Finite Group Databasebeta

← #14458277

Subgroups — #14458277

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
1C3×C5×#547C211yesmaximal, characteristic, complemented
2D4×D60C211yesmaximal, characteristic, complemented
3#11751189C211yesmaximal, characteristic, complemented
4#11751188C211yesmaximal, characteristic, complemented
5#11751139C212yesmaximal, complemented
6#11751137C212yesmaximal, complemented
7#11751116C211yesmaximal, characteristic, complemented
8C4×#6715C211yesmaximal, characteristic, complemented
9D4×C3⋊#168C211yesmaximal, characteristic, complemented
10#11751068C212yesmaximal, complemented
11C2×#91093C212yesmaximal, complemented
12#10615256—31nomaximal, Hall
13C2×C3×C5×#41—21no—
14C2×D4×D15—21no—
15C22×D60C2212yes—
16C3×C5×C4⋊Q8C2211yescharacteristic, complemented
17C3×C5×#123—22no—
18C3×C5×C22⋊Q8—21no—
19C3×C5×#121C2212yescomplemented
20C3×C4×C5×D4C2211yescharacteristic
21C3×C4×C5×D4C2211yescharacteristic, complemented
22C2×C3×C5×C4⋊C4C2212yescomplemented
23C2×#6715C2211yescharacteristic
24C2×#6715C2212yescomplemented
25#91819C2212yes—
26C22×C3⋊#168C2212yescomplemented
27#91784—21no—
28#91771—22no—
29#91763C2211yescharacteristic
30C4×D60C2211yescharacteristic
31#91116C2211yescharacteristic, complemented
32#91114C2212yescomplemented
33#91093C2212yes—
34#91093C2211yescharacteristic
35#91093C2212yescomplemented
36#91093C2212yescomplemented
37#91093C2212yescomplemented
38#91091C2212yescomplemented
39#91091C2211yescharacteristic, complemented
40C4⋊(C3⋊#168)C2211yescharacteristic, complemented
41C4×C3⋊#168C2211yescharacteristic, complemented
42#83525—51nomaximal, Hall
43C5×#547D311yescharacteristic, complemented
44D4×D20—31no—
45#65377—31no—
46#65376—31no—
47#65327—32no—
48#65325—32no—
49#65304—31no—
50C4×#686—31no—
51D4×#168—31no—
52#65256—32no—
53C2×#3941—32no—
54C23×D15—22no—
55C3×C5×#41—44no—
56C2×C3×C5×Q8—21no—
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4D411yescharacteristic
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5C2312yes—
63C2×#1220—22no—
64D4×D15—44no—
65C2×D60—24no—
66C2×D60—21no—
67C2×D60—22no—
68C2×D60—21no—
69C2×D60—42no—
70C2×D60C2312yes—
71C2×D60C2311yescharacteristic
72C2×C4×D15—21no—
73C3×C5×C4⋊C4—22no—
74C3×C5×C4⋊C4—21no—
75C3×C5×C4⋊C4D411yescharacteristic
76C3×C5×C4⋊C4C2311yescharacteristic
77C3×C5×C4⋊C4C2312yes—
78C3×C5×C22⋊C4—21no—
79C3×C5×C22⋊C4—22no—
80C3×C5×C22⋊C4D412yes—
81C3×C42×C5C2311yescharacteristic
82#6715—22no—
83#6715—42no—
84#6714—21no—
85#6714—22no—
86C4⋊Dic15C2311yescharacteristic
87C2×C3⋊#168—42no—
88C2×C3⋊#168C2311yescharacteristic
89C2×C3⋊#168C2312yes—
90C2×C3⋊#168C2311yescharacteristic
91C3×#547D511yescharacteristic, complemented
92D4×D12—51no—
93#5088—51no—
94#5087—51no—
95#5038—52no—
96#5036—52no—
97#5015—51no—
98C4×#213—51no—
99D4×C3⋊C8—51no—
100#4967—52no—
101C2×#809—52no—
102C2×C5×#41—21no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C5×C4⋊Q8C2×D311yescharacteristic, complemented
106C5×#123—22no—
107C5×C22⋊Q8—21no—
108C5×#121C2×D312yescomplemented
109C4×C5×D4C2×D311yescharacteristic
110C4×C5×D4C2×D311yescharacteristic, complemented
111C2×C5×C4⋊C4C2×D312yescomplemented
112C2×#686—31no—
113C2×#686—32no—
114#4076—32no—
115C22×#168—32no—
116#4041—61no—
117#4028—62no—
118#4020—31no—
119C4×D20—31no—
120#3964—31no—
121C22⋊#168—32no—
122#3941—31no—
123#3941—32no—
124#3941—32no—
125#3941—32no—
126#3941—32no—
127#3939—31no—
128#3939—32no—
129C4⋊#168—31no—
130C4×#168—31no—
131#3294—151noSylow
132C23×C3×C5—21no—
133C23×C3×C5C2×D412yes—
134C22×D15—21no—
135C22×D15—22no—
136C22×D15—44no—
137C22×D15—44no—
138C22×D15—82no—
139C3×C5×Q8—42no—
140C3×C5×D4—21no—
141C3×C5×D4—21no—
142C3×C5×D4—44no—
143C3×C5×D4D814yescomplemented
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—44no—
149C2×C3×C4×C5—42no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2411yescharacteristic
154#1220—44no—
155C2×Dic15—21no—
156D60—22no—
157D60—21no—
158D60—22no—
159D60—44no—
160D60—42no—
161D60—81no—
162D60#4112yescomplemented
163C4×D15—42no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C3⋊#168—22no—
167C2×C3×#41—21no—
168C2×D3×D4—101no—
169C22×D12—52no—
170C3×C4⋊Q8C2×D511yescharacteristic, complemented
171C3×#123—22no—
172C3×C22⋊Q8—21no—
173C3×#121C2×D512yescomplemented
174C3×C4×D4C2×D511yescharacteristic
175C3×C4×D4C2×D511yescharacteristic, complemented
176C2×C3×C4⋊C4C2×D512yescomplemented
177C2×#213—51no—
178C2×#213—52no—
179#930—52no—
180C22×C3⋊C8—52no—
181#895—101no—
182#882—102no—
183#874—51no—
184C4×D12—51no—
185#832—51no—
186#830—52no—
187#809—51no—
188#809—52no—
189#809—52no—
190#809—52no—
191#809—52no—
192#807—51no—
193#807—52no—
194C4⋊(C3⋊C8)—51no—
195C4×C3⋊C8—51no—
196C23×D5—62no—
197C5×#41—44no—
198C2×C5×Q8—21no—
199C2×C5×D4—22no—
200C2×C5×D4#6711yescharacteristic
201C2×C5×D4C22×D311yescharacteristic
202C22×C4×C5—22no—
203C22×C4×C5—21no—
204C22×C4×C5C22×D312yes—
205C2×C22⋊D5—62no—
206D4×D5—124no—
207C2×D20—31no—
208C2×D20—32no—
209C2×D20—64no—
210C2×D20—62no—
211C2×D20—61no—
212C2×D20—61no—
213C2×D20—122no—
214C2×C4×D5—61no—
215C5×C4⋊C4—22no—
216C5×C4⋊C4—21no—
217C5×C4⋊C4#6711yescharacteristic
218C5×C4⋊C4C22×D311yescharacteristic
219C5×C4⋊C4C22×D312yes—
220C5×C22⋊C4—21no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4#6712yes—
223C42×C5C22×D311yescharacteristic
224#686—62no—
225#686—122no—
226#685—61no—
227#685—62no—
228C4⋊Dic5—31no—
229C2×#168—31no—
230C2×#168—31no—
231C2×#168—32no—
232C2×#168—122no—
233#547D1511yescharacteristic, complemented
234D42—151no—
235#486—151no—
236#480—152no—
237#474—151no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—151no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5—22no—
245C22×C3×C5—22no—
246C22×C3×C5—41no—
247C22×C3×C5—41no—
248C22×C3×C5—41no—
249C22×C3×C5C2×D814yes—
250C22×C3×C5C22×D411yescharacteristic
251C2×D15—22no—
252C2×D15—41no—
253C2×D15—41no—
254C2×D15—42no—
255C2×D15—42no—
256C2×D15—42no—
257C2×D15—88no—
258C3×C4×C5—22no—
259C3×C4×C5—21no—
260C3×C4×C5—41no—
261C3×C4×C5—42no—
262C3×C4×C5C2×D812yes—
263C3×C4×C5C2×#4111yescharacteristic
264C3×C4×C5H(5,𝔽2)11yescharacteristic
265Dic15—41no—
266C23×D3—102no—
267C3×#41—44no—
268C2×C3×Q8—21no—
269C2×C3×D4—22no—
270C2×C3×D4C22⋊D511yescharacteristic
271C2×C3×D4C22×D511yescharacteristic
272C22×C3×C4—22no—
273C22×C3×C4—21no—
274C22×C3×C4C22×D512yes—
275C2×#67—102no—
276D3×D4—204no—
277C2×D12—51no—
278C2×D12—52no—
279C2×D12—104no—
280C2×D12—101no—
281C2×D12—102no—
282C2×D12—101no—
283C2×D12—202no—
284C2×C4×D3—101no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4C22⋊D511yescharacteristic
288C3×C4⋊C4C22×D511yescharacteristic
289C3×C4⋊C4C22×D512yes—
290C3×C22⋊C4—21no—
291C3×C22⋊C4—22no—
292C3×C22⋊C4C22⋊D512yes—
293C3×C42C22×D511yescharacteristic
294#213—102no—
295#213—202no—
296(C2×D3)⋊C4—101no—
297(C2×D3)⋊C4—102no—
298C4⋊Dic3—51no—
299C2×C3⋊C8—51no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—52no—
302C2×C3⋊C8—202no—
303C23×C5—21no—
304C23×C5C2×#6712yes—
305C22×D5—61no—
306C22×D5—62no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×Q8—42no—
311C5×D4—21no—
312C5×D4—21no—
313C5×D4—44no—
314C5×D4#21314yescomplemented
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—21no—
318C2×C4×C5—22no—
319C2×C4×C5—44no—
320C2×C4×C5—42no—
321C2×C4×C5C2×#6711yescharacteristic
322C2×C4×C5C2×#6712yes—
323C2×C4×C5C2×#6711yescharacteristic
324C2×C4×C5C23×D311yescharacteristic
325C22⋊D5—124no—
326C2×Dic5—61no—
327D20—32no—
328D20—62no—
329D20—61no—
330D20—62no—
331D20—124no—
332D20—122no—
333D20—241no—
334C4×D5—122no—
335#168—61no—
336#168—61no—
337#168—62no—
338C2×#41—21no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8C2×D1511yescharacteristic, complemented
345#127—151no—
346#123—22no—
347C22⋊Q8—21no—
348#121—152no—
349#121—301no—
350#121C2×D1512yescomplemented
351#120—302no—
352C4×D4—151no—
353C4×D4C2×D1511yescharacteristic
354C4×D4C2×D1511yescharacteristic, complemented
355C2×C4⋊C4C2×D1512yescomplemented
356#107—151no—
357#107—152no—
358C4⋊C8—151no—
359#102—151no—
360#102—151no—
361#102—152no—
362#102—152no—
363#102—152no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5—24no—
368C2×C3×C5—81no—
369C2×C3×C5#54611yescharacteristic
370C2×C3×C5C22×D811yescharacteristic
371C2×C3×C5#57711yescharacteristic
372D15—42no—
373D15—82no—
374C23×C3—21no—
375C23×C3C2×C22⋊D512yes—
376C22×D3—101no—
377C22×D3—102no—
378C22×D3—204no—
379C22×D3—204no—
380C22×D3—402no—
381C3×Q8—42no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—44no—
385C3×D4#68614yescomplemented
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—44no—
391C2×C3×C4—42no—
392C2×C3×C4C2×C22⋊D511yescharacteristic
393C2×C3×C4C2×C22⋊D512yes—
394C2×C3×C4C2×C22⋊D511yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—204no—
397C2×Dic3—101no—
398D12—52no—
399D12—102no—
400D12—101no—
401D12—102no—
402D12—204no—
403D12—202no—
404D12—401no—
405C4×D3—202no—
406C3⋊C8—101no—
407C3⋊C8—101no—
408C3⋊C8—102no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—41no—
414C22×C5C2×#21314yes—
415C22×C5C22×#6711yescharacteristic
416C2×D5—62no—
417C2×D5—121no—
418C2×D5—121no—
419C2×D5—122no—
420C2×D5—122no—
421C2×D5—122no—
422C2×D5—248no—
423C4×C5—22no—
424C4×C5—21no—
425C4×C5—41no—
426C4×C5—42no—
427C4×C5C2×#21312yes—
428C4×C5D3×#4111yescharacteristic
429C4×C5#100911yescharacteristic
430Dic5—121no—
431C24—302no—
432#41—44no—
433C2×Q8—21no—
434C2×D4—22no—
435C2×D4—151no—
436C2×D4—152no—
437C2×D4—304no—
438C2×D4—302no—
439C2×D4—302no—
440C2×D4—301no—
441C2×D4—301no—
442C2×D4—602no—
443C2×D4—604no—
444C2×D4#122011yescharacteristic
445C2×D4C22×D1511yescharacteristic
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—301no—
449C22×C4C22×D1512yes—
450D8—302no—
451D8—602no—
452C2×C8—151no—
453C2×C8—151no—
454C2×C8—152no—
455C2×C8—602no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—151no—
459C4⋊C4#122011yescharacteristic
460C4⋊C4C22×D1511yescharacteristic
461C4⋊C4C22×D1512yes—
462C22⋊C4—21no—
463C22⋊C4—22no—
464C22⋊C4—301no—
465C22⋊C4—302no—
466C22⋊C4#122012yes—
467C42C22×D1511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×#68614yes—
475C22×C3C22×C22⋊D511yescharacteristic
476C2×D3—102no—
477C2×D3—201no—
478C2×D3—201no—
479C2×D3—202no—
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—408no—
483C3×C4—22no—
484C3×C4—21no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4C2×#68612yes—
488C3×C4D5×#4111yescharacteristic
489C3×C4#414911yescharacteristic
490Dic3—201no—
491C2×C5—24no—
492C2×C5—81no—
493C2×C5C22×#21311yescharacteristic
494C2×C5#582911yescharacteristic
495C2×C5#583711yescharacteristic
496D5—122no—
497D5—242no—
498C23—21no—
499C23—301no—
500C23—302no—
501C23—604no—
502C23—604no—
503C23—1202no—
504C23C2×#122012yes—
505Q8—42no—
506D4—21no—
507D4—21no—
508D4—44no—
509D4—152no—
510D4—302no—
511D4—301no—
512D4—302no—
513D4—604no—
514D4—604no—
515D4—602no—
516D4—1201no—
517D4#671514yescomplemented
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—301no—
525C2×C4—602no—
526C2×C4C2×#122011yescharacteristic
527C2×C4C2×#122012yes—
528C2×C4C2×#122011yescharacteristic
529C2×C4C23×D1511yescharacteristic
530C8—301no—
531C8—301no—
532C8—302no—
533C2×C3—24no—
534C2×C3—81no—
535C2×C3C22×#68611yescharacteristic
536C2×C3#6616311yescharacteristic
537C2×C3#6617111yescharacteristic
538D3—202no—
539D3—402no—
540C5#8352511yescharacteristic, complemented, Sylow
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—302no—
547C22—601no—
548C22—601no—
549C22—602no—
550C22—602no—
551C22—602no—
552C22—1208no—
553C22C2×#671514yes—
554C22C22×#122011yescharacteristic
555C4—22no—
556C4—21no—
557C4—41no—
558C4—42no—
559C4—601no—
560C4C2×#671512yes—
561C4#41×D1511yescharacteristic
562C4#9210011yescharacteristic
563C3#1061525611yescharacteristic, complemented, Sylow
564C2—24no—
565C2—81no—
566C2—602no—
567C2—1202no—
568C2C22×#671511yescharacteristic
569C2#1175658811yescharacteristic
570C2#1175659611yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#547C2
socleC22×C3×C5C22×D4
hypercenter#547D15
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×C5×#547C2semidirect
2. D4×D60C2semidirect
3. #11751189C2semidirect
4. #11751188C2semidirect
5. #11751139C2semidirect
6. #11751137C2semidirect
7. #11751116C2semidirect
8. C4×#6715C2semidirect
9. D4×C3⋊#168C2semidirect
10. #11751068C2semidirect
11. C2×#91093C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
19. C3×C5×#121C22semidirect
21. C3×C4×C5×D4C22semidirect
22. C2×C3×C5×C4⋊C4C22semidirect
24. C2×#6715C22semidirect
26. C22×C3⋊#168C22semidirect
31. #91116C22semidirect
32. #91114C22semidirect
35. #91093C22semidirect
36. #91093C22semidirect
37. #91093C22semidirect
38. #91091C22semidirect
39. #91091C22semidirect
40. C4⋊(C3⋊#168)C22semidirect
41. C4×C3⋊#168C22semidirect
43. C5×#547D3semidirect
91. C3×#547D5semidirect
105. C5×C4⋊Q8C2×D3semidirect
108. C5×#121C2×D3semidirect
110. C4×C5×D4C2×D3semidirect
111. C2×C5×C4⋊C4C2×D3semidirect
143. C3×C5×D4D8semidirect
162. D60#41semidirect
170. C3×C4⋊Q8C2×D5semidirect
173. C3×#121C2×D5semidirect
175. C3×C4×D4C2×D5semidirect
176. C2×C3×C4⋊C4C2×D5semidirect
233. #547D15semidirect
314. C5×D4#213semidirect
344. C4⋊Q8C2×D15semidirect
350. #121C2×D15semidirect
354. C4×D4C2×D15semidirect
355. C2×C4⋊C4C2×D15semidirect
385. C3×D4#686semidirect
468. C3×C5#3294semidirect
517. D4#6715semidirect
540. C5#83525semidirect
563. C3#10615256semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5472
D4×D602
#117511892
#117511882
#117511392
#117511372
#117511162
C4×#67152
D4×C3⋊#1682
#117510682
C2×#910932
#106152563
#835255