Finite Group Databasebeta

← #14478602

Subgroups — #14478602

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
1#11753987C211yesmaximal, characteristic, complemented
2#11753963C211yesmaximal, characteristic, complemented
3#11751009C211yesmaximal, characteristic, complemented
4C5×#4909C211yesmaximal, characteristic, complemented
5C3×#65196C211yesmaximal, characteristic, complemented
6C2×#90945C211yesmaximal, characteristic, complemented
7#11747546C211yesmaximal, characteristic, complemented
8#11746834C212yesmaximal, complemented
9#11746827C212yesmaximal, complemented
10#11746817C212yesmaximal, complemented
11D5×#800C212yesmaximal, complemented
12#10613771—31nomaximal, Hall
13#92019—21no—
14#92000—21no—
15#91793C2211yescharacteristic, complemented
16C2×D120C2211yescharacteristic
17C2×C5×Dic12C2211yescharacteristic, complemented
18C5×#904C2211yescharacteristic
19C3×#4047C2211yescharacteristic
20#91321—21no—
21#91305—21no—
22#91271C2212yes—
23#91271C2212yescomplemented
24#91261—22no—
25D5×Dic12C2212yescomplemented
26#91249—22no—
27D5×D24C2212yes—
28C3×C5×M32C2211yescharacteristic, complemented
29#91084C2212yescomplemented
30D240C2212yescomplemented
31C5×Dic24C2212yescomplemented
32C5×#800C2212yescomplemented
33C2×C3×#672C2211yescharacteristic, complemented
34C3×#3930C2212yescomplemented
35C3×D5×C16C2212yescomplemented
36#90948C2211yescharacteristic, complemented
37#90945C2212yescomplemented
38#90945C2211yescharacteristic, complemented
39#90945C2212yescomplemented
40#90944C2211yescharacteristic, complemented
41#90939C2211yescharacteristic, complemented
42#82014—51nomaximal, Hall
43#66103—31no—
44#66100—31no—
45C5×#510—31no—
46#65483—31no—
47C2×#3960—31no—
48#65208—32no—
49#65207—32no—
50#65203—32no—
51D5×SD32—32no—
52#65197—31no—
53#65196D311yescharacteristic, complemented
54#6817—21no—
55C2×D60—21no—
56C5×#235—21no—
57C2×C5×Dic6—21no—
58C3×#709D411yescharacteristic
59#6790—42no—
60#6779—42no—
61D3×D20—41no—
62D5×D12—22no—
63#6773—22no—
64#6773—22no—
65D15⋊Q8—41no—
66#6769—41no—
67#6766—41no—
68D5×Dic6—22no—
69C2×C3×C5×C8C2311yescharacteristic
70Dic60C2311yescharacteristic
71D120—21no—
72D120C2311yescharacteristic
73D120C2312yes—
74#6706—21no—
75C5×Dic12—21no—
76C5×Dic12C2311yescharacteristic
77C5×Dic12C2312yes—
78C5×D24C2311yescharacteristic
79C5×#204—21no—
80C3×#681D411yescharacteristic
81C3×#676D412yes—
82C3×C8×D5C2312yes—
83#6663—22no—
84#6660—22no—
85#6660—22no—
86#6654—22no—
87C3×C5×C16C2312yes—
88C3×#672C2311yescharacteristic
89C3×#672C2311yescharacteristic
90#5753—51no—
91#5752—51no—
92C3×#504—51no—
93#4909—52no—
94#4909D511yescharacteristic, complemented
95#4908—51no—
96#4908—52no—
97#4904—51no—
98#4904—52no—
99C2×#800—51no—
100C2×#800—52no—
101#4149—61no—
102#4147—61no—
103C5×#135—31no—
104C2×C5×Q16—31no—
105#4095—61no—
106#4088—61no—
107#4065—32no—
108#4065—32no—
109#4064—62no—
110D5×Q16—32no—
111#4060—62no—
112D5×D8—32no—
113#4050—31no—
114C2×D40—31no—
115#4047C2×D311yescharacteristic
116C5×Q32—32no—
117C5×SD32—32no—
118C5×M32C2×D311yescharacteristic, complemented
119#3961—31no—
120#3960—32no—
121#3960—32no—
122#3960—31no—
123#3959—31no—
124#3958—31no—
125C2×#672C2×D311yescharacteristic, complemented
126#3932—32no—
127D80—32no—
128#3930C2×D312yescomplemented
129D5×C16C2×D312yescomplemented
130#3416—151noSylow
131C22×D15—41no—
132C2×D3×D5—42no—
133C2×C3×C4×C5C2×D411yescharacteristic
134#1220—41no—
135D60—21no—
136D60—22no—
137D60—41no—
138C4×D15—41no—
139Dic30—21no—
140C5×#67—41no—
141C2×C5×Dic3—41no—
142C5×D12—21no—
143C4×C5×D3—41no—
144C5×Dic6—21no—
145C5×Dic6—22no—
146C5×Dic6—41no—
147C3×C22⋊D5D812yes—
148C3×D20D811yescharacteristic
149C3×C4×D5C2×D412yes—
150C3×Dic10D811yescharacteristic
151#1204—42no—
152#1203—42no—
153#1202—42no—
154#1202—42no—
155#1200—42no—
156#1200—42no—
157D5×Dic3—42no—
158C3×C5×C8C2×D411yescharacteristic
159C3×C5×C8C2411yescharacteristic
160C3×#168C2×D412yes—
161#1010—101no—
162#1009—101no—
163C3×#131—51no—
164#909—102no—
165#909—101no—
166#908—102no—
167#908—101no—
168C2×Dic12—52no—
169C2×Dic12C2×D511yescharacteristic, complemented
170#904—51no—
171#904—52no—
172#904—52no—
173#904C2×D511yescharacteristic
174C2×D24—51no—
175C2×D24—52no—
176C3×M32—52no—
177C3×M32C2×D511yescharacteristic, complemented
178C2×C3×C16—51no—
179C2×C3×C16—52no—
180Dic24—51no—
181Dic24C2×D512yescomplemented
182#800—52no—
183#800—52no—
184#800—51no—
185#800—51no—
186#800—52no—
187#800C2×D512yescomplemented
188D48—51no—
189D48—52no—
190C5×#41—61no—
191C2×C5×Q8—61no—
192#713—62no—
193#713—62no—
194#713—121no—
195Q8×D5—62no—
196Q8×D5—121no—
197D4×D5—62no—
198D4×D5—122no—
199#709—61no—
200#709—122no—
201#709—121no—
202#709D1211yescharacteristic
203C2×D20—61no—
204C2×D20—121no—
205C5×Q16—32no—
206C5×Q16—31no—
207C5×Q16—61no—
208C5×SD16—61no—
209C5×D8—31no—
210C2×C5×C8C22×D311yescharacteristic
211#689—62no—
212#688—62no—
213#688—62no—
214#686—62no—
215#681D1211yescharacteristic
216Dic20—31no—
217D40—31no—
218D40—32no—
219D40—61no—
220#677—61no—
221#676D1212yes—
222C8×D5C22×D312yes—
223C5×C16C22×D312yes—
224#672C22×D311yescharacteristic
225#672C22×D311yescharacteristic
226#578—151no—
227#576—151no—
228#510—31no—
229#510—152no—
230#509—152no—
231#509—151no—
232#508—152no—
233#508—151no—
234C2×SD32—152no—
235C2×SD32—151no—
236#504—51no—
237C22×C3×C5C2×D811yescharacteristic
238C2×D15—41no—
239C2×D15—42no—
240C2×D15—161no—
241C2×C5×D3—41no—
242C2×C3×D5C2×D812yes—
243D3×D5—162no—
244C3×C4×C5C2×D811yescharacteristic
245C3×C4×C5C22×D411yescharacteristic
246Dic15—41no—
247C3×Dic5C2×D812yes—
248C5×Dic3—42no—
249C5×Dic3—41no—
250C3×#41—51no—
251#239—201no—
252D3×Q8—201no—
253#237—201no—
254#237—202no—
255D3×D4—201no—
256D3×D4—202no—
257#235—21no—
258#235—102no—
259#235—102no—
260#235—101no—
261C2×D12—102no—
262C2×D12—101no—
263C2×Dic6—21no—
264C2×Dic6—102no—
265C3×M16—51no—
266C3×M16—52no—
267C2×C3×C8—52no—
268C2×C3×C8C22×D511yescharacteristic
269Dic12—21no—
270Dic12—51no—
271Dic12—102no—
272Dic12C22×D512yes—
273Dic12C22×D511yescharacteristic
274D24—51no—
275D24—52no—
276D24—102no—
277D24—101no—
278D24C22×D511yescharacteristic
279#204—21no—
280#204—102no—
281#204—102no—
282#204—101no—
283C3×C16—51no—
284C3×C16—51no—
285C3×C16C22×D512yes—
286C22×D5—121no—
287C22×D5—122no—
288C5×Q8—61no—
289C5×Q8—62no—
290C5×Q8—121no—
291C5×D4—61no—
292C5×D4—121no—
293C2×C4×C5—121no—
294C2×C4×C5—121no—
295C2×C4×C5C2×D1211yescharacteristic
296C22⋊D5—121no—
297C22⋊D5—122no—
298C22⋊D5D2412yes—
299D20—61no—
300D20—62no—
301D20—122no—
302D20—122no—
303D20—121no—
304D20D2411yescharacteristic
305C4×D5—122no—
306C4×D5—121no—
307C4×D5—122no—
308C4×D5—122no—
309C4×D5C2×D1212yes—
310Dic10—61no—
311Dic10—122no—
312Dic10D2411yescharacteristic
313C5×C8C2×D1211yescharacteristic
314C5×C8C23×D311yescharacteristic
315#168C2×D1212yes—
3162−1+4—301no—
317H(5,𝔽2)—301no—
318#137—302no—
319#137—301no—
320AGL(1,ℤ/8)—302no—
321AGL(1,ℤ/8)—301no—
322#135—31no—
323#135—152no—
324#135—152no—
325#135—151no—
326C2×Q16—31no—
327C2×Q16—152no—
328C2×D8—152no—
329C2×D8—151no—
330#131—51no—
331Q32—32no—
332Q32—151no—
333SD32—32no—
334SD32—151no—
335SD32—152no—
336SD32—152no—
337SD32—151no—
338SD32—152no—
339D16—151no—
340D16—152no—
341M32—52no—
342M32D3×D511yescharacteristic, complemented
343C2×C16—51no—
344C2×C16—52no—
345C2×C3×C5—21no—
346C2×C3×C5C22×D811yescharacteristic
347D15—82no—
348D15—81no—
349C3×D5—22no—
350C5×D3—81no—
351C22×D3—202no—
352C22×D3—201no—
353C3×Q8—51no—
354C3×D4—52no—
355C3×D4—51no—
356C2×C3×C4—52no—
357C2×C3×C4D4×D511yescharacteristic
358#67—41no—
359#67—202no—
360#67—202no—
361#67—201no—
362C2×Dic3—41no—
363C2×Dic3—202no—
364D12—21no—
365D12—101no—
366D12—102no—
367D12—202no—
368D12—201no—
369C4×D3—41no—
370C4×D3—202no—
371C4×D3—202no—
372C4×D3—201no—
373Dic6—21no—
374Dic6—22no—
375Dic6—41no—
376Dic6—101no—
377Dic6—202no—
378C3×C8—52no—
379C3×C8D4×D511yescharacteristic
380C3×C8C23×D511yescharacteristic
381C22×C5—121no—
382C22×C5C2×D2411yescharacteristic
383C2×D5—121no—
384C2×D5—122no—
385C2×D5—482no—
386C2×D5—481no—
387C2×D5C2×D2412yes—
388C4×C5—122no—
389C4×C5—121no—
390C4×C5C2×D2411yescharacteristic
391C4×C5C22×D1211yescharacteristic
392Dic5—121no—
393Dic5C2×D2412yes—
394#41—51no—
395#41—61no—
396#41—302no—
397#41—302no—
398#41—301no—
399#41—602no—
400#41—601no—
401#41—601no—
402C2×Q8—61no—
403C2×Q8—302no—
404C2×Q8—601no—
405C2×D4—302no—
406C2×D4—301no—
407C2×D4—601no—
408C2×D4—602no—
409Q16—32no—
410Q16—31no—
411Q16—61no—
412Q16—151no—
413Q16—302no—
414SD16—61no—
415SD16—302no—
416SD16—302no—
417SD16—301no—
418D8—31no—
419D8—151no—
420D8—152no—
421D8—302no—
422D8—301no—
423M16—51no—
424M16—52no—
425C2×C8—52no—
426C2×C8C2×D3×D511yescharacteristic
427C16—51no—
428C16—51no—
429C16C2×D3×D512yes—
430C3×C5#341611yescharacteristic, complemented, Hall
431C22×C3—52no—
432C22×C3D5×D811yescharacteristic
433C2×D3—41no—
434C2×D3—201no—
435C2×D3—202no—
436C2×D3—802no—
437C2×D3—801no—
438C3×C4—52no—
439C3×C4D5×D811yescharacteristic
440C3×C4C2×D4×D511yescharacteristic
441Dic3—42no—
442Dic3—41no—
443Dic3—201no—
444C2×C5—21no—
445C2×C5—241no—
446C2×C5C22×D2411yescharacteristic
447D5—22no—
448D5—242no—
449D5—241no—
450C23—602no—
451C23—601no—
452Q8—51no—
453Q8—61no—
454Q8—62no—
455Q8—121no—
456Q8—301no—
457Q8—602no—
458D4—52no—
459D4—51no—
460D4—61no—
461D4—121no—
462D4—302no—
463D4—301no—
464D4—602no—
465D4—602no—
466D4—602no—
467D4—601no—
468D4—601no—
469C2×C4—52no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—602no—
473C2×C4—602no—
474C2×C4—602no—
475C2×C4—601no—
476C2×C4D5×D1211yescharacteristic
477C8—52no—
478C8D5×D1211yescharacteristic
479C8C22×D3×D511yescharacteristic
480C2×C3—21no—
481C2×C3—102no—
482C2×C3C2×D5×D811yescharacteristic
483D3—81no—
484D3—402no—
485D3—401no—
486C5#8201411yescharacteristic, complemented, Sylow
487C22—52no—
488C22—121no—
489C22—601no—
490C22—602no—
491C22—2401no—
492C22—2402no—
493C22D5×D2411yescharacteristic
494C4—52no—
495C4—121no—
496C4—122no—
497C4—601no—
498C4D5×D2411yescharacteristic
499C4C2×D5×D1211yescharacteristic
500C3#1061377111yescharacteristic, complemented, Sylow
501C2—21no—
502C2—102no—
503C2—241no—
504C2—1202no—
505C2—1201no—
506C2C2×D5×D2411yescharacteristic
subgroupG/H
centerC2C2×D5×D24
commutator, cosocleC3×C5×C8C24
FrattiniC8C22×D3×D5
FittingC3×C5×M32C22
socleC2×C3×C5C22×D8
hypercenterM32D3×D5
hypocenterC3×C5#3416
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
1. #11753987C2semidirect
2. #11753963C2semidirect
3. #11751009C2semidirect
4. C5×#4909C2semidirect
5. C3×#65196C2semidirect
6. C2×#90945C2semidirect
7. #11747546C2semidirect
8. #11746834C2semidirect
9. #11746827C2semidirect
10. #11746817C2semidirect
11. D5×#800C2semidirect
15. #91793C22semidirect
17. C2×C5×Dic12C22semidirect
23. #91271C22semidirect
25. D5×Dic12C22semidirect
28. C3×C5×M32C22semidirect
29. #91084C22semidirect
30. D240C22semidirect
31. C5×Dic24C22semidirect
32. C5×#800C22semidirect
33. C2×C3×#672C22semidirect
34. C3×#3930C22semidirect
35. C3×D5×C16C22semidirect
36. #90948C22semidirect
37. #90945C22semidirect
38. #90945C22semidirect
39. #90945C22semidirect
40. #90944C22semidirect
41. #90939C22semidirect
53. #65196D3semidirect
94. #4909D5semidirect
118. C5×M32C2×D3semidirect
125. C2×#672C2×D3semidirect
128. #3930C2×D3semidirect
129. D5×C16C2×D3semidirect
169. C2×Dic12C2×D5semidirect
177. C3×M32C2×D5semidirect
181. Dic24C2×D5semidirect
187. #800C2×D5semidirect
342. M32D3×D5semidirect
430. C3×C5#3416semidirect
486. C5#82014semidirect
500. C3#10613771semidirect

Maximal subgroups

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

subgroupindex
#117539872
#117539632
#117510092
C5×#49092
C3×#651962
C2×#909452
#117475462
#117468342
#117468272
#117468172
D5×#8002
#106137713
#820145