Finite Group Databasebeta

← #14302560

Subgroups — #14302560

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756507C211yesmaximal, characteristic
2C4×#6817C211yesmaximal, characteristic, complemented
3#11751107C211yesmaximal, characteristic, complemented
4#11751106C211yesmaximal, characteristic, complemented
5C2×#91089C211yesmaximal, characteristic, complemented
6C3×C5×#339C211yesmaximal, characteristic, complemented
7#11746380C211yesmaximal, characteristic, complemented
8#10599571—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C2×C3×C42×C5C2211yescharacteristic
14C4×#1220—22no—
15#91815C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17#91798—21no—
18#91798C412yescomplemented
19#91797—21no—
20#91797C412yescomplemented
21#91793—22no—
22C4×D60—21no—
23#91761—21no—
24C42×D15—21no—
25C4×Dic30—21no—
26#91090—22no—
27#91089—22no—
28#91089—21no—
29#91089—21no—
30#68778—51nomaximal, Hall
31#66082—31no—
32C4×#709—31no—
33#65295—31no—
34#65294—31no—
35C2×#3937—31no—
36C5×#339D311yescharacteristic, complemented
37#64779—31no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2311yescharacteristic
40C2×#1220—21no—
41C2×#1220—21no—
42#6817—21no—
43#6817—21no—
44#6817—21no—
45#6817—21no—
46#6817—21no—
47#6817—21no—
48#6817C2×C412yes—
49#6817D412yes—
50C2×D60C2×C411yescharacteristic, complemented
51C2×D60D411yescharacteristic
52C2×C4×D15—21no—
53C2×C4×D15—22no—
54C2×C4×D15—21no—
55C2×Dic30C2×C411yescharacteristic, complemented
56C2×Dic30D411yescharacteristic
57C3×C5×M16—21no—
58C3×C5×M16C2×C412yescomplemented
59C2×C3×C5×C8—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C42×C5—21no—
63C3×C42×C5—21no—
64C3×C42×C5D411yescharacteristic
65C3×C42×C5D411yescharacteristic
66#6719D412yes—
67#6714—22no—
68C4⋊Dic15—21no—
69#6712—22no—
70C4×Dic15—22no—
71C4×Dic15—21no—
72#6710—21no—
73#6710—21no—
74C2×C3⋊#168—21no—
75Dic60—22no—
76D120—22no—
77#6706—22no—
78#6706—22no—
79#5748—51no—
80C4×#235—51no—
81#5006—51no—
82#5005—51no—
83C2×#805—51no—
84C3×#339D511yescharacteristic, complemented
85#4559—51no—
86C2×#709—31no—
87C2×#709—31no—
88C2×C5×M16C2×D311yescharacteristic
89C5×#117C2×D311yescharacteristic
90C2×C42×C5C2×D311yescharacteristic
91C4×C22⋊D5—62no—
92#4072—31no—
93C2×#681—31no—
94#4055—32no—
95#4055—61no—
96#4054—32no—
97#4054—61no—
98#4050—62no—
99C4×D20—61no—
100#4018—61no—
101C42×D5—61no—
102C4×Dic10—61no—
103#3938—62no—
104#3937—62no—
105#3937—61no—
106#3937—61no—
107#1906—151noSylow
108C23×C3×C5C2×D411yescharacteristic
109C22×D15—21no—
110C22×D15—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—42no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5C22×C411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×C3×C4×C5#4111yescharacteristic
121#1220—22no—
122#1220—22no—
123#1220—41no—
124#1220—41no—
125C2×Dic15—21no—
126C2×Dic15—21no—
127C2×Dic15—22no—
128D60—21no—
129D60—21no—
130D60—21no—
131D60C22⋊C412yes—
132C4×D15—22no—
133C4×D15—22no—
134C4×D15—44no—
135C4×D15—41no—
136C4×D15—41no—
137Dic30—21no—
138Dic30—21no—
139Dic30—21no—
140Dic30C22⋊C412yes—
141C3×C5×C8—22no—
142C3⋊#168—41no—
143C2×#235—51no—
144C2×#235—51no—
145C2×C3×M16C2×D511yescharacteristic
146C3×#117C2×D511yescharacteristic
147C2×C3×C42C2×D511yescharacteristic
148C4×#67—102no—
149#926—51no—
150C2×#208—51no—
151#909—52no—
152#909—101no—
153#908—52no—
154#908—101no—
155#904—102no—
156C4×D12—101no—
157#872—101no—
158C42×D3—101no—
159C4×Dic6—101no—
160#806—102no—
161#805—102no—
162#805—101no—
163#805—101no—
164C22×C4×C5—21no—
165C22×C4×C5C22×D311yescharacteristic
166C2×C22⋊D5—61no—
167C2×C22⋊D5—61no—
168#709—32no—
169#709—32no—
170#709—61no—
171#709—61no—
172#709—61no—
173#709—61no—
174#709—61no—
175#709—61no—
176C2×D20—31no—
177C2×D20—31no—
178C2×C4×D5—61no—
179C2×C4×D5—62no—
180C2×C4×D5—61no—
181C2×Dic10—31no—
182C2×Dic10—31no—
183C5×M16—21no—
184C5×M16C4×D312yescomplemented
185C2×C5×C8—21no—
186C5×C4⋊C4—21no—
187C5×C22⋊C4—21no—
188C42×C5—21no—
189C42×C5—21no—
190C42×C5D1211yescharacteristic
191C42×C5D1211yescharacteristic
192C22⋊Dic5—32no—
193#685—62no—
194C4⋊Dic5—61no—
195#683—62no—
196C4×Dic5—62no—
197C4×Dic5—61no—
198#681—61no—
199#681—61no—
200C2×#168—61no—
201Dic20—62no—
202D40—62no—
203#677—62no—
204#677—62no—
205#575—151no—
206C4×#41—151no—
207#421—151no—
208C2×#104—151no—
209#410—151no—
210#344—151no—
211#339D1511yescharacteristic, complemented
212C22×C3×C5—41no—
213C22×C3×C5C4×D411yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C2×D15—22no—
217C2×D15—42no—
218C2×D15—41no—
219C2×D15—81no—
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5—81no—
223C3×C4×C5C2×C22⋊C411yescharacteristic
224C3×C4×C5C4×D411yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C3×C4×C5#12311yescharacteristic
227Dic15—22no—
228Dic15—42no—
229Dic15—41no—
230C22×C3×C4—21no—
231C22×C3×C4C22×D511yescharacteristic
232C2×#67—101no—
233C2×#67—101no—
234#235—52no—
235#235—52no—
236#235—101no—
237#235—101no—
238#235—101no—
239#235—101no—
240#235—101no—
241#235—101no—
242C2×D12—51no—
243C2×D12—51no—
244C2×C4×D3—101no—
245C2×C4×D3—102no—
246C2×C4×D3—101no—
247C2×Dic6—51no—
248C2×Dic6—51no—
249C3×M16—21no—
250C3×M16C4×D512yescomplemented
251C2×C3×C8—21no—
252C3×C4⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42D2011yescharacteristic
257C3×C42D2011yescharacteristic
258#217—52no—
259(C2×D3)⋊C4—102no—
260C4⋊Dic3—101no—
261Dic3⋊C4—102no—
262C4×Dic3—102no—
263C4×Dic3—101no—
264#208—101no—
265#208—101no—
266C2×C3⋊C8—101no—
267Dic12—102no—
268D24—102no—
269#204—102no—
270#204—102no—
271C23×C5D3×D411yescharacteristic
272C22×D5—61no—
273C22×D5—61no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—42no—
277C2×C4×C5—41no—
278C2×C4×C5C2×C4×D311yescharacteristic
279C2×C4×C5C2×D1211yescharacteristic
280C2×C4×C5#23511yescharacteristic
281C2×C4×C5D3×D411yescharacteristic
282C2×C4×C5D3×D411yescharacteristic
283C2×C4×C5#23911yescharacteristic
284C22⋊D5—62no—
285C22⋊D5—62no—
286C22⋊D5—121no—
287C22⋊D5—121no—
288C2×Dic5—61no—
289C2×Dic5—62no—
290C2×Dic5—61no—
291D20—32no—
292D20—61no—
293D20—61no—
294D20—61no—
295C4×D5—62no—
296C4×D5—62no—
297C4×D5—124no—
298C4×D5—121no—
299C4×D5—121no—
300Dic10—32no—
301Dic10—61no—
302Dic10—61no—
303Dic10—61no—
304C5×C8—22no—
305#168—121no—
306C2×#41—151no—
307C2×#41—151no—
308#137—152no—
309#137—301no—
310AGL(1,ℤ/8)—152no—
311AGL(1,ℤ/8)—301no—
312#135—302no—
313C2×M16—151no—
314C2×M16C2×D1511yescharacteristic
315C4×Q8—301no—
316C4×D4—301no—
317C4×D4—302no—
318#117—151no—
319#117—301no—
320#117C2×D1511yescharacteristic
321C2×C42—301no—
322C2×C42C2×D1511yescharacteristic
323#104—302no—
324#104—301no—
325#104—301no—
326#99—302no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5#38611yescharacteristic
331D15—42no—
332D15—81no—
333C23×C3D4×D511yescharacteristic
334C22×D3—101no—
335C22×D3—101no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4C2×C4×D511yescharacteristic
341C2×C3×C4C2×D2011yescharacteristic
342C2×C3×C4#70911yescharacteristic
343C2×C3×C4D4×D511yescharacteristic
344C2×C3×C4D4×D511yescharacteristic
345C2×C3×C4#71311yescharacteristic
346#67—102no—
347#67—102no—
348#67—201no—
349#67—201no—
350C2×Dic3—101no—
351C2×Dic3—102no—
352C2×Dic3—101no—
353D12—52no—
354D12—101no—
355D12—101no—
356D12—101no—
357C4×D3—102no—
358C4×D3—102no—
359C4×D3—204no—
360C4×D3—201no—
361C4×D3—201no—
362Dic6—52no—
363Dic6—101no—
364Dic6—101no—
365Dic6—101no—
366C3×C8—22no—
367C3⋊C8—201no—
368C22×C5—41no—
369C22×C5#88211yescharacteristic
370C22×C5#88411yescharacteristic
371C22×C5#89311yescharacteristic
372C2×D5—62no—
373C2×D5—122no—
374C2×D5—121no—
375C2×D5—241no—
376C4×C5—22no—
377C4×C5—41no—
378C4×C5—81no—
379C4×C5C4×D1211yescharacteristic
380C4×C5D3×C22⋊C411yescharacteristic
381C4×C5#89411yescharacteristic
382C4×C5#89511yescharacteristic
383Dic5—62no—
384Dic5—122no—
385Dic5—121no—
386#41—152no—
387#41—152no—
388#41—301no—
389#41—301no—
390#41—301no—
391#41—301no—
392#41—301no—
393#41—301no—
394C2×Q8—151no—
395C2×Q8—151no—
396C2×D4—151no—
397C2×D4—151no—
398C2×D4—301no—
399C2×D4—301no—
400C22×C4—21no—
401C22×C4—302no—
402C22×C4—301no—
403C22×C4—301no—
404C22×C4C22×D1511yescharacteristic
405Q16—302no—
406SD16—302no—
407SD16—302no—
408D8—302no—
409M16—21no—
410M16—301no—
411M16—301no—
412M16C4×D1512yescomplemented
413C2×C8—21no—
414C2×C8—301no—
415C4⋊C4—21no—
416C4⋊C4—302no—
417C4⋊C4—301no—
418C22⋊C4—21no—
419C22⋊C4—152no—
420C22⋊C4—302no—
421C42—21no—
422C42—21no—
423C42—302no—
424C42—301no—
425C42D6011yescharacteristic
426C42D6011yescharacteristic
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3#402811yescharacteristic
430C22×C3#403011yescharacteristic
431C22×C3#403911yescharacteristic
432C2×D3—102no—
433C2×D3—202no—
434C2×D3—201no—
435C2×D3—401no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—81no—
439C3×C4C4×D2011yescharacteristic
440C3×C4D5×C22⋊C411yescharacteristic
441C3×C4#404011yescharacteristic
442C3×C4#404111yescharacteristic
443Dic3—102no—
444Dic3—202no—
445Dic3—201no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5#466511yescharacteristic
450D5—122no—
451D5—241no—
452C23—301no—
453C23—301no—
454C23D4×D1511yescharacteristic
455Q8—152no—
456Q8—301no—
457Q8—301no—
458Q8—301no—
459D4—152no—
460D4—302no—
461D4—302no—
462D4—301no—
463D4—301no—
464D4—301no—
465D4—601no—
466D4—601no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—301no—
472C2×C4—302no—
473C2×C4—302no—
474C2×C4—302no—
475C2×C4—301no—
476C2×C4—604no—
477C2×C4—601no—
478C2×C4—601no—
479C2×C4C2×C4×D1511yescharacteristic
480C2×C4C2×D6011yescharacteristic
481C2×C4#681711yescharacteristic
482C2×C4D4×D1511yescharacteristic
483C2×C4D4×D1511yescharacteristic
484C2×C4#682111yescharacteristic
485C8—22no—
486C8—601no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3#6495411yescharacteristic
491D3—202no—
492D3—401no—
493C5#6877811yescharacteristic, complemented, Sylow
494C22—41no—
495C22—302no—
496C22—602no—
497C22—601no—
498C22—1201no—
499C22#9177111yescharacteristic
500C22#9177311yescharacteristic
501C22#9178211yescharacteristic
502C4—22no—
503C4—41no—
504C4—81no—
505C4—302no—
506C4—602no—
507C4—601no—
508C4C4×D6011yescharacteristic
509C4C22⋊C4×D1511yescharacteristic
510C4#9178311yescharacteristic
511C4#9178411yescharacteristic
512C3#1059957111yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—602no—
517C2—1201no—
518C2#1175076611yescharacteristic
subgroupG/H
centerC4#91784
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#339C2
socleC2×C3×C5#386
prime radical#11756507C2
cosocleC22×C3×C4×C5C23
hypercenter#339D15
hypocenterC3×C5#1906
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C4×#6817C2semidirect
3. #11751107C2semidirect
4. #11751106C2semidirect
5. C2×#91089C2semidirect
6. C3×C5×#339C2semidirect
7. #11746380C2semidirect
18. #91798C4semidirect
20. #91797C4semidirect
36. C5×#339D3semidirect
50. C2×D60C2×C4semidirect
55. C2×Dic30C2×C4semidirect
58. C3×C5×M16C2×C4semidirect
84. C3×#339D5semidirect
184. C5×M16C4×D3semidirect
211. #339D15semidirect
250. C3×M16C4×D5semidirect
412. M16C4×D15semidirect
427. C3×C5#1906semidirect
493. C5#68778semidirect
512. C3#10599571semidirect

Maximal subgroups

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

subgroupindex
#117565072
C4×#68172
#117511072
#117511062
C2×#910892
C3×C5×#3392
#117463802
#105995713
#687785