Finite Group Databasebeta

← #14439352

Subgroups — #14439352

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×#91836C211yesmaximal, characteristic, complemented
2C2×C5×#942C211yesmaximal, characteristic, complemented
3C2×C3×#4088C211yesmaximal, characteristic, complemented
4#11754586C211yesmaximal, characteristic, complemented
5C2×#91305C211yesmaximal, characteristic, complemented
6C2×#91301C211yesmaximal, characteristic, complemented
7#11749356C212yesmaximal, complemented
8#11749343C212yesmaximal, complemented
9#11749321C212yesmaximal, complemented
10#11749318C212yesmaximal, complemented
11#11748328C211yesmaximal, characteristic, complemented
12#10615704—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C2×C5×#235C2211yescharacteristic
16C2×C3×#709C2211yescharacteristic
17#91836—22no—
18#91836C2212yes—
19#91836C2212yes—
20C2×#6714—22no—
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×#216C2212yescomplemented
23C5×#942—24no—
24C2×C5×#215C2212yescomplemented
25C2×C5×#208C2211yescharacteristic
26C2×C3×#689C2212yescomplemented
27C3×#4088—24no—
28C2×C3×#688C2212yescomplemented
29C2×C3×#681C2211yescharacteristic
30#91571—22no—
31#91463C2211yescharacteristic, complemented
32C4×#1201—22no—
33#91385C2211yescharacteristic, complemented
34#91370C2211yescharacteristic, complemented
35#91368C2211yescharacteristic, complemented
36C2×#6660C2212yes—
37C2×#6658C2212yes—
38#91305—24no—
39C2×#6654C2212yes—
40#91301—24no—
41C2×#6653C2212yes—
42#90954C2212yescomplemented
43#90951C2212yescomplemented
44#83973—51nomaximal, Hall
45C2×C5×#137—31no—
46#66166—31no—
47C2×#4095—31no—
48C2×#4093—31no—
49C2×#4088D311yescharacteristic, complemented
50C2×#4054—31no—
51#65480—32no—
52#65479—32no—
53#65465—32no—
54#65457—32no—
55M16×Dic5—31no—
56C23×D15—21no—
57C2×C3×C5×Q8—21no—
58C2×C3×C5×Q8—22no—
59C2×C3×C5×Q8—22no—
60C2×C3×C5×Q8C2312yes—
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5C2311yescharacteristic
63C22×Dic15D412yes—
64C2×D60—22no—
65C2×D60—22no—
66C2×D60—21no—
67C2×D60C2312yes—
68C2×C5×#67—21no—
69C5×#235—22no—
70C5×#235—22no—
71C2×C5×D12C2311yescharacteristic
72C2×C4×C5×D3—21no—
73C2×C5×Dic6C2311yescharacteristic
74C2×C3×C22⋊D5—21no—
75C3×#709—22no—
76C3×#709—22no—
77C2×C3×D20C2311yescharacteristic
78C2×C3×C4×D5—21no—
79C2×C3×Dic10C2311yescharacteristic
80C2×#1201—22no—
81#6714—24no—
82#6714—24no—
83C4×Dic15D412yes—
84C4×Dic15C2311yescharacteristic
85C4×Dic15C2311yescharacteristic
86C5×#216—24no—
87C5×#215—24no—
88C5×#208D414yescomplemented
89C2×C5×C3⋊C8C2312yes—
90C3×#689—24no—
91C3×#688—24no—
92C3×#681D414yescomplemented
93C2×C3×#168C2312yes—
94#6669—22no—
95(C2×D3)⋊Dic5—22no—
96#6665—22no—
97#6660—24no—
98#6658—24no—
99#6654—24no—
100#6653—24no—
101C2×C3×#137—51no—
102#5832—51no—
103C2×#949—51no—
104C2×#947—51no—
105C2×#942D511yescharacteristic, complemented
106C2×#908—51no—
107#5191—52no—
108#5190—52no—
109#5176—52no—
110#5168—52no—
111Dic3×M16—51no—
112C2×C5×#41—31no—
113C22×C5×Q8C2×D311yescharacteristic
114C2×#709C2×D311yescharacteristic
115C22×D20—31no—
116C5×#137—64no—
117C2×C5×Q16—32no—
118C2×C5×SD16—32no—
119C2×C5×M16—31no—
120#4095—64no—
121#4093—32no—
122#4093—32no—
123#4093—31no—
124#4093—62no—
125#4090—31no—
126C2×#689C2×D312yescomplemented
127#4088—24no—
128C2×#688—32no—
129C2×#688C2×D312yescomplemented
130#4086—31no—
131#4085—62no—
132#4082—31no—
133C2×#686—32no—
134C4×C22⋊D5—62no—
135C2×#685—62no—
136C2×C4×Dic5—31no—
137C2×#681C2×D311yescharacteristic
138#4054—64no—
139C2×D40—32no—
140C2×#677—32no—
141#3947—32no—
142C8×Dic5—32no—
143#3153—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×D15—22no—
146C22×D15—44no—
147C22×D15—42no—
148C22×C5×D3—21no—
149C22×C3×D5—21no—
150C3×C5×Q8—22no—
151C3×C5×Q8—22no—
152C3×C5×Q8—42no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—42no—
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5C2×D412yes—
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5C2411yescharacteristic
159C2×Dic15—42no—
160C2×Dic15C2×D412yes—
161C2×Dic15C2×D412yes—
162D60—22no—
163D60—22no—
164D60—42no—
165C5×#67—42no—
166C2×C5×Dic3—21no—
167C5×D12—21no—
168C5×D12—21no—
169C4×C5×D3—42no—
170C5×Dic6—21no—
171C5×Dic6—21no—
172C3×C22⋊D5—42no—
173C2×C3×Dic5—21no—
174C3×D20—21no—
175C3×D20—21no—
176C3×C4×D5—42no—
177C3×Dic10—21no—
178C3×Dic10—21no—
179#1201—44no—
180C3×#168C2×D414yes—
181C5×C3⋊C8C2×D414yes—
182C2×C3×#41—51no—
183C22×C3×Q8C2×D511yescharacteristic
184C2×#235C2×D511yescharacteristic
185C22×D12—51no—
186C3×#137—104no—
187C2×C3×Q16—52no—
188C2×C3×SD16—52no—
189C2×C3×M16—51no—
190#949—104no—
191#947—52no—
192#947—52no—
193#947—51no—
194#947—102no—
195#944—51no—
196C2×#216C2×D512yescomplemented
197#942—24no—
198C2×#215—52no—
199C2×#215C2×D512yescomplemented
200#940—51no—
201#939—102no—
202#936—51no—
203C2×#213—52no—
204C4×#67—102no—
205C2×(C2×D3)⋊C4—102no—
206C2×C4×Dic3—51no—
207C2×#208C2×D511yescharacteristic
208#908—104no—
209C2×D24—52no—
210C2×#204—52no—
211#815—52no—
212C8×Dic3—52no—
213C23×D5—61no—
214C5×#41—62no—
215C5×#41—62no—
216C2×C5×Q8—21no—
217C2×C5×Q8—22no—
218C2×C5×Q8—22no—
219C2×C5×Q8—31no—
220C2×C5×Q8C22×D312yes—
221C2×C5×D4—31no—
222C2×C5×D4—61no—
223C22×C4×C5—21no—
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—32no—
232C2×D20—62no—
233C2×D20—62no—
234C2×D20—61no—
235C2×D20C22×D311yescharacteristic
236C2×C4×D5—21no—
237C2×Dic10C22×D311yescharacteristic
238C5×Q16—64no—
239C5×SD16—64no—
240C5×M16—34no—
241C2×C5×C8—32no—
242C22⋊Dic5—62no—
243#689—24no—
244#688—24no—
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#6714yescomplemented
255C2×#168C22×D312yes—
256D40—64no—
257#677—64no—
258C2×#137—31no—
259C2×#137—51no—
260C2×AGL(1,ℤ/8)—151no—
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×D15—42no—
273C2×D15—44no—
274C2×D15—84no—
275C2×C5×D3—41no—
276C2×C5×D3—42no—
277C2×C3×D5—41no—
278C2×C3×D5—42no—
279C3×C4×C5—42no—
280C3×C4×C5#12712yes—
281C3×C4×C5C22×D411yescharacteristic
282C3×C4×C5C22×D411yescharacteristic
283Dic15—22no—
284Dic15—22no—
285C3×Dic5—41no—
286C5×Dic3—41no—
287C23×D3—101no—
288C3×#41—102no—
289C3×#41—102no—
290C2×C3×Q8—21no—
291C2×C3×Q8—22no—
292C2×C3×Q8—22no—
293C2×C3×Q8—51no—
294C2×C3×Q8C22×D512yes—
295C2×C3×D4—51no—
296C2×C3×D4—101no—
297C22×C3×C4—21no—
298C22×C3×C4—101no—
299C22×C3×C4C22×D511yescharacteristic
300C2×#67—21no—
301C2×#67—102no—
302C22×Dic3—52no—
303#235—22no—
304#235—22no—
305C2×D12—52no—
306C2×D12—102no—
307C2×D12—102no—
308C2×D12—101no—
309C2×D12C22×D511yescharacteristic
310C2×C4×D3—21no—
311C2×Dic6C22×D511yescharacteristic
312C3×Q16—104no—
313C3×SD16—104no—
314C3×M16—54no—
315C2×C3×C8—52no—
316#217—102no—
317#216—24no—
318#215—24no—
319#215—104no—
320#213—104no—
321(C2×D3)⋊C4—104no—
322(C2×D3)⋊C4—104no—
323(C2×D3)⋊C4—102no—
324Dic3⋊C4—102no—
325C4×Dic3—52no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328#208C22⋊D514yescomplemented
329C2×C3⋊C8C22×D512yes—
330D24—104no—
331#204—104no—
332C23×C5—61no—
333C23×C5D3×D411yescharacteristic
334C22×D5—21no—
335C22×D5—62no—
336C22×D5—124no—
337C22×D5—122no—
338C5×Q8—22no—
339C5×Q8—22no—
340C5×Q8—42no—
341C5×Q8—61no—
342C5×Q8—61no—
343C5×D4—61no—
344C5×D4—61no—
345C5×D4—122no—
346C2×C4×C5—22no—
347C2×C4×C5—42no—
348C2×C4×C5—61no—
349C2×C4×C5—122no—
350C2×C4×C5D3×D412yes—
351C2×C4×C5D3×D411yescharacteristic
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—21no—
361D20—21no—
362D20—62no—
363D20—62no—
364D20—122no—
365C4×D5—42no—
366Dic10—21no—
367Dic10—21no—
368C5×C8—34no—
369#168C2×#6714yes—
370C2×#41—31no—
371C2×#41—51no—
372C22×Q8D3×D511yescharacteristic
373C22×D4—151no—
374#137—64no—
375#137—104no—
376AGL(1,ℤ/8)—304no—
377AGL(1,ℤ/8)—304no—
378C2×Q16—32no—
379C2×Q16—52no—
380C2×SD16—32no—
381C2×SD16—52no—
382C2×SD16—152no—
383C2×SD16—152no—
384C2×D8—152no—
385C2×D8—152no—
386C2×M16—31no—
387C2×M16—51no—
388C4⋊Q8—151no—
389#127—151no—
390#124—151no—
391#124—151no—
392#124—152no—
393#124—152no—
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—
404D15—82no—
405C3×D5—81no—
406C5×D3—81no—
407C23×C3—101no—
408C23×C3D4×D511yescharacteristic
409C22×D3—21no—
410C22×D3—102no—
411C22×D3—204no—
412C22×D3—202no—
413C3×Q8—22no—
414C3×Q8—22no—
415C3×Q8—42no—
416C3×Q8—101no—
417C3×Q8—101no—
418C3×D4—101no—
419C3×D4—101no—
420C3×D4—202no—
421C2×C3×C4—22no—
422C2×C3×C4—42no—
423C2×C3×C4—101no—
424C2×C3×C4—202no—
425C2×C3×C4D4×D512yes—
426C2×C3×C4D4×D511yescharacteristic
427C2×C3×C4C2×C22⋊D512yes—
428C2×C3×C4C23×D511yescharacteristic
429#67—42no—
430#67—204no—
431C2×Dic3—21no—
432C2×Dic3—52no—
433C2×Dic3—52no—
434C2×Dic3—202no—
435D12—21no—
436D12—21no—
437D12—102no—
438D12—102no—
439D12—202no—
440C4×D3—42no—
441Dic6—21no—
442Dic6—21no—
443C3×C8—54no—
444C3⋊C8C2×C22⋊D514yes—
445C22×C5—22no—
446C22×C5—122no—
447C22×C5—121no—
448C22×C5#94012yes—
449C22×C5C2×D3×D411yescharacteristic
450C2×D5—42no—
451C2×D5—41no—
452C2×D5—124no—
453C2×D5—122no—
454C2×D5—244no—
455C4×C5—42no—
456C4×C5—121no—
457C4×C5#94012yes—
458C4×C5C2×D3×D411yescharacteristic
459C4×C5C22×#6711yescharacteristic
460Dic5—41no—
461Dic5—62no—
462Dic5—62no—
463C24—301no—
464#41—62no—
465#41—62no—
466#41—102no—
467#41—102no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—31no—
472C2×Q8—51no—
473C2×Q8C2×D3×D512yes—
474C2×D4—31no—
475C2×D4—51no—
476C2×D4—61no—
477C2×D4—101no—
478C2×D4—152no—
479C2×D4—302no—
480C2×D4—302no—
481C2×D4—302no—
482C2×D4—301no—
483C22×C4—21no—
484C22×C4—61no—
485C22×C4—101no—
486C22×C4—152no—
487C22×C4C2×D3×D511yescharacteristic
488Q16—64no—
489Q16—104no—
490SD16—64no—
491SD16—104no—
492SD16—304no—
493SD16—304no—
494D8—304no—
495D8—304no—
496M16—34no—
497M16—54no—
498C2×C8—32no—
499C2×C8—52no—
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—201no—
512C22×C3#408612yes—
513C22×C3C2×D4×D511yescharacteristic
514C2×D3—42no—
515C2×D3—41no—
516C2×D3—204no—
517C2×D3—202no—
518C2×D3—404no—
519C3×C4—42no—
520C3×C4—201no—
521C3×C4#408612yes—
522C3×C4C2×D4×D511yescharacteristic
523C3×C4C22×C22⋊D511yescharacteristic
524Dic3—41no—
525Dic3—102no—
526Dic3—102no—
527C2×C5—22no—
528C2×C5—241no—
529C2×C5#577712yes—
530C2×C5C2×#94011yescharacteristic
531D5—81no—
532D5—242no—
533C23—61no—
534C23—101no—
535C23—302no—
536C23—604no—
537C23—602no—
538C23#679011yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—42no—
542Q8—61no—
543Q8—61no—
544Q8—101no—
545Q8—101no—
546D4—61no—
547D4—61no—
548D4—101no—
549D4—101no—
550D4—122no—
551D4—202no—
552D4—302no—
553D4—302no—
554D4—604no—
555D4—602no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—61no—
559C2×C4—101no—
560C2×C4—122no—
561C2×C4—152no—
562C2×C4—152no—
563C2×C4—202no—
564C2×C4—602no—
565C2×C4D5×#6712yes—
566C2×C4D3×C22⋊D512yes—
567C2×C4#679011yescharacteristic
568C2×C4C22×D3×D511yescharacteristic
569C8—34no—
570C8—54no—
571C2×C3—22no—
572C2×C3—401no—
573C2×C3#6611112yes—
574C2×C3C2×#408611yescharacteristic
575D3—81no—
576D3—402no—
577C5#8397311yescharacteristic, complemented, Sylow
578C22—22no—
579C22—122no—
580C22—121no—
581C22—202no—
582C22—201no—
583C22—604no—
584C22—602no—
585C22—1204no—
586C22#9156712yes—
587C22C2×#679011yescharacteristic
588C4—42no—
589C4—121no—
590C4—201no—
591C4—302no—
592C4—302no—
593C4#9156712yes—
594C4C2×D5×#6711yescharacteristic
595C4C2×D3×C22⋊D511yescharacteristic
596C3#1061570411yescharacteristic, complemented, Sylow
597C2—22no—
598C2—241no—
599C2—401no—
600C2—1202no—
601C2#1175509612yes—
602C2C2×#9156711yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
hypercenterC22×Q8D3×D5
hypocenterC3×C5#3153
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#91836C2semidirect
2. C2×C5×#942C2semidirect
3. C2×C3×#4088C2semidirect
4. #11754586C2semidirect
5. C2×#91305C2semidirect
6. C2×#91301C2semidirect
7. #11749356C2semidirect
8. #11749343C2semidirect
9. #11749321C2semidirect
10. #11749318C2semidirect
11. #11748328C2semidirect
22. C2×C5×#216C22semidirect
24. C2×C5×#215C22semidirect
26. C2×C3×#689C22semidirect
28. C2×C3×#688C22semidirect
31. #91463C22semidirect
33. #91385C22semidirect
34. #91370C22semidirect
35. #91368C22semidirect
42. #90954C22semidirect
43. #90951C22semidirect
49. C2×#4088D3semidirect
88. C5×#208D4semidirect
92. C3×#681D4semidirect
105. C2×#942D5semidirect
126. C2×#689C2×D3semidirect
129. C2×#688C2×D3semidirect
196. C2×#216C2×D5semidirect
199. C2×#215C2×D5semidirect
254. #681#67semidirect
328. #208C22⋊D5semidirect
508. C3×C5#3153semidirect
577. C5#83973semidirect
596. C3#10615704semidirect

Maximal subgroups

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

subgroupindex
C2×#918362
C2×C5×#9422
C2×C3×#40882
#117545862
C2×#913052
C2×#913012
#117493562
#117493432
#117493212
#117493182
#117483282
#106157043
#839735