Finite Group Databasebeta

← #14458389

Subgroups — #14458389

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
2D12×D20C211yesmaximal, characteristic, complemented
3#11751189C211yesmaximal, characteristic, complemented
4C5×#5087C211yesmaximal, characteristic, complemented
5#11747480C212yesmaximal, complemented
6D5×#809C212yesmaximal, complemented
7#11747442C212yesmaximal, complemented
8#11747164C212yesmaximal, complemented
9#11747028C211yesmaximal, characteristic, complemented
10C4×#6652C211yesmaximal, characteristic, complemented
11#11746996C211yesmaximal, characteristic, complemented
12#10612711—31nomaximal, Hall
13C2×C3×#713—21no—
14C2×D3×D20—21no—
15C2×D5×D12C2212yes—
16C3×C5×C4⋊Q8C2211yescharacteristic, complemented
17#91763C2211yescharacteristic
18C4×C5×D12C2211yescharacteristic
19C3×#4092—21no—
20C3×#4041C2212yescomplemented
21C3×#4040—22no—
22C3×D5×C4⋊C4C2212yescomplemented
23C3×C4×D20C2211yescharacteristic
24C3×C4×D20C2211yescharacteristic, complemented
25#91476—22no—
26#91460—21no—
27#91450C2212yes—
28C2×#6653C2212yescomplemented
29C2×#6652C2211yescharacteristic
30C2×D5×C3⋊C8C2212yescomplemented
31#91093C2212yescomplemented
32#91093C2212yescomplemented
33C4×C3⋊#168C2211yescharacteristic, complemented
34C5×#809C2212yes—
35C5×#807C2212yescomplemented
36C5×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
37#90989C2211yescharacteristic, complemented
38#90968C2212yescomplemented
39#90966C2211yescharacteristic
40#90964C2211yescharacteristic, complemented
41#90955C2212yescomplemented
42#83525—51nomaximal, Hall
43#66060D311yescharacteristic, complemented
44D4×D20—31no—
45C5×#474—31no—
46#65377—31no—
47C4×#686—31no—
48#65172—32no—
49#65171—32no—
50#65132—31no—
51#65123—31no—
52#65062—32no—
53D5×#102—32no—
54C22×D3×D5—22no—
55C2×C3×C5×Q8—21no—
56C2×D60—21no—
57C2×D60—21no—
58C2×D60C2312yes—
59C2×C5×D12C2311yescharacteristic
60C2×C4×C5×D3—21no—
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×#1202—22no—
70D3×D20—44no—
71D5×D12—24no—
72D5×D12—42no—
73C3×C5×C4⋊C4—21no—
74C3×C5×C4⋊C4C2312yes—
75C3×C42×C5C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2311yescharacteristic
78C5×(C2×D3)⋊C4—21no—
79C5×C4⋊Dic3C2311yescharacteristic
80C2×C5×C3⋊C8C2312yes—
81C3×#685—21no—
82C3×#685—22no—
83C3×#685D412yes—
84C3×C4⋊Dic5D411yescharacteristic
85C3×C4⋊Dic5C2311yescharacteristic
86C3×#683—22no—
87#6667—22no—
88#6653—42no—
89#6652—22no—
90D5×C3⋊C8—42no—
91C3×#547—51no—
92D4×D12—51no—
93#5088—51no—
94#5087D511yescharacteristic, complemented
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—
104C22×D20—61no—
105C5×C4⋊Q8C2×D311yescharacteristic, complemented
106C4×C5×D4—31no—
107#4092—21no—
108C2×#686—31no—
109#4076—61no—
110C2×D40—32no—
111C2×C8×D5—32no—
112#4041—32no—
113#4041C2×D312yescomplemented
114#4040—22no—
115D5×C4⋊C4C2×D312yescomplemented
116#4028—62no—
117#4020—31no—
118C4×D20C2×D311yescharacteristic
119C4×D20C2×D311yescharacteristic, complemented
120C5×#107—32no—
121C5×C4⋊C8—31no—
122C5×#102—32no—
123#3964—31no—
124#3953—32no—
125#3952—32no—
126#3941—31no—
127#3941—32no—
128#3941—32no—
129#3939—31no—
130C4×#168—31no—
131#3294—151noSylow
132C22×D15—22no—
133C22×C5×D3—21no—
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
144D60—22no—
145D60—22no—
146D60—42no—
147D60—81no—
148C2×C5×Dic3—21no—
149C5×D12—21no—
150C5×D12#4112yescomplemented
151C4×C5×D3—42no—
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#1202—44no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C5×C3⋊C8—22no—
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—51no—
178C2×#213—52no—
179#930—52no—
180C22×C3⋊C8—52no—
181#895—101no—
182#882—102no—
183#874—51no—
184C4×D12C2×D511yescharacteristic
185#832—51no—
186#830—52no—
187#809—51no—
188#809—52no—
189#809—52no—
190#809—52no—
191#809C2×D512yes—
192#807—51no—
193#807C2×D512yescomplemented
194C4⋊(C3⋊C8)C2×D511yescharacteristic, complemented
195C4×C3⋊C8—51no—
196C23×D5—62no—
197C2×C5×Q8—21no—
198C2×C5×D4—31no—
199C22×C4×C5—61no—
200C2×C22⋊D5—62no—
201#713—44no—
202D4×D5—64no—
203D4×D5—122no—
204C2×D20—22no—
205C2×D20—32no—
206C2×D20—62no—
207C2×D20—61no—
208C2×D20—61no—
209C2×D20—124no—
210C2×D20#6711yescharacteristic
211C2×D20C22×D311yescharacteristic
212C2×C4×D5—22no—
213C2×C4×D5—21no—
214C2×C4×D5C22×D312yes—
215C2×C5×C8—32no—
216C5×C4⋊C4—21no—
217C5×C4⋊C4—31no—
218C5×C4⋊C4C22×D312yes—
219C5×C22⋊C4—61no—
220C42×C5C22×D311yescharacteristic
221#686—62no—
222#685—21no—
223#685—22no—
224#685—62no—
225#685#6712yes—
226C4⋊Dic5#6711yescharacteristic
227C4⋊Dic5C22×D311yescharacteristic
228#683—22no—
229C2×#168—31no—
230C2×#168—31no—
231D40—122no—
232C8×D5—122no—
233#547—51no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—31no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—151no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5C22×D411yescharacteristic
245C2×D15—42no—
246C2×D15—42no—
247C2×D15—42no—
248C2×C5×D3—22no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
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
263C3×Dic5—22no—
264C3×Dic5—41no—
265C5×Dic3—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—52no—
278C2×D12—104no—
279C2×D12—102no—
280C2×D12—101no—
281C2×D12—101no—
282C2×D12—202no—
283C2×D12C22×D511yescharacteristic
284C2×C4×D3—21no—
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—21no—
297(C2×D3)⋊C4—102no—
298C4⋊Dic3C22×D511yescharacteristic
299C2×C3⋊C8—51no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—202no—
302C2×C3⋊C8C22×D512yes—
303C23×C5—61no—
304C22×D5—21no—
305C22×D5—62no—
306C22×D5—124no—
307C22×D5—124no—
308C22×D5—242no—
309C22×D5C2×#6712yes—
310C5×Q8—42no—
311C5×D4—32no—
312C5×D4—61no—
313C2×C4×C5—22no—
314C2×C4×C5—61no—
315C2×C4×C5—122no—
316C2×C4×C5C2×#6711yescharacteristic
317C2×C4×C5C2×#6711yescharacteristic
318C2×C4×C5C23×D311yescharacteristic
319C22⋊D5—124no—
320C2×Dic5—21no—
321C2×Dic5C2×#6712yes—
322D20—21no—
323D20—21no—
324D20—44no—
325D20—62no—
326D20—62no—
327D20—124no—
328D20—122no—
329D20—241no—
330D20#21314yescomplemented
331C4×D5—22no—
332C4×D5—22no—
333C4×D5—44no—
334C4×D5—42no—
335C5×C8—62no—
336#168—61no—
337#168—61no—
338C2×#41—101no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—152no—
342C2×D8—151no—
343C22×C8—152no—
344C4⋊Q8D3×D511yescharacteristic, complemented
345#127—151no—
346#123—102no—
347C22⋊Q8—101no—
348#121—52no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—51no—
354C4×D4—51no—
355C2×C4⋊C4—52no—
356#107—32no—
357#107—151no—
358C4⋊C8—31no—
359#102—32no—
360#102—151no—
361#102—152no—
362#102—151no—
363#102—152no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5C22×D811yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—82no—
371C3×D5—24no—
372C3×D5—81no—
373C5×D3—42no—
374C23×C3—52no—
375C23×C3—101no—
376C22×D3—21no—
377C22×D3—102no—
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—21no—
398D12—21no—
399D12—102no—
400D12—102no—
401D12—204no—
402D12—202no—
403D12—401no—
404D12#71312yescomplemented
405C4×D3—42no—
406C3⋊C8—22no—
407C3⋊C8—101no—
408C3⋊C8—101no—
409C22×C5—62no—
410C22×C5—121no—
411C22×C5—121no—
412C22×C5C22×#6711yescharacteristic
413C2×D5—22no—
414C2×D5—22no—
415C2×D5—41no—
416C2×D5—41no—
417C2×D5—41no—
418C2×D5—122no—
419C2×D5—122no—
420C2×D5—122no—
421C2×D5—248no—
422C2×D5C2×#21314yes—
423C4×C5—21no—
424C4×C5—42no—
425C4×C5—121no—
426C4×C5C2×#21312yes—
427C4×C5D3×#4111yescharacteristic
428C4×C5#100911yescharacteristic
429Dic5—22no—
430Dic5—41no—
431C24—302no—
432#41—204no—
433C2×Q8—21no—
434C2×D4—31no—
435C2×D4—51no—
436C2×D4—51no—
437C2×D4—102no—
438C2×D4—152no—
439C2×D4—302no—
440C2×D4—302no—
441C2×D4—304no—
442C2×D4—301no—
443C2×D4—301no—
444C2×D4—602no—
445C2×D4—604no—
446C22×C4—52no—
447C22×C4—61no—
448C22×C4—102no—
449C22×C4—101no—
450D8—302no—
451D8—602no—
452C2×C8—32no—
453C2×C8—151no—
454C2×C8—151no—
455C2×C8—602no—
456C4⋊C4—21no—
457C4⋊C4—31no—
458C4⋊C4—51no—
459C4⋊C4—51no—
460C4⋊C4—102no—
461C4⋊C4C2×D3×D512yes—
462C22⋊C4—52no—
463C22⋊C4—61no—
464C22⋊C4—102no—
465C22⋊C4—101no—
466C22⋊C4—302no—
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—22no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—202no—
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—408no—
483C3×C4—21no—
484C3×C4—42no—
485C3×C4—102no—
486C3×C4—201no—
487C3×C4D5×D812yes—
488C3×C4C2×#71311yescharacteristic
489C3×C4#414911yescharacteristic
490Dic3—41no—
491C2×C5—122no—
492C2×C5C22×#21311yescharacteristic
493C2×C5#582911yescharacteristic
494C2×C5#583711yescharacteristic
495D5—24no—
496D5—81no—
497D5—242no—
498C23—52no—
499C23—61no—
500C23—101no—
501C23—302no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—42no—
506D4—32no—
507D4—54no—
508D4—61no—
509D4—101no—
510D4—101no—
511D4—204no—
512D4—302no—
513D4—302no—
514D4—604no—
515D4—604no—
516D4—602no—
517D4—1201no—
518C2×C4—22no—
519C2×C4—52no—
520C2×C4—61no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—101no—
524C2×C4—122no—
525C2×C4—204no—
526C2×C4—202no—
527C2×C4D5×#6711yescharacteristic
528C2×C4D5×#6711yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—62no—
531C8—301no—
532C8—301no—
533C2×C3—104no—
534C2×C3—401no—
535C2×C3#6596411yescharacteristic
536C2×C3C2×D5×D811yescharacteristic
537C2×C3#6611211yescharacteristic
538D3—42no—
539D3—402no—
540C5#8352511yescharacteristic, complemented, Sylow
541C22—54no—
542C22—62no—
543C22—102no—
544C22—102no—
545C22—121no—
546C22—121no—
547C22—201no—
548C22—201no—
549C22—201no—
550C22—602no—
551C22—602no—
552C22—602no—
553C22—1208no—
554C22C2×D5×#6711yescharacteristic
555C4—21no—
556C4—42no—
557C4—102no—
558C4—121no—
559C4—201no—
560C4D5×#21312yes—
561C4D3×#71311yescharacteristic
562C4#9203511yescharacteristic
563C3#1061271111yescharacteristic, complemented, Sylow
564C2—104no—
565C2—122no—
566C2—401no—
567C2—1202no—
568C2C2×D5×#21311yescharacteristic
569C2#1175508611yescharacteristic
570C2#1175527311yescharacteristic
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. D12×D20C2semidirect
3. #11751189C2semidirect
4. C5×#5087C2semidirect
5. #11747480C2semidirect
6. D5×#809C2semidirect
7. #11747442C2semidirect
8. #11747164C2semidirect
9. #11747028C2semidirect
10. C4×#6652C2semidirect
11. #11746996C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
20. C3×#4041C22semidirect
22. C3×D5×C4⋊C4C22semidirect
24. C3×C4×D20C22semidirect
28. C2×#6653C22semidirect
30. C2×D5×C3⋊C8C22semidirect
31. #91093C22semidirect
32. #91093C22semidirect
33. C4×C3⋊#168C22semidirect
35. C5×#807C22semidirect
36. C5×C4⋊(C3⋊C8)C22semidirect
37. #90989C22semidirect
38. #90968C22semidirect
40. #90964C22semidirect
41. #90955C22semidirect
43. #66060D3semidirect
94. #5087D5semidirect
105. C5×C4⋊Q8C2×D3semidirect
113. #4041C2×D3semidirect
115. D5×C4⋊C4C2×D3semidirect
119. C4×D20C2×D3semidirect
150. C5×D12#41semidirect
157. C3×D20D8semidirect
170. C3×C4⋊Q8C2×D5semidirect
193. #807C2×D5semidirect
194. C4⋊(C3⋊C8)C2×D5semidirect
330. D20#213semidirect
344. C4⋊Q8D3×D5semidirect
404. D12#713semidirect
468. C3×C5#3294semidirect
540. C5#83525semidirect
563. C3#10612711semidirect

Maximal subgroups

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

subgroupindex
C3×#660602
D12×D202
#117511892
C5×#50872
#117474802
D5×#8092
#117474422
#117471642
#117470282
C4×#66522
#117469962
#106127113
#835255