Finite Group Databasebeta

← #14430855

Subgroups — #14430855

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
1#11756259C211yesmaximal, characteristic, complemented
2#11754503C211yesmaximal, characteristic
3C2×C5×#809C211yesmaximal, characteristic
4C3×#65253C211yesmaximal, characteristic, complemented
5#11748451C211yesmaximal, characteristic, complemented
6#11748436C211yesmaximal, characteristic, complemented
7#11748389C211yesmaximal, characteristic
8#11747260C212yesmaximal, complemented
9#11747258C212yesmaximal, complemented
10#11747256C212yesmaximal, complemented
11#11747254C212yesmaximal, complemented
12#10610462—31nomaximal, Hall
13C22×C5×D12C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15#91815C2211yescharacteristic
16#91815C2211yescharacteristic
17#91813—22no—
18#91779—21no—
19#91779C2212yescomplemented
20#91777—21no—
21#91777C2212yescomplemented
22C22×C5×C3⋊C8C2211yescharacteristic
23C3×#4072C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25D3×C22⋊Dic5—22no—
26#91435C2212yes—
27#91435C2212yes—
28Dic5×D12—21no—
29Dic5×D12—21no—
30C5×#809C2212yes—
31C5×#809C2212yes—
32C3×#3940C2212yescomplemented
33C3×#3939C2212yescomplemented
34#90990C2211yescharacteristic, complemented
35#90990C2211yescharacteristic, complemented
36#90987C2211yescharacteristic, complemented
37#90987C2211yescharacteristic, complemented
38#90983C2212yescomplemented
39#90983C2212yescomplemented
40#90967C2212yes—
41#90967C2212yes—
42#79242—51nomaximal, Hall
43#66132—31no—
44#65834—31no—
45C2×C5×#102—31no—
46#65504—31no—
47#65395—31no—
48#65391—31no—
49#65253D311yescharacteristic, complemented
50#65055—32no—
51#65052—32no—
52#65050—32no—
53#65048—32no—
54C23×C5×D3—21no—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×Dic30—22no—
58C2×C5×D12—21no—
59C2×C5×D12—22no—
60C2×C5×D12Q814yescomplemented
61C2×C5×D12C2312yes—
62C2×D3×Dic5—22no—
63C2×D3×Dic5—22no—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66#6719—21no—
67#6719D412yes—
68C4⋊Dic15—22no—
69C4⋊Dic15D411yescharacteristic
70C4⋊Dic15D411yescharacteristic
71C4⋊Dic15C2311yescharacteristic
72C4⋊Dic15C2311yescharacteristic
73#6712—22no—
74#6712—22no—
75C4×Dic15—21no—
76C4×Dic15C2312yes—
77C2×C5×C3⋊C8—22no—
78C2×C5×C3⋊C8C2311yescharacteristic
79C2×C5×C3⋊C8C2311yescharacteristic
80C3×C22⋊Dic5—21no—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×C4⋊Dic5C2311yescharacteristic
83C3×C4×Dic5—21no—
84C3×#681—22no—
85C2×C3×#168C2312yes—
86(C2×D3)⋊Dic5—22no—
87(C2×D3)⋊Dic5—22no—
88#5521—51no—
89#5500—51no—
90C3×#428—51no—
91#5131—51no—
92#5119—51no—
93#4998—51no—
94C2×#809D511yescharacteristic
95#4891—52no—
96#4890—52no—
97#4871—52no—
98#4870—52no—
99C22×C5×D4—31no—
100C22×C5×C8—31no—
101C2×C5×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊Dic5—62no—
103D4×Dic5—32no—
104D4×Dic5—32no—
105D4×Dic5—61no—
106D4×Dic5—61no—
107#4072—31no—
108#4072—31no—
109#4072C2×D311yescharacteristic
110#4070—62no—
111C2×#681C2×D311yescharacteristic
112#4036—32no—
113#4036—61no—
114#4034—32no—
115#4034—61no—
116C5×#102—32no—
117C5×#102—32no—
118#3964—32no—
119#3964—32no—
120#3950—31no—
121#3950—31no—
122#3949—31no—
123#3949—31no—
124#3946—32no—
125#3946—32no—
126#3940C2×D312yescomplemented
127#3939C2×D312yescomplemented
128#3081—151noSylow
129C23×C3×C5C2×D411yescharacteristic
130C22×C5×D3—22no—
131C22×C5×D3—24no—
132C22×C5×D3—81no—
133C22×C5×D3—81no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—81no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×Q812yes—
138C2×C3×C4×C5#4112yes—
139C2×C3×C4×C5C2411yescharacteristic
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15C2×D412yes—
143C2×Dic15C2×D412yes—
144Dic30—82no—
145C5×D12—24no—
146C5×D12—22no—
147C5×D12C2×Q814yes—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—21no—
150D3×Dic5—82no—
151D3×Dic5—82no—
152C3×#168—22no—
153C5×C3⋊C8—21no—
154C5×C3⋊C8—21no—
155C22×D12C2×D511yescharacteristic
156C2×C3×M16—51no—
157C3×#117—51no—
158C2×C3×C4⋊C4C2×D511yescharacteristic
159#926—51no—
160#926—51no—
161#924—102no—
162C22×C3⋊C8C2×D511yescharacteristic
163#893—52no—
164#893—52no—
165#890—52no—
166#890—101no—
167#888—52no—
168#888—101no—
169D3×C22⋊C4—102no—
170C4×D12—101no—
171C4×D12—101no—
172C3×#107—52no—
173C3×#106—52no—
174#821—52no—
175#821—52no—
176Dic3⋊C8—52no—
177Dic3⋊C8—52no—
178#809C2×D512yes—
179#809C2×D512yes—
180#808—51no—
181#808—51no—
182#807—51no—
183#807—51no—
184C24×C5—61no—
185C2×C5×D4—34no—
186C2×C5×D4—32no—
187C2×C5×D4—62no—
188C2×C5×D4—61no—
189C22×C4×C5—21no—
190C22×C4×C5C22×D311yescharacteristic
191C22×Dic5—62no—
192C22×Dic5—62no—
193C2×Dic10—62no—
194C2×C5×C8—31no—
195C2×C5×C8—31no—
196C2×C5×C8—62no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4C22×D312yes—
199C22⋊Dic5—21no—
200C22⋊Dic5—32no—
201C22⋊Dic5—62no—
202C22⋊Dic5—62no—
203C22⋊Dic5—61no—
204C4⋊Dic5—31no—
205C4⋊Dic5—31no—
206C4⋊Dic5—31no—
207C4⋊Dic5—31no—
208C4⋊Dic5—62no—
209C4⋊Dic5C22×D311yescharacteristic
210C4⋊Dic5C22×D311yescharacteristic
211#683—62no—
212#683—62no—
213C4×Dic5—21no—
214C4×Dic5—32no—
215C4×Dic5—61no—
216#681—22no—
217C2×#168C22×D312yes—
218#544—151no—
219#518—151no—
220#478—152no—
221#478—152no—
222#476—152no—
223#474—152no—
224#428—51no—
225#427—151no—
226#424—151no—
227#418—151no—
228C2×#102—31no—
229C22×C3×C5—21no—
230C22×C3×C5—21no—
231C22×C3×C5C22⋊Q812yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×C5×D3—24no—
234C2×C5×D3—42no—
235C2×C5×D3—42no—
236C2×C5×D3—84no—
237C3×C4×C5—42no—
238C3×C4×C5C22⋊Q812yes—
239C3×C4×C5C22×Q811yescharacteristic
240C3×C4×C5C2×#4111yescharacteristic
241Dic15—22no—
242Dic15—22no—
243Dic15—41no—
244Dic15—41no—
245C3×Dic5—41no—
246C3×Dic5—41no—
247C23×D3—21no—
248C22×C3×C4—21no—
249C22×C3×C4C22×D511yescharacteristic
250C2×D12—21no—
251C2×D12—22no—
252C2×D12Dic1014yescomplemented
253C2×D12C22×D512yes—
254C2×C4×D3—102no—
255C2×C4×D3—102no—
256C2×Dic6—102no—
257C3×M16—102no—
258C2×C3×C8—52no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—51no—
261C3×C4⋊C4—51no—
262C3×C4⋊C4C22×D512yes—
263C3×C22⋊C4—101no—
264C3×C42—101no—
265#217—52no—
266#217—101no—
267(C2×D3)⋊C4—102no—
268(C2×D3)⋊C4—102no—
269C4⋊Dic3—51no—
270C4⋊Dic3—51no—
271C4⋊Dic3—51no—
272C4⋊Dic3—51no—
273C4⋊Dic3—102no—
274Dic3⋊C4—102no—
275Dic3⋊C4—102no—
276C4×Dic3—52no—
277C4×Dic3—101no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8C22×D511yescharacteristic
280C2×C3⋊C8C22×D511yescharacteristic
281C23×C5—62no—
282C23×C5—64no—
283C23×C5—241no—
284C23×C5—241no—
285C23×C5D3×D411yescharacteristic
286C5×D4—34no—
287C5×D4—64no—
288C5×D4—62no—
289C2×C4×C5—22no—
290C2×C4×C5—81no—
291C2×C4×C5#23512yes—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5D3×Q812yes—
294C2×C4×C5C23×D311yescharacteristic
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—32no—
298C2×Dic5—32no—
299C2×Dic5—61no—
300C2×Dic5—61no—
301C2×Dic5—242no—
302C2×Dic5—242no—
303Dic10—242no—
304C5×C8—61no—
305C5×C8—61no—
306#168—22no—
307C22×D4—31no—
308C2×M16—51no—
309C22×C8—31no—
310C4⋊Q8—152no—
311C4⋊Q8—301no—
312#125—152no—
313#125—301no—
314C22⋊Q8—302no—
315C4×D4—152no—
316C4×D4—152no—
317C4×D4—301no—
318C4×D4—301no—
319#117—51no—
320#117—151no—
321#117—151no—
322C2×C4⋊C4D3×D511yescharacteristic
323C2×C22⋊C4—302no—
324#107—52no—
325#107—151no—
326#107—151no—
327#106—52no—
328#106—151no—
329#106—151no—
330C4⋊C8—152no—
331C4⋊C8—152no—
332#102—32no—
333#102—32no—
334#102—152no—
335#102—152no—
336C2×C3×C5—22no—
337C2×C3×C5C2×C22⋊Q811yescharacteristic
338C2×C3×C5#57611yescharacteristic
339C2×C3×C5#57711yescharacteristic
340C5×D3—44no—
341C23×C3D4×D511yescharacteristic
342C22×D3—22no—
343C22×D3—24no—
344C22×D3—81no—
345C22×D3—81no—
346C2×C3×C4—22no—
347C2×C3×C4—81no—
348C2×C3×C4—101no—
349C2×C3×C4—101no—
350C2×C3×C4C2×Dic1012yes—
351C2×C3×C4D4×D511yescharacteristic
352C2×C3×C4#71112yes—
353C2×C3×C4C23×D511yescharacteristic
354C2×Dic3—52no—
355C2×Dic3—52no—
356C2×Dic3—101no—
357C2×Dic3—101no—
358D12—22no—
359D12—24no—
360D12C2×Dic1014yes—
361C4×D3—402no—
362C4×D3—402no—
363Dic6—402no—
364C3×C8—102no—
365C3⋊C8—21no—
366C3⋊C8—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—64no—
370C22×C5—122no—
371C22×C5—122no—
372C22×C5—244no—
373C22×C5#89612yes—
374C22×C5C2×D3×D411yescharacteristic
375C4×C5—42no—
376C4×C5#89612yes—
377C4×C5C2×#23511yescharacteristic
378C4×C5C2×D3×Q811yescharacteristic
379Dic5—41no—
380Dic5—41no—
381Dic5—62no—
382Dic5—62no—
383Dic5—121no—
384Dic5—121no—
385C24—61no—
386C2×Q8—302no—
387C2×D4—34no—
388C2×D4—32no—
389C2×D4—62no—
390C2×D4—61no—
391C22×C4—21no—
392C22×C4—302no—
393C22×C4—302no—
394C22×C4C2×D3×D511yescharacteristic
395M16—102no—
396C2×C8—31no—
397C2×C8—31no—
398C2×C8—52no—
399C2×C8—62no—
400C4⋊C4—21no—
401C4⋊C4—51no—
402C4⋊C4—51no—
403C4⋊C4—151no—
404C4⋊C4—151no—
405C4⋊C4—151no—
406C4⋊C4—151no—
407C4⋊C4—302no—
408C4⋊C4—302no—
409C4⋊C4—302no—
410C4⋊C4C2×D3×D512yes—
411C22⋊C4—101no—
412C22⋊C4—152no—
413C22⋊C4—302no—
414C22⋊C4—302no—
415C22⋊C4—301no—
416C42—101no—
417C42—152no—
418C42—301no—
419C3×C5#308111yescharacteristic, complemented, Hall
420C22×C3—21no—
421C22×C3—21no—
422C22×C3#402412yes—
423C22×C3C2×D4×D511yescharacteristic
424C2×D3—24no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—84no—
428C3×C4—42no—
429C3×C4—201no—
430C3×C4—201no—
431C3×C4#402412yes—
432C3×C4C22×Dic1011yescharacteristic
433C3×C4C2×#71111yescharacteristic
434Dic3—102no—
435Dic3—102no—
436Dic3—201no—
437Dic3—201no—
438C2×C5—22no—
439C2×C5—124no—
440C2×C5C2×#89611yescharacteristic
441C2×C5#577411yescharacteristic
442C2×C5#577811yescharacteristic
443C23—62no—
444C23—64no—
445C23—241no—
446C23—241no—
447C23#679011yescharacteristic
448Q8—1202no—
449D4—34no—
450D4—64no—
451D4—62no—
452C2×C4—22no—
453C2×C4—81no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—152no—
457C2×C4—152no—
458C2×C4—301no—
459C2×C4—301no—
460C2×C4—1202no—
461C2×C4—1202no—
462C2×C4D3×Dic1012yes—
463C2×C4#677212yes—
464C2×C4#679011yescharacteristic
465C2×C4C22×D3×D511yescharacteristic
466C8—61no—
467C8—61no—
468C8—102no—
469C2×C3—22no—
470C2×C3C2×#402411yescharacteristic
471C2×C3#6609111yescharacteristic
472C2×C3#6609611yescharacteristic
473D3—44no—
474C5#7924211yescharacteristic, complemented, Sylow
475C22—21no—
476C22—21no—
477C22—64no—
478C22—122no—
479C22—122no—
480C22—244no—
481C22#9143212yes—
482C22C2×#679011yescharacteristic
483C4—42no—
484C4—201no—
485C4—201no—
486C4—302no—
487C4—302no—
488C4—601no—
489C4—601no—
490C4#9143212yes—
491C4C2×D3×Dic1011yescharacteristic
492C4C2×#677211yescharacteristic
493C3#1061046211yescharacteristic, complemented, Sylow
494C2—22no—
495C2—124no—
496C2C2×#9143211yescharacteristic
497C2#1175470411yescharacteristic
498C2#1175509811yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC22×C5×D12C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#3081
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11756259C2semidirect
4. C3×#65253C2semidirect
5. #11748451C2semidirect
6. #11748436C2semidirect
8. #11747260C2semidirect
9. #11747258C2semidirect
10. #11747256C2semidirect
11. #11747254C2semidirect
19. #91779C22semidirect
21. #91777C22semidirect
32. C3×#3940C22semidirect
33. C3×#3939C22semidirect
34. #90990C22semidirect
35. #90990C22semidirect
36. #90987C22semidirect
37. #90987C22semidirect
38. #90983C22semidirect
39. #90983C22semidirect
49. #65253D3semidirect
60. C2×C5×D12Q8semidirect
126. #3940C2×D3semidirect
127. #3939C2×D3semidirect
252. C2×D12Dic10semidirect
419. C3×C5#3081semidirect
474. C5#79242semidirect
493. C3#10610462semidirect

Maximal subgroups

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

subgroupindex
#117562592
#117545032
C2×C5×#8092
C3×#652532
#117484512
#117484362
#117483892
#117472602
#117472582
#117472562
#117472542
#106104623
#792425