Finite Group Databasebeta

← #14458394

Subgroups — #14458394

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×#66060C211yesmaximal, characteristic, complemented
2#11752737C211yesmaximal, characteristic, complemented
3#11751188C211yesmaximal, characteristic, complemented
4C5×#5088C211yesmaximal, characteristic, complemented
5#11747480C212yesmaximal, complemented
6D5×#809C212yesmaximal, complemented
7#11747307C212yesmaximal, complemented
8#11747306C212yesmaximal, complemented
9#11747026C211yesmaximal, characteristic, complemented
10C4×#6653C211yesmaximal, characteristic, complemented
11C3⋊C8×D20C211yesmaximal, characteristic, complemented
12#10613307—31nomaximal, Hall
13C2×C3×#713—21no—
14C2×#6777—21no—
15C2×D5×D12C2212yes—
16C3×C5×C4⋊Q8C2211yescharacteristic, complemented
17C4×D60C2211yescharacteristic
18C5×#874C2211yescharacteristic
19C3×#4092—21no—
20C3×#4041C2212yescomplemented
21C3×#4040—22no—
22C3×D5×C4⋊C4C2212yescomplemented
23C3×C4×D20C2211yescharacteristic
24C3×C4×D20C2211yescharacteristic, complemented
25#91475—22no—
26#91466—21no—
27#91449C2212yes—
28C2×#6653C2211yescharacteristic
29C2×#6652C2212yescomplemented
30C2×D5×C3⋊C8C2212yescomplemented
31#91093C2212yes—
32#91091C2212yescomplemented
33C4⋊(C3⋊#168)C2211yescharacteristic, complemented
34C5×#809C2212yescomplemented
35C5×#809C2212yescomplemented
36C4×C5×C3⋊C8C2211yescharacteristic, complemented
37#90990C2211yescharacteristic, complemented
38#90968C2211yescharacteristic
39#90966C2212yescomplemented
40#90965C2211yescharacteristic, complemented
41#90955C2212yescomplemented
42#83525—51nomaximal, Hall
43#66060D311yescharacteristic, complemented
44#66051—31no—
45C5×#486—31no—
46#65376—31no—
47#65361—31no—
48#65064—32no—
49#65063—32no—
50#65062—32no—
51D5×#102—32no—
52C4×D40—31no—
53C8×D20—31no—
54C22×D3×D5—22no—
55C2×C3×C5×Q8—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×C5×D12—21no—
59C2×C5×D12—21no—
60C2×C5×D12C2312yes—
61C3×#713—44no—
62C2×C3×D20—22no—
63C2×C3×D20D411yescharacteristic
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—21no—
67C2×C3×C4×D5C2312yes—
68C2×#1203—22no—
69C2×#1201—22no—
70#6777—44no—
71D5×D12—24no—
72D5×D12—42no—
73C3×C5×C4⋊C4—21no—
74C3×C5×C4⋊C4C2312yes—
75C3×C42×C5C2311yescharacteristic
76#6714—21no—
77C4⋊Dic15C2311yescharacteristic
78C2×C3⋊#168C2312yes—
79C2×C5×C3⋊C8C2311yescharacteristic
80C2×C5×C3⋊C8C2311yescharacteristic
81C3×#685—21no—
82C3×#685—22no—
83C3×#685D412yes—
84C3×C4⋊Dic5D411yescharacteristic
85C3×C4⋊Dic5C2311yescharacteristic
86C3×#683—22no—
87(C2×D3)⋊Dic5—22no—
88#6653—22no—
89#6652—42no—
90D5×C3⋊C8—42no—
91C3×#547—51no—
92D4×D12—51no—
93#5088D511yescharacteristic, complemented
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×#713—21no—
103C2×D4×D5—32no—
104C2×D4×D5—61no—
105C5×C4⋊Q8C2×D311yescharacteristic, complemented
106C5×#127—31no—
107#4092—21no—
108#4084—32no—
109#4084—61no—
110#4083—62no—
111C2×#686—32no—
112C2×D40—31no—
113C2×C8×D5—32no—
114#4041C2×D312yescomplemented
115#4040—22no—
116D5×C4⋊C4C2×D312yescomplemented
117C4×D20—31no—
118C4×D20C2×D311yescharacteristic
119C4×D20C2×D311yescharacteristic, complemented
120C5×#102—32no—
121C5×#102—32no—
122C4×C5×C8—31no—
123#3964—32no—
124#3953—31no—
125#3953—31no—
126#3952—32no—
127#3950—31no—
128#3941—32no—
129#3939—32no—
130C4⋊#168—31no—
131#3294—151noSylow
132C22×D15—21no—
133C22×C5×D3—22no—
134C22×C3×D5—21no—
135C22×C3×D5C2×D412yes—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C3×C5×Q8—42no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2411yescharacteristic
144C2×Dic15—21no—
145D60—21no—
146D60#4112yescomplemented
147C4×D15—42no—
148C5×D12—22no—
149C5×D12—22no—
150C5×D12—42no—
151C5×D12—81no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5C2×D412yes—
154C3×D20—21no—
155C3×D20—21no—
156C3×D20—44no—
157C3×D20D814yescomplemented
158C3×C4×D5—22no—
159C3×C4×D5—22no—
160C3×C4×D5—44no—
161C3×C4×D5—42no—
162#1203—44no—
163#1201—44no—
164C3⋊#168—22no—
165C5×C3⋊C8—21no—
166C5×C3⋊C8—21no—
167C2×C3×#41—101no—
168C2×D3×D4—101no—
169C22×D12—52no—
170C3×C4⋊Q8C2×D511yescharacteristic, complemented
171C3×#123—102no—
172C3×C22⋊Q8—101no—
173C3×#121—52no—
174C3×C4×D4—51no—
175C3×C4×D4—51no—
176C2×C3×C4⋊C4—52no—
177C2×#213—52no—
178C2×#213—51no—
179#930—52no—
180C22×C3⋊C8—52no—
181#895—101no—
182#882—102no—
183#874C2×D511yescharacteristic
184C4×D12—51no—
185#832—51no—
186#830—52no—
187#809—52no—
188#809—52no—
189#809—51no—
190#809C2×D512yescomplemented
191#809C2×D512yescomplemented
192#807—52no—
193#807—51no—
194C4⋊(C3⋊C8)—51no—
195C4×C3⋊C8C2×D511yescharacteristic, complemented
196C23×D5—62no—
197C2×C5×Q8—21no—
198C2×C5×D4—32no—
199C2×C5×D4—61no—
200C2×C5×D4—61no—
201C2×C22⋊D5—62no—
202C2×C22⋊D5—62no—
203#713—44no—
204D4×D5—64no—
205D4×D5—122no—
206D4×D5—124no—
207C2×D20—22no—
208C2×D20—31no—
209C2×D20#6711yescharacteristic
210C2×D20C22×D311yescharacteristic
211C2×C4×D5—22no—
212C2×C4×D5—21no—
213C2×C4×D5—61no—
214C2×C4×D5C22×D312yes—
215C2×C5×C8—31no—
216C2×C5×C8—31no—
217C5×C4⋊C4—21no—
218C5×C4⋊C4C22×D312yes—
219C42×C5C22×D311yescharacteristic
220C22⋊Dic5—62no—
221#686—122no—
222#685—21no—
223#685—22no—
224#685—61no—
225#685#6712yes—
226C4⋊Dic5—31no—
227C4⋊Dic5#6711yescharacteristic
228C4⋊Dic5C22×D311yescharacteristic
229#683—22no—
230C2×#168—32no—
231D40—62no—
232C8×D5—122no—
233#547—51no—
234D42—151no—
235#486—31no—
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×C5C22×D411yescharacteristic
245C2×D15—22no—
246C2×D15—41no—
247C2×D15—41no—
248C2×C5×D3—42no—
249C2×C5×D3—42no—
250C2×C5×D3—42no—
251C2×C3×D5—22no—
252C2×C3×D5—22no—
253C2×C3×D5—41no—
254C2×C3×D5—41no—
255C2×C3×D5—41no—
256C2×C3×D5C2×D814yes—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5—42no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263Dic15—41no—
264C3×Dic5—22no—
265C3×Dic5—41no—
266C23×D3—102no—
267C3×#41—204no—
268C2×C3×Q8—21no—
269C2×C3×D4—51no—
270C2×C3×D4—51no—
271C2×C3×D4—102no—
272C22×C3×C4—52no—
273C22×C3×C4—102no—
274C22×C3×C4—101no—
275C2×#67—102no—
276D3×D4—204no—
277C2×D12—21no—
278C2×D12—21no—
279C2×D12—51no—
280C2×D12—104no—
281C2×D12—102no—
282C2×D12—202no—
283C2×D12C22×D512yes—
284C2×C4×D3—101no—
285C3×C4⋊C4—21no—
286C3×C4⋊C4—51no—
287C3×C4⋊C4—51no—
288C3×C4⋊C4—102no—
289C3×C4⋊C4C22×D512yes—
290C3×C22⋊C4—52no—
291C3×C22⋊C4—102no—
292C3×C22⋊C4—101no—
293C3×C42C22×D511yescharacteristic
294#213—102no—
295#213—202no—
296(C2×D3)⋊C4—102no—
297(C2×D3)⋊C4—101no—
298C4⋊Dic3—51no—
299C2×C3⋊C8—52no—
300C2×C3⋊C8—202no—
301C2×C3⋊C8C22×D511yescharacteristic
302C2×C3⋊C8C22×D511yescharacteristic
303C23×C5—62no—
304C22×D5—21no—
305C22×D5—61no—
306C22×D5—124no—
307C22×D5—124no—
308C22×D5—242no—
309C22×D5C2×#6712yes—
310C5×Q8—42no—
311C5×D4—62no—
312C5×D4—62no—
313C5×D4—122no—
314C5×D4—241no—
315C2×C4×C5—22no—
316C2×C4×C5C2×#6711yescharacteristic
317C2×C4×C5C2×#6711yescharacteristic
318C2×C4×C5C23×D311yescharacteristic
319C22⋊D5—124no—
320C22⋊D5—124no—
321C2×Dic5—21no—
322C2×Dic5—61no—
323C2×Dic5C2×#6712yes—
324D20—21no—
325D20—21no—
326D20—32no—
327D20—44no—
328D20—61no—
329D20#21314yescomplemented
330C4×D5—22no—
331C4×D5—22no—
332C4×D5—44no—
333C4×D5—42no—
334C4×D5—122no—
335C5×C8—61no—
336C5×C8—61no—
337#168—62no—
338C2×#41—101no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8D3×D511yescharacteristic, complemented
345#127—31no—
346#123—102no—
347C22⋊Q8—101no—
348#121—52no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—51no—
353C4×D4—51no—
354C4×D4—151no—
355C2×C4⋊C4—52no—
356#107—151no—
357#107—152no—
358C4⋊C8—151no—
359#102—32no—
360#102—32no—
361#102—151no—
362#102—152no—
363#102—151no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—31no—
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5C22×D811yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—42no—
371C3×D5—24no—
372C3×D5—81no—
373C5×D3—82no—
374C23×C3—52no—
375C23×C3—101no—
376C22×D3—22no—
377C22×D3—101no—
378C22×D3—204no—
379C22×D3—204no—
380C22×D3—402no—
381C3×Q8—42no—
382C3×D4—54no—
383C3×D4—101no—
384C3×D4—101no—
385C3×D4—204no—
386C2×C3×C4—22no—
387C2×C3×C4—52no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—101no—
391C2×C3×C4—204no—
392C2×C3×C4—202no—
393C2×C3×C4D4×D511yescharacteristic
394C2×C3×C4D4×D511yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—204no—
397C2×Dic3—101no—
398D12—22no—
399D12—22no—
400D12—42no—
401D12—52no—
402D12—81no—
403D12—101no—
404D12—204no—
405C4×D3—202no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—102no—
409C22×C5—122no—
410C22×C5—122no—
411C22×C5—122no—
412C22×C5C22×#6711yescharacteristic
413C2×D5—22no—
414C2×D5—22no—
415C2×D5—41no—
416C2×D5—41no—
417C2×D5—41no—
418C2×D5—62no—
419C2×D5—121no—
420C2×D5—121no—
421C2×D5—248no—
422C2×D5C2×#21314yes—
423C4×C5—21no—
424C4×C5—42no—
425C4×C5C2×#21312yes—
426C4×C5D3×#4111yescharacteristic
427C4×C5#100911yescharacteristic
428Dic5—22no—
429Dic5—41no—
430Dic5—121no—
431C24—302no—
432#41—204no—
433C2×Q8—21no—
434C2×D4—32no—
435C2×D4—51no—
436C2×D4—51no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—102no—
440C2×D4—151no—
441C2×D4—302no—
442C2×D4—304no—
443C2×D4—302no—
444C2×D4—602no—
445C2×D4—604no—
446C22×C4—52no—
447C22×C4—102no—
448C22×C4—101no—
449C22×C4—301no—
450D8—302no—
451D8—602no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—152no—
455C2×C8—602no—
456C4⋊C4—21no—
457C4⋊C4—51no—
458C4⋊C4—51no—
459C4⋊C4—102no—
460C4⋊C4—151no—
461C4⋊C4C2×D3×D512yes—
462C22⋊C4—52no—
463C22⋊C4—102no—
464C22⋊C4—101no—
465C22⋊C4—302no—
466C22⋊C4—301no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—54no—
470C22×C3—102no—
471C22×C3—102no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3—201no—
475C22×C3C2×D4×D511yescharacteristic
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—102no—
480C2×D3—201no—
481C2×D3—201no—
482C2×D3—408no—
483C3×C4—21no—
484C3×C4—42no—
485C3×C4—102no—
486C3×C4—201no—
487C3×C4D5×D812yes—
488C3×C4D5×#4111yescharacteristic
489C3×C4#414911yescharacteristic
490Dic3—201no—
491C2×C5—242no—
492C2×C5C22×#21311yescharacteristic
493C2×C5#582911yescharacteristic
494C2×C5#583711yescharacteristic
495D5—24no—
496D5—81no—
497D5—122no—
498C23—52no—
499C23—62no—
500C23—101no—
501C23—301no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—42no—
506D4—54no—
507D4—62no—
508D4—62no—
509D4—101no—
510D4—101no—
511D4—122no—
512D4—152no—
513D4—204no—
514D4—241no—
515D4—301no—
516D4—604no—
517D4—604no—
518C2×C4—22no—
519C2×C4—52no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—101no—
523C2×C4—204no—
524C2×C4—202no—
525C2×C4—301no—
526C2×C4—602no—
527C2×C4D5×#6711yescharacteristic
528C2×C4D5×#6711yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—61no—
531C8—61no—
532C8—302no—
533C2×C3—104no—
534C2×C3—401no—
535C2×C3#6601011yescharacteristic
536C2×C3C2×D5×D811yescharacteristic
537C2×C3#6609611yescharacteristic
538D3—82no—
539D3—202no—
540C5#8352511yescharacteristic, complemented, Sylow
541C22—54no—
542C22—102no—
543C22—102no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—201no—
548C22—201no—
549C22—201no—
550C22—302no—
551C22—601no—
552C22—601no—
553C22—1208no—
554C22C2×D5×#6711yescharacteristic
555C4—21no—
556C4—42no—
557C4—102no—
558C4—201no—
559C4—601no—
560C4D5×#21312yes—
561C4#9203411yescharacteristic
562C4#9203511yescharacteristic
563C3#1061330711yescharacteristic, complemented, Sylow
564C2—104no—
565C2—242no—
566C2—401no—
567C2—602no—
568C2C2×D5×#21311yescharacteristic
569C2#1175508511yescharacteristic
570C2#1175527411yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66060C2semidirect
2. #11752737C2semidirect
3. #11751188C2semidirect
4. C5×#5088C2semidirect
5. #11747480C2semidirect
6. D5×#809C2semidirect
7. #11747307C2semidirect
8. #11747306C2semidirect
9. #11747026C2semidirect
10. C4×#6653C2semidirect
11. C3⋊C8×D20C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
20. C3×#4041C22semidirect
22. C3×D5×C4⋊C4C22semidirect
24. C3×C4×D20C22semidirect
29. C2×#6652C22semidirect
30. C2×D5×C3⋊C8C22semidirect
32. #91091C22semidirect
33. C4⋊(C3⋊#168)C22semidirect
34. C5×#809C22semidirect
35. C5×#809C22semidirect
36. C4×C5×C3⋊C8C22semidirect
37. #90990C22semidirect
39. #90966C22semidirect
40. #90965C22semidirect
41. #90955C22semidirect
43. #66060D3semidirect
93. #5088D5semidirect
105. C5×C4⋊Q8C2×D3semidirect
114. #4041C2×D3semidirect
116. D5×C4⋊C4C2×D3semidirect
119. C4×D20C2×D3semidirect
146. D60#41semidirect
157. C3×D20D8semidirect
170. C3×C4⋊Q8C2×D5semidirect
190. #809C2×D5semidirect
191. #809C2×D5semidirect
195. C4×C3⋊C8C2×D5semidirect
329. D20#213semidirect
344. C4⋊Q8D3×D5semidirect
468. C3×C5#3294semidirect
540. C5#83525semidirect
563. C3#10613307semidirect

Maximal subgroups

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

subgroupindex
C3×#660602
#117527372
#117511882
C5×#50882
#117474802
D5×#8092
#117473072
#117473062
#117470262
C4×#66532
C3⋊C8×D202
#106133073
#835255