Finite Group Databasebeta

← #14424288

Subgroups — #14424288

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92019C211yesmaximal, characteristic, complemented
2C2×C3×#4054C211yesmaximal, characteristic, complemented
3#11754600C211yesmaximal, characteristic, complemented
4C22×#6653C211yesmaximal, characteristic, complemented
5#11751232C211yesmaximal, characteristic, complemented
6C5×#5129C211yesmaximal, characteristic, complemented
7#11748365C211yesmaximal, characteristic, complemented
8#11748112C214yesmaximal, complemented
9#11747890C214yesmaximal, complemented
10#10610401—31nomaximal, Hall
11C22×D60C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C2×C3×#709C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic, complemented
15C2×#6790—22no—
16#92019—24no—
17#92019—24no—
18C2×D3×D20C2212yescomplemented
19C2×D5×D12—21no—
20C2×#6773—21no—
21C2×#6769C2212yescomplemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23#91815C2211yescharacteristic, complemented
24C22×C5×C3⋊C8C2211yescharacteristic, complemented
25C3×#4054—24no—
26C2×C3×D40C2212yescomplemented
27C2×C3×#677C2212yescomplemented
28#91563—22no—
29#91463—21no—
30#91456C2214yescomplemented
31#91368—21no—
32C2×#6653—24no—
33C2×#6653C2214yescomplemented
34#91105C2214yescomplemented
35C5×(C2×D3)⋊C8C2214yescomplemented
36#90971C2212yescomplemented
37#90965C2212yescomplemented
38#83947—51nomaximal, Hall
39C2×#4149—31no—
40#66164—31no—
41C2×#4054D311yescharacteristic, complemented
42C22×D40—31no—
43C5×#408—31no—
44#65420—31no—
45#65402—31no—
46#65023—34no—
47#65021—34no—
48C23×D15—21no—
49C23×C3×D5—21no—
50C22×D3×D5—22no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×D60—21no—
53C2×D60—22no—
54C2×D60D414yescomplemented
55C2×D60C2312yes—
56C2×C5×#67D412yes—
57C5×#235—22no—
58C5×#235—22no—
59C2×C5×D12D411yescharacteristic
60C2×C4×C5×D3C2312yes—
61C2×C5×Dic6D411yescharacteristic
62C2×C3×C22⋊D5—21no—
63C3×#709—22no—
64C3×#709D414yescomplemented
65C2×C3×D20—22no—
66C2×C3×D20—22no—
67C2×C3×D20—21no—
68C2×C3×D20C2312yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72#6790—216no—
73C2×#1203—22no—
74C2×#1202—22no—
75C2×#1201—24no—
76C2×#1200—22no—
77D3×D20—24no—
78D3×D20—24no—
79D5×D12—28no—
80#6773—28no—
81#6769—24no—
82#6769—24no—
83C3×C5×M16—22no—
84C2×C3×C5×C8C2312yes—
85#6719—21no—
86C4⋊Dic15C2312yes—
87C4×Dic15—21no—
88C2×C5×C3⋊C8—22no—
89C2×C5×C3⋊C8C2312yes—
90C3×D40—24no—
91C3×#677—24no—
92#6665—24no—
93#6653—28no—
94C2×#1009—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96#5805—51no—
97C22×#213—51no—
98#5233—51no—
99#5172—54no—
100#5156—54no—
101#5131—51no—
102#5129D511yescharacteristic, complemented
103C2×C5×#41—31no—
104#4149—64no—
105#4149—64no—
106C2×#713—61no—
107C2×D4×D5—61no—
108C2×D4×D5—62no—
109C2×#709—32no—
110C2×#709C2×D311yescharacteristic
111C22×D20—31no—
112C22×D20—32no—
113C22×D20C2×D311yescharacteristic, complemented
114C2×C5×M16C2×D311yescharacteristic, complemented
115C22×C5×C8—31no—
116#4093—61no—
117#4086—61no—
118#4083—62no—
119#4076—34no—
120#4072—31no—
121#4054—24no—
122C2×D40—34no—
123C2×D40—64no—
124C2×D40C2×D312yescomplemented
125C2×#677C2×D312yescomplemented
126C5×C22⋊C8—34no—
127#3953—32no—
128#3953—34no—
129#3948—32no—
130#3008—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—22no—
133C22×D15—24no—
134C22×D15—82no—
135C22×C5×D3C2×D412yes—
136C22×C3×D5—21no—
137C22×C3×D5—22no—
138C22×C3×D5—42no—
139C22×C3×D5—82no—
140C2×D3×D5—28no—
141C2×D3×D5—84no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—22no—
147D60—24no—
148D60—22no—
149D60C2×D414yes—
150C5×#67—24no—
151C2×C5×Dic3C2×D412yes—
152C5×D12—22no—
153C4×C5×D3—22no—
154C4×C5×D3—22no—
155C5×Dic6—22no—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C3×D20—21no—
159C3×D20—22no—
160C3×D20—22no—
161C3×D20—42no—
162C3×D20C2×D412yes—
163C3×C4×D5—24no—
164C3×Dic10—21no—
165C3×Dic10C2×D412yes—
166#1203—28no—
167#1202—28no—
168#1201—84no—
169#1200—28no—
170C3×C5×C8—22no—
171C5×C3⋊C8—22no—
172C2×C3×#41—51no—
173C22×C3×D4—51no—
174#1009—104no—
175#1009—104no—
176C2×#239—52no—
177C2×D3×D4—52no—
178C2×D3×D4—102no—
179C2×#235—101no—
180C2×#235C2×D511yescharacteristic
181C22×D12—51no—
182C22×D12—101no—
183C3×AGL(1,ℤ/8)—104no—
184C2×C3×SD16—52no—
185C2×C3×D8—52no—
186C2×C3×M16C2×D511yescharacteristic, complemented
187#953—102no—
188#940—101no—
189#938—54no—
190#936—101no—
191C2×#213—54no—
192C2×#213—104no—
193#926—51no—
194C22×C3⋊C8C2×D511yescharacteristic, complemented
195#835—52no—
196#832—52no—
197#821—54no—
198(C2×D3)⋊C8C2×D514yescomplemented
199C23×D5—21no—
200C23×D5—62no—
201C23×D5—61no—
202C5×#41—62no—
203C5×#41—62no—
204C2×C5×Q8—31no—
205C2×C5×D4—32no—
206C2×C5×D4—31no—
207C22×C4×C5—32no—
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—21no—
210C2×C22⋊D5—62no—
211C2×C22⋊D5—64no—
212#713—68no—
213D4×D5—68no—
214D4×D5—616no—
215#709—22no—
216#709—64no—
217#709—64no—
218#709#6714yescomplemented
219C2×D20—22no—
220C2×D20—22no—
221C2×D20—21no—
222C2×D20—34no—
223C2×D20—32no—
224C2×D20—64no—
225C2×D20—62no—
226C2×D20—61no—
227C2×D20—62no—
228C2×D20—64no—
229C2×D20C22×D312yes—
230C2×D20C22×D311yescharacteristic
231C2×C4×D5—21no—
232C2×C4×D5—62no—
233C2×Dic10C22×D311yescharacteristic
234C5×M16—22no—
235C2×C5×C8—32no—
236C2×C5×C8—62no—
237C2×C5×C8C22×D312yes—
238C22⋊Dic5—61no—
239#685—64no—
240C4⋊Dic5—32no—
241C4×Dic5—61no—
242D40—24no—
243D40—68no—
244#677—24no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—51no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—151no—
252#416—151no—
253#408—31no—
254C22×C3×C5—22no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—24no—
258C2×D15—44no—
259C2×D15—84no—
260C2×C5×D3—22no—
261C2×C5×D3—24no—
262C2×C3×D5—22no—
263C2×C3×D5—42no—
264C2×C3×D5—42no—
265C2×C3×D5—44no—
266C2×C3×D5—84no—
267D3×D5—88no—
268C3×C4×C5#12012yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271Dic15—42no—
272C3×Dic5—22no—
273C5×Dic3—22no—
274C24×C3—101no—
275C23×D3—101no—
276C23×D3—102no—
277C3×#41—54no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×D4—51no—
281C2×C3×D4—52no—
282C2×C3×D4—101no—
283C2×C3×D4—101no—
284C2×C3×D4—102no—
285C2×C3×D4—102no—
286C22×C3×C4—101no—
287C22×C3×C4C22×D511yescharacteristic
288C2×#67—104no—
289C2×#67—102no—
290C2×#67D2012yes—
291#239—104no—
292#239—104no—
293D3×D4—104no—
294D3×D4—104no—
295D3×D4—1016no—
296#235—22no—
297#235—22no—
298#235—108no—
299C2×D12—54no—
300C2×D12—52no—
301C2×D12—108no—
302C2×D12—102no—
303C2×D12—101no—
304C2×D12—102no—
305C2×D12D2011yescharacteristic
306C2×C4×D3—102no—
307C2×C4×D3C22×D512yes—
308C2×Dic6D2011yescharacteristic
309C3×SD16—104no—
310C3×D8—104no—
311C3×M16—22no—
312C2×C3×C8C22×D512yes—
313#217—101no—
314#217—104no—
315#213—108no—
316C4⋊Dic3—52no—
317C4×Dic3—101no—
318C2×C3⋊C8—22no—
319C2×C3⋊C8C22×D512yes—
320C23×C5—32no—
321C23×C5D3×D411yescharacteristic
322C22×D5—21no—
323C22×D5—22no—
324C22×D5—42no—
325C22×D5—68no—
326C22×D5—62no—
327C22×D5—64no—
328C22×D5—82no—
329C22×D5—244no—
330C22×D5—242no—
331C5×Q8—62no—
332C5×D4—64no—
333C5×D4—62no—
334C2×C4×C5—32no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5D3×D411yescharacteristic
338C2×C4×C5D3×D412yes—
339C2×C4×C5C2×#6712yes—
340C2×C4×C5C23×D311yescharacteristic
341C22⋊D5—24no—
342C22⋊D5—68no—
343C22⋊D5—244no—
344C2×Dic5—21no—
345C2×Dic5—62no—
346D20—21no—
347D20—22no—
348D20—22no—
349D20—34no—
350D20—42no—
351D20—68no—
352D20—64no—
353D20—62no—
354D20C2×#6712yes—
355C4×D5—24no—
356C4×D5—68no—
357Dic10—21no—
358Dic10C2×#6712yes—
359C5×C8—22no—
360C5×C8—62no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—31no—
364C2×#41—51no—
365C2×#41—152no—
366C2×#41—301no—
367C22×D4—51no—
368C22×D4—152no—
369C22×D4—151no—
370C22×D4—301no—
371C22×D4—302no—
372AGL(1,ℤ/8)—104no—
373C2×SD16—52no—
374C2×D8—52no—
375C2×D8—154no—
376C2×D8—304no—
377C2×M16D3×D511yescharacteristic, complemented
378C22×C8—31no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—151no—
384#103—152no—
385#102—152no—
386#102—154no—
387C22⋊C8—34no—
388C2×C3×C5—22no—
389C2×C3×C5C2×#12011yescharacteristic
390C2×C3×C5#57612yes—
391D15—44no—
392C3×D5—42no—
393C3×D5—82no—
394C5×D3—42no—
395C23×C3—101no—
396C23×C3—102no—
397C23×C3—202no—
398C23×C3—402no—
399C23×C3C2×D2011yescharacteristic
400C22×D3—108no—
401C22×D3—102no—
402C22×D3—104no—
403C22×D3—404no—
404C22×D3—402no—
405C22×D3C2×D2012yes—
406C3×Q8—52no—
407C3×Q8—101no—
408C3×D4—52no—
409C3×D4—104no—
410C3×D4—101no—
411C3×D4—102no—
412C3×D4—102no—
413C3×D4—202no—
414C2×C3×C4—101no—
415C2×C3×C4—104no—
416C2×C3×C4C2×D2011yescharacteristic
417C2×C3×C4D4×D512yes—
418C2×C3×C4D4×D512yes—
419C2×C3×C4C23×D511yescharacteristic
420#67—24no—
421#67—108no—
422#67—404no—
423C2×Dic3—102no—
424C2×Dic3C2×D2012yes—
425D12—22no—
426D12—54no—
427D12—108no—
428D12—102no—
429D12—104no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—108no—
433Dic6—22no—
434C3×C8—22no—
435C3⋊C8—22no—
436C22×C5—22no—
437C22×C5—64no—
438C22×C5—62no—
439C22×C5#93712yes—
440C22×C5C2×D3×D411yescharacteristic
441C2×D5—22no—
442C2×D5—42no—
443C2×D5—44no—
444C2×D5—42no—
445C2×D5—64no—
446C2×D5—84no—
447C2×D5—124no—
448C2×D5—248no—
449C2×D5—244no—
450C4×C5—62no—
451C4×C5#93712yes—
452C4×C5C2×D3×D411yescharacteristic
453C4×C5C22×#6711yescharacteristic
454Dic5—22no—
455Dic5—122no—
456C24—101no—
457C24—302no—
458C24—301no—
459#41—54no—
460#41—62no—
461#41—62no—
462#41—102no—
463#41—304no—
464#41—304no—
465#41—308no—
466C2×Q8—31no—
467C2×Q8—51no—
468C2×D4—32no—
469C2×D4—31no—
470C2×D4—51no—
471C2×D4—52no—
472C2×D4—101no—
473C2×D4—101no—
474C2×D4—102no—
475C2×D4—102no—
476C2×D4—154no—
477C2×D4—152no—
478C2×D4—302no—
479C2×D4—304no—
480C2×D4—304no—
481C2×D4—308no—
482C2×D4—302no—
483C2×D4—3016no—
484C2×D4—304no—
485C2×D4—301no—
486C2×D4—302no—
487C22×C4—32no—
488C22×C4—101no—
489C22×C4—302no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—104no—
492D8—104no—
493D8—308no—
494M16—22no—
495C2×C8—32no—
496C2×C8—62no—
497C2×C8C2×D3×D512yes—
498C4⋊C4—152no—
499C22⋊C4—304no—
500C22⋊C4—301no—
501C42—301no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—102no—
505C22×C3—202no—
506C22×C3—202no—
507C22×C3—204no—
508C22×C3—404no—
509C22×C3#402812yes—
510C22×C3C22×D2011yescharacteristic
511C2×D3—22no—
512C2×D3—24no—
513C2×D3—104no—
514C2×D3—204no—
515C2×D3—408no—
516C2×D3—404no—
517C3×C4—102no—
518C3×C4#402812yes—
519C3×C4C2×D4×D511yescharacteristic
520C3×C4C2×D4×D511yescharacteristic
521Dic3—22no—
522Dic3—202no—
523C2×C5—22no—
524C2×C5—122no—
525C2×C5#577412yes—
526C2×C5C2×#93711yescharacteristic
527D5—42no—
528D5—82no—
529D5—124no—
530C23—32no—
531C23—101no—
532C23—102no—
533C23—202no—
534C23—308no—
535C23—302no—
536C23—304no—
537C23—402no—
538C23—1202no—
539C23—1204no—
540C23D3×D2011yescharacteristic
541Q8—52no—
542Q8—62no—
543Q8—101no—
544D4—52no—
545D4—64no—
546D4—62no—
547D4—104no—
548D4—101no—
549D4—102no—
550D4—102no—
551D4—154no—
552D4—202no—
553D4—308no—
554D4—308no—
555D4—304no—
556D4—302no—
557D4—1204no—
558C2×C4—32no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—101no—
562C2×C4—104no—
563C2×C4—308no—
564C2×C4—302no—
565C2×C4D3×D2011yescharacteristic
566C2×C4#677712yes—
567C2×C4D5×#6712yes—
568C2×C4C22×D3×D511yescharacteristic
569C8—22no—
570C8—62no—
571C2×C3—22no—
572C2×C3—202no—
573C2×C3—402no—
574C2×C3C2×#402811yescharacteristic
575C2×C3#6608612yes—
576D3—42no—
577D3—204no—
578C5#8394711yescharacteristic, complemented, Sylow
579C22—22no—
580C22—64no—
581C22—62no—
582C22—102no—
583C22—202no—
584C22—204no—
585C22—202no—
586C22—304no—
587C22—404no—
588C22—604no—
589C22—1204no—
590C22—1208no—
591C22#9147412yes—
592C22C2×D3×D2011yescharacteristic
593C4—62no—
594C4—102no—
595C4—602no—
596C4#9147412yes—
597C4C2×#677711yescharacteristic
598C4C2×D5×#6711yescharacteristic
599C3#1061040111yescharacteristic, complemented, Sylow
600C2—22no—
601C2—122no—
602C2—202no—
603C2—402no—
604C2—604no—
605C2#1175403112yes—
606C2C2×#9147411yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
hypercenterC2×M16D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#92019C2semidirect
2. C2×C3×#4054C2semidirect
3. #11754600C2semidirect
4. C22×#6653C2semidirect
5. #11751232C2semidirect
6. C5×#5129C2semidirect
7. #11748365C2semidirect
8. #11748112C2semidirect
9. #11747890C2semidirect
14. C22×C3×D20C22semidirect
18. C2×D3×D20C22semidirect
21. C2×#6769C22semidirect
22. C2×C3×C5×M16C22semidirect
23. #91815C22semidirect
24. C22×C5×C3⋊C8C22semidirect
26. C2×C3×D40C22semidirect
27. C2×C3×#677C22semidirect
30. #91456C22semidirect
33. C2×#6653C22semidirect
34. #91105C22semidirect
35. C5×(C2×D3)⋊C8C22semidirect
36. #90971C22semidirect
37. #90965C22semidirect
41. C2×#4054D3semidirect
54. C2×D60D4semidirect
64. C3×#709D4semidirect
102. #5129D5semidirect
113. C22×D20C2×D3semidirect
114. C2×C5×M16C2×D3semidirect
124. C2×D40C2×D3semidirect
125. C2×#677C2×D3semidirect
186. C2×C3×M16C2×D5semidirect
194. C22×C3⋊C8C2×D5semidirect
198. (C2×D3)⋊C8C2×D5semidirect
218. #709#67semidirect
377. C2×M16D3×D5semidirect
502. C3×C5#3008semidirect
578. C5#83947semidirect
599. C3#10610401semidirect

Maximal subgroups

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

subgroupindex
C2×#920192
C2×C3×#40542
#117546002
C22×#66532
#117512322
C5×#51292
#117483652
#117481122
#117478902
#106104013
#839475