Finite Group Databasebeta

← #14430870

Subgroups — #14430870

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
1C5×#5519C211yesmaximal, characteristic, complemented
2#11754488C211yesmaximal, characteristic
3#11751099C211yesmaximal, characteristic, complemented
4C3×#65290C211yesmaximal, characteristic
5#11748459C211yesmaximal, characteristic, complemented
6#11748441C211yesmaximal, characteristic, complemented
7C2×#90964C211yesmaximal, characteristic
8#11747251C212yesmaximal, complemented
9#11747249C212yesmaximal, complemented
10#11747247C212yesmaximal, complemented
11#11747245C212yesmaximal, complemented
12#10615205—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C3×C5×#117C2211yescharacteristic
15#91815C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C5×#926C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C5×#890C2212yescomplemented
20C5×#888C2212yescomplemented
21C5×#878—22no—
22C5×#870—21no—
23C5×#869—21no—
24C2×C3×#681C2211yescharacteristic
25D5×#217—22no—
26#91434—21no—
27#91434—21no—
28Dic3×D20C2212yes—
29Dic3×D20C2212yes—
30#91092C2211yescharacteristic, complemented
31#91092C2211yescharacteristic, complemented
32#91091C2211yescharacteristic, complemented
33#91091C2211yescharacteristic, complemented
34C3×#3941C2212yes—
35C3×#3941C2212yes—
36#90988C2212yescomplemented
37#90985C2212yescomplemented
38Dic3⋊#168C2212yescomplemented
39Dic3⋊#168C2212yescomplemented
40#90964C2212yes—
41#90964C2212yes—
42#78918—51nomaximal, Hall
43C5×#544—31no—
44#65855—31no—
45#65376—32no—
46#65373—32no—
47#65355—32no—
48#65355—32no—
49#65290D311yescharacteristic
50#65288—31no—
51#65287—31no—
52C2×#3941—31no—
53#65253—31no—
54C23×C3×D5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C5×Dic3—21no—
57C2×C5×Dic6—22no—
58C2×C3×D20—21no—
59C2×C3×D20—22no—
60C2×C3×D20Q814yescomplemented
61C2×C3×D20C2312yes—
62C2×D5×Dic3—22no—
63C2×D5×Dic3—22no—
64C3×C5×C4⋊C4C2311yescharacteristic
65C3×C5×C4⋊C4C2311yescharacteristic
66C3×C5×C22⋊C4—21no—
67C3×C42×C5—21no—
68#6719—21no—
69C4⋊Dic15C2311yescharacteristic
70C4⋊Dic15C2311yescharacteristic
71C4×Dic15—21no—
72C2×C3⋊#168—22no—
73C2×C3⋊#168C2311yescharacteristic
74C2×C3⋊#168C2311yescharacteristic
75C5×#217D412yes—
76C5×C4⋊Dic3—21no—
77C5×C4⋊Dic3—22no—
78C5×C4⋊Dic3D411yescharacteristic
79C5×C4⋊Dic3D411yescharacteristic
80C5×C4⋊Dic3C2312yes—
81C5×Dic3⋊C4—22no—
82C5×Dic3⋊C4—22no—
83C4×C5×Dic3C2312yes—
84C3×#681—22no—
85C2×C3×#168C2312yes—
86#6665—22no—
87#6665—22no—
88#5798—51no—
89#5519D511yescharacteristic, complemented
90C3×#418—51no—
91C2×#832—51no—
92#5121—51no—
93#5119—51no—
94#4998—51no—
95#4766—52no—
96#4763—52no—
97#4761—52no—
98#4759—52no—
99C22×D20C2×D311yescharacteristic
100C5×C4⋊Q8—32no—
101C5×C4⋊Q8—61no—
102C5×#125—32no—
103C5×#125—61no—
104C5×C22⋊Q8—62no—
105C5×#117—31no—
106C5×#117C2×D311yescharacteristic
107C2×C5×C4⋊C4—31no—
108#4072—31no—
109C2×#681C2×D311yescharacteristic
110C22×#168—31no—
111#4039—61no—
112#4039—61no—
113D5×C22⋊C4—62no—
114C4×D20—32no—
115C4×D20—32no—
116#3941—32no—
117#3941—32no—
118#3941C2×D312yes—
119#3941C2×D312yes—
120#3940—31no—
121#3940—31no—
122#3940—32no—
123#3939—31no—
124#3939—31no—
125#3939—32no—
126C4⋊#168—32no—
127C4⋊#168—32no—
128#3081—151noSylow
129C23×C3×C5C2×D411yescharacteristic
130C22×C3×D5—22no—
131C22×C3×D5—24no—
132C22×C3×D5—81no—
133C22×C3×D5—81no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
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×C5×Dic3—22no—
143C2×C5×Dic3—81no—
144C2×C5×Dic3C2×D412yes—
145C2×C5×Dic3C2×D412yes—
146C5×Dic6—82no—
147C3×D20—24no—
148C3×D20—22no—
149C3×D20C2×Q814yes—
150D5×Dic3—82no—
151D5×Dic3—82no—
152C3⋊#168—21no—
153C3⋊#168—21no—
154C3×#168—22no—
155C22×C3×D4—51no—
156C2×C3×M16—51no—
157C3×#117C2×D511yescharacteristic
158C2×#217—102no—
159D4×Dic3—52no—
160D4×Dic3—52no—
161D4×Dic3—101no—
162D4×Dic3—101no—
163#926—51no—
164#926C2×D511yescharacteristic
165C2×C4⋊Dic3C2×D511yescharacteristic
166C22×C3⋊C8—51no—
167#890C2×D512yescomplemented
168#888C2×D512yescomplemented
169#878—22no—
170#870—21no—
171#869—21no—
172C3×#102—52no—
173C3×#102—52no—
174#832—52no—
175#832—52no—
176#818—52no—
177#817—52no—
178Dic3⋊C8—52no—
179Dic3⋊C8—52no—
180#808—51no—
181#808—51no—
182#807—51no—
183#807—51no—
184C23×D5—21no—
185C2×C5×Q8—62no—
186C22×C4×C5—61no—
187C22×C4×C5C22×D311yescharacteristic
188C2×D20—21no—
189C2×D20—22no—
190C2×D20Dic614yescomplemented
191C2×D20C22×D312yes—
192C2×C4×D5—62no—
193C2×C4×D5—62no—
194C5×C4⋊C4—31no—
195C5×C4⋊C4—31no—
196C5×C4⋊C4—32no—
197C5×C4⋊C4—62no—
198C5×C4⋊C4—62no—
199C5×C4⋊C4—62no—
200C5×C4⋊C4—61no—
201C5×C4⋊C4C22×D311yescharacteristic
202C5×C4⋊C4C22×D311yescharacteristic
203C5×C22⋊C4—21no—
204C5×C22⋊C4—32no—
205C42×C5—21no—
206C42×C5—32no—
207C22⋊Dic5—61no—
208#685—62no—
209#685—62no—
210C4⋊Dic5—31no—
211C4⋊Dic5—31no—
212C4×Dic5—61no—
213#681—22no—
214C2×#168—31no—
215C2×#168—31no—
216C2×#168—62no—
217C2×#168C22×D312yes—
218#544—31no—
219#518—151no—
220#478—152no—
221#478—152no—
222#476—152no—
223#474—152no—
224#428—151no—
225#427—151no—
226#424—151no—
227#418—51no—
228C2×#102—151no—
229C22×C3×C5—21no—
230C22×C3×C5—21no—
231C22×C3×C5C22⋊Q812yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×C3×D5—24no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—84no—
237C3×C4×C5—41no—
238C3×C4×C5—41no—
239C3×C4×C5C22⋊Q812yes—
240C3×C4×C5C22×Q811yescharacteristic
241C3×C4×C5C2×#4111yescharacteristic
242Dic15—41no—
243Dic15—41no—
244C5×Dic3—22no—
245C5×Dic3—22no—
246C5×Dic3—42no—
247C24×C3—101no—
248C2×C3×D4—54no—
249C2×C3×D4—52no—
250C2×C3×D4—102no—
251C2×C3×D4—101no—
252C22×C3×C4C22×D511yescharacteristic
253C22×Dic3—21no—
254C22×Dic3—102no—
255C22×Dic3—102no—
256C2×Dic6—22no—
257C3×M16—102no—
258C2×C3×C8—52no—
259C3×C4⋊C4C22×D511yescharacteristic
260C3×C4⋊C4C22×D511yescharacteristic
261C3×C22⋊C4—21no—
262C3×C42—21no—
263#217—101no—
264#217—102no—
265#217—102no—
266#217C22⋊D512yes—
267C4⋊Dic3—21no—
268C4⋊Dic3—22no—
269C4⋊Dic3—51no—
270C4⋊Dic3—51no—
271C4⋊Dic3C22⋊D511yescharacteristic
272C4⋊Dic3C22⋊D511yescharacteristic
273C4⋊Dic3C22×D512yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—101no—
277C4×Dic3C22×D512yes—
278C2×C3⋊C8—51no—
279C2×C3⋊C8—51no—
280C2×C3⋊C8—102no—
281C23×C5D3×D411yescharacteristic
282C22×D5—22no—
283C22×D5—24no—
284C22×D5—81no—
285C22×D5—81no—
286C5×Q8—242no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—32no—
290C2×C4×C5—32no—
291C2×C4×C5—62no—
292C2×C4×C5—241no—
293C2×C4×C5C2×Dic612yes—
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5#23712yes—
296C2×C4×C5C23×D311yescharacteristic
297C2×Dic5—61no—
298C2×Dic5—61no—
299D20—24no—
300D20—22no—
301D20C2×Dic614yes—
302C4×D5—242no—
303C4×D5—242no—
304#168—22no—
305#168—61no—
306#168—61no—
307C22×D4—51no—
308C2×M16—51no—
309C22×C8—151no—
310C4⋊Q8—32no—
311C4⋊Q8—61no—
312#125—32no—
313#125—61no—
314C22⋊Q8—62no—
315C4×D4—152no—
316C4×D4—152no—
317C4×D4—301no—
318C4×D4—301no—
319#117—31no—
320#117—151no—
321#117D3×D511yescharacteristic
322C2×C4⋊C4—31no—
323C2×C22⋊C4—302no—
324#107—152no—
325#107—151no—
326#107—151no—
327#106—152no—
328#106—151no—
329#106—151no—
330C4⋊C8—152no—
331C4⋊C8—152no—
332#102—52no—
333#102—52no—
334#102—152no—
335#102—152no—
336C2×C3×C5—22no—
337C2×C3×C5C2×C22⋊Q811yescharacteristic
338C2×C3×C5#57611yescharacteristic
339C2×C3×C5#57711yescharacteristic
340C3×D5—44no—
341C23×C3—102no—
342C23×C3—104no—
343C23×C3—401no—
344C23×C3—401no—
345C23×C3C2×C22⋊D511yescharacteristic
346C3×D4—54no—
347C3×D4—102no—
348C3×D4—104no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4Q8×D512yes—
352C2×C3×C4#71312yes—
353C2×C3×C4C2×C22⋊D511yescharacteristic
354C2×C3×C4C23×D511yescharacteristic
355C2×Dic3—22no—
356C2×Dic3—81no—
357C2×Dic3—101no—
358C2×Dic3—101no—
359C2×Dic3—402no—
360C2×Dic3—402no—
361C2×Dic3C2×C22⋊D512yes—
362C2×Dic3C2×C22⋊D512yes—
363Dic6—82no—
364C3×C8—102no—
365C3⋊C8—101no—
366C3⋊C8—101no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5#87812yes—
370C22×C5C2×D3×D411yescharacteristic
371C2×D5—24no—
372C2×D5—42no—
373C2×D5—42no—
374C2×D5—84no—
375C4×C5—41no—
376C4×C5—41no—
377C4×C5—62no—
378C4×C5—62no—
379C4×C5—122no—
380C4×C5#87812yes—
381C4×C5C22×Dic611yescharacteristic
382C4×C5C2×#23711yescharacteristic
383Dic5—121no—
384Dic5—121no—
385C24—101no—
386C2×Q8—62no—
387C2×D4—54no—
388C2×D4—52no—
389C2×D4—102no—
390C2×D4—101no—
391C22×C4—61no—
392C22×C4—302no—
393C22×C4—302no—
394C22×C4C2×D3×D511yescharacteristic
395M16—102no—
396C2×C8—52no—
397C2×C8—151no—
398C2×C8—151no—
399C2×C8—302no—
400C4⋊C4—31no—
401C4⋊C4—31no—
402C4⋊C4—32no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—151no—
408C4⋊C4—151no—
409C4⋊C4C2×D3×D511yescharacteristic
410C4⋊C4C2×D3×D511yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—32no—
413C22⋊C4—302no—
414C22⋊C4—302no—
415C22⋊C4—301no—
416C42—21no—
417C42—32no—
418C42—301no—
419C3×C5#308111yescharacteristic, complemented, Hall
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—104no—
423C22×C3—202no—
424C22×C3—202no—
425C22×C3—404no—
426C22×C3#409212yes—
427C22×C3C22×C22⋊D511yescharacteristic
428C3×C4—41no—
429C3×C4—41no—
430C3×C4#409212yes—
431C3×C4C2×Q8×D511yescharacteristic
432C3×C4C2×#71311yescharacteristic
433Dic3—22no—
434Dic3—22no—
435Dic3—42no—
436Dic3—201no—
437Dic3—201no—
438C2×C5—22no—
439C2×C5C2×#87811yescharacteristic
440C2×C5#577411yescharacteristic
441C2×C5#577511yescharacteristic
442D5—44no—
443C23—102no—
444C23—104no—
445C23—401no—
446C23—401no—
447C23D3×C22⋊D511yescharacteristic
448Q8—242no—
449D4—54no—
450D4—104no—
451D4—102no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—62no—
457C2×C4—241no—
458C2×C4—301no—
459C2×C4—301no—
460C2×C4—1202no—
461C2×C4—1202no—
462C2×C4D5×Dic612yes—
463C2×C4#676612yes—
464C2×C4D3×C22⋊D511yescharacteristic
465C2×C4C22×D3×D511yescharacteristic
466C8—102no—
467C8—301no—
468C8—301no—
469C2×C3—22no—
470C2×C3—204no—
471C2×C3C2×#409211yescharacteristic
472C2×C3#6616911yescharacteristic
473C2×C3#6617111yescharacteristic
474C5#7891811yescharacteristic, complemented, Sylow
475C22—21no—
476C22—21no—
477C22—104no—
478C22—202no—
479C22—202no—
480C22—404no—
481C22#9142112yes—
482C22C2×D3×C22⋊D511yescharacteristic
483C4—41no—
484C4—41no—
485C4—62no—
486C4—62no—
487C4—122no—
488C4—601no—
489C4—601no—
490C4#9142112yes—
491C4C2×D5×Dic611yescharacteristic
492C4C2×#676611yescharacteristic
493C3#1061520511yescharacteristic, complemented, Sylow
494C2—22no—
495C2—204no—
496C2C2×#9142111yescharacteristic
497C2#1175532311yescharacteristic
498C2#1175532411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC22×C3×D20C22
hypercenter#117D3×D5
hypocenterC3×C5#3081
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5519C2semidirect
3. #11751099C2semidirect
5. #11748459C2semidirect
6. #11748441C2semidirect
8. #11747251C2semidirect
9. #11747249C2semidirect
10. #11747247C2semidirect
11. #11747245C2semidirect
19. C5×#890C22semidirect
20. C5×#888C22semidirect
30. #91092C22semidirect
31. #91092C22semidirect
32. #91091C22semidirect
33. #91091C22semidirect
36. #90988C22semidirect
37. #90985C22semidirect
38. Dic3⋊#168C22semidirect
39. Dic3⋊#168C22semidirect
60. C2×C3×D20Q8semidirect
89. #5519D5semidirect
167. #890C2×D5semidirect
168. #888C2×D5semidirect
190. C2×D20Dic6semidirect
419. C3×C5#3081semidirect
474. C5#78918semidirect
493. C3#10615205semidirect

Maximal subgroups

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

subgroupindex
C5×#55192
#117544882
#117510992
C3×#652902
#117484592
#117484412
C2×#909642
#117472512
#117472492
#117472472
#117472452
#106152053
#789185