Finite Group Databasebeta

← #14306014

Subgroups — #14306014

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66166C211yesmaximal, characteristic
2C2×#91121C211yesmaximal, characteristic, complemented
3C2×C5×#837C211yesmaximal, characteristic, complemented
4C4×Dic3×Dic5C211yesmaximal, characteristic, complemented
5#11748431C211yesmaximal, characteristic, complemented
6C2×#90977C211yesmaximal, characteristic, complemented
7C2×#90976C211yesmaximal, characteristic, complemented
8#10601149—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C4×Dic15C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4093C411yescharacteristic, complemented
16C3×#4090C411yescharacteristic, complemented
17C3×#4086C411yescharacteristic, complemented
18C3×#4085—22no—
19C3×#4082C411yescharacteristic, complemented
20C3×C4×C22⋊D5—22no—
21C2×C3×C4×Dic5C2211yescharacteristic
22C2×Dic3×Dic5—22no—
23#91121—22no—
24C5×#837—22no—
25#90982—22no—
26#90977—22no—
27#90976—22no—
28#71815—51nomaximal, Hall
29#66166D311yescharacteristic
30C2×C5×#104—31no—
31C2×#3969—31no—
32C2×#3957—31no—
33#65407—31no—
34C42×Dic5—31no—
35C2×#3937—31no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4C2×C411yescharacteristic
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2311yescharacteristic
45C22×Dic15—21no—
46C22×C5×Dic3—21no—
47C2×C3×C22⋊D5—21no—
48C2×C3×C22⋊D5—22no—
49C22×C3×Dic5D412yes—
50C3×#709—21no—
51C3×#709—21no—
52C3×#709—21no—
53C3×#709—21no—
54C2×C3×D20C2×C411yescharacteristic
55C2×C3×C4×D5—21no—
56C2×C3×Dic10C2×C411yescharacteristic
57C4×Dic15—21no—
58C4×Dic15D412yescomplemented
59#6710—21no—
60#6710—21no—
61C2×C3⋊#168—21no—
62C4×C5×Dic3—21no—
63C4×C5×Dic3D412yescomplemented
64C5×#208—21no—
65C5×#208—21no—
66C2×C5×C3⋊C8—21no—
67C3×C22⋊Dic5—22no—
68C3×#685—22no—
69C3×#683—22no—
70C3×C4×Dic5C2×C411yescharacteristic
71C3×C4×Dic5C2×C411yescharacteristic
72C3×C4×Dic5D412yes—
73Dic3×Dic5—28no—
74C3×#532—51no—
75C2×#837—51no—
76C2×#837—51no—
77C2×#837—51no—
78C2×#837D511yescharacteristic, complemented
79C42×Dic3—51no—
80#4924—51no—
81C2×C5×#41C2×D311yescharacteristic
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C42×C5—31no—
85#4093Dic311yescharacteristic, complemented
86#4090Dic311yescharacteristic, complemented
87#4086Dic311yescharacteristic, complemented
88#4085—22no—
89#4082Dic311yescharacteristic, complemented
90C4×C22⋊D5—22no—
91C2×C4×Dic5—31no—
92C2×C4×Dic5—62no—
93C2×C4×Dic5C2×D311yescharacteristic
94C2×#681—31no—
95C5×#104—62no—
96#3969—62no—
97#3957—62no—
98#3946—62no—
99#3937—62no—
100#1962—151noSylow
101C23×C3×C5—21no—
102C23×C3×C5C2×D411yescharacteristic
103C22×C3×D5—21no—
104C3×C5×Q8—21no—
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×Dic15—24no—
120C2×Dic15—22no—
121C2×C5×Dic3—24no—
122C2×C5×Dic3—22no—
123C3×C22⋊D5—41no—
124C3×C22⋊D5—41no—
125C3×C22⋊D5—44no—
126C2×C3×Dic5—24no—
127C2×C3×Dic5—21no—
128C2×C3×Dic5C22⋊C412yes—
129C2×C3×Dic5C22⋊C412yes—
130C3×D20—21no—
131C3×D20—21no—
132C3×C4×D5—41no—
133C3×C4×D5—41no—
134C3×Dic10—21no—
135C3×Dic10—21no—
136C3⋊#168—41no—
137C5×C3⋊C8—41no—
138C2×C3×#41—51no—
139C2×C3×#41C2×D511yescharacteristic
140C3×C4⋊Q8—51no—
141C3×#127—51no—
142C3×#124—51no—
143C3×#124—51no—
144C3×#121—102no—
145C3×C4×D4—102no—
146C2×C3×C42—51no—
147C2×C4×Dic3—51no—
148C2×C4×Dic3—102no—
149C2×C4×Dic3C2×D511yescharacteristic
150C2×#208—51no—
151C2×#208C2×D511yescharacteristic
152#837—22no—
153#837—102no—
154#837—102no—
155#837—102no—
156C4⋊(C3⋊C8)—102no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41—21no—
160C5×#41—21no—
161C2×C5×Q8C2×Dic311yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4C2×Dic311yescharacteristic
164C22×C4×C5—21no—
165C22×C4×C5—61no—
166C22×C4×C5C22×D311yescharacteristic
167C2×C22⋊D5—21no—
168C2×C22⋊D5—22no—
169C22×Dic5—61no—
170C22×Dic5#6712yes—
171#709—21no—
172#709—21no—
173#709—21no—
174#709—21no—
175C2×D20C2×Dic311yescharacteristic
176C2×C4×D5—21no—
177C2×Dic10C2×Dic311yescharacteristic
178C5×M16—61no—
179C5×M16—61no—
180C2×C5×C8—61no—
181C42×C5—32no—
182C42×C5—61no—
183C22⋊Dic5—22no—
184#685—22no—
185#683—22no—
186C4×Dic5—32no—
187C4×Dic5—68no—
188C4×Dic5—61no—
189C4×Dic5C2×Dic311yescharacteristic
190C4×Dic5C2×Dic311yescharacteristic
191C4×Dic5#6712yes—
192#681—61no—
193#681—61no—
194C2×#168—61no—
195#532—51no—
196#423—151no—
197C2×#104—31no—
198C2×#104—151no—
199C2×#104—151no—
200C2×#104—151no—
201C43—151no—
202C22×C3×C5—22no—
203C22×C3×C5—41no—
204C22×C3×C5—42no—
205C22×C3×C5C2×C22⋊C411yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C22×C3×C5#12111yescharacteristic
208C2×C3×D5—41no—
209C2×C3×D5—42no—
210C3×C4×C5—41no—
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12411yescharacteristic
214C3×C4×C5#12711yescharacteristic
215Dic15—24no—
216C3×Dic5—22no—
217C3×Dic5—41no—
218C3×Dic5#10414yescomplemented
219C5×Dic3—24no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—101no—
225C3×#41—101no—
226C3×#41—101no—
227C3×#41—101no—
228C2×C3×Q8—51no—
229C2×C3×Q8C4×D511yescharacteristic
230C2×C3×D4—21no—
231C2×C3×D4—51no—
232C2×C3×D4—102no—
233C2×C3×D4—101no—
234C2×C3×D4C4×D511yescharacteristic
235C22×C3×C4—21no—
236C22×C3×C4—52no—
237C22×C3×C4—101no—
238C22×C3×C4C22×D511yescharacteristic
239C22×Dic3—21no—
240C22×Dic3—101no—
241C3×C4⋊C4—102no—
242C3×C22⋊C4—102no—
243C3×C22⋊C4—102no—
244C3×C42—51no—
245C3×C42—51no—
246C3×C42—52no—
247C4×Dic3—21no—
248C4×Dic3—52no—
249C4×Dic3—108no—
250C4×Dic3—101no—
251C4×Dic3C22⋊D512yescomplemented
252#208—21no—
253#208—21no—
254#208—101no—
255#208—101no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—101no—
258C23×C5—21no—
259C23×C5C2×#6711yescharacteristic
260C22×D5—21no—
261C5×Q8—21no—
262C5×Q8—21no—
263C5×D4—21no—
264C5×D4—21no—
265C5×D4—41no—
266C5×D4—41no—
267C2×C4×C5—21no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—64no—
271C2×C4×C5—62no—
272C2×C4×C5D3×D411yescharacteristic
273C2×C4×C5D3×D411yescharacteristic
274C2×C4×C5#23711yescharacteristic
275C2×C4×C5#23711yescharacteristic
276C2×C4×C5C22×Dic311yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C22⋊D5—41no—
279C22⋊D5—41no—
280C22⋊D5—44no—
281C2×Dic5—24no—
282C2×Dic5—21no—
283C2×Dic5—64no—
284C2×Dic5—62no—
285C2×Dic5#21712yes—
286C2×Dic5#21712yes—
287D20—21no—
288D20—21no—
289C4×D5—41no—
290C4×D5—41no—
291Dic10—21no—
292Dic10—21no—
293C5×C8—121no—
294#168—121no—
295C2×#41—51no—
296C2×#41D3×D511yescharacteristic
297C2×M16—31no—
298C2×M16—151no—
299C4⋊Q8—51no—
300#127—51no—
301#124—51no—
302#124—51no—
303#121—102no—
304C4×D4—102no—
305C2×C42—31no—
306C2×C42—51no—
307C2×C42—151no—
308C2×C42—302no—
309C4⋊C8—302no—
310#104—62no—
311#104—302no—
312#104—302no—
313#104—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5—81no—
317C2×C3×C5#39011yescharacteristic
318C2×C3×C5C2×#10412yes—
319C3×D5—81no—
320C23×C3—21no—
321C23×C3—101no—
322C23×C3D4×D511yescharacteristic
323C3×Q8—21no—
324C3×Q8—21no—
325C3×Q8—101no—
326C3×Q8—101no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—101no—
332C3×D4—101no—
333C3×D4—204no—
334C3×D4—201no—
335C3×D4—201no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—52no—
340C2×C3×C4—52no—
341C2×C3×C4—104no—
342C2×C3×C4—101no—
343C2×C3×C4—201no—
344C2×C3×C4—201no—
345C2×C3×C4C2×C4×D511yescharacteristic
346C2×C3×C4#70911yescharacteristic
347C2×C3×C4D4×D511yescharacteristic
348C2×C3×C4D4×D511yescharacteristic
349C2×C3×C4#71111yescharacteristic
350C2×C3×C4C2×C22⋊D511yescharacteristic
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—104no—
354C2×Dic3—102no—
355C3⋊C8—41no—
356C3⋊C8—201no—
357C22×C5—22no—
358C22×C5—41no—
359C22×C5—42no—
360C22×C5#93811yescharacteristic
361C22×C5#93811yescharacteristic
362C22×C5C2×#21711yescharacteristic
363C2×D5—41no—
364C2×D5—42no—
365C4×C5—41no—
366C4×C5—64no—
367C4×C5D4×Dic311yescharacteristic
368C4×C5D4×Dic311yescharacteristic
369C4×C5#93611yescharacteristic
370C4×C5#94011yescharacteristic
371Dic5—22no—
372Dic5—41no—
373Dic5—64no—
374Dic5#83714yescomplemented
375#41—21no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—101no—
380#41—101no—
381#41—101no—
382#41—101no—
383C2×Q8—51no—
384C2×Q8D5×Dic311yescharacteristic
385C2×D4—21no—
386C2×D4—51no—
387C2×D4—102no—
388C2×D4—101no—
389C2×D4D5×Dic311yescharacteristic
390C22×C4—21no—
391C22×C4—52no—
392C22×C4—61no—
393C22×C4—101no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—61no—
397M16—61no—
398M16—301no—
399M16—301no—
400C2×C8—61no—
401C2×C8—301no—
402C4⋊C4—102no—
403C22⋊C4—102no—
404C22⋊C4—102no—
405C42—32no—
406C42—51no—
407C42—51no—
408C42—52no—
409C42—61no—
410C42—152no—
411C42—308no—
412C42—301no—
413C3×C5#196211yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—42no—
416C22×C3—41no—
417C22×C3—202no—
418C22×C3—201no—
419C22×C3D5×C22⋊C411yescharacteristic
420C22×C3#403011yescharacteristic
421C22×C3#408511yescharacteristic
422C3×C4—41no—
423C3×C4—54no—
424C3×C4—102no—
425C3×C4—201no—
426C3×C4#402711yescharacteristic
427C3×C4#403111yescharacteristic
428C3×C4C4×C22⋊D511yescharacteristic
429C3×C4#408611yescharacteristic
430Dic3—24no—
431Dic3—104no—
432C2×C5—21no—
433C2×C5—21no—
434C2×C5—81no—
435C2×C5#522111yescharacteristic
436C2×C5C2×#83712yes—
437D5—81no—
438C23—21no—
439C23—101no—
440C23D5×#6711yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—101no—
444Q8—101no—
445D4—21no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—101no—
450D4—101no—
451D4—204no—
452D4—201no—
453D4—201no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—52no—
458C2×C4—52no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—104no—
462C2×C4—101no—
463C2×C4—201no—
464C2×C4—201no—
465C2×C4—304no—
466C2×C4—302no—
467C2×C4C2×D5×Dic311yescharacteristic
468C2×C4#677911yescharacteristic
469C2×C4#678011yescharacteristic
470C2×C4D5×#6711yescharacteristic
471C2×C4D3×C22⋊D511yescharacteristic
472C2×C4#679011yescharacteristic
473C8—121no—
474C8—601no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—81no—
478C2×C3—401no—
479C2×C3D5×#10412yes—
480C2×C3#6524611yescharacteristic
481C5#7181511yescharacteristic, complemented, Sylow
482C22—22no—
483C22—42no—
484C22—41no—
485C22—202no—
486C22—201no—
487C22#9153311yescharacteristic
488C22D5×#21711yescharacteristic
489C22#9156511yescharacteristic
490C4—41no—
491C4—54no—
492C4—64no—
493C4—102no—
494C4—201no—
495C4—304no—
496C4#9153411yescharacteristic
497C4Dic3×C22⋊D511yescharacteristic
498C4#9155911yescharacteristic
499C4#9156711yescharacteristic
500C3#1060114911yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—81no—
504C2—401no—
505C2D5×#83712yes—
506C2#1174948911yescharacteristic
subgroupG/H
centerC2×C4D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66166C2
cosocleC22×C3×C4×C5C23
hypercenterC2×#41D3×D5
hypocenterC3×C5#1962
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×#91121C2semidirect
3. C2×C5×#837C2semidirect
4. C4×Dic3×Dic5C2semidirect
5. #11748431C2semidirect
6. C2×#90977C2semidirect
7. C2×#90976C2semidirect
15. C3×#4093C4semidirect
16. C3×#4090C4semidirect
17. C3×#4086C4semidirect
19. C3×#4082C4semidirect
58. C4×Dic15D4semidirect
63. C4×C5×Dic3D4semidirect
78. C2×#837D5semidirect
85. #4093Dic3semidirect
86. #4090Dic3semidirect
87. #4086Dic3semidirect
89. #4082Dic3semidirect
218. C3×Dic5#104semidirect
251. C4×Dic3C22⋊D5semidirect
374. Dic5#837semidirect
413. C3×C5#1962semidirect
481. C5#71815semidirect
500. C3#10601149semidirect

Maximal subgroups

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

subgroupindex
C3×#661662
C2×#911212
C2×C5×#8372
C4×Dic3×Dic52
#117484312
C2×#909772
C2×#909762
#106011493
#718155