Finite Group Databasebeta

← #14306006

Subgroups — #14306006

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
1#11754586C211yesmaximal, characteristic
2C42×Dic15C211yesmaximal, characteristic, complemented
3C2×C5×#805C211yesmaximal, characteristic, complemented
4C2×C3×#3937C211yesmaximal, characteristic, complemented
5#11748437C211yesmaximal, characteristic, complemented
6C2×#90978C211yesmaximal, characteristic, complemented
7C2×#90976C211yesmaximal, characteristic, complemented
8#10601149—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C2×C4×Dic15—22no—
13C2×C4×Dic15C2211yescharacteristic
14C2×C4×Dic15C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17#91571—22no—
18#91463C411yescharacteristic, complemented
19C4×#1201—22no—
20#91385C411yescharacteristic, complemented
21#91370C411yescharacteristic, complemented
22#91368C411yescharacteristic, complemented
23C5×#805—22no—
24C3×#3937—22no—
25#90983—22no—
26#90978—22no—
27#90976—22no—
28#70356—51nomaximal, Hall
29#66166—31no—
30C2×C5×#104—31no—
31C2×#3969—31no—
32C2×#3957—31no—
33#65407—31no—
34C42×Dic5—31no—
35C2×#3937D311yescharacteristic, complemented
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C22×Dic15D412yes—
40C2×C5×#67—21no—
41C5×#235—21no—
42C5×#235—21no—
43C5×#235—21no—
44C5×#235—21no—
45C2×C5×D12C2×C411yescharacteristic
46C2×C4×C5×D3—21no—
47C2×C5×Dic6C2×C411yescharacteristic
48C2×C3×C22⋊D5—21no—
49C3×#709—21no—
50C3×#709—21no—
51C3×#709—21no—
52C3×#709—21no—
53C2×C3×D20C2×C411yescharacteristic
54C2×C3×C4×D5—21no—
55C2×C3×Dic10C2×C411yescharacteristic
56C2×#1201—22no—
57C3×C42×C5—21no—
58C3×C42×C5D412yescomplemented
59C4×Dic15—28no—
60C4×Dic15—21no—
61C4×Dic15C2×C411yescharacteristic
62C4×Dic15C2×C411yescharacteristic
63C4×Dic15D412yes—
64C4×Dic15D412yescomplemented
65C5×#208—21no—
66C5×#208—21no—
67C2×C5×C3⋊C8—21no—
68C3×#681—21no—
69C3×#681—21no—
70C2×C3×#168—21no—
71#6669—22no—
72(C2×D3)⋊Dic5—22no—
73#6665—22no—
74#5832—51no—
75C2×C3×#104—51no—
76C2×#837—51no—
77C2×#825—51no—
78#5118—51no—
79C42×Dic3—51no—
80C2×#805D511yescharacteristic, complemented
81C2×C5×#41—31no—
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C42×C5C2×D311yescharacteristic
85#4093—31no—
86#4090—31no—
87#4086—31no—
88#4085—62no—
89#4082—31no—
90C4×C22⋊D5—62no—
91C2×C4×Dic5—31no—
92C2×C4×Dic5—31no—
93C2×C4×Dic5—62no—
94C2×#681C2×D311yescharacteristic
95C5×#104—62no—
96#3969—62no—
97#3957—62no—
98#3946—62no—
99#3937—22no—
100#1962—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C22×C3×D5—21no—
104C2×C3×C4×C5—24no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112C2×Dic15—24no—
113C2×Dic15—22no—
114C2×Dic15—24no—
115C2×Dic15C22⋊C412yes—
116C2×Dic15C22⋊C412yes—
117C5×#67—41no—
118C5×#67—41no—
119C2×C5×Dic3—21no—
120C5×D12—21no—
121C5×D12—21no—
122C4×C5×D3—41no—
123C4×C5×D3—41no—
124C5×Dic6—21no—
125C5×Dic6—21no—
126C3×C22⋊D5—41no—
127C3×C22⋊D5—41no—
128C2×C3×Dic5—21no—
129C3×D20—21no—
130C3×D20—21no—
131C3×C4×D5—41no—
132C3×C4×D5—41no—
133C3×Dic10—21no—
134C3×Dic10—21no—
135#1201—44no—
136C3×#168—41no—
137C5×C3⋊C8—41no—
138C2×C3×#41—51no—
139C2×#235C2×D511yescharacteristic
140C2×C3×M16—51no—
141C2×C3×C42C2×D511yescharacteristic
142#947—51no—
143#944—51no—
144#940—51no—
145#939—102no—
146#936—51no—
147C4×#67—102no—
148C2×C4×Dic3—51no—
149C2×C4×Dic3—51no—
150C2×C4×Dic3—102no—
151C2×#208C2×D511yescharacteristic
152C3×#104—102no—
153#837—102no—
154#825—102no—
155Dic3⋊C8—102no—
156#805—22no—
157C5×#41—61no—
158C5×#41—61no—
159C5×#41—61no—
160C5×#41—61no—
161C2×C5×Q8—31no—
162C2×C5×D4—31no—
163C2×C5×D4—61no—
164C22×C4×C5—21no—
165C22×C4×C5—61no—
166C22×C4×C5C22×D311yescharacteristic
167C2×C22⋊D5—21no—
168C2×C22⋊D5—62no—
169C22×Dic5—32no—
170C22×Dic5—61no—
171#709—21no—
172#709—21no—
173#709—21no—
174#709—21no—
175C2×D20C4×D311yescharacteristic
176C2×C4×D5—21no—
177C2×Dic10C4×D311yescharacteristic
178C5×M16—61no—
179C5×M16—61no—
180C2×C5×C8—61no—
181C42×C5—21no—
182C42×C5#6712yescomplemented
183C22⋊Dic5—62no—
184#685—62no—
185#683—62no—
186C4×Dic5—31no—
187C4×Dic5—31no—
188C4×Dic5—32no—
189C4×Dic5—32no—
190C4×Dic5—68no—
191C4×Dic5—61no—
192#681—21no—
193#681—21no—
194C2×#168—21no—
195#532—151no—
196#423—151no—
197C2×#104—31no—
198C2×#104—51no—
199C2×#104—151no—
200C2×#104—151no—
201C43—151no—
202C22×C3×C5—22no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5#12111yescharacteristic
205C22×C3×C5#12111yescharacteristic
206C2×C5×D3—41no—
207C2×C5×D3—42no—
208C2×C3×D5—41no—
209C2×C3×D5—42no—
210C3×C4×C5—24no—
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12411yescharacteristic
214C3×C4×C5#12711yescharacteristic
215Dic15—24no—
216Dic15—22no—
217Dic15#10414yescomplemented
218C3×Dic5—41no—
219C5×Dic3—41no—
220C3×#41—101no—
221C3×#41—101no—
222C3×#41—101no—
223C3×#41—101no—
224C2×C3×Q8—51no—
225C2×C3×D4—51no—
226C2×C3×D4—101no—
227C22×C3×C4—21no—
228C22×C3×C4—101no—
229C22×C3×C4C22×D511yescharacteristic
230C2×#67—21no—
231C2×#67—102no—
232C22×Dic3—52no—
233C22×Dic3—101no—
234#235—21no—
235#235—21no—
236#235—21no—
237#235—21no—
238C2×D12C4×D511yescharacteristic
239C2×C4×D3—21no—
240C2×Dic6C4×D511yescharacteristic
241C3×M16—101no—
242C3×M16—101no—
243C2×C3×C8—101no—
244C3×C42—21no—
245C3×C42C22⋊D512yescomplemented
246#217—102no—
247(C2×D3)⋊C4—102no—
248Dic3⋊C4—102no—
249C4×Dic3—51no—
250C4×Dic3—51no—
251C4×Dic3—52no—
252C4×Dic3—52no—
253C4×Dic3—108no—
254C4×Dic3—101no—
255#208—21no—
256#208—21no—
257C2×C3⋊C8—21no—
258C23×C5—61no—
259C23×C5D3×D411yescharacteristic
260C22×D5—21no—
261C5×Q8—61no—
262C5×Q8—61no—
263C5×D4—61no—
264C5×D4—61no—
265C5×D4—121no—
266C5×D4—121no—
267C2×C4×C5—24no—
268C2×C4×C5—22no—
269C2×C4×C5—61no—
270C2×C4×C5—121no—
271C2×C4×C5—121no—
272C2×C4×C5C2×C4×D311yescharacteristic
273C2×C4×C5#23511yescharacteristic
274C2×C4×C5D3×D411yescharacteristic
275C2×C4×C5D3×D411yescharacteristic
276C2×C4×C5#23711yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C22⋊D5—41no—
279C22⋊D5—41no—
280C22⋊D5—124no—
281C2×Dic5—21no—
282C2×Dic5—32no—
283C2×Dic5—32no—
284C2×Dic5—64no—
285C2×Dic5—64no—
286C2×Dic5—62no—
287D20—21no—
288D20—21no—
289C4×D5—41no—
290C4×D5—41no—
291Dic10—21no—
292Dic10—21no—
293C5×C8—121no—
294#168—41no—
295C2×#41—31no—
296C2×#41—51no—
297C2×M16—31no—
298C2×M16—51no—
299C4⋊Q8—151no—
300#127—151no—
301#124—151no—
302#124—151no—
303#121—302no—
304C4×D4—302no—
305C2×C42—151no—
306C2×C42—151no—
307C2×C42—302no—
308C2×C42D3×D511yescharacteristic
309C4⋊C8—302no—
310#104—62no—
311#104—102no—
312#104—302no—
313#104—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#39011yescharacteristic
317C2×C3×C5C2×#10412yes—
318C3×D5—81no—
319C5×D3—81no—
320C23×C3—101no—
321C23×C3D4×D511yescharacteristic
322C22×D3—21no—
323C3×Q8—101no—
324C3×Q8—101no—
325C3×D4—101no—
326C3×D4—101no—
327C3×D4—201no—
328C3×D4—201no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—101no—
332C2×C3×C4—201no—
333C2×C3×C4—201no—
334C2×C3×C4C2×C4×D511yescharacteristic
335C2×C3×C4#70911yescharacteristic
336C2×C3×C4D4×D511yescharacteristic
337C2×C3×C4D4×D511yescharacteristic
338C2×C3×C4#71111yescharacteristic
339C2×C3×C4C2×C22⋊D511yescharacteristic
340#67—41no—
341#67—41no—
342#67—204no—
343C2×Dic3—21no—
344C2×Dic3—52no—
345C2×Dic3—52no—
346C2×Dic3—104no—
347C2×Dic3—104no—
348C2×Dic3—102no—
349D12—21no—
350D12—21no—
351C4×D3—41no—
352C4×D3—41no—
353Dic6—21no—
354Dic6—21no—
355C3×C8—201no—
356C3⋊C8—41no—
357C22×C5—22no—
358C22×C5—121no—
359C22×C5—122no—
360C22×C5D3×C22⋊C411yescharacteristic
361C22×C5#88411yescharacteristic
362C22×C5#93911yescharacteristic
363C2×D5—41no—
364C2×D5—42no—
365C4×C5—24no—
366C4×C5—121no—
367C4×C5#67⋊C411yescharacteristic
368C4×C5#88511yescharacteristic
369C4×C5C4×#6711yescharacteristic
370C4×C5#94011yescharacteristic
371Dic5—34no—
372Dic5—41no—
373Dic5—64no—
374Dic5—62no—
375#41—61no—
376#41—61no—
377#41—61no—
378#41—61no—
379#41—101no—
380#41—101no—
381#41—101no—
382#41—101no—
383C2×Q8—31no—
384C2×Q8—51no—
385C2×D4—31no—
386C2×D4—51no—
387C2×D4—61no—
388C2×D4—101no—
389C2×D4—302no—
390C22×C4—21no—
391C22×C4—61no—
392C22×C4—101no—
393C22×C4—152no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—61no—
397M16—61no—
398M16—101no—
399M16—101no—
400C2×C8—61no—
401C2×C8—101no—
402C4⋊C4—302no—
403C22⋊C4—302no—
404C22⋊C4—302no—
405C42—21no—
406C42—151no—
407C42—151no—
408C42—152no—
409C42—152no—
410C42—308no—
411C42—301no—
412C42#120112yescomplemented
413C3×C5#196211yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—202no—
416C22×C3—201no—
417C22×C3D5×C22⋊C411yescharacteristic
418C22×C3#403011yescharacteristic
419C22×C3#408511yescharacteristic
420C2×D3—42no—
421C2×D3—41no—
422C3×C4—24no—
423C3×C4—201no—
424C3×C4#402711yescharacteristic
425C3×C4#403111yescharacteristic
426C3×C4C4×C22⋊D511yescharacteristic
427C3×C4#408611yescharacteristic
428Dic3—41no—
429Dic3—54no—
430Dic3—104no—
431Dic3—102no—
432C2×C5—21no—
433C2×C5—21no—
434C2×C5—241no—
435C2×C5D3×#10412yes—
436C2×C5#495711yescharacteristic
437D5—81no—
438C23—61no—
439C23—101no—
440C23#679011yescharacteristic
441Q8—61no—
442Q8—61no—
443Q8—101no—
444Q8—101no—
445D4—61no—
446D4—61no—
447D4—101no—
448D4—101no—
449D4—121no—
450D4—121no—
451D4—201no—
452D4—201no—
453D4—604no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—61no—
457C2×C4—101no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—152no—
461C2×C4—152no—
462C2×C4—201no—
463C2×C4—201no—
464C2×C4—304no—
465C2×C4—304no—
466C2×C4—302no—
467C2×C4#677111yescharacteristic
468C2×C4C4×D3×D511yescharacteristic
469C2×C4#677711yescharacteristic
470C2×C4#677711yescharacteristic
471C2×C4#678011yescharacteristic
472C2×C4C2×#120111yescharacteristic
473C8—121no—
474C8—201no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—401no—
478C2×C3D5×#10412yes—
479C2×C3#6524611yescharacteristic
480D3—81no—
481C5#7035611yescharacteristic, complemented, Sylow
482C22—22no—
483C22—122no—
484C22—121no—
485C22—202no—
486C22—201no—
487C22#9146111yescharacteristic
488C22#9147311yescharacteristic
489C22#9157111yescharacteristic
490C4—24no—
491C4—121no—
492C4—154no—
493C4—201no—
494C4—304no—
495C4—302no—
496C4C4×#120111yescharacteristic
497C4#9144211yescharacteristic
498C4#9146311yescharacteristic
499C4#9146411yescharacteristic
500C3#1060114911yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—241no—
504C2—401no—
505C2#1174711212yes—
506C2#1174891511yescharacteristic
subgroupG/H
centerC2×C4#6777
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754586C2
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1962
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C42×Dic15C2semidirect
3. C2×C5×#805C2semidirect
4. C2×C3×#3937C2semidirect
5. #11748437C2semidirect
6. C2×#90978C2semidirect
7. C2×#90976C2semidirect
18. #91463C4semidirect
20. #91385C4semidirect
21. #91370C4semidirect
22. #91368C4semidirect
35. C2×#3937D3semidirect
58. C3×C42×C5D4semidirect
64. C4×Dic15D4semidirect
80. C2×#805D5semidirect
182. C42×C5#67semidirect
217. Dic15#104semidirect
245. C3×C42C22⋊D5semidirect
412. C42#1201semidirect
413. C3×C5#1962semidirect
481. C5#70356semidirect
500. C3#10601149semidirect

Maximal subgroups

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

subgroupindex
#117545862
C42×Dic152
C2×C5×#8052
C2×C3×#39372
#117484372
C2×#909782
C2×#909762
#106011493
#703565