Finite Group Databasebeta

← #14301409

Subgroups — #14301409

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5832C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3#11751337C211yesmaximal, characteristic, complemented
4C3×#65525C211yesmaximal, characteristic, complemented
5#11748625C211yesmaximal, characteristic, complemented
6#11745976C211yesmaximal, characteristic, complemented
7#11745974C211yesmaximal, characteristic, complemented
8#10602610—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#947C411yescharacteristic, complemented
14C5×#944C411yescharacteristic, complemented
15C5×#940C411yescharacteristic, complemented
16C5×#939—22no—
17C5×#936C411yescharacteristic, complemented
18C4×C5×#67—22no—
19C2×C4×C5×Dic3C2211yescharacteristic
20C3×#4072C2211yescharacteristic
21C2×C3×#681C2211yescharacteristic
22#91288—22no—
23D3×#681—22no—
24#91286—22no—
25#91121—22no—
26C3×#3969—22no—
27#90996—22no—
28#70417—51nomaximal, Hall
29C5×#532—31no—
30#66153—31no—
31#65529—31no—
32#65525—31no—
33#65525D311yescharacteristic, complemented
34#64743—31no—
35#64743—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
45C2×C5×#67—22no—
46C2×C5×#67D412yes—
47C22×C5×Dic3—21no—
48C5×#235—21no—
49C5×#235—21no—
50C5×#235—21no—
51C5×#235—21no—
52C2×C5×D12C2×C411yescharacteristic
53C2×C4×C5×D3D412yes—
54C2×C5×Dic6C2×C411yescharacteristic
55#6719—21no—
56C4⋊Dic15—21no—
57C4×Dic15—21no—
58#6710—21no—
59#6710D412yescomplemented
60C2×C3⋊#168—21no—
61C5×#217—22no—
62C5×(C2×D3)⋊C4—22no—
63C5×Dic3⋊C4—22no—
64C4×C5×Dic3—21no—
65C4×C5×Dic3C2×C411yescharacteristic
66C4×C5×Dic3C2×C411yescharacteristic
67C3×C22⋊Dic5—21no—
68C3×C4⋊Dic5—21no—
69C3×C4×Dic5—21no—
70C3×#681—21no—
71C3×#681D412yescomplemented
72C2×C3×#168—21no—
73#6650—24no—
74D3×#168—24no—
75#5832D511yescharacteristic
76#5745—51no—
77C3×#421—51no—
78#5236—51no—
79#5006—51no—
80#4555—51no—
81#4532—51no—
82C2×C5×#41—31no—
83C2×C5×#41C2×D311yescharacteristic
84C5×C4⋊Q8—31no—
85C5×#127—31no—
86C5×#124—31no—
87C5×#124—31no—
88C5×#121—62no—
89C4×C5×D4—62no—
90C2×C42×C5—31no—
91#4094—62no—
92#4094—62no—
93#4072—31no—
94#4072C2×D311yescharacteristic
95C2×#681—31no—
96C2×#681—62no—
97C2×#681C2×D311yescharacteristic
98#3969—22no—
99#3969—62no—
100#3966—62no—
101#1884—151noSylow
102C23×C3×C5—21no—
103C23×C3×C5C2×D411yescharacteristic
104C22×C5×D3C22⋊C412yes—
105C3×C5×Q8—21no—
106C3×C5×Q8—21no—
107C3×C5×D4—21no—
108C3×C5×D4—21no—
109C3×C5×D4—41no—
110C3×C5×D4—41no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×Dic15—41no—
121C5×#67—22no—
122C5×#67—22no—
123C5×#67—44no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—42no—
127C2×C5×Dic3C22⋊C412yes—
128C5×D12—21no—
129C5×D12—21no—
130C4×C5×D3—22no—
131C4×C5×D3—22no—
132C5×Dic6—21no—
133C5×Dic6—21no—
134C2×C3×Dic5—41no—
135C3⋊#168—22no—
136C3×#168—22no—
137C2×C3×#41C2×D511yescharacteristic
138C2×#235C2×D511yescharacteristic
139C2×C3×M16—51no—
140C3×#117—51no—
141#947Dic511yescharacteristic, complemented
142#944Dic511yescharacteristic, complemented
143#940Dic511yescharacteristic, complemented
144#939—22no—
145#936Dic511yescharacteristic, complemented
146C4×#67—22no—
147#926—51no—
148C2×C4×Dic3C2×D511yescharacteristic
149C2×#208—51no—
150#907—102no—
151D3×M16—102no—
152#901—102no—
153C3×#104—102no—
154#837—102no—
155#806—102no—
156C5×#41—21no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41—21no—
160C5×#41—61no—
161C5×#41—61no—
162C5×#41—61no—
163C5×#41—61no—
164C2×C5×Q8—31no—
165C2×C5×Q8C4×D311yescharacteristic
166C2×C5×D4—21no—
167C2×C5×D4—32no—
168C2×C5×D4—31no—
169C2×C5×D4—62no—
170C2×C5×D4C4×D311yescharacteristic
171C22×C4×C5—21no—
172C22×C4×C5—32no—
173C22×C4×C5—61no—
174C22×C4×C5C22×D311yescharacteristic
175C5×C4⋊C4—62no—
176C5×C22⋊C4—62no—
177C5×C22⋊C4—62no—
178C42×C5—31no—
179C42×C5—31no—
180C42×C5—61no—
181C22⋊Dic5—21no—
182C22⋊Dic5—61no—
183C4⋊Dic5—21no—
184C4⋊Dic5—61no—
185C4×Dic5—21no—
186C4×Dic5—61no—
187#681—21no—
188#681—32no—
189#681—64no—
190#681—61no—
191#681#6712yescomplemented
192C2×#168—21no—
193C2×#168—64no—
194C2×#168—61no—
195#568—151no—
196#532—31no—
197#421—51no—
198#421—151no—
199#410—151no—
200#339—151no—
201#339—151no—
202C22×C3×C5—41no—
203C22×C3×C5—42no—
204C22×C3×C5—41no—
205C22×C3×C5C2×C22⋊C411yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C22×C3×C5#12111yescharacteristic
208C2×C5×D3—22no—
209C2×C5×D3—42no—
210C3×C4×C5—41no—
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12011yescharacteristic
214C3×C4×C5#12311yescharacteristic
215Dic15—81no—
216C3×Dic5—81no—
217C5×Dic3—22no—
218C5×Dic3—22no—
219C5×Dic3—41no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C2×C3×Q8C2×Dic511yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4C2×Dic511yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4C22×D511yescharacteristic
229C2×#67—22no—
230C2×#67C22⋊D512yes—
231C22×Dic3—21no—
232#235—21no—
233#235—21no—
234#235—21no—
235#235—21no—
236C2×D12C2×Dic511yescharacteristic
237C2×C4×D3C22⋊D512yes—
238C2×Dic6C2×Dic511yescharacteristic
239C3×M16—52no—
240C3×M16—101no—
241C2×C3×C8—101no—
242C3×C4⋊C4—101no—
243C3×C22⋊C4—101no—
244C3×C42—101no—
245#217—22no—
246#217—101no—
247(C2×D3)⋊C4—22no—
248C4⋊Dic3—101no—
249Dic3⋊C4—22no—
250C4×Dic3—21no—
251C4×Dic3—101no—
252C4×Dic3C2×Dic511yescharacteristic
253C4×Dic3C2×Dic511yescharacteristic
254#208—52no—
255#208—101no—
256C2×C3⋊C8—101no—
257ΓL(1,𝔽25)—104no—
258D3×C8—104no—
259C23×C5—21no—
260C23×C5—32no—
261C23×C5D3×D411yescharacteristic
262C5×Q8—21no—
263C5×Q8—21no—
264C5×Q8—61no—
265C5×Q8—61no—
266C5×D4—21no—
267C5×D4—21no—
268C5×D4—41no—
269C5×D4—41no—
270C5×D4—62no—
271C5×D4—62no—
272C5×D4—61no—
273C5×D4—61no—
274C5×D4—124no—
275C2×C4×C5—21no—
276C2×C4×C5—32no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—61no—
280C2×C4×C5—62no—
281C2×C4×C5—62no—
282C2×C4×C5—61no—
283C2×C4×C5—122no—
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×C4×C5#23511yescharacteristic
286C2×C4×C5D3×D411yescharacteristic
287C2×C4×C5D3×D411yescharacteristic
288C2×C4×C5#23711yescharacteristic
289C2×C4×C5C2×#6711yescharacteristic
290C2×Dic5—41no—
291C2×Dic5—121no—
292#168—22no—
293#168—62no—
294C2×#41—31no—
295C2×#41D3×D511yescharacteristic
296#131—302no—
297#131—302no—
298C2×M16—51no—
299C2×M16—151no—
300C2×M16—302no—
301C4⋊Q8—31no—
302#127—31no—
303#124—31no—
304#124—31no—
305#121—62no—
306C4×D4—62no—
307#117—51no—
308#117—151no—
309C2×C42—31no—
310#104—102no—
311#104—302no—
312#99—302no—
313C2×C3×C5—21no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5—81no—
317C2×C3×C5#38611yescharacteristic
318C5×D3—42no—
319C23×C3—21no—
320C23×C3C2×C22⋊D511yescharacteristic
321C22×D3C22⋊Dic512yes—
322C3×Q8—21no—
323C3×Q8—21no—
324C3×D4—21no—
325C3×D4—21no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—201no—
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4#71111yescharacteristic
335C2×C3×C4#71111yescharacteristic
336C2×C3×C4C22×Dic511yescharacteristic
337C2×C3×C4C2×C22⋊D511yescharacteristic
338#67—22no—
339#67—22no—
340#67—44no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—42no—
344C2×Dic3—201no—
345C2×Dic3C22⋊Dic512yes—
346D12—21no—
347D12—21no—
348C4×D3—22no—
349C4×D3—22no—
350Dic6—21no—
351Dic6—21no—
352C3×C8—102no—
353C3⋊C8—102no—
354C22×C5—41no—
355C22×C5—42no—
356C22×C5—41no—
357C22×C5—62no—
358C22×C5—122no—
359C22×C5D3×C22⋊C411yescharacteristic
360C22×C5#88411yescharacteristic
361C22×C5#93811yescharacteristic
362C4×C5—41no—
363C4×C5—62no—
364C4×C5—62no—
365C4×C5—121no—
366C4×C5#67⋊C411yescharacteristic
367C4×C5#88311yescharacteristic
368C4×C5C4×#6711yescharacteristic
369C4×C5#93711yescharacteristic
370Dic5—81no—
371Dic5—241no—
372#41—21no—
373#41—21no—
374#41—21no—
375#41—21no—
376#41—61no—
377#41—61no—
378#41—61no—
379#41—61no—
380C2×Q8—31no—
381C2×Q8D3×Dic511yescharacteristic
382C2×D4—21no—
383C2×D4—32no—
384C2×D4—31no—
385C2×D4—62no—
386C2×D4D3×Dic511yescharacteristic
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—61no—
390C22×C4C2×D3×D511yescharacteristic
391M16—52no—
392M16—101no—
393M16—152no—
394M16—304no—
395M16—301no—
396C2×C8—101no—
397C2×C8—304no—
398C2×C8—301no—
399C4⋊C4—62no—
400C4⋊C4—101no—
401C4⋊C4—301no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—101no—
405C22⋊C4—301no—
406C42—31no—
407C42—31no—
408C42—61no—
409C42—101no—
410C42—301no—
411C3×C5#188411yescharacteristic, complemented, Hall
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—41no—
415C22×C3#408511yescharacteristic
416C22×C3#408511yescharacteristic
417C22×C3C2×C22⋊Dic511yescharacteristic
418C2×D3—22no—
419C2×D3—42no—
420C3×C4—41no—
421C3×C4—401no—
422C3×C4D4×Dic511yescharacteristic
423C3×C4D4×Dic511yescharacteristic
424C3×C4#408111yescharacteristic
425C3×C4#408311yescharacteristic
426Dic3—22no—
427Dic3—22no—
428Dic3—41no—
429Dic3—401no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—21no—
433C2×C5—81no—
434C2×C5—122no—
435C2×C5#495611yescharacteristic
436C23—21no—
437C23—32no—
438C23D3×C22⋊D511yescharacteristic
439Q8—21no—
440Q8—21no—
441Q8—61no—
442Q8—61no—
443D4—21no—
444D4—21no—
445D4—41no—
446D4—41no—
447D4—62no—
448D4—62no—
449D4—61no—
450D4—61no—
451D4—124no—
452C2×C4—21no—
453C2×C4—32no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—122no—
461C2×C4—201no—
462C2×C4—601no—
463C2×C4#678011yescharacteristic
464C2×C4C2×D3×Dic511yescharacteristic
465C2×C4#678211yescharacteristic
466C2×C4D5×#6711yescharacteristic
467C2×C4D3×C22⋊D511yescharacteristic
468C2×C4#679011yescharacteristic
469C8—102no—
470C8—302no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—81no—
475C2×C3#6550911yescharacteristic
476D3—42no—
477C5#7041711yescharacteristic, complemented, Sylow
478C22—41no—
479C22—42no—
480C22—41no—
481C22—62no—
482C22—122no—
483C22D3×C22⋊Dic511yescharacteristic
484C22#9155811yescharacteristic
485C22#9156511yescharacteristic
486C4—41no—
487C4—62no—
488C4—62no—
489C4—121no—
490C4—401no—
491C4—1201no—
492C4Dic5×#6711yescharacteristic
493C4#9155511yescharacteristic
494C4#67⋊Dic511yescharacteristic
495C4#9157711yescharacteristic
496C3#1060261011yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—81no—
501C2—122no—
502C2#1174951911yescharacteristic
subgroupG/H
centerC4#91577
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5#386
prime radicalC5×#5832C2
cosocleC22×C3×C4×C5C23
hypercenterC2×#41D3×D5
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #11754125C2semidirect
3. #11751337C2semidirect
4. C3×#65525C2semidirect
5. #11748625C2semidirect
6. #11745976C2semidirect
7. #11745974C2semidirect
13. C5×#947C4semidirect
14. C5×#944C4semidirect
15. C5×#940C4semidirect
17. C5×#936C4semidirect
33. #65525D3semidirect
59. #6710D4semidirect
71. C3×#681D4semidirect
141. #947Dic5semidirect
142. #944Dic5semidirect
143. #940Dic5semidirect
145. #936Dic5semidirect
191. #681#67semidirect
411. C3×C5#1884semidirect
477. C5#70417semidirect
496. C3#10602610semidirect

Maximal subgroups

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

subgroupindex
C5×#58322
#117541252
#117513372
C3×#655252
#117486252
#117459762
#117459742
#106026103
#704175