Finite Group Databasebeta

← #14458399

Subgroups — #14458399

Order 1920.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11752737C211yesmaximal, characteristic, complemented
2#11752561C211yesmaximal, characteristic, complemented
3#11750935C211yesmaximal, characteristic, complemented
4C5×#4843C211yesmaximal, characteristic, complemented
5C3×#65135C211yesmaximal, characteristic, complemented
6#11748119C212yesmaximal, complemented
7#11748108C212yesmaximal, complemented
8#11747933C212yesmaximal, complemented
9#11747637C212yesmaximal, complemented
10C4×#6652C211yesmaximal, characteristic, complemented
11#11746901C211yesmaximal, characteristic, complemented
12#10612083—31nomaximal, Hall
13C2×#6777C2212yes—
14C2×D5×D12—21no—
15C2×#6769—21no—
16C2×C8×D15C2212yescomplemented
17C4×D60C2211yescharacteristic
18C2×C5×D24C2212yescomplemented
19C5×#874C2211yescharacteristic
20C4×C5×D12C2211yescharacteristic, complemented
21C3×C4×D20C2211yescharacteristic
22C3×#4015C2211yescharacteristic, complemented
23#91475—22no—
24#91466C2212yes—
25#91466C2212yescomplemented
26#91449—21no—
27#91438—22no—
28#91437C2212yescomplemented
29#91369—21no—
30C2×#6652C2211yescharacteristic
31C3×C5×C4⋊C8C2211yescharacteristic, complemented
32#91104C2212yescomplemented
33C4×C3⋊#168C2211yescharacteristic, complemented
34C5×#821C2212yescomplemented
35C3×#3953C2212yes—
36C3×#3950C2212yescomplemented
37#90989C2211yescharacteristic, complemented
38#90966C2212yescomplemented
39#90966C2212yescomplemented
40#90966C2211yescharacteristic, complemented
41#90964C2211yescharacteristic
42#80980—51nomaximal, Hall
43#66051—31no—
44#65885—31no—
45C5×#459—31no—
46#65446—32no—
47#65445—32no—
48#65359—31no—
49C4×#686—31no—
50#65135D311yescharacteristic, complemented
51#65123—31no—
52#65063—32no—
53D5×#102—32no—
54C22×D3×D5—22no—
55C2×D60D411yescharacteristic
56C2×C4×D15C2312yes—
57C2×C5×D12—21no—
58C2×C5×D12—21no—
59C2×C5×D12C2311yescharacteristic
60C2×C5×D12C2312yes—
61C2×C4×C5×D3—21no—
62C2×C3×D20C2311yescharacteristic
63C2×C3×C4×D5—21no—
64C2×C3×Dic10—21no—
65C2×#1203—22no—
66C2×#1203—22no—
67C2×#1201—22no—
68C2×#1200—22no—
69#6777—24no—
70#6777—42no—
71D5×D12—44no—
72#6769—44no—
73C2×C3×C5×C8C2312yes—
74C3×C42×C5C2311yescharacteristic
75#6714D412yes—
76C4⋊Dic15D411yescharacteristic
77C2×C3⋊#168C2311yescharacteristic
78C2×C3⋊#168C2311yescharacteristic
79C8×D15—42no—
80C5×(C2×D3)⋊C4—21no—
81C5×C4⋊Dic3C2311yescharacteristic
82C5×D24—42no—
83C3×#685—21no—
84C3×C4⋊Dic5—21no—
85C3×C4⋊Dic5C2312yes—
86C3×C4⋊Dic5C2311yescharacteristic
87#6669—22no—
88(C2×D3)⋊Dic5—22no—
89(C2×D3)⋊Dic5—22no—
90#6652—22no—
91#5726—51no—
92D4×D12—51no—
93C3×#474—51no—
94#5088—51no—
95C4×#213—51no—
96#4883—52no—
97#4882—52no—
98#4843D511yescharacteristic, complemented
99#4834—51no—
100#4773—52no—
101D3×#102—52no—
102C2×D4×D5—32no—
103C2×D4×D5—61no—
104C2×#709—61no—
105C2×C5×D8—32no—
106C5×#127—31no—
107C4×C5×D4—31no—
108#4084—32no—
109#4084—32no—
110#4084—61no—
111#4083—62no—
112C2×#686—31no—
113#4070—61no—
114C2×C8×D5—32no—
115D5×C4⋊C4—32no—
116#4029—62no—
117C4×D20—31no—
118C4×D20C2×D311yescharacteristic
119#4015C2×D311yescharacteristic, complemented
120C5×C4⋊C8C2×D311yescharacteristic, complemented
121C5×#102—32no—
122#3964—31no—
123#3964—32no—
124#3964—32no—
125#3953C2×D312yes—
126#3952—32no—
127#3950C2×D312yescomplemented
128#3941—31no—
129#3939—31no—
130C4×#168—31no—
131#3294—151noSylow
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C22×C5×D3—22no—
135C22×C3×D5—21no—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2411yescharacteristic
142C2×Dic15C2×D412yes—
143D60D814yescomplemented
144C4×D15—22no—
145C4×D15—22no—
146C2×C5×Dic3—21no—
147C5×D12—22no—
148C5×D12—21no—
149C5×D12—21no—
150C5×D12—22no—
151C5×D12—42no—
152C5×D12—81no—
153C4×C5×D3—42no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—22no—
156C3×D20—21no—
157C3×D20#4112yescomplemented
158C3×C4×D5—42no—
159C3×Dic10—42no—
160#1203—44no—
161#1203—44no—
162#1201—44no—
163#1200—44no—
164C3×C5×C8—22no—
165C3⋊#168—21no—
166C3⋊#168—21no—
167C2×#239—101no—
168C2×D3×D4—52no—
169C22×D12—101no—
170C3×C4⋊Q8—51no—
171C3×C4×D4—51no—
172#946—101no—
173C2×#213—51no—
174#930—101no—
175C2×D24C2×D512yescomplemented
176C2×D3×C8—52no—
177#895—52no—
178#895—52no—
179#894—102no—
180D3×C4⋊C4—52no—
181#882—102no—
182#874C2×D511yescharacteristic
183C4×D12—51no—
184C4×D12C2×D511yescharacteristic, complemented
185C3×#107—52no—
186C3×C4⋊C8C2×D511yescharacteristic, complemented
187C3×#102—52no—
188#832—51no—
189#821C2×D512yescomplemented
190(C2×D3)⋊C8—52no—
191#809—52no—
192#809—51no—
193#809—52no—
194#807—51no—
195C4×C3⋊C8—51no—
196C23×D5—62no—
197C2×C5×D4—31no—
198C2×C5×D4—32no—
199C2×C5×D4—61no—
200C2×C5×D4—61no—
201C22×C4×C5—61no—
202C2×C22⋊D5—62no—
203C2×C22⋊D5—62no—
204C2×C22⋊D5—62no—
205D4×D5—64no—
206D4×D5—124no—
207D4×D5—122no—
208#709—124no—
209C2×D20—31no—
210C2×D20C22×D311yescharacteristic
211C2×C4×D5—21no—
212C2×C4×D5—32no—
213C2×C4×D5—62no—
214C2×Dic10—21no—
215C5×D8—122no—
216C2×C5×C8C22×D312yes—
217C5×C4⋊C4—31no—
218C5×C22⋊C4—61no—
219C42×C5C22×D311yescharacteristic
220C22⋊Dic5—62no—
221C22⋊Dic5—62no—
222#686—62no—
223#685—21no—
224#685—32no—
225C4⋊Dic5—21no—
226C4⋊Dic5—31no—
227C4⋊Dic5C22×D311yescharacteristic
228C4⋊Dic5C22×D312yes—
229#683—62no—
230C2×#168—31no—
231C2×#168—31no—
232C8×D5—122no—
233#547—151no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—51no—
238#466—152no—
239#459—31no—
240#447—152no—
241C4×D8—151no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5C22×D411yescharacteristic
245C2×D15—22no—
246C2×D15—22no—
247C2×D15C2×D814yes—
248C2×C5×D3—42no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
251C2×C5×D3—42no—
252C2×C5×D3—42no—
253C2×C5×D3—41no—
254C2×C3×D5—22no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5C2×D812yes—
260C3×C4×C5C2×#4111yescharacteristic
261C3×C4×C5H(5,𝔽2)11yescharacteristic
262Dic15—22no—
263C3×Dic5—42no—
264C3×Dic5—41no—
265C5×Dic3—41no—
266C23×D3—102no—
267C2×C3×Q8—101no—
268C2×C3×D4—51no—
269C22×C3×C4—101no—
270C2×#67—102no—
271#239—204no—
272D3×D4—104no—
273D3×D4—202no—
274C2×D12—21no—
275C2×D12—21no—
276C2×D12—51no—
277C2×D12—102no—
278C2×D12—102no—
279C2×D12—204no—
280C2×D12C22×D512yes—
281C2×D12C22×D511yescharacteristic
282C2×C4×D3—21no—
283C2×C4×D3—52no—
284C2×C4×D3—102no—
285C2×C3×C8C22×D512yes—
286C3×C4⋊C4—51no—
287C3×C4⋊C4—52no—
288C3×C4⋊C4—101no—
289C3×C22⋊C4—101no—
290C3×C42C22×D511yescharacteristic
291#213—102no—
292(C2×D3)⋊C4—21no—
293(C2×D3)⋊C4—52no—
294(C2×D3)⋊C4—102no—
295(C2×D3)⋊C4—102no—
296C4⋊Dic3—51no—
297C4⋊Dic3C22×D511yescharacteristic
298Dic3⋊C4—102no—
299C2×C3⋊C8—51no—
300C2×C3⋊C8—51no—
301D24—42no—
302D3×C8—202no—
303C23×C5—61no—
304C23×C5—62no—
305C22×D5—21no—
306C22×D5—32no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×D4—61no—
311C5×D4—62no—
312C5×D4—61no—
313C5×D4—62no—
314C5×D4—122no—
315C5×D4—241no—
316C2×C4×C5—61no—
317C2×C4×C5—122no—
318C2×C4×C5D3×D411yescharacteristic
319C2×C4×C5D3×D411yescharacteristic
320C2×C4×C5C23×D311yescharacteristic
321C22⋊D5—124no—
322C22⋊D5—124no—
323C22⋊D5—124no—
324C2×Dic5—22no—
325C2×Dic5—21no—
326C2×Dic5—32no—
327D20—21no—
328D20—34no—
329D20#23912yescomplemented
330C4×D5—42no—
331C4×D5—62no—
332C4×D5—62no—
333C4×D5—124no—
334Dic10—42no—
335C5×C8—22no—
336#168—61no—
337#168—61no—
338C2×#41—301no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—32no—
342C2×D8—151no—
343C22×C8—152no—
344C4⋊Q8—51no—
345#127—31no—
346#123—302no—
347C22⋊Q8—301no—
348#121—152no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—51no—
354C4×D4—151no—
355C2×C4⋊C4—152no—
356#107—52no—
357#107—151no—
358C4⋊C8D3×D511yescharacteristic, complemented
359#102—32no—
360#102—52no—
361#102—151no—
362#102—152no—
363#102—151no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5C22×D811yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—24no—
371C3×D5—42no—
372C5×D3—82no—
373C5×D3—81no—
374C23×C3—101no—
375C22×D3—21no—
376C22×D3—22no—
377C22×D3—52no—
378C22×D3—204no—
379C22×D3—204no—
380C22×D3—402no—
381C3×Q8—202no—
382C3×D4—52no—
383C3×D4—101no—
384C2×C3×C4—102no—
385C2×C3×C4—101no—
386C2×C3×C4—202no—
387C2×C3×C4D4×D511yescharacteristic
388C2×C3×C4D4×D511yescharacteristic
389C2×C3×C4C23×D511yescharacteristic
390#67—204no—
391C2×Dic3—21no—
392C2×Dic3—52no—
393D12—21no—
394D12—22no—
395D12—21no—
396D12—22no—
397D12—42no—
398D12—54no—
399D12—81no—
400D12—204no—
401D12—204no—
402C4×D3—42no—
403C4×D3—102no—
404C4×D3—102no—
405C4×D3—204no—
406C3×C8—22no—
407C3⋊C8—101no—
408C3⋊C8—101no—
409C22×C5—121no—
410C22×C5—122no—
411C22×C5—122no—
412C22×C5—121no—
413C22×C5—122no—
414C22×C5—121no—
415C22×C5C2×D3×D411yescharacteristic
416C2×D5—22no—
417C2×D5—34no—
418C2×D5—41no—
419C2×D5—41no—
420C2×D5—62no—
421C2×D5—62no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—121no—
425C4×C5D3×D812yes—
426C4×C5C2×#23911yescharacteristic
427C4×C5#100911yescharacteristic
428Dic5—42no—
429Dic5—41no—
430Dic5—62no—
431C24—302no—
432#41—604no—
433C2×Q8—101no—
434C2×D4—32no—
435C2×D4—31no—
436C2×D4—51no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—151no—
440C2×D4—302no—
441C2×D4—302no—
442C2×D4—302no—
443C2×D4—304no—
444C2×D4—604no—
445C2×D4—602no—
446C22×C4—61no—
447C22×C4—101no—
448C22×C4—152no—
449C22×C4—302no—
450D8—122no—
451D8—302no—
452C2×C8—151no—
453C2×C8—151no—
454C2×C8—602no—
455C2×C8C2×D3×D512yes—
456C4⋊C4—31no—
457C4⋊C4—52no—
458C4⋊C4—51no—
459C4⋊C4—101no—
460C4⋊C4—151no—
461C4⋊C4—302no—
462C22⋊C4—61no—
463C22⋊C4—101no—
464C22⋊C4—152no—
465C22⋊C4—302no—
466C22⋊C4—302no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—102no—
470C22×C3—201no—
471C22×C3—201no—
472C22×C3C2×D4×D511yescharacteristic
473C2×D3—41no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—41no—
477C2×D3—42no—
478C2×D3—41no—
479C2×D3—54no—
480C2×D3—102no—
481C2×D3—102no—
482C2×D3—408no—
483C3×C4—21no—
484C3×C4—202no—
485C3×C4—201no—
486C3×C4D5×D812yes—
487C3×C4#414411yescharacteristic
488C3×C4D5×#4111yescharacteristic
489Dic3—41no—
490Dic3—102no—
491C2×C5—242no—
492C2×C5—241no—
493C2×C5#563011yescharacteristic
494C2×C5C2×D3×D811yescharacteristic
495C2×C5#577811yescharacteristic
496D5—42no—
497D5—64no—
498C23—61no—
499C23—62no—
500C23—101no—
501C23—152no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—202no—
506D4—52no—
507D4—61no—
508D4—62no—
509D4—61no—
510D4—62no—
511D4—101no—
512D4—122no—
513D4—154no—
514D4—241no—
515D4—604no—
516D4—604no—
517D4—604no—
518C2×C4—61no—
519C2×C4—101no—
520C2×C4—102no—
521C2×C4—122no—
522C2×C4—152no—
523C2×C4—202no—
524C2×C4—302no—
525C2×C4—302no—
526C2×C4—604no—
527C2×C4#677711yescharacteristic
528C2×C4#677711yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—22no—
531C8—301no—
532C8—301no—
533C2×C3—202no—
534C2×C3#6594611yescharacteristic
535C2×C3C2×D5×D811yescharacteristic
536C2×C3#6610911yescharacteristic
537D3—82no—
538D3—81no—
539D3—104no—
540C5#8098011yescharacteristic, complemented, Sylow
541C22—102no—
542C22—122no—
543C22—121no—
544C22—122no—
545C22—121no—
546C22—121no—
547C22—122no—
548C22—154no—
549C22—201no—
550C22—201no—
551C22—302no—
552C22—302no—
553C22—1208no—
554C22C2×#677711yescharacteristic
555C4—21no—
556C4—121no—
557C4—201no—
558C4—202no—
559C4—302no—
560C4#9125612yes—
561C4#9202711yescharacteristic
562C4D5×#23911yescharacteristic
563C3#1061208311yescharacteristic, complemented, Sylow
564C2—202no—
565C2—242no—
566C2—241no—
567C2—304no—
568C2#1175388711yescharacteristic
569C2C2×#9125611yescharacteristic
570C2#1175403911yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊C8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊C8D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #11752737C2semidirect
2. #11752561C2semidirect
3. #11750935C2semidirect
4. C5×#4843C2semidirect
5. C3×#65135C2semidirect
6. #11748119C2semidirect
7. #11748108C2semidirect
8. #11747933C2semidirect
9. #11747637C2semidirect
10. C4×#6652C2semidirect
11. #11746901C2semidirect
16. C2×C8×D15C22semidirect
18. C2×C5×D24C22semidirect
20. C4×C5×D12C22semidirect
22. C3×#4015C22semidirect
25. #91466C22semidirect
28. #91437C22semidirect
31. C3×C5×C4⋊C8C22semidirect
32. #91104C22semidirect
33. C4×C3⋊#168C22semidirect
34. C5×#821C22semidirect
36. C3×#3950C22semidirect
37. #90989C22semidirect
38. #90966C22semidirect
39. #90966C22semidirect
40. #90966C22semidirect
50. #65135D3semidirect
98. #4843D5semidirect
119. #4015C2×D3semidirect
120. C5×C4⋊C8C2×D3semidirect
127. #3950C2×D3semidirect
143. D60D8semidirect
157. C3×D20#41semidirect
175. C2×D24C2×D5semidirect
184. C4×D12C2×D5semidirect
186. C3×C4⋊C8C2×D5semidirect
189. #821C2×D5semidirect
329. D20#239semidirect
358. C4⋊C8D3×D5semidirect
468. C3×C5#3294semidirect
540. C5#80980semidirect
563. C3#10612083semidirect

Maximal subgroups

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

subgroupindex
#117527372
#117525612
#117509352
C5×#48432
C3×#651352
#117481192
#117481082
#117479332
#117476372
C4×#66522
#117469012
#106120833
#809805