Finite Group Databasebeta

← #14424308

Subgroups — #14424308

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92095C211yesmaximal, characteristic
2C22×C3×C5×SD16C211yesmaximal, characteristic, complemented
3#11756575C211yesmaximal, characteristic, complemented
4C2×#91831C211yesmaximal, characteristic, complemented
5#11751316C211yesmaximal, characteristic, complemented
6#11751274C214yesmaximal, complemented
7#11751270C214yesmaximal, complemented
8#11751214C211yesmaximal, characteristic, complemented
9#11751211C211yesmaximal, characteristic, complemented
10#10615590—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic, complemented
12C22×C3×C5×D4C2211yescharacteristic
13C22×#1220—22no—
14#92095—24no—
15#92095—24no—
16C2×#6819—21no—
17C2×#6819C2212yes—
18C2×D4×D15—21no—
19C2×D4×D15C2212yes—
20C2×#6817C2211yescharacteristic
21C2×#6817C2211yescharacteristic
22C2×C3×C5×SD16—24no—
23C2×C3×C5×SD16C2214yescomplemented
24C22×C3×C5×C8C2211yescharacteristic, complemented
25#91836—21no—
26#91835C2214yescomplemented
27#91833—21no—
28C2×#6718C2212yescomplemented
29#91831—24no—
30C2×#6717C2212yescomplemented
31#91818—22no—
32#91815C2211yescharacteristic, complemented
33C2×#6710C2211yescharacteristic, complemented
34#91116C2214yescomplemented
35#91105C2212yescomplemented
36#91104C2214yescomplemented
37#91100C2212yescomplemented
38#83859—51nomaximal, Hall
39C2×#4144—31no—
40C22×C5×SD16D311yescharacteristic, complemented
41#66150—31no—
42C2×#4088—31no—
43#65504—31no—
44#65462—34no—
45#65458—34no—
46#65402—31no—
47#65399—31no—
48C24×C3×C5—21no—
49C23×D15—22no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8C2312yes—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4D414yes—
57C2×C3×C5×D4C2312yes—
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—216no—
61C2×#1220—21no—
62C2×#1220—22no—
63C2×#1220—22no—
64C2×#1220D412yes—
65C22×Dic15—22no—
66#6819—28no—
67#6819—24no—
68#6819—24no—
69D4×D15—28no—
70D4×D15—24no—
71D4×D15—24no—
72#6817—22no—
73#6817—22no—
74#6817—22no—
75#6817D414yes—
76C2×D60D411yescharacteristic
77C2×D60C2311yescharacteristic
78C2×C4×D15—21no—
79C2×C4×D15C2312yes—
80C2×Dic30D411yescharacteristic
81C2×Dic30C2311yescharacteristic
82C3×C5×SD16—28no—
83C2×C3×C5×C8—22no—
84C2×C3×C5×C8C2312yes—
85#6719—21no—
86#6718—24no—
87#6717—24no—
88#6714—24no—
89C4⋊Dic15C2312yes—
90#6712—24no—
91C4×Dic15—21no—
92#6710—22no—
93C2×C3⋊#168C2312yes—
94C2×#1004—51no—
95C22×C3×SD16D511yescharacteristic, complemented
96#5816—51no—
97C2×#942—51no—
98#5215—51no—
99#5173—54no—
100#5169—54no—
101#5113—51no—
102#5110—51no—
103C22×C5×Q8C2×D311yescharacteristic, complemented
104C22×C5×D4C2×D311yescharacteristic
105C22×C22⋊D5—62no—
106#4144—64no—
107#4144—64no—
108C2×#711—32no—
109C2×#711—61no—
110C2×D4×D5—32no—
111C2×D4×D5—61no—
112C2×#709—31no—
113C2×#709—31no—
114C2×C5×SD16—24no—
115C2×C5×SD16C2×D314yescomplemented
116C22×C5×C8C2×D311yescharacteristic, complemented
117#4093—61no—
118#4092—34no—
119#4090—61no—
120C2×#689—32no—
121#4088—64no—
122C2×#688—32no—
123#4075—62no—
124#4072—31no—
125C2×#681—31no—
126#3964—34no—
127#3953—32no—
128#3952—34no—
129#3948—32no—
130#3009—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—21no—
136C22×D15—28no—
137C22×D15—84no—
138C22×D15C2×D412yes—
139C3×C5×Q8—22no—
140C3×C5×Q8—22no—
141C3×C5×Q8—42no—
142C3×C5×D4—24no—
143C3×C5×D4—22no—
144C3×C5×D4C2×D414yes—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—42no—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2411yescharacteristic
151#1220—28no—
152#1220—28no—
153#1220—24no—
154#1220—24no—
155C2×Dic15—28no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15C2×D412yes—
159D60—22no—
160D60—21no—
161D60C2×D412yes—
162C4×D15—24no—
163C4×D15—22no—
164C4×D15—22no—
165Dic30—22no—
166Dic30—21no—
167Dic30C2×D412yes—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C22×C3×Q8C2×D511yescharacteristic, complemented
171C22×C3×D4C2×D511yescharacteristic
172C22×#67—102no—
173#1004—104no—
174#1004—104no—
175C2×#237—52no—
176C2×#237—101no—
177C2×D3×D4—52no—
178C2×D3×D4—101no—
179C2×#235—51no—
180C2×#235—51no—
181C2×C3×SD16—24no—
182C2×C3×SD16C2×D514yescomplemented
183C22×C3×C8C2×D511yescharacteristic, complemented
184#947—101no—
185#946—54no—
186#944—101no—
187C2×#216—52no—
188#942—104no—
189C2×#215—52no—
190#929—102no—
191#926—51no—
192C2×#208—51no—
193#832—54no—
194#821—52no—
195(C2×D3)⋊C8—54no—
196#816—52no—
197C24×C5—21no—
198C23×D5—62no—
199C2×C5×Q8—21no—
200C2×C5×Q8—22no—
201C2×C5×Q8—22no—
202C2×C5×Q8C22×D312yes—
203C2×C5×D4—21no—
204C2×C5×D4—22no—
205C2×C5×D4#6714yes—
206C2×C5×D4C22×D312yes—
207C22×C4×C5—21no—
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—32no—
210C2×C22⋊D5—62no—
211C2×C22⋊D5—62no—
212C2×C22⋊D5—616no—
213C2×C22⋊D5—61no—
214C22×Dic5—62no—
215#711—64no—
216#711—68no—
217#711—64no—
218D4×D5—64no—
219D4×D5—68no—
220D4×D5—64no—
221#709—34no—
222#709—62no—
223#709—62no—
224#709—62no—
225C2×D20—31no—
226C2×D20—31no—
227C2×C4×D5—32no—
228C2×C4×D5—61no—
229C2×Dic10—31no—
230C2×Dic10—31no—
231C5×SD16—28no—
232C2×C5×C8—22no—
233C2×C5×C8C22×D312yes—
234C22⋊Dic5—61no—
235#689—64no—
236#688—64no—
237#685—64no—
238C4⋊Dic5—32no—
239#683—64no—
240C4×Dic5—61no—
241#681—62no—
242C2×#168—32no—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16D1511yescharacteristic, complemented
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—44no—
255C22×C3×C5—84no—
256C22×C3×C5#12012yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×D15—22no—
259C2×D15—22no—
260C2×D15—24no—
261C2×D15—42no—
262C2×D15—88no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268Dic15—22no—
269Dic15—42no—
270C24×C3—21no—
271C23×D3—102no—
272C2×C3×Q8—21no—
273C2×C3×Q8—22no—
274C2×C3×Q8—22no—
275C2×C3×Q8C22×D512yes—
276C2×C3×D4—21no—
277C2×C3×D4—22no—
278C2×C3×D4C22⋊D514yes—
279C2×C3×D4C22×D512yes—
280C22×C3×C4—21no—
281C22×C3×C4C22×D511yescharacteristic
282C2×#67—52no—
283C2×#67—102no—
284C2×#67—102no—
285C2×#67—1016no—
286C2×#67—101no—
287C22×Dic3—102no—
288#237—104no—
289#237—108no—
290#237—104no—
291D3×D4—104no—
292D3×D4—108no—
293D3×D4—104no—
294#235—54no—
295#235—102no—
296#235—102no—
297#235—102no—
298C2×D12—51no—
299C2×D12—51no—
300C2×C4×D3—52no—
301C2×C4×D3—101no—
302C2×Dic6—51no—
303C2×Dic6—51no—
304C3×SD16—28no—
305C2×C3×C8—22no—
306C2×C3×C8C22×D512yes—
307#217—101no—
308#216—104no—
309#215—104no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—52no—
312Dic3⋊C4—104no—
313C4×Dic3—101no—
314#208—102no—
315C2×C3⋊C8—52no—
316C23×C5—22no—
317C23×C5—24no—
318C23×C5—82no—
319C23×C5D3×D411yescharacteristic
320C22×D5—32no—
321C22×D5—61no—
322C22×D5—68no—
323C22×D5—244no—
324C5×Q8—22no—
325C5×Q8—22no—
326C5×Q8—42no—
327C5×D4—24no—
328C5×D4—22no—
329C5×D4C2×#6714yes—
330C2×C4×C5—22no—
331C2×C4×C5—42no—
332C2×C4×C5D3×D411yescharacteristic
333C2×C4×C5D3×D412yes—
334C2×C4×C5C2×#6712yes—
335C2×C4×C5C23×D311yescharacteristic
336C22⋊D5—68no—
337C22⋊D5—68no—
338C22⋊D5—64no—
339C22⋊D5—64no—
340C2×Dic5—32no—
341C2×Dic5—68no—
342C2×Dic5—61no—
343C2×Dic5—62no—
344D20—32no—
345D20—62no—
346D20—61no—
347C4×D5—64no—
348C4×D5—62no—
349C4×D5—62no—
350Dic10—32no—
351Dic10—62no—
352Dic10—61no—
353C5×C8—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—151no—
360C2×#41—301no—
361C22×Q8C2×D1511yescharacteristic, complemented
362C22×D4—152no—
363C22×D4—301no—
364C22×D4—302no—
365C22×D4C2×D1511yescharacteristic
366#137—304no—
367C2×Q16—152no—
368C2×SD16—24no—
369C2×SD16—152no—
370C2×SD16C2×D1514yescomplemented
371C2×M16—151no—
372C22×C8C2×D1511yescharacteristic, complemented
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—151no—
378#103—152no—
379#102—154no—
380#102—152no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—44no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—42no—
387D15—42no—
388C23×C3—22no—
389C23×C3—24no—
390C23×C3—82no—
391C23×C3D4×D511yescharacteristic
392C22×D3—52no—
393C22×D3—101no—
394C22×D3—108no—
395C22×D3—404no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—42no—
399C3×D4—22no—
400C3×D4—24no—
401C3×D4C2×C22⋊D514yes—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4D4×D512yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4C2×C22⋊D512yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—108no—
409#67—108no—
410#67—104no—
411#67—104no—
412C2×Dic3—52no—
413C2×Dic3—108no—
414C2×Dic3—101no—
415C2×Dic3—102no—
416D12—52no—
417D12—101no—
418D12—102no—
419C4×D3—102no—
420C4×D3—102no—
421C4×D3—104no—
422Dic6—52no—
423Dic6—101no—
424Dic6—102no—
425C3×C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—44no—
430C22×C5—84no—
431C22×C5#93712yes—
432C22×C5C2×D3×D411yescharacteristic
433C2×D5—64no—
434C2×D5—62no—
435C2×D5—62no—
436C2×D5—122no—
437C2×D5—248no—
438C4×C5—42no—
439C4×C5#93712yes—
440C4×C5C2×D3×D411yescharacteristic
441C4×C5C22×#6711yescharacteristic
442Dic5—62no—
443Dic5—62no—
444Dic5—122no—
445C24—21no—
446C24—302no—
447#41—154no—
448#41—304no—
449#41—308no—
450#41—304no—
451#41—302no—
452#41—302no—
453#41—302no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—151no—
458C2×Q8—151no—
459C2×Q8C22×D1512yes—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—152no—
463C2×D4—151no—
464C2×D4—151no—
465C2×D4—302no—
466C2×D4—304no—
467C2×D4—308no—
468C2×D4—302no—
469C2×D4—3016no—
470C2×D4—301no—
471C2×D4—304no—
472C2×D4#122014yes—
473C2×D4C22×D1512yes—
474C22×C4—21no—
475C22×C4—152no—
476C22×C4—302no—
477C22×C4—301no—
478C22×C4C22×D1511yescharacteristic
479Q16—304no—
480SD16—28no—
481SD16—304no—
482M16—302no—
483C2×C8—22no—
484C2×C8—152no—
485C2×C8C22×D1512yes—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—304no—
489C22⋊C4—301no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—84no—
496C22×C3#408312yes—
497C22×C3C2×D4×D511yescharacteristic
498C2×D3—104no—
499C2×D3—102no—
500C2×D3—102no—
501C2×D3—202no—
502C2×D3—408no—
503C3×C4—42no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C22×C22⋊D511yescharacteristic
507Dic3—102no—
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—44no—
512C2×C5#576212yes—
513C2×C5C2×#93711yescharacteristic
514D5—122no—
515D5—122no—
516C23—22no—
517C23—24no—
518C23—82no—
519C23—152no—
520C23—308no—
521C23—301no—
522C23—1204no—
523C23D4×D1511yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—42no—
527Q8—152no—
528Q8—302no—
529Q8—301no—
530D4—24no—
531D4—22no—
532D4—152no—
533D4—308no—
534D4—308no—
535D4—304no—
536D4—304no—
537D4—302no—
538D4—301no—
539D4C2×#122014yes—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—152no—
543C2×C4—308no—
544C2×C4—301no—
545C2×C4—304no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4D4×D1511yescharacteristic
550C2×C4D4×D1512yes—
551C2×C4C2×#122012yes—
552C2×C4C23×D1511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—22no—
556C2×C3—44no—
557C2×C3#6609612yes—
558C2×C3C2×#408311yescharacteristic
559D3—202no—
560D3—202no—
561C5#8385911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—44no—
565C22—84no—
566C22—302no—
567C22—302no—
568C22—304no—
569C22—602no—
570C22—1208no—
571C22#9182612yes—
572C22C2×D4×D1511yescharacteristic
573C4—42no—
574C4—302no—
575C4—302no—
576C4—602no—
577C4#9182612yes—
578C4C2×D4×D1511yescharacteristic
579C4C22×#122011yescharacteristic
580C3#1061559011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—44no—
583C2—602no—
584C2—602no—
585C2#1175652112yes—
586C2C2×#9182611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC22×C3×C5×SD16C2
socleC22×C3×C5C22×D4
prime radicalC2×#92095C2
hypercenterC22×SD16D15
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×SD16C2semidirect
3. #11756575C2semidirect
4. C2×#91831C2semidirect
5. #11751316C2semidirect
6. #11751274C2semidirect
7. #11751270C2semidirect
8. #11751214C2semidirect
9. #11751211C2semidirect
11. C22×C3×C5×Q8C22semidirect
23. C2×C3×C5×SD16C22semidirect
24. C22×C3×C5×C8C22semidirect
26. #91835C22semidirect
28. C2×#6718C22semidirect
30. C2×#6717C22semidirect
32. #91815C22semidirect
33. C2×#6710C22semidirect
34. #91116C22semidirect
35. #91105C22semidirect
36. #91104C22semidirect
37. #91100C22semidirect
40. C22×C5×SD16D3semidirect
95. C22×C3×SD16D5semidirect
103. C22×C5×Q8C2×D3semidirect
115. C2×C5×SD16C2×D3semidirect
116. C22×C5×C8C2×D3semidirect
170. C22×C3×Q8C2×D5semidirect
182. C2×C3×SD16C2×D5semidirect
183. C22×C3×C8C2×D5semidirect
245. C22×SD16D15semidirect
361. C22×Q8C2×D15semidirect
370. C2×SD16C2×D15semidirect
372. C22×C8C2×D15semidirect
491. C3×C5#3009semidirect
561. C5#83859semidirect
580. C3#10615590semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C22×C3×C5×SD162
#117565752
C2×#918312
#117513162
#117512742
#117512702
#117512142
#117512112
#106155903
#838595