Finite Group Databasebeta

← #14424382

Subgroups — #14424382

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×#92017C211yesmaximal, characteristic
2C22×C3×#677C211yesmaximal, characteristic, complemented
3#11754278C211yesmaximal, characteristic, complemented
4C2×#91309C211yesmaximal, characteristic, complemented
5#11751211C211yesmaximal, characteristic, complemented
6C5×#5113C211yesmaximal, characteristic, complemented
7#11748366C211yesmaximal, characteristic, complemented
8#11748148C214yesmaximal, complemented
9#11748111C214yesmaximal, complemented
10#10610566—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C22×C3×D20C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic, complemented
15C2×D5×#67—22no—
16#92017—24no—
17#92017—24no—
18C2×#6777C2212yes—
19C2×D3×D20—21no—
20C2×#6766C2212yes—
21C2×#6765—21no—
22C22×C3×C5×C8C2211yescharacteristic, complemented
23C2×#6710C2211yescharacteristic, complemented
24C5×#926C2211yescharacteristic, complemented
25C2×C3×#677—24no—
26C2×C3×#677C2214yescomplemented
27#91546—22no—
28#91390C2214yescomplemented
29#91389—21no—
30#91374—21no—
31C2×#6661C2212yescomplemented
32#91309—24no—
33C2×#6656C2212yescomplemented
34#91104C2214yescomplemented
35C5×#821C2212yescomplemented
36C5×#816C2212yescomplemented
37#90964C2214yescomplemented
38#83859—51nomaximal, Hall
39C2×#4149—31no—
40C2×#4097—31no—
41C22×#677D311yescharacteristic, complemented
42#65860—31no—
43C5×#416—31no—
44#65399—31no—
45#65290—31no—
46#65099—34no—
47#65065—34no—
48C23×C3×D5—21no—
49C22×D3×D5—22no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220D412yes—
52#6817—22no—
53#6817—22no—
54C2×D60D411yescharacteristic
55C2×C4×D15C2312yes—
56C2×Dic30D411yescharacteristic
57C2×C5×#67—21no—
58C5×#235—22no—
59C5×#235D414yes—
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C5×Dic6C2311yescharacteristic
63C22×C3×Dic5—21no—
64C2×C3×D20—21no—
65C2×C3×D20—22no—
66C2×C3×D20D414yes—
67C2×C3×D20C2312yes—
68C2×C3×Dic10—22no—
69C2×C3×Dic10—22no—
70C2×C3×Dic10—21no—
71C2×C3×Dic10C2312yes—
72D5×#67—216no—
73C2×#1202—22no—
74C2×#1201—22no—
75C2×D5×Dic3—22no—
76#6777—24no—
77#6777—24no—
78D3×D20—28no—
79#6766—24no—
80#6766—24no—
81#6765—28no—
82C2×C3×C5×C8—22no—
83C2×C3×C5×C8C2312yes—
84#6710—22no—
85C2×C3⋊#168C2312yes—
86C5×#217—21no—
87C5×C4⋊Dic3C2312yes—
88C4×C5×Dic3—21no—
89C3×#677—28no—
90Dic3⋊Dic5—24no—
91#6667—24no—
92#6661—24no—
93#6656—24no—
94C2×#1004—51no—
95C22×C3×SD16—51no—
96#5816—51no—
97C2×#942—51no—
98#5215—51no—
99#5173—54no—
100#5169—54no—
101#5113D511yescharacteristic, complemented
102#5110—51no—
103C2×C5×#41—31no—
104#4149—64no—
105#4149—64no—
106C2×#713—32no—
107C2×D4×D5—32no—
108C2×D4×D5—62no—
109C2×#709—31no—
110C2×#709—61no—
111C22×D20—61no—
112C22×D20C2×D311yescharacteristic
113C22×Dic10C2×D311yescharacteristic, complemented
114C22×C5×C8C2×D311yescharacteristic, complemented
115C5×#117—31no—
116#4097—64no—
117C2×#689—32no—
118C2×#687—32no—
119C2×#681—31no—
120C2×#677—24no—
121C2×#677C2×D314yescomplemented
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#103—32no—
127C5×#102—32no—
128#3952—34no—
129#3941—34no—
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C22×C3×D5—22no—
135C22×C3×D5—24no—
136C22×C3×D5—82no—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—24no—
144C2×Dic15C2×D412yes—
145D60—22no—
146C4×D15—22no—
147C4×D15—22no—
148Dic30—22no—
149C5×#67—24no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C5×D12—21no—
153C5×D12C2×D412yes—
154C4×C5×D3—24no—
155C5×Dic6—21no—
156C5×Dic6C2×D412yes—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—42no—
159C3×D20—24no—
160C3×D20—22no—
161C3×D20C2×D414yes—
162C3×Dic10—22no—
163C3×Dic10—22no—
164C3×Dic10—42no—
165#1202—28no—
166#1201—28no—
167D5×Dic3—28no—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C22×C3×Q8—51no—
171C22×C3×D4—51no—
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×#235C2×D511yescharacteristic
181C2×C3×SD16—54no—
182C2×C3×SD16—104no—
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#926C2×D511yescharacteristic, complemented
192C2×#208—51no—
193#832—54no—
194#821C2×D512yescomplemented
195(C2×D3)⋊C8—54no—
196#816C2×D512yescomplemented
197C23×D5—21no—
198C23×D5—62no—
199C5×#41—34no—
200C5×#41—62no—
201C2×C5×Q8—31no—
202C2×C5×D4—31no—
203C2×C5×D4—61no—
204C22×C4×C5—61no—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—32no—
207C2×C22⋊D5—62no—
208C22×Dic5—21no—
209#713—64no—
210#713—64no—
211D4×D5—64no—
212D4×D5—616no—
213D4×D5—64no—
214#709—68no—
215#709—62no—
216#709—62no—
217C2×D20—21no—
218C2×D20—22no—
219C2×D20—31no—
220C2×D20—68no—
221C2×D20—62no—
222C2×D20#6714yes—
223C2×D20C22×D312yes—
224C2×C4×D5—32no—
225C2×C4×D5—62no—
226C2×Dic10—22no—
227C2×Dic10—22no—
228C2×Dic10—21no—
229C2×Dic10—31no—
230C2×Dic10C22×D312yes—
231C2×C5×C8—22no—
232C2×C5×C8C22×D312yes—
233C5×C4⋊C4—32no—
234C5×C22⋊C4—61no—
235C42×C5—61no—
236#689—64no—
237#687—64no—
238#685—64no—
239C4⋊Dic5—64no—
240#681—62no—
241C2×#168—32no—
242#677—28no—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16—51no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—31no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C2×C5×D3—22no—
258C2×C5×D3—42no—
259C2×C3×D5—24no—
260C2×C3×D5—44no—
261C2×C3×D5—84no—
262D3×D5—88no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—22no—
267C3×Dic5—42no—
268C5×Dic3—22no—
269C5×Dic3—42no—
270C24×C3—101no—
271C23×D3—102no—
272C2×C3×Q8—52no—
273C2×C3×Q8—101no—
274C2×C3×Q8—102no—
275C2×C3×Q8—102no—
276C2×C3×D4—54no—
277C2×C3×D4—52no—
278C2×C3×D4—102no—
279C2×C3×D4—101no—
280C22×C3×C4—101no—
281C22×C3×C4C22×D511yescharacteristic
282C2×#67—21no—
283C2×#67—52no—
284C2×#67—102no—
285C2×#67—102no—
286C2×#67—1016no—
287C22×Dic3—102no—
288#237—104no—
289#237—108no—
290#237—104no—
291D3×D4—108no—
292D3×D4—104no—
293D3×D4—104no—
294#235—22no—
295#235—102no—
296#235—102no—
297#235D2014yes—
298C2×D12—51no—
299C2×D12C22×D511yescharacteristic
300C2×C4×D3—21no—
301C2×C4×D3—52no—
302C2×Dic6—51no—
303C2×Dic6C22×D511yescharacteristic
304C3×SD16—108no—
305C2×C3×C8—22no—
306C2×C3×C8C22×D512yes—
307#217—21no—
308#216—104no—
309#215—104no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3C22×D512yes—
312Dic3⋊C4—104no—
313C4×Dic3—21no—
314#208—102no—
315C2×C3⋊C8—52no—
316C23×C5—61no—
317C23×C5D3×D411yescharacteristic
318C22×D5—22no—
319C22×D5—24no—
320C22×D5—32no—
321C22×D5—68no—
322C22×D5—82no—
323C22×D5—244no—
324C5×Q8—32no—
325C5×Q8—61no—
326C5×D4—32no—
327C5×D4—64no—
328C5×D4—61no—
329C2×C4×C5—61no—
330C2×C4×C5—64no—
331C2×C4×C5—62no—
332C2×C4×C5D3×D412yes—
333C2×C4×C5D3×D411yescharacteristic
334C2×C4×C5C2×#6712yes—
335C2×C4×C5C23×D311yescharacteristic
336C22⋊D5—68no—
337C22⋊D5—64no—
338C2×Dic5—22no—
339C2×Dic5—32no—
340C2×Dic5—42no—
341D20—24no—
342D20—22no—
343D20—68no—
344D20—62no—
345D20C2×#6714yes—
346C4×D5—68no—
347C4×D5—62no—
348C4×D5—62no—
349Dic10—22no—
350Dic10—22no—
351Dic10—42no—
352Dic10—62no—
353C5×C8—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—31no—
358C2×#41—151no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8—51no—
362C22×D4—51no—
363C22×D4—152no—
364C22×D4—301no—
365C22×D4—302no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—54no—
369C2×SD16—104no—
370C2×SD16—152no—
371C2×M16—151no—
372C22×C8D3×D511yescharacteristic, complemented
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—31no—
378#103—32no—
379#102—32no—
380#102—154no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—42no—
386C3×D5—44no—
387C5×D3—42no—
388C23×C3—102no—
389C23×C3—104no—
390C23×C3—402no—
391C23×C3D4×D511yescharacteristic
392C22×D3—21no—
393C22×D3—52no—
394C22×D3—108no—
395C22×D3—404no—
396C3×Q8—102no—
397C3×Q8—102no—
398C3×Q8—202no—
399C3×D4—54no—
400C3×D4—102no—
401C3×D4—104no—
402C2×C3×C4—102no—
403C2×C3×C4—202no—
404C2×C3×C4C2×D2012yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4D4×D512yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—24no—
409#67—108no—
410#67—108no—
411#67—104no—
412C2×Dic3—21no—
413C2×Dic3—22no—
414C2×Dic3—52no—
415C2×Dic3—108no—
416D12—21no—
417D12—102no—
418D12C2×D2012yes—
419C4×D3—24no—
420C4×D3—102no—
421C4×D3—102no—
422Dic6—21no—
423Dic6—102no—
424Dic6C2×D2012yes—
425C3×C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—62no—
429C22×C5—122no—
430C22×C5#93712yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—24no—
433C2×D5—44no—
434C2×D5—64no—
435C2×D5—62no—
436C2×D5—84no—
437C2×D5—248no—
438C4×C5—62no—
439C4×C5—122no—
440C4×C5#93712yes—
441C4×C5C2×D3×D411yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—42no—
444Dic5—62no—
445C24—101no—
446C24—302no—
447#41—34no—
448#41—62no—
449#41—308no—
450#41—304no—
451#41—304no—
452#41—302no—
453#41—302no—
454C2×Q8—31no—
455C2×Q8—52no—
456C2×Q8—101no—
457C2×Q8—102no—
458C2×Q8—102no—
459C2×Q8—151no—
460C2×D4—31no—
461C2×D4—54no—
462C2×D4—52no—
463C2×D4—61no—
464C2×D4—102no—
465C2×D4—101no—
466C2×D4—152no—
467C2×D4—151no—
468C2×D4—308no—
469C2×D4—302no—
470C2×D4—302no—
471C2×D4—304no—
472C2×D4—304no—
473C2×D4—3016no—
474C22×C4—61no—
475C22×C4—101no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—108no—
481SD16—304no—
482M16—302no—
483C2×C8—22no—
484C2×C8—152no—
485C2×C8C2×D3×D512yes—
486C4⋊C4—32no—
487C4⋊C4—304no—
488C22⋊C4—61no—
489C22⋊C4—304no—
490C42—61no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—104no—
494C22×C3—204no—
495C22×C3—404no—
496C22×C3#402812yes—
497C22×C3C2×D4×D511yescharacteristic
498C2×D3—22no—
499C2×D3—42no—
500C2×D3—102no—
501C2×D3—104no—
502C2×D3—408no—
503C3×C4—202no—
504C3×C4#402812yes—
505C3×C4C22×D2011yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507Dic3—22no—
508Dic3—42no—
509Dic3—102no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5#576212yes—
513C2×C5C2×#93711yescharacteristic
514D5—44no—
515D5—122no—
516C23—61no—
517C23—102no—
518C23—104no—
519C23—152no—
520C23—308no—
521C23—402no—
522C23—1204no—
523C23#677711yescharacteristic
524Q8—32no—
525Q8—61no—
526Q8—102no—
527Q8—102no—
528Q8—202no—
529Q8—302no—
530D4—32no—
531D4—54no—
532D4—64no—
533D4—61no—
534D4—104no—
535D4—102no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—61no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—152no—
545C2×C4—202no—
546C2×C4—308no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4D3×D2012yes—
550C2×C4#677711yescharacteristic
551C2×C4D5×#6712yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—22no—
556C2×C3—204no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6610412yes—
559D3—42no—
560D3—202no—
561C5#8385911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—62no—
564C22—104no—
565C22—122no—
566C22—204no—
567C22—302no—
568C22—304no—
569C22—404no—
570C22—1208no—
571C22#9147412yes—
572C22C2×#677711yescharacteristic
573C4—62no—
574C4—122no—
575C4—202no—
576C4—302no—
577C4#9147412yes—
578C4C2×D3×D2011yescharacteristic
579C4C2×D5×#6711yescharacteristic
580C3#1061056611yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—204no—
584C2—602no—
585C2#1175398012yes—
586C2C2×#9147411yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92017C2
hypercenterC22×C8D3×D5
hypocenterC3×C5#3009
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
2. C22×C3×#677C2semidirect
3. #11754278C2semidirect
4. C2×#91309C2semidirect
5. #11751211C2semidirect
6. C5×#5113C2semidirect
7. #11748366C2semidirect
8. #11748148C2semidirect
9. #11748111C2semidirect
14. C22×C3×Dic10C22semidirect
22. C22×C3×C5×C8C22semidirect
23. C2×#6710C22semidirect
24. C5×#926C22semidirect
26. C2×C3×#677C22semidirect
28. #91390C22semidirect
31. C2×#6661C22semidirect
33. C2×#6656C22semidirect
34. #91104C22semidirect
35. C5×#821C22semidirect
36. C5×#816C22semidirect
37. #90964C22semidirect
41. C22×#677D3semidirect
101. #5113D5semidirect
113. C22×Dic10C2×D3semidirect
114. C22×C5×C8C2×D3semidirect
121. C2×#677C2×D3semidirect
183. C22×C3×C8C2×D5semidirect
191. #926C2×D5semidirect
194. #821C2×D5semidirect
196. #816C2×D5semidirect
372. C22×C8D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#83859semidirect
580. C3#10610566semidirect

Maximal subgroups

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

subgroupindex
C2×#920172
C22×C3×#6772
#117542782
C2×#913092
#117512112
C5×#51132
#117483662
#117481482
#117481112
#106105663
#838595