Finite Group Databasebeta

← #14478523

Subgroups — #14478523

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5769C211yesmaximal, characteristic, complemented
2#11753971C211yesmaximal, characteristic, complemented
3#11751295C211yesmaximal, characteristic, complemented
4C3×#65482C211yesmaximal, characteristic, complemented
5#11748304C212yesmaximal, complemented
6#11748291C212yesmaximal, complemented
7#11748281C212yesmaximal, complemented
8D3×#3958C212yesmaximal, complemented
9#11747547C211yesmaximal, characteristic, complemented
10C2×#90937C211yesmaximal, characteristic, complemented
11#11747529C211yesmaximal, characteristic, complemented
12#10615738—31nomaximal, Hall
13C5×#1009—21no—
14#92017—21no—
15C3×C5×#135C2211yescharacteristic, complemented
16#91793C2211yescharacteristic, complemented
17C5×#949—21no—
18C5×#919C2212yescomplemented
19C5×#914—22no—
20C5×D3×D8C2212yescomplemented
21C2×C5×D24C2211yescharacteristic, complemented
22C5×#901C2211yescharacteristic
23C2×C3×D40C2211yescharacteristic, complemented
24#91312—21no—
25#91273C2212yescomplemented
26#91254—22no—
27D3×D40C2212yescomplemented
28#91113C2211yescharacteristic, complemented
29#91112C2211yescharacteristic, complemented
30#91111C2211yescharacteristic, complemented
31#91110C2211yescharacteristic, complemented
32C2×C3⋊#672C2211yescharacteristic, complemented
33C3×#3960C2212yescomplemented
34C3×#3958C2212yescomplemented
35C3×#3944C2211yescharacteristic, complemented
36#90944C2212yescomplemented
37#90939C2212yescomplemented
38#90937C2212yescomplemented
39#90937C2212yescomplemented
40#90935C2212yescomplemented
41D3×#672C2212yescomplemented
42#82036—51nomaximal, Hall
43C5×#576—31no—
44#66086—31no—
45#65483—32no—
46#65483—31no—
47#65482—32no—
48#65482D311yescharacteristic, complemented
49#65436—32no—
50#65436—31no—
51C2×#3958—31no—
52C2×#3958—32no—
53#65429—31no—
54C3×C5×#41—21no—
55#6817—21no—
56C5×#239—22no—
57C5×D3×D4—22no—
58C5×D3×D4—42no—
59C5×#235—41no—
60C5×#235D411yescharacteristic
61C2×C5×D12—21no—
62C2×C5×D12—41no—
63C2×C3×D20—21no—
64D5×#67—42no—
65#6777—41no—
66D3×D20—22no—
67#6766—41no—
68#6765—22no—
69C3×C5×Q16C2311yescharacteristic
70C3×C5×SD16—21no—
71C3×C5×D8C2311yescharacteristic
72C2×C3×C5×C8C2311yescharacteristic
73Dic60C2311yescharacteristic
74D120C2311yescharacteristic
75#6706—21no—
76C5×#215—22no—
77C5×#213—22no—
78C5×#208D411yescharacteristic
79C5×D24—21no—
80C5×D24C2312yes—
81C5×ΓL(1,𝔽25)D412yes—
82C5×D3×C8C2312yes—
83C3×D40—21no—
84C3×D40C2312yes—
85#6657—22no—
86#6653—22no—
87C3⋊#672C2311yescharacteristic
88C3⋊#672C2311yescharacteristic
89C3×#672C2312yes—
90#5769D511yescharacteristic, complemented
91#5757—51no—
92C3×#509—51no—
93#5194—51no—
94C2×#826—51no—
95#4916—52no—
96#4914—52no—
97#4911—52no—
98D3×D16—52no—
99#4908—51no—
100#4907—51no—
101C5×H(5,𝔽2)—61no—
102#4149—61no—
103C5×AGL(1,ℤ/8)—62no—
104C5×AGL(1,ℤ/8)—61no—
105C5×#135—32no—
106C5×#135C2×D311yescharacteristic, complemented
107C2×C5×D8—32no—
108C2×C5×D8—31no—
109C5×#131—31no—
110#4054—62no—
111#4054—61no—
112#4050—31no—
113#4050—32no—
114C2×D40—32no—
115C2×D40C2×D311yescharacteristic, complemented
116#3961—31no—
117#3960—31no—
118#3960C2×D312yescomplemented
119#3959—31no—
120#3959—32no—
121#3958—31no—
122#3958—32no—
123#3958—32no—
124#3958—32no—
125#3958C2×D312yescomplemented
126#3944—32no—
127#3944C2×D311yescharacteristic, complemented
128C2×#672—31no—
129C2×#672—32no—
130#3415—151noSylow
131C22×C5×D3—41no—
132C22×C5×D3—42no—
133C22×C3×D5—41no—
134C2×D3×D5—42no—
135C3×C5×Q8—21no—
136C3×C5×D4—21no—
137C3×C5×D4—41no—
138C2×C3×C4×C5—41no—
139C2×C3×C4×C5C2×D411yescharacteristic
140#1220—41no—
141D60—21no—
142C4×D15—41no—
143Dic30—21no—
144C5×#67—42no—
145C5×#67D812yes—
146C5×D12—22no—
147C5×D12—42no—
148C5×D12—41no—
149C5×D12D811yescharacteristic
150C4×C5×D3—42no—
151C4×C5×D3C2×D412yes—
152C5×Dic6D811yescharacteristic
153C3×D20—22no—
154C3×D20—41no—
155#1202—42no—
156#1201—42no—
157D5×Dic3—42no—
158C3×C5×C8C2×D411yescharacteristic
159C3×C5×C8C2411yescharacteristic
160C5×C3⋊C8C2×D412yes—
161#1009—21no—
162#1004—101no—
163C3×#135C2×D511yescharacteristic, complemented
164C2×C3×D8—51no—
165#949—21no—
166#932—101no—
167#919C2×D512yescomplemented
168#914—22no—
169#912—52no—
170#911—102no—
171D3×D8—52no—
172D3×D8C2×D512yescomplemented
173#904—51no—
174C2×D24C2×D511yescharacteristic, complemented
175#901C2×D511yescharacteristic
176C3×SD32—52no—
177C3×D16—52no—
178C3×M32—51no—
179#829—51no—
180#828—51no—
181#827—51no—
182#826—51no—
183#826—52no—
184#826—52no—
185C2×C3⋊C16—51no—
186#800—52no—
187D48—52no—
188#798—52no—
189D3×C16—52no—
190C5×#41—21no—
191C5×#41—31no—
192C5×#41—62no—
193C5×#41—121no—
194C2×C5×D4—62no—
195C2×C5×D4—61no—
196C2×C5×D4—121no—
197C2×C5×D4—122no—
198#713—121no—
199D4×D5—121no—
200D4×D5—122no—
201#709—62no—
202#709—61no—
203C2×D20—21no—
204C2×D20—62no—
205C5×Q16C22×D311yescharacteristic
206C5×SD16—21no—
207C5×SD16—62no—
208C5×D8—32no—
209C5×D8—62no—
210C5×D8—61no—
211C5×D8C22×D311yescharacteristic
212C5×M16—31no—
213C5×M16—32no—
214C2×C5×C8—32no—
215C2×C5×C8C22×D311yescharacteristic
216Dic20—31no—
217D40—21no—
218D40—31no—
219D40—62no—
220D40C22×D312yes—
221#677—62no—
222#677—61no—
223#672—31no—
224#672—31no—
225#672C22×D312yes—
226#576—31no—
227#576—151no—
228#509—51no—
229#509—151no—
230#509—152no—
231#509—152no—
232#508—152no—
233#508—151no—
234C2×D16—152no—
235C2×D16—151no—
236#504—151no—
237C22×C3×C5—41no—
238C22×C3×C5C2×D811yescharacteristic
239C2×D15—41no—
240C2×C5×D3—42no—
241C2×C5×D3—162no—
242C2×C5×D3—161no—
243C2×C5×D3C2×D812yes—
244C2×C3×D5—42no—
245C2×C3×D5—161no—
246D3×D5—162no—
247C3×C4×C5—41no—
248C3×C4×C5C2×D811yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250Dic15—41no—
251C5×Dic3C2×D812yes—
252C3×#41—21no—
253C2×C3×D4—101no—
254C2×#67—202no—
255#239—22no—
256#237—102no—
257#237—201no—
258D3×D4—22no—
259D3×D4—42no—
260D3×D4—102no—
261D3×D4—201no—
262#235—41no—
263#235—101no—
264#235C22⋊D511yescharacteristic
265C2×D12—21no—
266C2×D12—41no—
267C3×Q16C22×D511yescharacteristic
268C3×SD16—21no—
269C3×D8—52no—
270C3×D8—101no—
271C3×D8C22×D511yescharacteristic
272C2×C3×C8C22×D511yescharacteristic
273#215—22no—
274#214—102no—
275#213—22no—
276#213—102no—
277#208C22⋊D511yescharacteristic
278Dic12—51no—
279D24—21no—
280D24—51no—
281D24C22×D512yes—
282#204—101no—
283ΓL(1,𝔽25)C22⋊D512yes—
284D3×C8C22×D512yes—
285C3×C16—52no—
286C3⋊C16—51no—
287C3⋊C16—51no—
288C23×C5—122no—
289C23×C5—121no—
290C22×D5—41no—
291C22×D5—122no—
292C5×Q8—21no—
293C5×Q8—31no—
294C5×D4—21no—
295C5×D4—32no—
296C5×D4—31no—
297C5×D4—41no—
298C5×D4—62no—
299C5×D4—122no—
300C5×D4—122no—
301C5×D4—121no—
302C2×C4×C5—32no—
303C2×C4×C5—41no—
304C2×C4×C5—122no—
305C2×C4×C5D3×D411yescharacteristic
306C22⋊D5—122no—
307C22⋊D5—121no—
308D20—22no—
309D20—41no—
310D20—61no—
311D20—122no—
312C4×D5—122no—
313C4×D5—121no—
314Dic10—61no—
315C5×C8—32no—
316C5×C8D3×D411yescharacteristic
317C5×C8C23×D311yescharacteristic
318H(5,𝔽2)—61no—
319H(5,𝔽2)—301no—
320AGL(1,ℤ/8)—62no—
321AGL(1,ℤ/8)—61no—
322AGL(1,ℤ/8)—302no—
323AGL(1,ℤ/8)—301no—
324#135—32no—
325#135—152no—
326#135—151no—
327#135D3×D511yescharacteristic, complemented
328C2×D8—32no—
329C2×D8—31no—
330C2×D8—51no—
331C2×D8—152no—
332#131—31no—
333Q32—151no—
334SD32—52no—
335SD32—152no—
336SD32—151no—
337SD32—151no—
338D16—52no—
339D16—152no—
340D16—152no—
341D16—152no—
342D16—151no—
343M32—51no—
344M32—152no—
345C2×C16—152no—
346C2×C16—151no—
347C2×C3×C5—21no—
348C2×C3×C5—81no—
349C2×C3×C5C22×D811yescharacteristic
350D15—81no—
351C3×D5—82no—
352C5×D3—22no—
353C5×D3—82no—
354C23×C3—201no—
355C22×D3—41no—
356C22×D3—42no—
357C22×D3—202no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—41no—
361C3×D4—102no—
362C3×D4—201no—
363C2×C3×C4—41no—
364C2×C3×C4C2×C22⋊D511yescharacteristic
365#67—42no—
366#67—201no—
367#67—202no—
368#67—202no—
369#67#68612yes—
370C2×Dic3—202no—
371D12—22no—
372D12—42no—
373D12—41no—
374D12—101no—
375D12#68611yescharacteristic
376C4×D3—42no—
377C4×D3—201no—
378C4×D3C2×C22⋊D512yes—
379Dic6—101no—
380Dic6#68611yescharacteristic
381C3×C8C2×C22⋊D511yescharacteristic
382C3×C8C23×D511yescharacteristic
383C3⋊C8C2×C22⋊D512yes—
384C22×C5—32no—
385C22×C5—41no—
386C22×C5—122no—
387C22×C5—481no—
388C22×C5—482no—
389C22×C5D3×D811yescharacteristic
390C2×D5—42no—
391C2×D5—121no—
392C2×D5—161no—
393C2×D5—482no—
394C4×C5—32no—
395C4×C5—41no—
396C4×C5D3×D811yescharacteristic
397C4×C5C2×D3×D411yescharacteristic
398Dic5—121no—
399#41—21no—
400#41—31no—
401#41—62no—
402#41—121no—
403#41—302no—
404#41—301no—
405#41—601no—
406C2×D4—62no—
407C2×D4—61no—
408C2×D4—101no—
409C2×D4—121no—
410C2×D4—122no—
411C2×D4—302no—
412C2×D4—601no—
413C2×D4—602no—
414Q16—151no—
415Q16C2×D3×D511yescharacteristic
416SD16—21no—
417SD16—62no—
418SD16—302no—
419SD16—301no—
420D8—32no—
421D8—52no—
422D8—62no—
423D8—61no—
424D8—101no—
425D8—151no—
426D8—302no—
427D8C2×D3×D511yescharacteristic
428M16—31no—
429M16—32no—
430C2×C8—32no—
431C2×C8C2×D3×D511yescharacteristic
432C16—52no—
433C16—151no—
434C16—151no—
435C3×C5#341511yescharacteristic, complemented, Hall
436C22×C3—41no—
437C22×C3—202no—
438C22×C3—801no—
439C22×C3C2×#68611yescharacteristic
440C2×D3—42no—
441C2×D3—162no—
442C2×D3—161no—
443C2×D3—201no—
444C2×D3—802no—
445C2×D3C2×#68612yes—
446C3×C4—41no—
447C3×C4C2×#68611yescharacteristic
448C3×C4C22×C22⋊D511yescharacteristic
449Dic3—201no—
450Dic3C2×#68612yes—
451C2×C5—21no—
452C2×C5—62no—
453C2×C5—81no—
454C2×C5—242no—
455C2×C5C2×D3×D811yescharacteristic
456D5—82no—
457D5—241no—
458C23—122no—
459C23—121no—
460C23—201no—
461C23—602no—
462Q8—21no—
463Q8—31no—
464Q8—301no—
465D4—21no—
466D4—32no—
467D4—31no—
468D4—41no—
469D4—62no—
470D4—102no—
471D4—122no—
472D4—122no—
473D4—121no—
474D4—201no—
475D4—301no—
476D4—602no—
477D4—602no—
478D4—601no—
479C2×C4—32no—
480C2×C4—41no—
481C2×C4—122no—
482C2×C4—602no—
483C2×C4—601no—
484C2×C4D3×C22⋊D511yescharacteristic
485C8—32no—
486C8D3×C22⋊D511yescharacteristic
487C8C22×D3×D511yescharacteristic
488C2×C3—21no—
489C2×C3—81no—
490C2×C3—402no—
491C2×C3C22×#68611yescharacteristic
492D3—22no—
493D3—82no—
494D3—401no—
495C5#8203611yescharacteristic, complemented, Sylow
496C22—32no—
497C22—41no—
498C22—122no—
499C22—202no—
500C22—481no—
501C22—482no—
502C22—601no—
503C22—801no—
504C22—2402no—
505C22D3×#68611yescharacteristic
506C4—32no—
507C4—41no—
508C4—601no—
509C4D3×#68611yescharacteristic
510C4C2×D3×C22⋊D511yescharacteristic
511C3#1061573811yescharacteristic, complemented, Sylow
512C2—21no—
513C2—62no—
514C2—81no—
515C2—242no—
516C2—402no—
517C2—1201no—
518C2C2×D3×#68611yescharacteristic
subgroupG/H
centerC2C2×D3×#686
commutator, cosocleC3×C5×C8C24
FrattiniC8C22×D3×D5
FittingC3×C5×#135C22
socleC2×C3×C5C22×D8
hypercenter#135D3×D5
hypocenterC3×C5#3415
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135

Complements

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

NH ≅ G/Ntype
1. C5×#5769C2semidirect
2. #11753971C2semidirect
3. #11751295C2semidirect
4. C3×#65482C2semidirect
5. #11748304C2semidirect
6. #11748291C2semidirect
7. #11748281C2semidirect
8. D3×#3958C2semidirect
9. #11747547C2semidirect
10. C2×#90937C2semidirect
11. #11747529C2semidirect
15. C3×C5×#135C22semidirect
16. #91793C22semidirect
18. C5×#919C22semidirect
20. C5×D3×D8C22semidirect
21. C2×C5×D24C22semidirect
23. C2×C3×D40C22semidirect
25. #91273C22semidirect
27. D3×D40C22semidirect
28. #91113C22semidirect
29. #91112C22semidirect
30. #91111C22semidirect
31. #91110C22semidirect
32. C2×C3⋊#672C22semidirect
33. C3×#3960C22semidirect
34. C3×#3958C22semidirect
35. C3×#3944C22semidirect
36. #90944C22semidirect
37. #90939C22semidirect
38. #90937C22semidirect
39. #90937C22semidirect
40. #90935C22semidirect
41. D3×#672C22semidirect
48. #65482D3semidirect
90. #5769D5semidirect
106. C5×#135C2×D3semidirect
115. C2×D40C2×D3semidirect
118. #3960C2×D3semidirect
125. #3958C2×D3semidirect
127. #3944C2×D3semidirect
163. C3×#135C2×D5semidirect
167. #919C2×D5semidirect
172. D3×D8C2×D5semidirect
174. C2×D24C2×D5semidirect
327. #135D3×D5semidirect
435. C3×C5#3415semidirect
495. C5#82036semidirect
511. C3#10615738semidirect

Maximal subgroups

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

subgroupindex
C5×#57692
#117539712
#117512952
C3×#654822
#117483042
#117482912
#117482812
D3×#39582
#117475472
C2×#909372
#117475292
#106157383
#820365