Finite Group Databasebeta

← #14439281

Subgroups — #14439281

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91807C212yesmaximal, complemented
2C2×#91803C212yesmaximal, complemented
3#11756484C211yesmaximal, characteristic, complemented
4D15×#124C211yesmaximal, characteristic, complemented
5C3×C5×#497C211yesmaximal, characteristic, complemented
6#11751292C212yesmaximal, complemented
7#11751269C212yesmaximal, complemented
8#11751187C211yesmaximal, characteristic, complemented
9#11751159C211yesmaximal, characteristic, complemented
10#11750813C211yesmaximal, characteristic, complemented
11#11750808C211yesmaximal, characteristic, complemented
12#10613602—31nomaximal, Hall
13C2×#6821C2212yes—
14C2×Q8×D15C2211yescharacteristic
15C2×D4×D15C2211yescharacteristic
16C2×C3×C5×Q16C2212yescomplemented
17C2×C3×C5×SD16C2212yes—
18C3×C5×C4⋊Q8C2211yescharacteristic, complemented
19C3×C5×#124C2211yescharacteristic
20#91836C2211yescharacteristic
21#91836C2212yescomplemented
22C2×#6718C2212yescomplemented
23C2×#6717C2212yes—
24C2×#6717C2212yescomplemented
25#91825C2211yescharacteristic
26C2×#6715C2212yes—
27#91807—28no—
28#91803—28no—
29C2×D120C2212yes—
30C2×#6706C2212yescomplemented
31C2×#6705C2212yes—
32#91784—22no—
33#91782—22no—
34#91774—22no—
35C22⋊C4×D15—22no—
36#91764C2211yescharacteristic
37#91763C2211yescharacteristic, complemented
38C42×D15C2211yescharacteristic
39C3×C5×#97C2211yescharacteristic, complemented
40C8×Dic15C2212yescomplemented
41#91087C2211yescharacteristic, complemented
42#81871—51nomaximal, Hall
43C2×#4064—32no—
44C2×#4060—32no—
45#66059—31no—
46D5×#124—31no—
47C5×#497D311yescharacteristic, complemented
48#65480—32no—
49#65457—32no—
50#65375—31no—
51#65347—31no—
52#65001—31no—
53#64996—31no—
54C23×D15—21no—
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×Q8C2311yescharacteristic
57C2×C3×C5×D4C2311yescharacteristic
58C2×#1220—21no—
59#6821—24no—
60#6821—24no—
61Q8×D15—24no—
62D4×D15—24no—
63C2×D60—22no—
64C2×D60—22no—
65C2×D60C2312yes—
66C2×D60C2311yescharacteristic
67C2×C4×D15—21no—
68C2×C4×D15—22no—
69C2×C4×D15D412yes—
70C2×C4×D15C2311yescharacteristic
71C2×Dic30—21no—
72C2×Dic30C2311yescharacteristic
73C3×C5×Q16—24no—
74C3×C5×SD16—24no—
75C2×C3×C5×C8C2312yes—
76C3×C5×C4⋊C4—22no—
77C3×C5×C22⋊C4—22no—
78C3×C42×C5C2311yescharacteristic
79#6719—22no—
80#6718—24no—
81#6717—24no—
82#6717—24no—
83#6715—24no—
84#6714—22no—
85#6714—22no—
86#6714—24no—
87C4×Dic15D412yes—
88C4×Dic15C2311yescharacteristic
89C2×C3⋊#168C2312yes—
90D120—24no—
91#6706—24no—
92#6705D418yescomplemented
93C2×#918—52no—
94C2×#914—52no—
95#5725—51no—
96D3×#124—51no—
97C3×#497D511yescharacteristic, complemented
98#5191—52no—
99#5168—52no—
100#5086—51no—
101#5058—51no—
102#4712—51no—
103#4707—51no—
104C2×#713—32no—
105C2×Q8×D5—31no—
106C2×D4×D5—31no—
107C2×C5×Q16C2×D312yescomplemented
108C2×C5×SD16C2×D312yes—
109C5×C4⋊Q8C2×D311yescharacteristic, complemented
110C5×#124C2×D311yescharacteristic
111#4093—32no—
112#4093—31no—
113C2×#689—32no—
114C2×#688—32no—
115C2×#688—32no—
116#4082—31no—
117C2×#686—32no—
118#4064—68no—
119#4060—68no—
120C2×D40—32no—
121C2×#677—32no—
122C2×#676—32no—
123#4041—62no—
124#4039—62no—
125#4031—62no—
126D5×C22⋊C4—62no—
127#4021—31no—
128#4020—31no—
129C42×D5—31no—
130C5×#97C2×D311yescharacteristic, complemented
131C8×Dic5—32no—
132#3935—31no—
133#3153—151noSylow
134C23×C3×C5—21no—
135C22×D15—21no—
136C22×D15—22no—
137C22×D15—44no—
138C22×D15—42no—
139C22×D15C2×D411yescharacteristic
140C3×C5×Q8—22no—
141C3×C5×Q8—22no—
142C3×C5×Q8—22no—
143C3×C5×D4—22no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2411yescharacteristic
148#1220—42no—
149C2×Dic15—21no—
150C2×Dic15C2×D412yes—
151C2×Dic15C2×D411yescharacteristic
152D60—22no—
153D60—22no—
154D60—22no—
155D60—44no—
156D60—44no—
157C4×D15—42no—
158C4×D15—42no—
159C4×D15—44no—
160C4×D15C2×D414yes—
161Dic30—22no—
162Dic30—42no—
163C3×C5×C8C2×D414yes—
164C3⋊#168C2×D414yes—
165C2×#239—52no—
166C2×D3×Q8—51no—
167C2×D3×D4—51no—
168C2×C3×Q16C2×D512yescomplemented
169C2×C3×SD16C2×D512yes—
170C3×C4⋊Q8C2×D511yescharacteristic, complemented
171C3×#124C2×D511yescharacteristic
172#947—51no—
173#947—52no—
174C2×#216—52no—
175C2×#215—52no—
176C2×#215—52no—
177#936—51no—
178C2×#213—52no—
179#918—108no—
180#914—108no—
181C2×D24—52no—
182C2×#204—52no—
183C2×ΓL(1,𝔽25)—52no—
184#895—102no—
185#893—102no—
186#885—102no—
187D3×C22⋊C4—102no—
188#875—51no—
189#874—51no—
190C42×D3—51no—
191C3×#97C2×D511yescharacteristic, complemented
192C8×Dic3—52no—
193#803—51no—
194C23×D5—61no—
195C2×C5×Q8C22×D311yescharacteristic
196C2×C5×Q8C22×D312yes—
197C2×C5×D4C22×D311yescharacteristic
198C2×C22⋊D5—61no—
199#713—64no—
200#713—64no—
201Q8×D5—64no—
202D4×D5—64no—
203C2×D20—31no—
204C2×D20—32no—
205C2×D20—62no—
206C2×D20—62no—
207C2×C4×D5—31no—
208C2×C4×D5—32no—
209C2×C4×D5—62no—
210C2×C4×D5—61no—
211C2×Dic10—31no—
212C2×Dic10—61no—
213C5×Q16—24no—
214C5×SD16—24no—
215C2×C5×C8C22×D312yes—
216C5×C4⋊C4—22no—
217C5×C22⋊C4—22no—
218C42×C5C22×D311yescharacteristic
219C22⋊Dic5—62no—
220#689—64no—
221#688—64no—
222#688—64no—
223#686—64no—
224#685—62no—
225#685—64no—
226#685—62no—
227C4×Dic5—32no—
228C4×Dic5—31no—
229C2×#168—32no—
230D40—64no—
231#677—64no—
232#676—38no—
233C2×#137—152no—
234C2×AGL(1,ℤ/8)—152no—
235#532—151no—
236C2×#124—151no—
237#497—151no—
238#497D1511yescharacteristic, complemented
239#496—151no—
240#496—151no—
241#495—152no—
242#495—152no—
243C4×M16—151no—
244C22×C3×C5—41no—
245C22×C3×C5—42no—
246C22×C3×C5C22×D411yescharacteristic
247C2×D15—22no—
248C2×D15—42no—
249C2×D15—41no—
250C2×D15—44no—
251C2×D15—42no—
252C2×D15—84no—
253C2×D15#12712yes—
254C3×C4×C5—22no—
255C3×C4×C5—41no—
256C3×C4×C5—42no—
257C3×C4×C5C22×D412yes—
258Dic15—22no—
259Dic15—41no—
260Dic15#12712yes—
261C23×D3—101no—
262C2×C3×Q8C22×D511yescharacteristic
263C2×C3×Q8C22×D512yes—
264C2×C3×D4C22×D511yescharacteristic
265C2×#67—101no—
266#239—104no—
267#239—104no—
268D3×Q8—104no—
269D3×D4—104no—
270C2×D12—52no—
271C2×D12—51no—
272C2×D12—102no—
273C2×D12—102no—
274C2×C4×D3—51no—
275C2×C4×D3—52no—
276C2×C4×D3—102no—
277C2×C4×D3—101no—
278C2×Dic6—51no—
279C2×Dic6—101no—
280C3×Q16—24no—
281C3×SD16—24no—
282C2×C3×C8C22×D512yes—
283C3×C4⋊C4—22no—
284C3×C22⋊C4—22no—
285C3×C42C22×D511yescharacteristic
286#217—102no—
287#216—104no—
288#215—104no—
289#215—104no—
290#213—104no—
291(C2×D3)⋊C4—102no—
292(C2×D3)⋊C4—104no—
293(C2×D3)⋊C4—102no—
294C4×Dic3—52no—
295C4×Dic3—51no—
296C2×C3⋊C8—52no—
297D24—104no—
298#204—104no—
299ΓL(1,𝔽25)—58no—
300C23×C5—21no—
301C22×D5—31no—
302C22×D5—61no—
303C22×D5—62no—
304C22×D5—122no—
305C22×D5—124no—
306C5×Q8—22no—
307C5×Q8—22no—
308C5×Q8—22no—
309C5×D4—22no—
310C2×C4×C5—21no—
311C2×C4×C5—22no—
312C2×C4×C5D3×D412yes—
313C2×C4×C5C23×D311yescharacteristic
314C22⋊D5—122no—
315C2×Dic5—31no—
316C2×Dic5—32no—
317C2×Dic5—61no—
318D20—62no—
319D20—62no—
320D20—62no—
321D20—124no—
322D20—124no—
323C4×D5—34no—
324C4×D5—124no—
325C4×D5—122no—
326C4×D5—122no—
327Dic10—62no—
328Dic10—122no—
329C5×C8D3×D414yes—
330#168—34no—
331C2×#41—152no—
332C22×Q8—151no—
333C22×D4—151no—
334#137—308no—
335AGL(1,ℤ/8)—308no—
336C2×Q16—152no—
337C2×Q16C2×D1512yescomplemented
338C2×SD16—152no—
339C2×SD16—152no—
340C2×SD16—152no—
341C2×SD16C2×D1512yes—
342C2×D8—152no—
343C2×D8—152no—
344C2×M16—152no—
345C4⋊Q8C2×D1511yescharacteristic, complemented
346#127—151no—
347#124—151no—
348#124—152no—
349#124—151no—
350#124—151no—
351#124—302no—
352#124C2×D1511yescharacteristic
353#121—302no—
354C4×D4—302no—
355C2×C22⋊C4—302no—
356C2×C42—151no—
357#97—151no—
358#97C2×D1511yescharacteristic, complemented
359C4×C8—152no—
360C2×C3×C5—81no—
361C2×C3×C5C2×#12711yescharacteristic
362C2×C3×C5#57512yes—
363D15—22no—
364D15—82no—
365D15—81no—
366C23×C3—21no—
367C22×D3—51no—
368C22×D3—101no—
369C22×D3—102no—
370C22×D3—202no—
371C22×D3—204no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×D4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4D4×D512yes—
379C2×C3×C4C23×D511yescharacteristic
380#67—202no—
381C2×Dic3—51no—
382C2×Dic3—52no—
383C2×Dic3—101no—
384D12—102no—
385D12—102no—
386D12—102no—
387D12—204no—
388D12—204no—
389C4×D3—54no—
390C4×D3—204no—
391C4×D3—202no—
392C4×D3—202no—
393Dic6—102no—
394Dic6—202no—
395C3×C8D4×D514yes—
396C3⋊C8—54no—
397C22×C5—41no—
398C22×C5—42no—
399C22×C5C2×D3×D411yescharacteristic
400C2×D5—32no—
401C2×D5—62no—
402C2×D5—122no—
403C2×D5—121no—
404C2×D5—124no—
405C2×D5—122no—
406C2×D5—244no—
407C4×C5—22no—
408C4×C5—41no—
409C4×C5—42no—
410C4×C5C2×D3×D412yes—
411Dic5—32no—
412Dic5—62no—
413Dic5—121no—
414C24—301no—
415#41—304no—
416#41—304no—
417C2×Q8—151no—
418C2×Q8—301no—
419C2×Q8—304no—
420C2×Q8C22×D1511yescharacteristic
421C2×Q8C22×D1512yes—
422C2×D4—151no—
423C2×D4—152no—
424C2×D4—302no—
425C2×D4—304no—
426C2×D4—301no—
427C2×D4—302no—
428C2×D4C22×D1511yescharacteristic
429C22×C4—151no—
430C22×C4—152no—
431C22×C4—302no—
432C22×C4—301no—
433Q16—24no—
434Q16—304no—
435SD16—24no—
436SD16—304no—
437SD16—304no—
438SD16—304no—
439D8—304no—
440D8—304no—
441M16—158no—
442C2×C8—152no—
443C2×C8C22×D1512yes—
444C4⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—302no—
447C22⋊C4—304no—
448C22⋊C4—302no—
449C22⋊C4—302no—
450C42—152no—
451C42—151no—
452C42C22×D1511yescharacteristic
453C3×C5#315311yescharacteristic, complemented, Hall
454C22×C3—41no—
455C22×C3—42no—
456C22×C3C2×D4×D511yescharacteristic
457C2×D3—52no—
458C2×D3—102no—
459C2×D3—202no—
460C2×D3—201no—
461C2×D3—204no—
462C2×D3—202no—
463C2×D3—404no—
464C3×C4—22no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4C2×D4×D512yes—
468Dic3—52no—
469Dic3—102no—
470Dic3—201no—
471C2×C5—81no—
472C2×C5D3×#12711yescharacteristic
473C2×C5#577712yes—
474D5—62no—
475D5—241no—
476D5—242no—
477C23—21no—
478C23—151no—
479C23—301no—
480C23—302no—
481C23—602no—
482C23—604no—
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—302no—
487Q8—602no—
488D4—22no—
489D4—302no—
490D4—302no—
491D4—302no—
492D4—604no—
493D4—604no—
494D4—602no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—151no—
498C2×C4—154no—
499C2×C4—152no—
500C2×C4—301no—
501C2×C4—602no—
502C2×C4—604no—
503C2×C4—602no—
504C2×C4D4×D1512yes—
505C2×C4C23×D1511yescharacteristic
506C8—154no—
507C8D4×D1514yes—
508C2×C3—81no—
509C2×C3D5×#12711yescharacteristic
510C2×C3#6611112yes—
511D3—102no—
512D3—401no—
513D3—402no—
514C5#8187111yescharacteristic, complemented, Sylow
515C22—41no—
516C22—42no—
517C22—152no—
518C22—302no—
519C22—602no—
520C22—601no—
521C22—604no—
522C22—602no—
523C22—1204no—
524C22C2×D4×D1511yescharacteristic
525C4—22no—
526C4—42no—
527C4—41no—
528C4—152no—
529C4—302no—
530C4—601no—
531C4C2×D4×D1512yes—
532C3#1061360211yescharacteristic, complemented, Sylow
533C2—81no—
534C2—302no—
535C2—1201no—
536C2—1202no—
537C2D15×#12711yescharacteristic
538C2#1175653612yes—
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#497C2
socleC22×C3×C5C22×D4
hypercenter#497D15
hypocenterC3×C5#3153
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#91807C2semidirect
2. C2×#91803C2semidirect
3. #11756484C2semidirect
4. D15×#124C2semidirect
5. C3×C5×#497C2semidirect
6. #11751292C2semidirect
7. #11751269C2semidirect
8. #11751187C2semidirect
9. #11751159C2semidirect
10. #11750813C2semidirect
11. #11750808C2semidirect
16. C2×C3×C5×Q16C22semidirect
18. C3×C5×C4⋊Q8C22semidirect
21. #91836C22semidirect
22. C2×#6718C22semidirect
24. C2×#6717C22semidirect
30. C2×#6706C22semidirect
37. #91763C22semidirect
39. C3×C5×#97C22semidirect
40. C8×Dic15C22semidirect
41. #91087C22semidirect
47. C5×#497D3semidirect
92. #6705D4semidirect
97. C3×#497D5semidirect
107. C2×C5×Q16C2×D3semidirect
109. C5×C4⋊Q8C2×D3semidirect
130. C5×#97C2×D3semidirect
168. C2×C3×Q16C2×D5semidirect
170. C3×C4⋊Q8C2×D5semidirect
191. C3×#97C2×D5semidirect
238. #497D15semidirect
337. C2×Q16C2×D15semidirect
345. C4⋊Q8C2×D15semidirect
358. #97C2×D15semidirect
453. C3×C5#3153semidirect
514. C5#81871semidirect
532. C3#10613602semidirect

Maximal subgroups

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

subgroupindex
C2×#918072
C2×#918032
#117564842
D15×#1242
C3×C5×#4972
#117512922
#117512692
#117511872
#117511592
#117508132
#117508082
#106136023
#818715