Finite Group Databasebeta

← #14478567

Subgroups — #14478567

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66100C211yesmaximal, characteristic, complemented
2#11753961C211yesmaximal, characteristic, complemented
3#11751280C211yesmaximal, characteristic, complemented
4C2×C5×#828C211yesmaximal, characteristic, complemented
5#11748301C212yesmaximal, complemented
6#11748300C212yesmaximal, complemented
7#11748289C212yesmaximal, complemented
8D5×#828C212yesmaximal, complemented
9#11747542C211yesmaximal, characteristic, complemented
10#11747540C211yesmaximal, characteristic, complemented
11#11747524C211yesmaximal, characteristic, complemented
12#10613755—31nomaximal, Hall
13C3×#4147—21no—
14#92018—21no—
15C2×C3×C5×Q16C2211yescharacteristic, complemented
16#91793C2211yescharacteristic
17C2×C5×D24C2211yescharacteristic
18C3×#4088—21no—
19C3×#4065C2212yescomplemented
20C3×#4064—22no—
21C3×D5×Q16C2212yescomplemented
22C3×#4050C2211yescharacteristic, complemented
23C3×#4047C2211yescharacteristic
24#91304—21no—
25#91270C2212yes—
26#91250—22no—
27D5×D24C2212yes—
28#91113C2212yescomplemented
29#91112C2212yescomplemented
30#91096C2211yescharacteristic, complemented
31C5×#828C2212yescomplemented
32C5×#828C2212yescomplemented
33C2×C5×C3⋊C16C2211yescharacteristic, complemented
34#90947C2211yescharacteristic, complemented
35#90943C2211yescharacteristic, complemented
36#90942C2211yescharacteristic, complemented
37#90941C2212yescomplemented
38#90938C2211yescharacteristic, complemented
39#90937C2212yescomplemented
40#90934C2212yescomplemented
41D5×C3⋊C16C2212yescomplemented
42#84016—51nomaximal, Hall
43#66100D311yescharacteristic, complemented
44#66091—31no—
45C2×C5×SD32—31no—
46#65468—31no—
47#65436—31no—
48#65205—32no—
49#65204—32no—
50#65203—32no—
51D5×SD32—32no—
52#65193—31no—
53#65190—31no—
54C2×C3×C5×Q8—21no—
55#6817—21no—
56C2×C5×D12—21no—
57C3×#713—22no—
58C3×#713—41no—
59C3×Q8×D5—22no—
60C3×Q8×D5—41no—
61C3×#709—21no—
62C3×#709—42no—
63C3×#709D411yescharacteristic
64D3×C22⋊D5—42no—
65#6777—41no—
66D5×D12—22no—
67#6772—22no—
68#6768—41no—
69C3×C5×Q16—21no—
70C3×C5×Q16C2312yes—
71C2×C3×C5×C8C2311yescharacteristic
72Dic60C2311yescharacteristic
73D120C2311yescharacteristic
74#6706—21no—
75C5×D24—21no—
76C5×D24C2312yes—
77C3×#689—22no—
78C3×#688—22no—
79C3×#681D411yescharacteristic
80C3×Dic20C2311yescharacteristic
81C3×D40C2311yescharacteristic
82C3×#677—21no—
83C3×#676D412yes—
84C3×C8×D5C2312yes—
85#6659—22no—
86#6654—22no—
87C3⋊#672C2312yes—
88C5×C3⋊C16C2311yescharacteristic
89C5×C3⋊C16C2311yescharacteristic
90C3×#578—51no—
91#5752—51no—
92#5194—52no—
93#5194—51no—
94#5193—51no—
95#5193—52no—
96#5179—51no—
97#5179—52no—
98C2×#828—52no—
99C2×#828D511yescharacteristic, complemented
100#5140—51no—
101#4147—21no—
102#4144—61no—
103C2×C5×Q16C2×D311yescharacteristic, complemented
104C2×C5×D8—31no—
105#4088—21no—
106#4078—61no—
107#4065C2×D312yescomplemented
108#4064—22no—
109D5×Q16C2×D312yescomplemented
110#4058—32no—
111#4057—62no—
112D5×D8—32no—
113#4050—31no—
114#4050C2×D311yescharacteristic, complemented
115#4047C2×D311yescharacteristic
116C5×SD32—32no—
117C5×SD32—32no—
118C2×C5×C16—31no—
119#3961—32no—
120#3960—32no—
121#3959—32no—
122#3958—32no—
123#3944—31no—
124Dic40—31no—
125#3932—31no—
126#3932—31no—
127D80—31no—
128#3930—32no—
129D5×C16—32no—
130#3416—151noSylow
131C22×C5×D3—41no—
132C2×D3×D5—42no—
133C3×C5×Q8—22no—
134C3×C5×Q8—41no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5C2×D411yescharacteristic
137#1220—41no—
138D60—21no—
139C4×D15—41no—
140Dic30—21no—
141C5×D12—22no—
142C5×D12—41no—
143C3×C22⋊D5—41no—
144C3×C22⋊D5D812yes—
145C3×D20—21no—
146C3×D20—42no—
147C3×D20D811yescharacteristic
148C3×C4×D5—42no—
149C3×C4×D5—41no—
150C3×C4×D5—42no—
151C3×C4×D5C2×D412yes—
152C3×Dic10—21no—
153C3×Dic10—42no—
154C3×Dic10D811yescharacteristic
155#1203—42no—
156#1201—42no—
157D3×Dic5—42no—
158C3×C5×C8C2×D411yescharacteristic
159C3×C5×C8C2411yescharacteristic
160C3×#168C2×D412yes—
161C3×2−1+4—101no—
162#1009—101no—
163C3×#137—102no—
164C3×#137—101no—
165C3×#135—52no—
166C3×#135—51no—
167C2×C3×Q16—52no—
168C2×C3×Q16C2×D511yescharacteristic, complemented
169C3×#131—51no—
170#908—102no—
171#908—101no—
172#904—51no—
173#904—52no—
174C2×D24—52no—
175C2×D24C2×D511yescharacteristic
176#829—52no—
177#829—51no—
178#828—52no—
179#828—52no—
180#828—51no—
181#828C2×D512yescomplemented
182#828C2×D512yescomplemented
183#827—51no—
184#826—52no—
185#826—51no—
186#812—51no—
187#812—52no—
188C2×C3⋊C16—52no—
189C2×C3⋊C16C2×D511yescharacteristic, complemented
190C2×C5×Q8—21no—
191C2×C5×D4—61no—
192C2×C22⋊D5—122no—
193#713—22no—
194#713—41no—
195Q8×D5—22no—
196Q8×D5—41no—
197#711—62no—
198#711—121no—
199D4×D5—62no—
200D4×D5—121no—
201#709—21no—
202#709—42no—
203#709—61no—
204#709#6711yescharacteristic
205C5×Q16—21no—
206C5×Q16C22×D312yes—
207C5×D8—32no—
208C5×D8—61no—
209C2×C5×C8C22×D311yescharacteristic
210#689—22no—
211#688—22no—
212#687—62no—
213#686—62no—
214#681#6711yescharacteristic
215Dic20—31no—
216Dic20C22×D311yescharacteristic
217D40—31no—
218D40C22×D311yescharacteristic
219#677—21no—
220#677—61no—
221#676#6712yes—
222C8×D5C22×D312yes—
223C5×C16—31no—
224C5×C16—31no—
225#672—32no—
226#578—51no—
227#576—151no—
228#510—152no—
229#510—151no—
230#509—152no—
231#509—151no—
232#508—151no—
233#508—152no—
234C2×SD32—31no—
235C2×SD32—152no—
236#504—151no—
237C22×C3×C5C2×D811yescharacteristic
238C2×D15—41no—
239C2×C5×D3—42no—
240C2×C5×D3—161no—
241C2×C3×D5—41no—
242C2×C3×D5C2×D812yes—
243D3×D5—162no—
244C3×C4×C5—42no—
245C3×C4×C5C2×D811yescharacteristic
246C3×C4×C5C22×D411yescharacteristic
247Dic15—41no—
248C3×Dic5—41no—
249C3×Dic5C2×D812yes—
250C3×#41—51no—
251C3×#41—102no—
252C3×#41—101no—
253C3×#41—202no—
254C3×#41—201no—
255C2×C3×Q8—21no—
256C2×C3×Q8—102no—
257C2×C3×Q8—201no—
258#239—201no—
259D3×D4—201no—
260D3×D4—202no—
261#235—102no—
262#235—101no—
263C2×D12—21no—
264C2×D12—102no—
265C3×Q16—21no—
266C3×Q16—51no—
267C3×Q16—102no—
268C3×Q16C22×D512yes—
269C3×SD16—102no—
270C3×SD16—101no—
271C3×D8—51no—
272C3×M16—51no—
273C3×M16—52no—
274C2×C3×C8—52no—
275C2×C3×C8C22×D511yescharacteristic
276Dic12—51no—
277D24—21no—
278D24—51no—
279D24—102no—
280D24C22×D512yes—
281#204—102no—
282#204—101no—
283C3⋊C16—52no—
284C3⋊C16C22×D511yescharacteristic
285C3⋊C16C22×D511yescharacteristic
286C23×C5—121no—
287C22×D5—122no—
288C5×Q8—22no—
289C5×Q8—41no—
290C5×D4—62no—
291C5×D4—121no—
292C2×C4×C5—41no—
293C2×C4×C5C2×#6711yescharacteristic
294C22⋊D5—41no—
295C22⋊D5—121no—
296C22⋊D5—122no—
297C22⋊D5—122no—
298C22⋊D5#21312yes—
299C2×Dic5—122no—
300D20—21no—
301D20—42no—
302D20—61no—
303D20#21311yescharacteristic
304C4×D5—42no—
305C4×D5—41no—
306C4×D5—42no—
307C4×D5—121no—
308C4×D5C2×#6712yes—
309Dic10—21no—
310Dic10—42no—
311Dic10—61no—
312Dic10#21311yescharacteristic
313C5×C8C2×#6711yescharacteristic
314C5×C8C23×D311yescharacteristic
315#168C2×#6712yes—
3162−1+4—101no—
317H(5,𝔽2)—301no—
318#137—102no—
319#137—101no—
320AGL(1,ℤ/8)—302no—
321AGL(1,ℤ/8)—301no—
322#135—52no—
323#135—51no—
324#135—152no—
325#135—151no—
326C2×Q16—52no—
327C2×Q16D3×D511yescharacteristic, complemented
328C2×D8—31no—
329C2×D8—152no—
330#131—51no—
331Q32—151no—
332Q32—152no—
333SD32—32no—
334SD32—32no—
335SD32—151no—
336SD32—151no—
337SD32—152no—
338SD32—152no—
339D16—151no—
340D16—152no—
341M32—152no—
342M32—151no—
343C2×C16—31no—
344C2×C16—152no—
345C2×C3×C5—21no—
346C2×C3×C5C22×D811yescharacteristic
347D15—81no—
348C3×D5—22no—
349C3×D5—81no—
350C5×D3—82no—
351C22×D3—41no—
352C22×D3—202no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—51no—
356C3×Q8—101no—
357C3×Q8—202no—
358C3×D4—52no—
359C3×D4—51no—
360C3×D4—101no—
361C3×D4—202no—
362C3×D4—201no—
363C2×C3×C4—41no—
364C2×C3×C4—52no—
365C2×C3×C4—202no—
366C2×C3×C4—202no—
367C2×C3×C4—201no—
368C2×C3×C4D4×D511yescharacteristic
369#67—202no—
370#67—201no—
371D12—22no—
372D12—41no—
373D12—101no—
374D12—202no—
375C4×D3—202no—
376C4×D3—201no—
377Dic6—101no—
378C3×C8—52no—
379C3×C8D4×D511yescharacteristic
380C3×C8C23×D511yescharacteristic
381C22×C5—122no—
382C22×C5—481no—
383C22×C5C2×#21311yescharacteristic
384C2×D5—41no—
385C2×D5—121no—
386C2×D5—482no—
387C2×D5C2×#21312yes—
388C4×C5—42no—
389C4×C5C2×#21311yescharacteristic
390C4×C5C22×#6711yescharacteristic
391Dic5—41no—
392Dic5—121no—
393Dic5C2×#21312yes—
394#41—51no—
395#41—102no—
396#41—101no—
397#41—202no—
398#41—201no—
399#41—302no—
400#41—301no—
401#41—601no—
402C2×Q8—21no—
403C2×Q8—102no—
404C2×Q8—201no—
405C2×D4—61no—
406C2×D4—302no—
407C2×D4—601no—
408C2×D4—602no—
409Q16—21no—
410Q16—51no—
411Q16—102no—
412Q16—151no—
413Q16C2×D3×D512yes—
414SD16—102no—
415SD16—101no—
416SD16—302no—
417SD16—301no—
418D8—32no—
419D8—51no—
420D8—61no—
421D8—151no—
422D8—302no—
423M16—51no—
424M16—52no—
425C2×C8—52no—
426C2×C8C2×D3×D511yescharacteristic
427C16—31no—
428C16—31no—
429C16—152no—
430C3×C5#341611yescharacteristic, complemented, Hall
431C22×C3—52no—
432C22×C3—201no—
433C22×C3D5×D811yescharacteristic
434C2×D3—42no—
435C2×D3—161no—
436C2×D3—201no—
437C2×D3—802no—
438C3×C4—42no—
439C3×C4—52no—
440C3×C4—201no—
441C3×C4D5×D811yescharacteristic
442C3×C4C2×D4×D511yescharacteristic
443Dic3—201no—
444C2×C5—21no—
445C2×C5—242no—
446C2×C5C22×#21311yescharacteristic
447D5—22no—
448D5—81no—
449D5—241no—
450C23—121no—
451C23—602no—
452Q8—22no—
453Q8—41no—
454Q8—51no—
455Q8—101no—
456Q8—202no—
457Q8—301no—
458D4—52no—
459D4—51no—
460D4—62no—
461D4—101no—
462D4—121no—
463D4—202no—
464D4—201no—
465D4—301no—
466D4—602no—
467D4—602no—
468D4—601no—
469C2×C4—41no—
470C2×C4—52no—
471C2×C4—202no—
472C2×C4—202no—
473C2×C4—201no—
474C2×C4—602no—
475C2×C4—601no—
476C2×C4D5×#6711yescharacteristic
477C8—52no—
478C8D5×#6711yescharacteristic
479C8C22×D3×D511yescharacteristic
480C2×C3—21no—
481C2×C3—102no—
482C2×C3—401no—
483C2×C3C2×D5×D811yescharacteristic
484D3—82no—
485D3—401no—
486C5#8401611yescharacteristic, complemented, Sylow
487C22—52no—
488C22—122no—
489C22—201no—
490C22—481no—
491C22—601no—
492C22—2402no—
493C22D5×#21311yescharacteristic
494C4—42no—
495C4—52no—
496C4—201no—
497C4—601no—
498C4D5×#21311yescharacteristic
499C4C2×D5×#6711yescharacteristic
500C3#1061375511yescharacteristic, complemented, Sylow
501C2—21no—
502C2—102no—
503C2—242no—
504C2—401no—
505C2—1201no—
506C2C2×D5×#21311yescharacteristic
subgroupG/H
centerC2C2×D5×#213
commutator, cosocleC3×C5×C8C24
FrattiniC8C22×D3×D5
FittingC2×C3×C5×Q16C22
socleC2×C3×C5C22×D8
hypercenterC2×Q16D3×D5
hypocenterC3×C5#3416
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C3×#66100C2semidirect
2. #11753961C2semidirect
3. #11751280C2semidirect
4. C2×C5×#828C2semidirect
5. #11748301C2semidirect
6. #11748300C2semidirect
7. #11748289C2semidirect
8. D5×#828C2semidirect
9. #11747542C2semidirect
10. #11747540C2semidirect
11. #11747524C2semidirect
15. C2×C3×C5×Q16C22semidirect
19. C3×#4065C22semidirect
21. C3×D5×Q16C22semidirect
22. C3×#4050C22semidirect
28. #91113C22semidirect
29. #91112C22semidirect
30. #91096C22semidirect
31. C5×#828C22semidirect
32. C5×#828C22semidirect
33. C2×C5×C3⋊C16C22semidirect
34. #90947C22semidirect
35. #90943C22semidirect
36. #90942C22semidirect
37. #90941C22semidirect
38. #90938C22semidirect
39. #90937C22semidirect
40. #90934C22semidirect
41. D5×C3⋊C16C22semidirect
43. #66100D3semidirect
99. C2×#828D5semidirect
103. C2×C5×Q16C2×D3semidirect
107. #4065C2×D3semidirect
109. D5×Q16C2×D3semidirect
114. #4050C2×D3semidirect
168. C2×C3×Q16C2×D5semidirect
181. #828C2×D5semidirect
182. #828C2×D5semidirect
189. C2×C3⋊C16C2×D5semidirect
327. C2×Q16D3×D5semidirect
430. C3×C5#3416semidirect
486. C5#84016semidirect
500. C3#10613755semidirect

Maximal subgroups

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

subgroupindex
C3×#661002
#117539612
#117512802
C2×C5×#8282
#117483012
#117483002
#117482892
D5×#8282
#117475422
#117475402
#117475242
#106137553
#840165