesmaximal, characteristic, complemented
5C5×#4964C211yesmaximal, characteristic, complemented
6C2×C3×#3942C211yesmaximal, characteristic, complemented
7#11748393C211yesmaximal, characteristic, complemented
8#11747359C212yesmaximal, complemented
9#11747358C212yesmaximal, complemented
10#11747357C212yesmaximal, complemented
11#11747356C212yesmaximal, complemented
12#10610945—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×#6817C2211yescharacteristic
15C22×C3×Dic10C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17#91819—22no—
18#91786—21no—
19#91786C2212yes—
20#91784—21no—
21#91784C2212yes—
22C5×#926C2211yescharacteristic
23C2×C5×#208C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25#91546—22no—
26#91390C2212yes—
27#91390C2212yes—
28#91389—21no—
29#91374—21no—
30C2×#6662C2212yes—
31#91315—24no—
32C2×#6658C2212yes—
33C2×#6651C2212yes—
34#91290—24no—
35C2×#6648C2212yes—
36C5×#808C2212yes—
37C5×#807C2212yes—
38C3×#3942C2212yes—
39C3×#3942C2212yes—
40#90975C2211yescharacteristic
41#90975C2211yescharacteristic
42#90969C2211yescharacteristic
43#90969C2211yescharacteristic
44#78800—51nomaximal, Hall
45C2×#4055—31no—
46C2×#4053—31no—
47#65860—31no—
48#65844—31no—
49C5×#428—31no—
50#65291—31no—
51C2×#3942D311yescharacteristic, complemented
52#65176—32no—
53#65175—32no—
54#65157—32no—
55#65153—32no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—21no—
59C2×#1220D412yes—
60#6817—22no—
61#6817—22no—
62#6817—22no—
63#6817—22no—
64C2×D60—22no—
65C2×D60D411yescharacteristic
66C2×D60C2311yescharacteristic
67C2×C4×D15—21no—
68C2×C4×D15C2312yes—
69C2×Dic30D411yescharacteristic
70C2×Dic30C2311yescharacteristic
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#6714—22no—
79#6714—22no—
80C4⋊Dic15—22no—
81C5×#217—21no—
82C5×C4⋊Dic3C2311yescharacteristic
83C5×C4⋊Dic3C2311yescharacteristic
84C4×C5×Dic3—21no—
85C5×#208D414yescomplemented
86C2×C5×C3⋊C8C2312yes—
87C2×C3×#168—22no—
88C2×C3×#168C2311yescharacteristic
89C2×C3×#168C2311yescharacteristic
90Dic3⋊Dic5—22no—
91Dic3⋊Dic5—22no—
92#6667—22no—
93#6667—22no—
94#6662—24no—
95#6658—24no—
96#6651—24no—
97#6648—24no—
98#5816—51no—
99C2×#942—51no—
100C2×#901—51no—
101#5510—51no—
102C2×C3×#103—51no—
103#5113—51no—
104#4964D511yescharacteristic, complemented
105#4816—52no—
106#4813—52no—
107#4812—52no—
108#4807—52no—
109C2×#709—31no—
110C2×#709—31no—
111C22×Dic10C2×D311yescharacteristic
112C2×C5×M16—31no—
113C5×#117—31no—
114C2×C5×C4⋊C4C2×D311yescharacteristic
115#4076—62no—
116C22×#168C2×D311yescharacteristic
117#4055—64no—
118#4053—64no—
119C2×Dic20—32no—
120C2×#677—32no—
121C2×#676—32no—
122C2×C8×D5—32no—
123#4043—32no—
124#4043—32no—
125#4043—32no—
126#4043—61no—
127#4041—32no—
128#4041—61no—
129#4032—62no—
130#4021—61no—
131#4015—61no—
132C5×#107—32no—
133C5×#106—32no—
134#3942—31no—
135#3942—31no—
136#3942C2×D312yes—
137#3942C2×D312yes—
138#3941—31no—
139#3941—31no—
140#3063—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×D15—21no—
143C22×D15C2×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
150#1220—24no—
151#1220—42no—
152C2×Dic15—21no—
153C2×Dic15C2×D412yes—
154D60—22no—
155D60—21no—
156D60—21no—
157D60—82no—
158C4×D15—22no—
159C4×D15—22no—
160C4×D15—42no—
161Dic30—22no—
162Dic30—21no—
163Dic30—21no—
164C2×C5×Dic3—21no—
165C2×C5×Dic3—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—42no—
168C3×Dic10—22no—
169C3×Dic10—22no—
170C3×Dic10—42no—
171C3×#168—21no—
172C3×#168—21no—
173C5×C3⋊C8C2×D414yes—
174C22×C3×Q8—51no—
175C2×#235—51no—
176C2×#235—51no—
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—102no—
187#929—102no—
188#926C2×D511yescharacteristic
189C2×#208C2×D511yescharacteristic
190#901—104no—
191C2×ΓL(1,𝔽25)—52no—
192C2×D3×C8—52no—
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#808C2×D512yes—
204#807C2×D512yes—
205C22×C4×C5—21no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—32no—
208C2×C22⋊D5—61no—
209C22×Dic5—21no—
210#709—62no—
211#709—62no—
212#709—62no—
213#709—62no—
214C2×D20—31no—
215C2×D20—31no—
216C2×D20—62no—
217C2×C4×D5—32no—
218C2×C4×D5—61no—
219C2×Dic10—22no—
220C2×Dic10—22no—
221C2×Dic10—21no—
222C2×Dic10—31no—
223C2×Dic10—31no—
224C2×Dic10C22×D312yes—
225C5×M16—34no—
226C2×C5×C8—32no—
227C5×C4⋊C4—21no—
228C5×C4⋊C4—31no—
229C5×C4⋊C4—31no—
230C5×C4⋊C4C22×D312yes—
231C5×C22⋊C4—61no—
232C42×C5—61no—
233#685—62no—
234#685—62no—
235#685—62no—
236#685—62no—
237C4⋊Dic5—62no—
238C4⋊Dic5—62no—
239C4⋊Dic5—62no—
240C2×#168—22no—
241C2×#168C22×D311yescharacteristic
242C2×#168C22×D311yescharacteristic
243Dic20—64no—
244#677—64no—
245#676—64no—
246C8×D5—64no—
247C2×#137—151no—
248C2×#131—151no—
249#537—151no—
250#536—151no—
251#470—152no—
252#468—152no—
253#467—152no—
254#465—152no—
255#428—31no—
256#416—151no—
257C2×#103—51no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5#12112yes—
261C22×C3×C5C22×D411yescharacteristic
262C2×D15—22no—
263C2×D15—22no—
264C2×D15—22no—
265C2×D15—41no—
266C2×D15—41no—
267C2×D15—41no—
268C3×C4×C5—42no—
269C3×C4×C5#12112yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C2×#4111yescharacteristic
272Dic15—22no—
273Dic15—41no—
274C3×Dic5—42no—
275C5×Dic3—41no—
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—52no—
285C2×#67—101no—
286#235—102no—
287#235—102no—
288#235—102no—
289#235—102no—
290C2×D12—51no—
291C2×D12—51no—
292C2×D12—102no—
293C2×C4×D3—52no—
294C2×C4×D3—101no—
295C2×Dic6—51no—
296C2×Dic6—51no—
297C2×C3×C8—51no—
298C2×C3×C8—51no—
299C2×C3×C8—102no—
300C3×C4⋊C4—21no—
301C3×C4⋊C4C22×D512yes—
302#217—21no—
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—102no—
310C4⋊Dic3C22×D511yescharacteristic
311C4⋊Dic3C22×D511yescharacteristic
312Dic3⋊C4—102no—
313Dic3⋊C4—102no—
314C4×Dic3—21no—
315#208D2014yescomplemented
316C2×C3⋊C8C22×D512yes—
317ΓL(1,𝔽25)—104no—
318D3×C8—104no—
319C23×C5D3×D411yescharacteristic
320C22×D5—32no—
321C22×D5—61no—
322C2×C4×C5—22no—
323C2×C4×C5—61no—
324C2×C4×C5—61no—
325C2×C4×C5—81no—
326C2×C4×C5#23512yes—
327C2×C4×C5D3×D412yes—
328C2×C4×C5D3×D411yescharacteristic
329C2×C4×C5C23×D311yescharacteristic
330C22⋊D5—64no—
331C22⋊D5—122no—
332C2×Dic5—22no—
333C2×Dic5—32no—
334C2×Dic5—42no—
335C2×Dic5—61no—
336D20—62no—
337D20—61no—
338D20—61no—
339D20—242no—
340C4×D5—62no—
341C4×D5—62no—
342C4×D5—122no—
343Dic10—22no—
344Dic10—22no—
345Dic10—42no—
346Dic10—62no—
347Dic10—61no—
348Dic10—61no—
349C5×C8—34no—
350#168—21no—
351#168—21no—
352C2×#41—151no—
353C2×#41—151no—
354C22×Q8—51no—
355#137—304no—
356C2×Q16—152no—
357C2×SD16—152no—
358#131—304no—
359C2×M16—31no—
360C2×M16—152no—
361C22×C8—51no—
362C22×C8—152no—
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—31no—
374C2×C4⋊C4D3×D511yescharacteristic
375#107—32no—
376#106—32no—
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—42no—
388D15—81no—
389C23×C3D4×D511yescharacteristic
390C22×D3—52no—
391C22×D3—101no—
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×C4C2×D2012yes—
400C2×C3×C4D4×D511yescharacteristic
401C2×C3×C4#71312yes—
402C2×C3×C4C23×D511yescharacteristic
403#67—104no—
404#67—202no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—52no—
408C2×Dic3—101no—
409D12—101no—
410D12—101no—
411D12—102no—
412D12—402no—
413C4×D3—102no—
414C4×D3—102no—
415C4×D3—202no—
416Dic6—102no—
417Dic6—101no—
418Dic6—101no—
419C3×C8—101no—
420C3×C8—101no—
421C3⋊C8C2×D2014yes—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5#88412yes—
425C22×C5C2×D3×D411yescharacteristic
426C2×D5—62no—
427C2×D5—62no—
428C2×D5—62no—
429C2×D5—121no—
430C2×D5—121no—
431C2×D5—121no—
432C4×C5—42no—
433C4×C5—121no—
434C4×C5—121no—
435C4×C5#88412yes—
436C4×C5C2×#23511yescharacteristic
437C4×C5C2×D3×D411yescharacteristic
438Dic5—42no—
439Dic5—62no—
440Dic5—121no—
441#41—302no—
442#41—302no—
443#41—302no—
444#41—302no—
445C2×Q8—52no—
446C2×Q8—101no—
447C2×Q8—102no—
448C2×Q8—102no—
449C2×Q8—151no—
450C2×Q8—151no—
451C2×D4—152no—
452C2×D4—151no—
453C2×D4—151no—
454C2×D4—302no—
455C2×D4—301no—
456C22×C4—21no—
457C22×C4—101no—
458C22×C4—152no—
459C22×C4—301no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—304no—
462SD16—304no—
463M16—34no—
464M16—304no—
465C2×C8—32no—
466C2×C8—51no—
467C2×C8—51no—
468C2×C8—102no—
469C2×C8—304no—
470C4⋊C4—21no—
471C4⋊C4—31no—
472C4⋊C4—31no—
473C4⋊C4—302no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4C2×D3×D512yes—
477C22⋊C4—61no—
478C22⋊C4—302no—
479C22⋊C4—302no—
480C22⋊C4—302no—
481C22⋊C4—302no—
482C42—61no—
483C3×C5#306311yescharacteristic, complemented, Hall
484C22×C3—21no—
485C22×C3—21no—
486C22×C3#404112yes—
487C22×C3C2×D4×D511yescharacteristic
488C2×D3—102no—
489C2×D3—102no—
490C2×D3—102no—
491C2×D3—201no—
492C2×D3—201no—
493C2×D3—201no—
494C3×C4—42no—
495C3×C4—202no—
496C3×C4#404112yes—
497C3×C4C22×D2011yescharacteristic
498C3×C4C2×#71311yescharacteristic
499Dic3—41no—
500Dic3—41no—
501Dic3—102no—
502Dic3—201no—
503C2×C5—22no—
504C2×C5C2×#88411yescharacteristic
505C2×C5#576211yescharacteristic
506C2×C5#576611yescharacteristic
507D5—122no—
508D5—241no—
509C23—152no—
510C23—301no—
511C23#679011yescharacteristic
512Q8—102no—
513Q8—102no—
514Q8—202no—
515Q8—302no—
516Q8—301no—
517Q8—301no—
518D4—304no—
519D4—302no—
520D4—301no—
521D4—301no—
522D4—602no—
523D4—1202no—
524C2×C4—22no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—81no—
528C2×C4—102no—
529C2×C4—152no—
530C2×C4—202no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—301no—
534C2×C4—602no—
535C2×C4#677312yes—
536C2×C4D3×D2012yes—
537C2×C4#679011yescharacteristic
538C2×C4C22×D3×D511yescharacteristic
539C8—34no—
540C8—101no—
541C8—101no—
542C2×C3—22no—
543C2×C3C2×#404111yescharacteristic
544C2×C3#6610911yescharacteristic
545C2×C3#6611311yescharacteristic
546D3—202no—
547D3—401no—
548C5#7880011yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—302no—
552C22—302no—
553C22—302no—
554C22—601no—
555C22—601no—
556C22—601no—
557C22#9145912yes—
558C22C2×#679011yescharacteristic
559C4—42no—
560C4—121no—
561C4—121no—
562C4—202no—
563C4—302no—
564C4—601no—
565C4#9145912yes—
566C4C2×#677311yescharacteristic
567C4C2×D3×D2011yescharacteristic
568C3#1061094511yescharacteristic, complemented, Sylow
569C2—22no—
570C2—602no—
571C2—1201no—
572C2C2×#9145911yescharacteristic
573C2#1175472811yescharacteristic
574C2#1175512211yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radical#11756269C2
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. #11754278C2semidirect
3. C2×#91315C2semidirect
4. C2×#91290C2semidirect
5. C5×#4964C2semidirect
6. C2×C3×#3942C2semidirect
7. #11748393C2semidirect
8. #11747359C2semidirect
9. #11747358C2semidirect
10. #11747357C2semidirect
11. #11747356C2semidirect
51. C2×#3942D3semidirect
85. C5×#208D4semidirect
104. #4964D5semidirect
315. #208D20semidirect
483. C3×C5#3063semidirect
548. C5#78800semidirect
568. C3#10610945semidirect

Maximal subgroups

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

subgroupindex
#117562692
#117542782
C2×#913152
C2×#912902
C5×#49642
C2×C3×#39422
#117483932
#117473592
#117473582
#117473572
#117473562
#106109453
#788005