Finite Group Databasebeta

← #14430869

Subgroups — #14430869

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66132C211yesmaximal, characteristic
2#11754620C211yesmaximal, characteristic, complemented
3C2×#91116C211yesmaximal, characteristic
4C5×#5215C211yesmaximal, characteristic
5#11749104C212yesmaximal, complemented
6#11749101C212yesmaximal, complemented
7#11749086C212yesmaximal, complemented
8#11749083C212yesmaximal, complemented
9#11748469C211yesmaximal, characteristic, complemented
10#11748455C211yesmaximal, characteristic, complemented
11#11748433C211yesmaximal, characteristic, complemented
12#10610649—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C2×C4⋊Dic15C2211yescharacteristic
15C22×C3⋊#168C2211yescharacteristic
16C5×#926C2211yescharacteristic
17C2×C5×#208C2211yescharacteristic
18C2×C3×C22⋊Dic5—22no—
19C3×D4×Dic5—21no—
20C3×D4×Dic5—21no—
21C3×D4×Dic5C2212yes—
22C3×D4×Dic5C2212yes—
23C3×#4072C2211yescharacteristic
24C3×#4072C2211yescharacteristic
25#91576—22no—
26#91469—21no—
27#91468C2212yescomplemented
28#91388C2212yescomplemented
29#91381—21no—
30#91116C2212yes—
31#91116C2212yes—
32C5×#832C2212yes—
33C5×#832C2212yes—
34#90990C2212yescomplemented
35#90989C2211yescharacteristic, complemented
36#90989C2211yescharacteristic, complemented
37#90987C2212yescomplemented
38#90986C2211yescharacteristic, complemented
39#90986C2211yescharacteristic, complemented
40#90982C2212yescomplemented
41#90982C2212yescomplemented
42#82539—51nomaximal, Hall
43#66132D311yescharacteristic
44#65853—31no—
45C5×#418—31no—
46C2×#3964—31no—
47#65410—31no—
48#65408—31no—
49#65287—31no—
50#65055—32no—
51#65052—32no—
52#65050—32no—
53#65048—32no—
54C24×C3×C5—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4Q814yescomplemented
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C22×Dic15—21no—
61C22×C3×Dic5—22no—
62C22×C3×Dic5—22no—
63C2×#1204—22no—
64C4⋊Dic15—21no—
65C4⋊Dic15C2312yes—
66C2×C3⋊#168—22no—
67C2×C3⋊#168C2311yescharacteristic
68C2×C3⋊#168C2311yescharacteristic
69C5×#217—21no—
70C5×C4⋊Dic3C2311yescharacteristic
71C5×C4⋊Dic3C2311yescharacteristic
72C4×C5×Dic3—21no—
73C5×#208—22no—
74C2×C5×C3⋊C8C2312yes—
75C3×C22⋊Dic5—21no—
76C3×C22⋊Dic5—22no—
77C3×C22⋊Dic5—22no—
78C3×C22⋊Dic5D412yes—
79C3×C4⋊Dic5D411yescharacteristic
80C3×C4⋊Dic5D411yescharacteristic
81C3×C4⋊Dic5C2311yescharacteristic
82C3×C4⋊Dic5C2311yescharacteristic
83C3×C4×Dic5—21no—
84C3×C4×Dic5C2312yes—
85Dic3⋊Dic5—22no—
86#6669—22no—
87#6668—22no—
88C3×#518—51no—
89#5519—51no—
90#5215D511yescharacteristic
91C2×#832—51no—
92#5009—52no—
93#5009—52no—
94#5008—52no—
95#5007—52no—
96#4999—51no—
97#4998—51no—
98#4995—51no—
99C22×C5×D4C2×D311yescharacteristic
100C2×C5×M16—31no—
101C5×#117—31no—
102C2×C22⋊Dic5—22no—
103D4×Dic5—21no—
104D4×Dic5—21no—
105D4×Dic5C2×D312yes—
106D4×Dic5C2×D312yes—
107#4072C2×D311yescharacteristic
108#4072C2×D311yescharacteristic
109C2×C4⋊Dic5—31no—
110C22×#168—31no—
111#4036—32no—
112#4034—32no—
113#4024—62no—
114#4016—61no—
115#4015—61no—
116C5×#102—32no—
117C5×#102—32no—
118#3964—32no—
119#3964—32no—
120#3950—32no—
121#3949—32no—
122#3946—32no—
123#3946—32no—
124#3940—31no—
125#3940—31no—
126#3939—31no—
127#3939—31no—
128#3081—151noSylow
129C23×C3×C5—22no—
130C23×C3×C5—24no—
131C23×C3×C5—81no—
132C23×C3×C5—81no—
133C23×C3×C5C2×D411yescharacteristic
134C3×C5×D4—24no—
135C3×C5×D4—22no—
136C3×C5×D4C2×Q814yes—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×Q812yes—
139C2×C3×C4×C5#4112yes—
140C2×C3×C4×C5C2411yescharacteristic
141C2×Dic15—22no—
142C2×Dic15—81no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—82no—
148C2×C3×Dic5—82no—
149C2×C3×Dic5C2×D412yes—
150C2×C3×Dic5C2×D412yes—
151#1204—82no—
152C3⋊#168—21no—
153C3⋊#168—21no—
154C5×C3⋊C8—22no—
155C22×C3×D4C2×D511yescharacteristic
156C3×C4×D4—52no—
157C3×C4×D4—52no—
158C3×C4×D4—101no—
159C3×C4×D4—101no—
160C3×#117—51no—
161C3×#117—51no—
162C2×C3×C22⋊C4—102no—
163#926C2×D511yescharacteristic
164C2×C4⋊Dic3—51no—
165C2×#208C2×D511yescharacteristic
166C22×C3⋊C8—51no—
167#890—101no—
168#888—101no—
169#878—102no—
170#870—52no—
171#869—52no—
172#832—52no—
173#832—52no—
174#832C2×D512yes—
175#832C2×D512yes—
176#808—51no—
177#808—51no—
178#808—52no—
179#807—51no—
180#807—51no—
181#807—52no—
182C4⋊(C3⋊C8)—52no—
183C4⋊(C3⋊C8)—52no—
184C24×C5—21no—
185C2×C5×D4—22no—
186C2×C5×D4—21no—
187C2×C5×D4Dic614yescomplemented
188C2×C5×D4C22×D312yes—
189C22×C4×C5C22×D311yescharacteristic
190C22×Dic5—22no—
191C22×Dic5—22no—
192C22×Dic5—61no—
193C2×Dic10—62no—
194C5×M16—62no—
195C2×C5×C8—32no—
196C5×C4⋊C4—31no—
197C5×C4⋊C4—31no—
198C5×C22⋊C4—61no—
199C42×C5—61no—
200C22⋊Dic5—21no—
201C22⋊Dic5—22no—
202C22⋊Dic5—22no—
203C22⋊Dic5#6712yes—
204C4⋊Dic5—32no—
205C4⋊Dic5—61no—
206C4⋊Dic5—62no—
207C4⋊Dic5#6711yescharacteristic
208C4⋊Dic5#6711yescharacteristic
209C4⋊Dic5C22×D311yescharacteristic
210C4⋊Dic5C22×D311yescharacteristic
211#683—62no—
212#683—62no—
213C4×Dic5—21no—
214C4×Dic5C22×D312yes—
215C2×#168—31no—
216C2×#168—31no—
217C2×#168—62no—
218#544—151no—
219#518—51no—
220#478—152no—
221#478—152no—
222#476—152no—
223#474—152no—
224#428—151no—
225#427—151no—
226#424—151no—
227#418—31no—
228C2×#102—151no—
229C22×C3×C5—21no—
230C22×C3×C5—21no—
231C22×C3×C5—24no—
232C22×C3×C5—42no—
233C22×C3×C5—42no—
234C22×C3×C5—84no—
235C22×C3×C5C22⋊Q812yes—
236C22×C3×C5C22×D411yescharacteristic
237C3×C4×C5C22⋊Q812yes—
238C3×C4×C5C22×Q811yescharacteristic
239C3×C4×C5C2×#4111yescharacteristic
240Dic15—42no—
241C3×Dic5—22no—
242C3×Dic5—22no—
243C3×Dic5—41no—
244C3×Dic5—41no—
245C5×Dic3—41no—
246C5×Dic3—41no—
247C24×C3—21no—
248C2×C3×D4—21no—
249C2×C3×D4—22no—
250C2×C3×D4Dic1014yescomplemented
251C2×C3×D4C22×D512yes—
252C22×C3×C4—102no—
253C22×C3×C4—102no—
254C22×C3×C4C22×D511yescharacteristic
255C22×Dic3—101no—
256C2×Dic6—102no—
257C3×C4⋊C4—51no—
258C3×C4⋊C4—51no—
259C3×C4⋊C4—51no—
260C3×C4⋊C4—51no—
261C3×C22⋊C4—52no—
262C3×C22⋊C4—102no—
263C3×C22⋊C4—102no—
264C3×C22⋊C4—101no—
265C3×C42—52no—
266C3×C42—101no—
267#217—21no—
268C4⋊Dic3—52no—
269C4⋊Dic3—102no—
270C4⋊Dic3—101no—
271C4⋊Dic3C22×D511yescharacteristic
272C4⋊Dic3C22×D511yescharacteristic
273Dic3⋊C4—102no—
274Dic3⋊C4—102no—
275C4×Dic3—21no—
276#208—22no—
277C2×C3⋊C8—51no—
278C2×C3⋊C8—51no—
279C2×C3⋊C8—102no—
280C2×C3⋊C8C22×D512yes—
281C23×C5—22no—
282C23×C5—24no—
283C23×C5—81no—
284C23×C5—81no—
285C23×C5C2×#6711yescharacteristic
286C5×D4—24no—
287C5×D4—22no—
288C5×D4C2×Dic614yes—
289C2×C4×C5—61no—
290C2×C4×C5—61no—
291C2×C4×C5C2×Dic612yes—
292C2×C4×C5#23512yes—
293C2×C4×C5C2×#6711yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—62no—
298C2×Dic5—82no—
299C2×Dic5—82no—
300C2×Dic5—241no—
301C2×Dic5C2×#6712yes—
302C2×Dic5C2×#6712yes—
303Dic10—242no—
304C5×C8—62no—
305#168—61no—
306#168—61no—
307C22×D4D3×D511yescharacteristic
308C2×M16—31no—
309C22×C8—151no—
310C4⋊Q8—152no—
311C4⋊Q8—301no—
312#125—152no—
313#125—301no—
314C22⋊Q8—302no—
315C4×D4—52no—
316C4×D4—52no—
317C4×D4—101no—
318C4×D4—101no—
319#117—31no—
320#117—51no—
321#117—51no—
322C2×C4⋊C4—151no—
323C2×C22⋊C4—102no—
324#107—151no—
325#107—151no—
326#107—152no—
327#106—151no—
328#106—151no—
329#106—152no—
330C4⋊C8—152no—
331C4⋊C8—152no—
332#102—32no—
333#102—32no—
334#102—152no—
335#102—152no—
336C2×C3×C5—22no—
337C2×C3×C5—44no—
338C2×C3×C5C2×C22⋊Q811yescharacteristic
339C2×C3×C5#57611yescharacteristic
340C2×C3×C5#57711yescharacteristic
341C23×C3—22no—
342C23×C3—24no—
343C23×C3—81no—
344C23×C3—81no—
345C23×C3D4×D511yescharacteristic
346C3×D4—22no—
347C3×D4—24no—
348C3×D4C2×Dic1014yes—
349C2×C3×C4—52no—
350C2×C3×C4—52no—
351C2×C3×C4—101no—
352C2×C3×C4—101no—
353C2×C3×C4—402no—
354C2×C3×C4—402no—
355C2×C3×C4C2×Dic1012yes—
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4#71112yes—
358C2×C3×C4C23×D511yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—102no—
362C2×Dic3—401no—
363Dic6—402no—
364C3⋊C8—22no—
365C3⋊C8—101no—
366C3⋊C8—101no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—24no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—84no—
373C22×C5#92412yes—
374C22×C5C22×#6711yescharacteristic
375C4×C5—121no—
376C4×C5—121no—
377C4×C5#92412yes—
378C4×C5C22×Dic611yescharacteristic
379C4×C5C2×#23511yescharacteristic
380Dic5—22no—
381Dic5—22no—
382Dic5—41no—
383Dic5—41no—
384Dic5—122no—
385C24—21no—
386C2×Q8—302no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4#120414yescomplemented
390C2×D4C2×D3×D512yes—
391C22×C4—102no—
392C22×C4—102no—
393C22×C4—301no—
394C22×C4C2×D3×D511yescharacteristic
395M16—62no—
396C2×C8—32no—
397C2×C8—151no—
398C2×C8—151no—
399C2×C8—302no—
400C4⋊C4—31no—
401C4⋊C4—31no—
402C4⋊C4—51no—
403C4⋊C4—51no—
404C4⋊C4—51no—
405C4⋊C4—51no—
406C4⋊C4—152no—
407C4⋊C4—302no—
408C4⋊C4—302no—
409C4⋊C4—302no—
410C4⋊C4—301no—
411C22⋊C4—52no—
412C22⋊C4—61no—
413C22⋊C4—102no—
414C22⋊C4—102no—
415C22⋊C4—101no—
416C42—52no—
417C42—61no—
418C42—101no—
419C3×C5#308111yescharacteristic, complemented, Hall
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—84no—
426C22×C3#402412yes—
427C22×C3C2×D4×D511yescharacteristic
428C3×C4—102no—
429C3×C4—102no—
430C3×C4—201no—
431C3×C4—201no—
432C3×C4#402412yes—
433C3×C4C22×Dic1011yescharacteristic
434C3×C4C2×#71111yescharacteristic
435Dic3—41no—
436Dic3—41no—
437Dic3—202no—
438C2×C5—22no—
439C2×C5—44no—
440C2×C5C2×#92411yescharacteristic
441C2×C5#583511yescharacteristic
442C2×C5#583611yescharacteristic
443C23—22no—
444C23—24no—
445C23—81no—
446C23—81no—
447C23D5×#6711yescharacteristic
448Q8—1202no—
449D4—22no—
450D4—24no—
451D4C2×#120414yes—
452C2×C4—52no—
453C2×C4—52no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—101no—
457C2×C4—101no—
458C2×C4—302no—
459C2×C4—402no—
460C2×C4—402no—
461C2×C4—1201no—
462C2×C4#678212yes—
463C2×C4C2×#120412yes—
464C2×C4D5×#6711yescharacteristic
465C2×C4C22×D3×D511yescharacteristic
466C8—62no—
467C8—301no—
468C8—301no—
469C2×C3—22no—
470C2×C3—44no—
471C2×C3C2×#402411yescharacteristic
472C2×C3#6610811yescharacteristic
473C2×C3#6610911yescharacteristic
474C5#8253911yescharacteristic, complemented, Sylow
475C22—21no—
476C22—21no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22#9157412yes—
482C22C2×D5×#6711yescharacteristic
483C4—102no—
484C4—102no—
485C4—121no—
486C4—121no—
487C4—201no—
488C4—201no—
489C4—602no—
490C4#9157412yes—
491C4C2×#678211yescharacteristic
492C4C22×#120411yescharacteristic
493C3#1061064911yescharacteristic, complemented, Sylow
494C2—22no—
495C2—44no—
496C2#1175469211yescharacteristic
497C2#1175470811yescharacteristic
498C2C2×#9157411yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
Fitting, prime radicalC22×C3×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC22×D4D3×D5
hypocenterC3×C5#3081
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #11754620C2semidirect
5. #11749104C2semidirect
6. #11749101C2semidirect
7. #11749086C2semidirect
8. #11749083C2semidirect
9. #11748469C2semidirect
10. #11748455C2semidirect
11. #11748433C2semidirect
27. #91468C22semidirect
28. #91388C22semidirect
34. #90990C22semidirect
35. #90989C22semidirect
36. #90989C22semidirect
37. #90987C22semidirect
38. #90986C22semidirect
39. #90986C22semidirect
40. #90982C22semidirect
41. #90982C22semidirect
57. C2×C3×C5×D4Q8semidirect
187. C2×C5×D4Dic6semidirect
250. C2×C3×D4Dic10semidirect
389. C2×D4#1204semidirect
419. C3×C5#3081semidirect
474. C5#82539semidirect
493. C3#10610649semidirect

Maximal subgroups

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

subgroupindex
C3×#661322
#117546202
C2×#911162
C5×#52152
#117491042
#117491012
#117490862
#117490832
#117484692
#117484552
#117484332
#106106493
#825395