Finite Group Databasebeta

← #14302569

Subgroups — #14302569

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
1C3×C5×#575C211yesmaximal, characteristic
2#41×Dic15C211yesmaximal, characteristic, complemented
3#11751341C211yesmaximal, characteristic, complemented
4#11751337C211yesmaximal, characteristic, complemented
5C2×#91121C211yesmaximal, characteristic, complemented
6#11746380C211yesmaximal, characteristic, complemented
7#11746376C211yesmaximal, characteristic, complemented
8#10602907—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C3×C5×#137—21no—
12C3×C5×#137C412yescomplemented
13C3×C5×AGL(1,ℤ/8)—21no—
14C3×C5×AGL(1,ℤ/8)C412yescomplemented
15C3×C5×#135—22no—
16C2×C3×C5×M16C2211yescharacteristic
17Q8×Dic15—21no—
18D4×Dic15—21no—
19D4×Dic15—22no—
20#91815—21no—
21#91815C2211yescharacteristic
22#91815C2211yescharacteristic
23C2×C4×Dic15—21no—
24C2×C4×Dic15C2211yescharacteristic
25C2×#6710C2211yescharacteristic
26#91121—22no—
27#91121—21no—
28#91121—21no—
29#91118—22no—
30#72114—51nomaximal, Hall
31C5×#575D311yescharacteristic
32#41×Dic5—31no—
33#65529—31no—
34#65525—31no—
35C2×#3969—31no—
36#64779—31no—
37#64775—31no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41C2×C412yes—
45C3×C5×#41D412yes—
46C2×C3×C5×Q8C2×C411yescharacteristic, complemented
47C2×C3×C5×Q8D411yescharacteristic
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4C2×C411yescharacteristic, complemented
51C2×C3×C5×D4D411yescharacteristic
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15—22no—
56C22×Dic15—21no—
57C3×C5×Q16—22no—
58C3×C5×SD16—22no—
59C3×C5×SD16—22no—
60C3×C5×D8—22no—
61C3×C5×M16—21no—
62C3×C5×M16C2×C412yescomplemented
63C2×C3×C5×C8—21no—
64#6719—22no—
65#6719—21no—
66#6719D412yes—
67C4⋊Dic15—22no—
68C4⋊Dic15—21no—
69C4⋊Dic15—21no—
70C4×Dic15—22no—
71C4×Dic15—21no—
72C4×Dic15—21no—
73C4×Dic15—21no—
74C4×Dic15D411yescharacteristic
75C4×Dic15D411yescharacteristic
76#6710—21no—
77#6710—21no—
78C2×C3⋊#168—21no—
79C3×#575D511yescharacteristic
80Dic3×#41—51no—
81#5240—51no—
82#5236—51no—
83C2×#837—51no—
84#4559—51no—
85#4555—51no—
86C2×C5×#41C2×D311yescharacteristic
87C2×C5×#41C2×D311yescharacteristic
88C5×#137—21no—
89C5×#137Dic312yescomplemented
90C5×AGL(1,ℤ/8)—21no—
91C5×AGL(1,ℤ/8)Dic312yescomplemented
92C5×#135—22no—
93C2×C5×M16C2×D311yescharacteristic
94Q8×Dic5—61no—
95D4×Dic5—62no—
96D4×Dic5—61no—
97#4072—31no—
98#4072—31no—
99#4072—61no—
100C2×C4×Dic5—31no—
101C2×C4×Dic5—61no—
102C2×#681—31no—
103#3969—61no—
104#3969—61no—
105#3969—62no—
106#3966—62no—
107#1906—151noSylow
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5C2×D411yescharacteristic
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×Q8—21no—
114C3×C5×Q8C22⋊C412yes—
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—21no—
118C3×C5×D4—21no—
119C3×C5×D4—21no—
120C3×C5×D4—41no—
121C3×C5×D4—41no—
122C3×C5×D4C22⋊C412yes—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5#4111yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C2×Dic15—21no—
136C2×Dic15—22no—
137C2×Dic15—21no—
138C2×Dic15—42no—
139C2×Dic15—44no—
140C2×Dic15—41no—
141C3×C5×C8—22no—
142C3⋊#168—41no—
143C2×C3×#41C2×D511yescharacteristic
144C2×C3×#41C2×D511yescharacteristic
145C3×#137—21no—
146C3×#137Dic512yescomplemented
147C3×AGL(1,ℤ/8)—21no—
148C3×AGL(1,ℤ/8)Dic512yescomplemented
149C3×#135—22no—
150C2×C3×M16C2×D511yescharacteristic
151Q8×Dic3—101no—
152D4×Dic3—102no—
153D4×Dic3—101no—
154#926—51no—
155#926—51no—
156#926—101no—
157C2×C4×Dic3—51no—
158C2×C4×Dic3—101no—
159C2×#208—51no—
160#837—101no—
161#837—101no—
162#837—102no—
163#834—102no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C5×#41C2×Dic312yes—
171C5×#41#6712yes—
172C2×C5×Q8C2×Dic311yescharacteristic, complemented
173C2×C5×Q8#6711yescharacteristic
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4C2×Dic311yescharacteristic, complemented
177C2×C5×D4#6711yescharacteristic
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5C22×D311yescharacteristic
181C22×Dic5—62no—
182C22×Dic5—61no—
183C5×Q16—22no—
184C5×SD16—22no—
185C5×SD16—22no—
186C5×D8—22no—
187C5×M16—21no—
188C5×M16C2×Dic312yescomplemented
189C2×C5×C8—21no—
190C22⋊Dic5—32no—
191C22⋊Dic5—62no—
192C22⋊Dic5—61no—
193C4⋊Dic5—62no—
194C4⋊Dic5—61no—
195C4⋊Dic5—61no—
196C4×Dic5—31no—
197C4×Dic5—31no—
198C4×Dic5—62no—
199C4×Dic5—61no—
200C4×Dic5—61no—
201C4×Dic5—61no—
202#681—61no—
203#681—61no—
204C2×#168—61no—
205#575D1511yescharacteristic
206C4×#41—151no—
207#421—151no—
208C2×#104—151no—
209#410—151no—
210#344—151no—
211#339—151no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5—41no—
216C22×C3×C5—81no—
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5#12011yescharacteristic
219C22×C3×C5#12111yescharacteristic
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5C2×C22⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12311yescharacteristic
226Dic15—22no—
227Dic15—42no—
228Dic15—41no—
229Dic15—81no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C2×Dic512yes—
237C3×#41C22⋊D512yes—
238C2×C3×Q8C2×Dic511yescharacteristic, complemented
239C2×C3×Q8C22⋊D511yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4C2×Dic511yescharacteristic, complemented
243C2×C3×D4C22⋊D511yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—102no—
248C22×Dic3—101no—
249C3×Q16—22no—
250C3×SD16—22no—
251C3×SD16—22no—
252C3×D8—22no—
253C3×M16—21no—
254C3×M16C2×Dic512yescomplemented
255C2×C3×C8—21no—
256#217—52no—
257#217—102no—
258#217—101no—
259C4⋊Dic3—102no—
260C4⋊Dic3—101no—
261C4⋊Dic3—101no—
262C4×Dic3—51no—
263C4×Dic3—51no—
264C4×Dic3—102no—
265C4×Dic3—101no—
266C4×Dic3—101no—
267C4×Dic3—101no—
268#208—101no—
269#208—101no—
270C2×C3⋊C8—101no—
271C23×C5—21no—
272C23×C5—21no—
273C23×C5D3×D411yescharacteristic
274C5×Q8—21no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8#21712yes—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—21no—
281C5×D4—21no—
282C5×D4—21no—
283C5×D4—41no—
284C5×D4—41no—
285C5×D4#21712yes—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5#23711yescharacteristic
294C2×C4×C5#23711yescharacteristic
295C2×C4×C5C22×Dic311yescharacteristic
296C2×C4×C5C2×#6711yescharacteristic
297C2×C4×C5C2×#6711yescharacteristic
298C2×Dic5—61no—
299C2×Dic5—61no—
300C2×Dic5—62no—
301C2×Dic5—122no—
302C2×Dic5—124no—
303C2×Dic5—121no—
304C5×C8—22no—
305#168—121no—
306C2×#41C2×D1511yescharacteristic
307C2×#41C2×D1511yescharacteristic
308#137—21no—
309#137Dic1512yescomplemented
310AGL(1,ℤ/8)—21no—
311AGL(1,ℤ/8)Dic1512yescomplemented
312#135—22no—
313C2×M16—151no—
314C2×M16C2×D1511yescharacteristic
315C4×Q8—301no—
316C4×D4—302no—
317C4×D4—301no—
318#117—151no—
319#117—151no—
320#117—301no—
321C2×C42—151no—
322C2×C42—301no—
323#104—301no—
324#104—301no—
325#104—302no—
326#99—302no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—42no—
331C2×C3×C5—81no—
332C2×C3×C5#38611yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3D4×D511yescharacteristic
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8C22⋊Dic512yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4C22⋊Dic512yes—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4D4×D511yescharacteristic
355C2×C3×C4#71111yescharacteristic
356C2×C3×C4#71111yescharacteristic
357C2×C3×C4C22×Dic511yescharacteristic
358C2×C3×C4C2×C22⋊D511yescharacteristic
359C2×C3×C4C2×C22⋊D511yescharacteristic
360C2×Dic3—101no—
361C2×Dic3—101no—
362C2×Dic3—102no—
363C2×Dic3—202no—
364C2×Dic3—204no—
365C2×Dic3—201no—
366C3×C8—22no—
367C3⋊C8—201no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—81no—
373C22×C5D4×Dic311yescharacteristic
374C22×C5#93711yescharacteristic
375C22×C5#93911yescharacteristic
376C4×C5—22no—
377C4×C5—41no—
378C4×C5D4×Dic311yescharacteristic
379C4×C5#93511yescharacteristic
380C4×C5#93911yescharacteristic
381C4×C5C2×#21711yescharacteristic
382Dic5—62no—
383Dic5—121no—
384Dic5—122no—
385Dic5—241no—
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41C2×Dic1512yes—
393#41#122012yes—
394C2×Q8C2×Dic1511yescharacteristic, complemented
395C2×Q8#122011yescharacteristic
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×Dic1511yescharacteristic, complemented
399C2×D4#122011yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—302no—
403C22×C4—301no—
404C22×C4C22×D1511yescharacteristic
405Q16—22no—
406SD16—22no—
407SD16—22no—
408D8—22no—
409M16—21no—
410M16—301no—
411M16—301no—
412M16C2×Dic1512yescomplemented
413C2×C8—21no—
414C2×C8—301no—
415C4⋊C4—302no—
416C4⋊C4—301no—
417C4⋊C4—301no—
418C22⋊C4—152no—
419C22⋊C4—302no—
420C22⋊C4—301no—
421C42—151no—
422C42—151no—
423C42—302no—
424C42—301no—
425C42—301no—
426C42—301no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—81no—
433C22×C3D4×Dic511yescharacteristic
434C22×C3#408311yescharacteristic
435C22×C3#408511yescharacteristic
436C3×C4—22no—
437C3×C4—41no—
438C3×C4D4×Dic511yescharacteristic
439C3×C4#408111yescharacteristic
440C3×C4#408511yescharacteristic
441C3×C4C2×C22⋊Dic511yescharacteristic
442Dic3—102no—
443Dic3—201no—
444Dic3—202no—
445Dic3—401no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—42no—
450C2×C5—81no—
451C2×C5#522011yescharacteristic
452C23—21no—
453C23—21no—
454C23D4×D1511yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8#671912yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4#671912yes—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—301no—
474C2×C4—301no—
475C2×C4—302no—
476C2×C4—602no—
477C2×C4—604no—
478C2×C4—601no—
479C2×C4D4×D1511yescharacteristic
480C2×C4#681911yescharacteristic
481C2×C4#681911yescharacteristic
482C2×C4C22×Dic1511yescharacteristic
483C2×C4C2×#122011yescharacteristic
484C2×C4C2×#122011yescharacteristic
485C8—22no—
486C8—601no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—42no—
491C2×C3—81no—
492C2×C3#6550911yescharacteristic
493C5#7211411yescharacteristic, complemented, Sylow
494C22—22no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—81no—
499C22D4×Dic1511yescharacteristic
500C22#9182611yescharacteristic
501C22#9182811yescharacteristic
502C4—22no—
503C4—41no—
504C4—302no—
505C4—601no—
506C4—602no—
507C4—1201no—
508C4D4×Dic1511yescharacteristic
509C4#9182411yescharacteristic
510C4#9182811yescharacteristic
511C4C2×#671911yescharacteristic
512C3#1060290711yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—81no—
518C2#1175132111yescharacteristic
subgroupG/H
centerC4#91828
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
Fitting, prime radicalC3×C5×#575C2
socleC2×C3×C5#386
cosocleC22×C3×C4×C5C23
hypercenter#575D15
hypocenterC3×C5#1906
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
2. #41×Dic15C2semidirect
3. #11751341C2semidirect
4. #11751337C2semidirect
5. C2×#91121C2semidirect
6. #11746380C2semidirect
7. #11746376C2semidirect
12. C3×C5×#137C4semidirect
14. C3×C5×AGL(1,ℤ/8)C4semidirect
46. C2×C3×C5×Q8C2×C4semidirect
50. C2×C3×C5×D4C2×C4semidirect
62. C3×C5×M16C2×C4semidirect
89. C5×#137Dic3semidirect
91. C5×AGL(1,ℤ/8)Dic3semidirect
146. C3×#137Dic5semidirect
148. C3×AGL(1,ℤ/8)Dic5semidirect
172. C2×C5×Q8C2×Dic3semidirect
176. C2×C5×D4C2×Dic3semidirect
188. C5×M16C2×Dic3semidirect
238. C2×C3×Q8C2×Dic5semidirect
242. C2×C3×D4C2×Dic5semidirect
254. C3×M16C2×Dic5semidirect
309. #137Dic15semidirect
311. AGL(1,ℤ/8)Dic15semidirect
394. C2×Q8C2×Dic15semidirect
398. C2×D4C2×Dic15semidirect
412. M16C2×Dic15semidirect
427. C3×C5#1906semidirect
493. C5#72114semidirect
512. C3#10602907semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5752
#41×Dic152
#117513412
#117513372
C2×#911212
#117463802
#117463762
#106029073
#721145