Finite Group Databasebeta

← #14429255

Subgroups — #14429255

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
1C3×#65844C211yesmaximal, characteristic
2#11754251C211yesmaximal, characteristic, complemented
3C2×#91307C211yesmaximal, characteristic, complemented
4C2×#91283C211yesmaximal, characteristic, complemented
5#11751065C211yesmaximal, characteristic, complemented
6C2×C5×#810C211yesmaximal, characteristic, complemented
7#11748365C211yesmaximal, characteristic, complemented
8#11747345C212yesmaximal, complemented
9#11747343C212yesmaximal, complemented
10#11747341C212yesmaximal, complemented
11#11747339C212yesmaximal, complemented
12#10610531—31nomaximal, Hall
13C22×C5×Dic6C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17#91815C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C22×C5×C3⋊C8C2211yescharacteristic
20C3×#4076—22no—
21C3×#4043—21no—
22C3×#4043C2212yes—
23C3×#4041—21no—
24C3×#4041C2212yes—
25#91536—22no—
26#91385—21no—
27#91368—21no—
28#91359C2212yes—
29#91359C2212yes—
30C2×#6661C2212yes—
31#91307—24no—
32C2×#6655C2212yes—
33C2×#6649C2212yes—
34#91283—24no—
35C2×D5×C3⋊C8C2212yes—
36#91092C2212yes—
37#91091C2212yes—
38C5×#810C2212yes—
39C5×#810C2212yes—
40#90971C2211yescharacteristic
41#90971C2211yescharacteristic
42#90965C2211yescharacteristic
43#90965C2211yescharacteristic
44#83423—51nomaximal, Hall
45#66150—31no—
46C2×#4088—31no—
47C2×#4047—31no—
48#65844D311yescharacteristic
49C2×C5×#103—31no—
50#65402—31no—
51#65253—31no—
52#65105—32no—
53#65102—32no—
54#65101—32no—
55#65096—32no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C5×Dic3—21no—
59C2×C5×Dic6—22no—
60C2×C5×Dic6—22no—
61C2×C5×Dic6—21no—
62C2×C5×Dic6C2312yes—
63C2×C3×C22⋊D5—21no—
64C2×C3×C22⋊D5D412yes—
65C3×#709—22no—
66C3×#709—22no—
67C3×#709—22no—
68C3×#709—22no—
69C2×C3×D20—22no—
70C2×C3×D20D411yescharacteristic
71C2×C3×D20C2311yescharacteristic
72C2×C3×C4×D5—21no—
73C2×C3×C4×D5C2312yes—
74C2×C3×Dic10D411yescharacteristic
75C2×C3×Dic10C2311yescharacteristic
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78#6719—21no—
79C4⋊Dic15C2311yescharacteristic
80C4⋊Dic15C2311yescharacteristic
81C4×Dic15—21no—
82#6710D414yescomplemented
83C2×C3⋊#168C2312yes—
84C2×C5×C3⋊C8—22no—
85C2×C5×C3⋊C8C2311yescharacteristic
86C2×C5×C3⋊C8C2311yescharacteristic
87C3×#685—22no—
88C3×#685—22no—
89C3×C4⋊Dic5—22no—
90#6669—22no—
91#6669—22no—
92#6665—22no—
93#6665—22no—
94#6661—24no—
95#6655—24no—
96#6649—24no—
97D5×C3⋊C8—24no—
98C3×#537—51no—
99#5825—51no—
100C2×#951—51no—
101C2×#948—51no—
102#5233—51no—
103#5052—52no—
104#5051—52no—
105#5042—52no—
106#5041—52no—
107C2×#810D511yescharacteristic, complemented
108#4964—51no—
109C22×C5×Q8—31no—
110C2×#709C2×D311yescharacteristic
111C2×#709C2×D311yescharacteristic
112C22×C5×C8—31no—
113C2×C5×C4⋊C4C2×D311yescharacteristic
114#4093—61no—
115#4092—32no—
116#4092—32no—
117#4090—61no—
118C2×#689—32no—
119#4088—64no—
120C2×#688—32no—
121#4076—22no—
122#4075—62no—
123#4072—31no—
124C2×#681—31no—
125#4047—64no—
126C2×#676—32no—
127C2×C8×D5—32no—
128#4043—21no—
129#4043C2×D312yes—
130#4041—21no—
131#4041C2×D312yes—
132C5×#103—32no—
133C5×#103—32no—
134#3953—31no—
135#3953—31no—
136#3948—31no—
137#3948—31no—
138#3940—32no—
139#3939—32no—
140#3063—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×C3×D5—21no—
143C22×C3×D5C2×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—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—42no—
154C5×Dic6—22no—
155C5×Dic6—22no—
156C5×Dic6—42no—
157C3×C22⋊D5—24no—
158C3×C22⋊D5—42no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5C2×D412yes—
161C3×D20—22no—
162C3×D20—21no—
163C3×D20—21no—
164C3×D20—82no—
165C3×C4×D5—22no—
166C3×C4×D5—22no—
167C3×C4×D5—42no—
168C3×Dic10—22no—
169C3×Dic10—21no—
170C3×Dic10—21no—
171C3⋊#168C2×D414yes—
172C5×C3⋊C8—21no—
173C5×C3⋊C8—21no—
174C2×C3×#41—51no—
175C2×C3×#41—51no—
176C22×Dic6C2×D511yescharacteristic
177C3×C22⋊Q8—52no—
178C3×C22⋊Q8—101no—
179C3×#121—52no—
180C3×#121—101no—
181C3×#121—102no—
182C2×C3×C4⋊C4C2×D511yescharacteristic
183#951—104no—
184#948—104no—
185#944—101no—
186C2×#216—52no—
187#936—101no—
188#935—102no—
189C2×#214—52no—
190#926—51no—
191#924—52no—
192#924—52no—
193C2×#208—51no—
194C2×#208—52no—
195C22×C3⋊C8—52no—
196C22×C3⋊C8C2×D511yescharacteristic
197#835—51no—
198#835—51no—
199#832—51no—
200#832—51no—
201#810C2×D512yes—
202#810C2×D512yes—
203#808—52no—
204#807—52no—
205C2×C5×Q8—32no—
206C2×C5×Q8—61no—
207C2×C5×Q8—62no—
208C2×C5×Q8—62no—
209C22×C4×C5—21no—
210C22×C4×C5—61no—
211C22×C4×C5C22×D311yescharacteristic
212C2×C22⋊D5—21no—
213C2×C22⋊D5#6712yes—
214#709—22no—
215#709—22no—
216#709—22no—
217#709—22no—
218C2×D20—22no—
219C2×D20#6711yescharacteristic
220C2×D20C22×D311yescharacteristic
221C2×C4×D5—21no—
222C2×C4×D5C22×D312yes—
223C2×Dic10#6711yescharacteristic
224C2×Dic10C22×D311yescharacteristic
225C2×C5×C8—31no—
226C2×C5×C8—31no—
227C2×C5×C8—62no—
228C5×C4⋊C4—21no—
229C5×C4⋊C4C22×D312yes—
230C22⋊Dic5—61no—
231#689—64no—
232#688—64no—
233#685—22no—
234#685—22no—
235#685—62no—
236#685—62no—
237C4⋊Dic5—22no—
238C4⋊Dic5—31no—
239C4⋊Dic5—31no—
240#683—62no—
241#683—62no—
242C4×Dic5—61no—
243#681—34no—
244C2×#168—32no—
245#676—64no—
246C8×D5—64no—
247C2×#137—151no—
248C2×#131—151no—
249#537—51no—
250#536—151no—
251#470—152no—
252#468—152no—
253#467—152no—
254#465—152no—
255#428—151no—
256#416—151no—
257C2×#103—31no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5#12112yes—
261C22×C3×C5C22×D411yescharacteristic
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5—22no—
265C2×C3×D5—41no—
266C2×C3×D5—41no—
267C2×C3×D5—41no—
268C3×C4×C5—42no—
269C3×C4×C5#12112yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C2×#4111yescharacteristic
272Dic15—41no—
273Dic15—41no—
274C3×Dic5—22no—
275C3×Dic5—41no—
276C5×Dic3—42no—
277C3×#41—102no—
278C3×#41—102no—
279C3×#41—102no—
280C3×#41—102no—
281C2×C3×Q8—51no—
282C2×C3×Q8—51no—
283C2×C3×D4—52no—
284C2×C3×D4—51no—
285C2×C3×D4—51no—
286C2×C3×D4—102no—
287C2×C3×D4—101no—
288C22×C3×C4—21no—
289C22×C3×C4—52no—
290C22×C3×C4—101no—
291C22×C3×C4C22×D511yescharacteristic
292C22×Dic3—21no—
293C2×Dic6—22no—
294C2×Dic6—22no—
295C2×Dic6—21no—
296C2×Dic6C22×D512yes—
297C3×C4⋊C4—21no—
298C3×C4⋊C4—102no—
299C3×C4⋊C4C22×D512yes—
300C3×C22⋊C4—102no—
301C3×C22⋊C4—102no—
302#217—102no—
303#217—102no—
304#217—101no—
305#216—104no—
306#214—104no—
307C4⋊Dic3—51no—
308C4⋊Dic3—51no—
309Dic3⋊C4—102no—
310Dic3⋊C4—102no—
311C4×Dic3—101no—
312#208—54no—
313#208—104no—
314C2×C3⋊C8—22no—
315C2×C3⋊C8—52no—
316C2×C3⋊C8—104no—
317C2×C3⋊C8C22×D511yescharacteristic
318C2×C3⋊C8C22×D511yescharacteristic
319C23×C5C2×#6711yescharacteristic
320C22×D5—21no—
321C22×D5C2×#6712yes—
322C5×Q8—62no—
323C5×Q8—62no—
324C5×Q8—122no—
325C2×C4×C5—22no—
326C2×C4×C5—62no—
327C2×C4×C5—81no—
328C2×C4×C5—122no—
329C2×C4×C5D3×D412yes—
330C2×C4×C5#23712yes—
331C2×C4×C5C2×#6711yescharacteristic
332C2×C4×C5C23×D311yescharacteristic
333C22⋊D5—24no—
334C22⋊D5—42no—
335C2×Dic5—21no—
336C2×Dic5—61no—
337C2×Dic5—61no—
338C2×Dic5C2×#6712yes—
339D20—22no—
340D20—21no—
341D20—21no—
342D20—82no—
343C4×D5—22no—
344C4×D5—22no—
345C4×D5—42no—
346Dic10—22no—
347Dic10—21no—
348Dic10—21no—
349C5×C8—61no—
350C5×C8—61no—
351#168—34no—
352C2×#41—51no—
353C2×#41—51no—
354C22×Q8—31no—
355#137—304no—
356C2×Q16—152no—
357C2×SD16—152no—
358#131—304no—
359C2×M16—152no—
360C2×M16—151no—
361C22×C8—31no—
362C22×C8—152no—
363C4⋊Q8—301no—
364#124—301no—
365#123—302no—
366C22⋊Q8—52no—
367C22⋊Q8—101no—
368C22⋊Q8—152no—
369C22⋊Q8—152no—
370#121—52no—
371#121—101no—
372#121—102no—
373#117—151no—
374C2×C4⋊C4D3×D511yescharacteristic
375#107—152no—
376#106—152no—
377#103—32no—
378#103—32no—
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
387C3×D5—42no—
388C3×D5—81no—
389C23×C3—52no—
390C23×C3—101no—
391C23×C3D4×D511yescharacteristic
392C3×Q8—102no—
393C3×Q8—101no—
394C3×Q8—101no—
395C3×D4—104no—
396C3×D4—101no—
397C3×D4—101no—
398C3×D4—102no—
399C3×D4—202no—
400C3×D4—402no—
401C2×C3×C4—22no—
402C2×C3×C4—52no—
403C2×C3×C4—81no—
404C2×C3×C4—102no—
405C2×C3×C4—102no—
406C2×C3×C4—101no—
407C2×C3×C4—202no—
408C2×C3×C4#70912yes—
409C2×C3×C4D4×D512yes—
410C2×C3×C4D4×D511yescharacteristic
411C2×C3×C4C23×D511yescharacteristic
412C2×Dic3—22no—
413C2×Dic3—42no—
414C2×Dic3—101no—
415C2×Dic3—101no—
416Dic6—22no—
417Dic6—22no—
418Dic6—42no—
419C3⋊C8—21no—
420C3⋊C8—21no—
421C3⋊C8—54no—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5#93912yes—
425C22×C5C22×#6711yescharacteristic
426C2×D5—22no—
427C2×D5—22no—
428C2×D5—22no—
429C2×D5—41no—
430C2×D5—41no—
431C2×D5—41no—
432C4×C5—42no—
433C4×C5—122no—
434C4×C5#93912yes—
435C4×C5C2×D3×D411yescharacteristic
436C4×C5C2×#23711yescharacteristic
437Dic5—22no—
438Dic5—41no—
439Dic5—121no—
440Dic5—121no—
441#41—102no—
442#41—102no—
443#41—102no—
444#41—102no—
445C2×Q8—32no—
446C2×Q8—51no—
447C2×Q8—51no—
448C2×Q8—61no—
449C2×Q8—62no—
450C2×Q8—62no—
451C2×D4—52no—
452C2×D4—51no—
453C2×D4—51no—
454C2×D4—102no—
455C2×D4—101no—
456C22×C4—21no—
457C22×C4—52no—
458C22×C4—61no—
459C22×C4—101no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—304no—
462SD16—304no—
463M16—154no—
464M16—304no—
465C2×C8—31no—
466C2×C8—31no—
467C2×C8—62no—
468C2×C8—152no—
469C2×C8—304no—
470C4⋊C4—21no—
471C4⋊C4—102no—
472C4⋊C4—151no—
473C4⋊C4—151no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4C2×D3×D512yes—
477C22⋊C4—102no—
478C22⋊C4—102no—
479C22⋊C4—302no—
480C22⋊C4—302no—
481C22⋊C4—301no—
482C42—301no—
483C3×C5#306311yescharacteristic, complemented, Hall
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—102no—
487C22×C3—102no—
488C22×C3—102no—
489C22×C3—201no—
490C22×C3—201no—
491C22×C3—201no—
492C22×C3#403012yes—
493C22×C3C2×D4×D511yescharacteristic
494C3×C4—42no—
495C3×C4—102no—
496C3×C4—201no—
497C3×C4#403012yes—
498C3×C4C2×#70911yescharacteristic
499C3×C4C2×D4×D511yescharacteristic
500Dic3—42no—
501Dic3—201no—
502Dic3—201no—
503C2×C5—22no—
504C2×C5C2×#93911yescharacteristic
505C2×C5#583611yescharacteristic
506C2×C5#583811yescharacteristic
507D5—42no—
508D5—81no—
509C23—52no—
510C23—101no—
511C23D5×#6711yescharacteristic
512Q8—62no—
513Q8—62no—
514Q8—102no—
515Q8—101no—
516Q8—101no—
517Q8—122no—
518D4—104no—
519D4—102no—
520D4—101no—
521D4—101no—
522D4—202no—
523D4—402no—
524C2×C4—22no—
525C2×C4—52no—
526C2×C4—62no—
527C2×C4—81no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—101no—
531C2×C4—122no—
532C2×C4—202no—
533C2×C4—301no—
534C2×C4—301no—
535C2×C4#676512yes—
536C2×C4#677712yes—
537C2×C4D5×#6711yescharacteristic
538C2×C4C22×D3×D511yescharacteristic
539C8—61no—
540C8—61no—
541C8—154no—
542C2×C3—22no—
543C2×C3—202no—
544C2×C3—401no—
545C2×C3C2×#403011yescharacteristic
546C2×C3#6609611yescharacteristic
547C2×C3#6610011yescharacteristic
548C5#8342311yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—102no—
552C22—102no—
553C22—102no—
554C22—201no—
555C22—201no—
556C22—201no—
557C22#9140812yes—
558C22C2×D5×#6711yescharacteristic
559C4—42no—
560C4—102no—
561C4—122no—
562C4—201no—
563C4—601no—
564C4—601no—
565C4#9140812yes—
566C4C2×#676511yescharacteristic
567C4C2×#677711yescharacteristic
568C3#1061053111yescharacteristic, complemented, Sylow
569C2—22no—
570C2—202no—
571C2—401no—
572C2C2×#9140811yescharacteristic
573C2#1175470711yescharacteristic
574C2#1175510111yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC3×#65844C2
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. #11754251C2semidirect
3. C2×#91307C2semidirect
4. C2×#91283C2semidirect
5. #11751065C2semidirect
6. C2×C5×#810C2semidirect
7. #11748365C2semidirect
8. #11747345C2semidirect
9. #11747343C2semidirect
10. #11747341C2semidirect
11. #11747339C2semidirect
82. #6710D4semidirect
107. C2×#810D5semidirect
483. C3×C5#3063semidirect
548. C5#83423semidirect
568. C3#10610531semidirect

Maximal subgroups

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

subgroupindex
C3×#658442
#117542512
C2×#913072
C2×#912832
#117510652
C2×C5×#8102
#117483652
#117473452
#117473432
#117473412
#117473392
#106105313
#834235