Finite Group Databasebeta

← #14439337

Subgroups — #14439337

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#909C211yesmaximal, characteristic, complemented
2C2×C3×#4054C211yesmaximal, characteristic, complemented
3#11754586C211yesmaximal, characteristic, complemented
4C2×#91368C211yesmaximal, characteristic, complemented
5C2×#91307C211yesmaximal, characteristic, complemented
6C2×#91298C211yesmaximal, characteristic, complemented
7M16×Dic15C211yesmaximal, characteristic, complemented
8#11748015C212yesmaximal, complemented
9#11747981C212yesmaximal, complemented
10#11747980C212yesmaximal, complemented
11#11747977C212yesmaximal, complemented
12#10615704—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C22×C3×D20C2211yescharacteristic
17C2×C3×C5×M16C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20C5×#909—24no—
21C2×C5×Dic12C2212yescomplemented
22C2×C5×#204C2212yescomplemented
23C3×#4054—24no—
24C2×C3×D40C2212yes—
25C2×C3×#677C2212yes—
26#91571—22no—
27C2×#6665—22no—
28#91463C2211yescharacteristic, complemented
29C4×#1201—22no—
30#91385C2211yescharacteristic, complemented
31#91370C2211yescharacteristic, complemented
32#91368—22no—
33#91368C2212yes—
34#91368C2212yes—
35#91368C2211yescharacteristic, complemented
36C2×#6661C2212yescomplemented
37#91307—24no—
38C2×#6655C2212yes—
39C2×#6655C2212yescomplemented
40#91298—24no—
41C2×#6652C2212yes—
42#91099C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83938—51nomaximal, Hall
45C2×C5×#137—31no—
46#66166—31no—
47C2×#4095—31no—
48C2×#4093—31no—
49C2×#4088—31no—
50C2×#4054D311yescharacteristic, complemented
51#65480—32no—
52#65479—32no—
53#65465—32no—
54#65457—32no—
55M16×Dic5—31no—
56C23×C3×D5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15D412yes—
59C2×C5×#67—21no—
60C22×C5×Dic3—21no—
61C5×#235—22no—
62C5×#235—22no—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3—21no—
65C2×C5×Dic6—22no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6—21no—
68C2×C5×Dic6C2312yes—
69C2×C5×Dic6C2311yescharacteristic
70C2×C3×C22⋊D5—21no—
71C3×#709—22no—
72C3×#709—22no—
73C2×C3×D20—22no—
74C2×C3×D20—22no—
75C2×C3×D20—21no—
76C2×C3×D20C2311yescharacteristic
77C2×C3×D20C2312yes—
78C2×C3×C4×D5—21no—
79C2×C3×Dic10C2311yescharacteristic
80C2×#1201—22no—
81C3×C5×M16D414yescomplemented
82C2×C3×C5×C8C2312yes—
83C4×Dic15D412yes—
84C4×Dic15C2311yescharacteristic
85C4×Dic15C2311yescharacteristic
86#6710D414yescomplemented
87C2×C3⋊#168C2312yes—
88C5×Dic12—24no—
89C5×#204—24no—
90C3×D40—24no—
91C3×#677—24no—
92#6669—22no—
93(C2×D3)⋊Dic5—22no—
94#6665—22no—
95#6665—24no—
96#6665—24no—
97#6661—24no—
98#6655—24no—
99#6655—24no—
100#6652—24no—
101C2×C3×AGL(1,ℤ/8)—51no—
102#5832—51no—
103C2×#951—51no—
104C2×#936—51no—
105C2×#932—51no—
106C2×#909D511yescharacteristic, complemented
107#5168—52no—
108#5167—52no—
109#5155—52no—
110#5154—52no—
111Dic3×M16—51no—
112C2×C5×#41—31no—
113C22×C5×Q8—31no—
114C2×#709C2×D311yescharacteristic
115C22×D20C2×D311yescharacteristic
116C5×#137—64no—
117C2×C5×Q16—32no—
118C2×C5×SD16—32no—
119C2×C5×M16C2×D311yescharacteristic
120#4095—64no—
121#4093—32no—
122#4093—32no—
123#4093—31no—
124#4093—62no—
125#4090—31no—
126C2×#689—32no—
127#4088—64no—
128C2×#688—32no—
129C2×#688—32no—
130#4086—31no—
131#4085—62no—
132#4082—31no—
133C2×#686—32no—
134C4×C22⋊D5—62no—
135C2×#685—62no—
136C2×C4×Dic5—31no—
137C2×#681—31no—
138#4054—24no—
139C2×D40C2×D312yes—
140C2×#677C2×D312yes—
141#3947—32no—
142C8×Dic5—32no—
143#3153—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C5×D3—21no—
146C22×C3×D5—22no—
147C22×C3×D5—21no—
148C22×C3×D5—44no—
149C22×C3×D5—42no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D412yes—
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5C2411yescharacteristic
154C2×Dic15—42no—
155C2×Dic15C2×D412yes—
156C2×Dic15C2×D412yes—
157C5×#67—42no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—42no—
161C5×D12—21no—
162C5×D12—21no—
163C4×C5×D3—42no—
164C5×Dic6—22no—
165C5×Dic6—22no—
166C5×Dic6—21no—
167C5×Dic6—21no—
168C5×Dic6—42no—
169C3×C22⋊D5—42no—
170C2×C3×Dic5—21no—
171C3×D20—22no—
172C3×D20—21no—
173C3×D20—22no—
174C3×D20—21no—
175C3×D20—42no—
176C3×C4×D5—42no—
177C3×Dic10—21no—
178C3×Dic10—21no—
179#1201—44no—
180C3×C5×C8C2×D414yes—
181C3⋊#168C2×D414yes—
182C2×C3×#41—51no—
183C22×C3×D4—51no—
184C2×#235C2×D511yescharacteristic
185C22×Dic6C2×D511yescharacteristic
186C3×AGL(1,ℤ/8)—104no—
187C2×C3×SD16—52no—
188C2×C3×D8—52no—
189C2×C3×M16C2×D511yescharacteristic
190C2×#217—102no—
191#951—104no—
192#947—51no—
193#944—51no—
194C2×#216—52no—
195#940—51no—
196#939—102no—
197#936—52no—
198#936—51no—
199#936—52no—
200#936—102no—
201C2×#214—52no—
202C2×#214—52no—
203#932—104no—
204C2×#213—52no—
205C4×#67—102no—
206C2×C4×Dic3—51no—
207C2×#208—51no—
208#909—24no—
209C2×Dic12C2×D512yescomplemented
210C2×#204C2×D512yescomplemented
211#815—52no—
212C8×Dic3—52no—
213C23×D5—21no—
214C5×#41—62no—
215C5×#41—62no—
216C2×C5×Q8—32no—
217C2×C5×Q8—31no—
218C2×C5×Q8—61no—
219C2×C5×Q8—62no—
220C2×C5×Q8—62no—
221C2×C5×D4—31no—
222C2×C5×D4—61no—
223C22×C4×C5—61no—
224C22×C4×C5—61no—
225C22×C4×C5C22×D311yescharacteristic
226C2×C22⋊D5—21no—
227C2×C22⋊D5—62no—
228C22×Dic5—32no—
229#709—22no—
230#709—22no—
231C2×D20—22no—
232C2×D20—22no—
233C2×D20—21no—
234C2×D20C22×D312yes—
235C2×D20C22×D311yescharacteristic
236C2×C4×D5—21no—
237C2×Dic10C22×D311yescharacteristic
238C5×Q16—64no—
239C5×SD16—64no—
240C5×M16#6714yescomplemented
241C2×C5×C8C22×D312yes—
242C22⋊Dic5—62no—
243#689—64no—
244#688—64no—
245#688—64no—
246#686—64no—
247#685—64no—
248#685—64no—
249#685—62no—
250#683—62no—
251C4×Dic5—32no—
252C4×Dic5—31no—
253C4×Dic5—31no—
254#681—34no—
255C2×#168—32no—
256D40—24no—
257#677—24no—
258C2×#137—31no—
259C2×#137—151no—
260C2×AGL(1,ℤ/8)—51no—
261C2×AGL(1,ℤ/8)—151no—
262#532—151no—
263C2×#124—151no—
264#497—152no—
265#496—152no—
266#495—152no—
267#495—152no—
268C4×M16—151no—
269C22×C3×C5—22no—
270C22×C3×C5#12712yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×C5×D3—41no—
273C2×C5×D3—42no—
274C2×C3×D5—41no—
275C2×C3×D5—42no—
276C2×C3×D5—42no—
277C2×C3×D5—44no—
278C2×C3×D5—84no—
279C3×C4×C5#12712yes—
280C3×C4×C5C22×D411yescharacteristic
281C3×C4×C5C22×D411yescharacteristic
282Dic15—22no—
283Dic15—22no—
284C3×Dic5—41no—
285C5×Dic3—42no—
286C5×Dic3—41no—
287C24×C3—101no—
288C3×#41—102no—
289C3×#41—102no—
290C2×C3×Q8—51no—
291C2×C3×D4—52no—
292C2×C3×D4—51no—
293C2×C3×D4—101no—
294C2×C3×D4—102no—
295C2×C3×D4—101no—
296C2×C3×D4—102no—
297C22×C3×C4—101no—
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—21no—
300C2×#67—102no—
301C22×Dic3—21no—
302C22×Dic3—52no—
303#235—22no—
304#235—22no—
305C2×D12C22×D511yescharacteristic
306C2×C4×D3—21no—
307C2×Dic6—22no—
308C2×Dic6—22no—
309C2×Dic6—21no—
310C2×Dic6C22×D512yes—
311C2×Dic6C22×D511yescharacteristic
312C3×SD16—104no—
313C3×D8—104no—
314C3×M16C22⋊D514yescomplemented
315C2×C3×C8C22×D512yes—
316#217—102no—
317#217—104no—
318#217—104no—
319#216—104no—
320#214—104no—
321#214—104no—
322#213—104no—
323(C2×D3)⋊C4—102no—
324Dic3⋊C4—102no—
325C4×Dic3—52no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328#208—54no—
329C2×C3⋊C8—52no—
330Dic12—24no—
331#204—24no—
332C23×C5—61no—
333C23×C5D3×D411yescharacteristic
334C22×D5—22no—
335C22×D5—21no—
336C22×D5—44no—
337C22×D5—42no—
338C5×Q8—61no—
339C5×Q8—62no—
340C5×Q8—62no—
341C5×Q8—61no—
342C5×Q8—122no—
343C5×D4—61no—
344C5×D4—61no—
345C5×D4—122no—
346C2×C4×C5—61no—
347C2×C4×C5—62no—
348C2×C4×C5—122no—
349C2×C4×C5—122no—
350C2×C4×C5D3×D411yescharacteristic
351C2×C4×C5D3×D412yes—
352C2×C4×C5C2×#6712yes—
353C2×C4×C5C23×D311yescharacteristic
354C22⋊D5—42no—
355C22⋊D5—124no—
356C2×Dic5—21no—
357C2×Dic5—32no—
358C2×Dic5—32no—
359C2×Dic5—122no—
360D20—22no—
361D20—21no—
362D20—22no—
363D20—21no—
364D20—42no—
365C4×D5—42no—
366Dic10—21no—
367Dic10—21no—
368C5×C8C2×#6714yes—
369#168—34no—
370C2×#41—31no—
371C2×#41—51no—
372C22×Q8—31no—
373C22×D4—51no—
374#137—64no—
375#137—304no—
376AGL(1,ℤ/8)—104no—
377AGL(1,ℤ/8)—304no—
378C2×Q16—32no—
379C2×Q16—152no—
380C2×SD16—32no—
381C2×SD16—52no—
382C2×SD16—152no—
383C2×SD16—152no—
384C2×D8—52no—
385C2×D8—152no—
386C2×M16—151no—
387C2×M16D3×D511yescharacteristic
388C4⋊Q8—151no—
389#127—151no—
390#124—151no—
391#124—152no—
392#124—152no—
393#124—151no—
394#124—302no—
395#121—302no—
396C4×D4—302no—
397C2×C22⋊C4—302no—
398C2×C42—151no—
399#97—152no—
400C4×C8—152no—
401C2×C3×C5—22no—
402C2×C3×C5C2×#12711yescharacteristic
403C2×C3×C5#57512yes—
404C3×D5—81no—
405C3×D5—82no—
406C5×D3—81no—
407C23×C3—102no—
408C23×C3—101no—
409C23×C3—204no—
410C23×C3—202no—
411C23×C3D4×D511yescharacteristic
412C22×D3—21no—
413C3×Q8—101no—
414C3×Q8—101no—
415C3×D4—102no—
416C3×D4—101no—
417C3×D4—101no—
418C3×D4—102no—
419C3×D4—202no—
420C3×D4—202no—
421C2×C3×C4—101no—
422C2×C3×C4—202no—
423C2×C3×C4D4×D512yes—
424C2×C3×C4D4×D511yescharacteristic
425C2×C3×C4C2×C22⋊D512yes—
426C2×C3×C4C23×D511yescharacteristic
427#67—42no—
428#67—204no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—42no—
432C2×Dic3—52no—
433C2×Dic3—52no—
434C2×Dic3—202no—
435D12—21no—
436D12—21no—
437C4×D3—42no—
438Dic6—21no—
439Dic6—22no—
440Dic6—22no—
441Dic6—21no—
442Dic6—42no—
443C3×C8C2×C22⋊D514yes—
444C3⋊C8—54no—
445C22×C5—22no—
446C22×C5—122no—
447C22×C5—121no—
448C22×C5#94012yes—
449C22×C5C2×D3×D411yescharacteristic
450C2×D5—42no—
451C2×D5—44no—
452C2×D5—41no—
453C2×D5—42no—
454C2×D5—84no—
455C4×C5—121no—
456C4×C5—122no—
457C4×C5#94012yes—
458C4×C5C2×D3×D411yescharacteristic
459C4×C5C22×#6711yescharacteristic
460Dic5—41no—
461Dic5—62no—
462Dic5—62no—
463C24—101no—
464#41—62no—
465#41—62no—
466#41—102no—
467#41—102no—
468C2×Q8—32no—
469C2×Q8—31no—
470C2×Q8—51no—
471C2×Q8—61no—
472C2×Q8—62no—
473C2×Q8—62no—
474C2×D4—31no—
475C2×D4—52no—
476C2×D4—51no—
477C2×D4—61no—
478C2×D4—101no—
479C2×D4—102no—
480C2×D4—101no—
481C2×D4—102no—
482C2×D4—302no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—101no—
486C22×C4—152no—
487C22×C4C2×D3×D511yescharacteristic
488Q16—64no—
489Q16—304no—
490SD16—64no—
491SD16—104no—
492SD16—304no—
493SD16—304no—
494D8—104no—
495D8—304no—
496M16—154no—
497M16#120114yescomplemented
498C2×C8—152no—
499C2×C8C2×D3×D512yes—
500C4⋊C4—302no—
501C22⋊C4—304no—
502C22⋊C4—304no—
503C22⋊C4—302no—
504C22⋊C4—302no—
505C42—152no—
506C42—151no—
507C42—151no—
508C3×C5#315311yescharacteristic, complemented, Hall
509C22×C3—22no—
510C22×C3—202no—
511C22×C3—204no—
512C22×C3—201no—
513C22×C3—202no—
514C22×C3—404no—
515C22×C3#408612yes—
516C22×C3C2×D4×D511yescharacteristic
517C2×D3—42no—
518C2×D3—41no—
519C3×C4—201no—
520C3×C4#408612yes—
521C3×C4C2×D4×D511yescharacteristic
522C3×C4C22×C22⋊D511yescharacteristic
523Dic3—42no—
524Dic3—41no—
525Dic3—102no—
526Dic3—102no—
527C2×C5—22no—
528C2×C5—241no—
529C2×C5#577312yes—
530C2×C5C2×#94011yescharacteristic
531D5—82no—
532D5—81no—
533C23—61no—
534C23—102no—
535C23—101no—
536C23—204no—
537C23—202no—
538C23#677711yescharacteristic
539Q8—61no—
540Q8—62no—
541Q8—62no—
542Q8—61no—
543Q8—101no—
544Q8—101no—
545Q8—122no—
546D4—61no—
547D4—61no—
548D4—102no—
549D4—101no—
550D4—101no—
551D4—102no—
552D4—122no—
553D4—202no—
554D4—202no—
555D4—604no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—101no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—152no—
562C2×C4—152no—
563C2×C4—202no—
564C2×C4—602no—
565C2×C4#677711yescharacteristic
566C2×C4#677712yes—
567C2×C4C2×#120112yes—
568C2×C4C22×D3×D511yescharacteristic
569C8—154no—
570C8C2×#120114yes—
571C2×C3—22no—
572C2×C3—402no—
573C2×C3—401no—
574C2×C3#6611112yes—
575C2×C3C2×#408611yescharacteristic
576D3—81no—
577C5#8393811yescharacteristic, complemented, Sylow
578C22—22no—
579C22—122no—
580C22—121no—
581C22—202no—
582C22—204no—
583C22—201no—
584C22—202no—
585C22—404no—
586C22#9146312yes—
587C22C2×#677711yescharacteristic
588C4—121no—
589C4—122no—
590C4—201no—
591C4—302no—
592C4—302no—
593C4#9146312yes—
594C4C2×#677711yescharacteristic
595C4C22×#120111yescharacteristic
596C3#1061570411yescharacteristic, complemented, Sylow
597C2—22no—
598C2—241no—
599C2—402no—
600C2—401no—
601C2#1175403612yes—
602C2C2×#9146311yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
hypercenterC2×M16D3×D5
hypocenterC3×C5#3153
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×C5×#909C2semidirect
2. C2×C3×#4054C2semidirect
3. #11754586C2semidirect
4. C2×#91368C2semidirect
5. C2×#91307C2semidirect
6. C2×#91298C2semidirect
7. M16×Dic15C2semidirect
8. #11748015C2semidirect
9. #11747981C2semidirect
10. #11747980C2semidirect
11. #11747977C2semidirect
21. C2×C5×Dic12C22semidirect
22. C2×C5×#204C22semidirect
28. #91463C22semidirect
30. #91385C22semidirect
31. #91370C22semidirect
35. #91368C22semidirect
36. C2×#6661C22semidirect
39. C2×#6655C22semidirect
42. #91099C22semidirect
43. C8×Dic15C22semidirect
50. C2×#4054D3semidirect
81. C3×C5×M16D4semidirect
86. #6710D4semidirect
106. C2×#909D5semidirect
209. C2×Dic12C2×D5semidirect
210. C2×#204C2×D5semidirect
240. C5×M16#67semidirect
314. C3×M16C22⋊D5semidirect
497. M16#1201semidirect
508. C3×C5#3153semidirect
577. C5#83938semidirect
596. C3#10615704semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9092
C2×C3×#40542
#117545862
C2×#913682
C2×#913072
C2×#912982
M16×Dic152
#117480152
#117479812
#117479802
#117479772
#106157043
#839385