Finite Group Databasebeta

← #14301740

Subgroups — #14301740

Order 1920.

684 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
1#11756283C211yesmaximal, characteristic
2#11754130C211yesmaximal, characteristic, complemented
3#11748357C211yesmaximal, characteristic, complemented
4#11748351C211yesmaximal, characteristic, complemented
5#11748351C211yesmaximal, characteristic, complemented
6C5×#4533C211yesmaximal, characteristic, complemented
7C3×#64753C211yesmaximal, characteristic, complemented
8#10599829—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C4×D15C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91786—21no—
13#91784—21no—
14#91775C412yes—
15#91771C412yes—
16C4×D60—22no—
17C2×C5×#208C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21#91292—22no—
22#91291—22no—
23#91290—22no—
24#90963—21no—
25#90960—21no—
26#90957—22no—
27#90957—22no—
28#69036—51nomaximal, Hall
29#66079—31no—
30#65858—31no—
31#65421—31no—
32#65415—31no—
33#65415—31no—
34C5×#344—31no—
35#64753D311yescharacteristic, complemented
36C23×D15C2×C411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C2×#1220C2×C412yes—
39C22×Dic15C2×C411yescharacteristic
40#6817—21no—
41#6817—21no—
42#6817—21no—
43#6817—21no—
44C2×D60—22no—
45C2×D60D411yescharacteristic
46C2×C4×D15—21no—
47C2×C4×D15—21no—
48C2×C4×D15—21no—
49C2×C4×D15—21no—
50C2×C4×D15—21no—
51C2×C4×D15—21no—
52C2×C4×D15D412yes—
53C2×Dic30D411yescharacteristic
54C3×C5×C4⋊C4—21no—
55C3×C5×C22⋊C4C2×C412yes—
56C3×C42×C5—21no—
57#6714—22no—
58#6714—22no—
59C4⋊Dic15—22no—
60C5×#208—21no—
61C5×#208—21no—
62C5×#208—21no—
63C5×#208—21no—
64C2×C5×C3⋊C8—21no—
65C2×C5×C3⋊C8D412yescomplemented
66C3×#681—21no—
67C3×#681—21no—
68C3×#681—21no—
69C3×#681—21no—
70C2×C3×#168—21no—
71C2×C3×#168D412yescomplemented
72#6651—24no—
73#6648—24no—
74#5745—51no—
75#5524—51no—
76#5132—51no—
77#5126—51no—
78#5126—51no—
79C3×#344—51no—
80#4533D511yescharacteristic, complemented
81C2×#709—31no—
82C22×C4×D5—31no—
83C2×C5×M16—31no—
84C2×C5×M16—31no—
85C5×#117C2×D311yescharacteristic
86C2×#681C2×D311yescharacteristic
87C2×#681C2×D311yescharacteristic
88#4053—62no—
89D5×M16—62no—
90#4047—62no—
91#4043—61no—
92#4041—61no—
93#4032—32no—
94#4028—32no—
95C4×D20—62no—
96#3956—61no—
97#3955—61no—
98#3952—62no—
99#3952—62no—
100#1891—151noSylow
101C23×C3×C5C22×C411yescharacteristic
102C22×D15—21no—
103C22×D15—21no—
104C22×D15—21no—
105C22×D15—42no—
106C22×D15C22⋊C412yes—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×C3×C4×C5#4111yescharacteristic
114#1220—22no—
115#1220—22no—
116C2×Dic15—21no—
117C2×Dic15—21no—
118C2×Dic15—21no—
119C2×Dic15C22⋊C412yes—
120D60—21no—
121D60—21no—
122D60—44no—
123C4×D15—22no—
124C4×D15—22no—
125C4×D15—41no—
126C4×D15—41no—
127C4×D15—41no—
128C4×D15—41no—
129Dic30—21no—
130Dic30—21no—
131C3×#168—22no—
132C3×#168—41no—
133C5×C3⋊C8—22no—
134C5×C3⋊C8—41no—
135C2×#235—51no—
136C22×C4×D3—51no—
137C2×C3×M16—51no—
138C2×C3×M16—51no—
139C3×#117C2×D511yescharacteristic
140C2×#208C2×D511yescharacteristic
141C2×#208C2×D511yescharacteristic
142#907—102no—
143D3×M16—102no—
144#901—102no—
145#897—101no—
146#895—101no—
147#886—52no—
148#882—52no—
149C4×D12—102no—
150#824—101no—
151#823—101no—
152(C2×D3)⋊C8—102no—
153(C2×D3)⋊C8—102no—
154C23×D5—31no—
155C22×C4×C5C22×D311yescharacteristic
156C2×C22⋊D5—32no—
157C22×Dic5—31no—
158#709—61no—
159#709—61no—
160#709—61no—
161#709—61no—
162C2×D20—31no—
163C2×D20—62no—
164C2×C4×D5—32no—
165C2×C4×D5—61no—
166C2×C4×D5—61no—
167C2×C4×D5—61no—
168C2×C4×D5—61no—
169C2×C4×D5—61no—
170C2×C4×D5—61no—
171C2×Dic10—31no—
172C5×M16—61no—
173C5×M16—61no—
174C5×M16—61no—
175C5×M16—61no—
176C2×C5×C8—32no—
177C2×C5×C8—61no—
178C5×C4⋊C4—21no—
179C5×C22⋊C4C4×D312yes—
180C42×C5—21no—
181#685—62no—
182#685—62no—
183C4⋊Dic5—62no—
184#681—21no—
185#681—21no—
186#681—21no—
187#681—21no—
188C2×#168—21no—
189C2×#168D1212yescomplemented
190#676—64no—
191C8×D5—64no—
192#568—151no—
193#525—151no—
194#413—151no—
195#407—151no—
196#407—151no—
197#344—31no—
198#344—51no—
199C22×C3×C5—41no—
200C22×C3×C5C2×C22⋊C411yescharacteristic
201C22×C3×C5C4×D411yescharacteristic
202C22×C3×C5C4×D411yescharacteristic
203C2×D15—22no—
204C2×D15—22no—
205C2×D15—22no—
206C2×D15—22no—
207C2×D15—41no—
208C2×D15—41no—
209C2×D15—41no—
210C2×D15—41no—
211C2×D15—42no—
212C3×C4×C5—42no—
213C3×C4×C5#12011yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12311yescharacteristic
217Dic15—22no—
218Dic15—41no—
219C23×D3—51no—
220C22×C3×C4C22×D511yescharacteristic
221C2×#67—52no—
222C22×Dic3—51no—
223#235—101no—
224#235—101no—
225#235—101no—
226#235—101no—
227C2×D12—51no—
228C2×D12—102no—
229C2×C4×D3—52no—
230C2×C4×D3—101no—
231C2×C4×D3—101no—
232C2×C4×D3—101no—
233C2×C4×D3—101no—
234C2×C4×D3—101no—
235C2×C4×D3—101no—
236C2×Dic6—51no—
237C3×M16—101no—
238C3×M16—101no—
239C3×M16—101no—
240C3×M16—101no—
241C2×C3×C8—52no—
242C2×C3×C8—101no—
243C3×C4⋊C4—21no—
244C3×C22⋊C4C4×D512yes—
245C3×C42—21no—
246(C2×D3)⋊C4—102no—
247(C2×D3)⋊C4—102no—
248C4⋊Dic3—102no—
249#208—21no—
250#208—21no—
251#208—21no—
252#208—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8D2012yescomplemented
255ΓL(1,𝔽25)—104no—
256D3×C8—104no—
257C23×C5C2×C4×D311yescharacteristic
258C22×D5—32no—
259C22×D5—61no—
260C22×D5—61no—
261C22×D5—61no—
262C22×D5—122no—
263C2×C4×C5—22no—
264C2×C4×C5C2×D1211yescharacteristic
265C2×C4×C5#23511yescharacteristic
266C2×C4×C5D3×D411yescharacteristic
267C2×C4×C5D3×D411yescharacteristic
268C2×C4×C5D3×D411yescharacteristic
269C2×C4×C5#23911yescharacteristic
270C22⋊D5—62no—
271C22⋊D5—62no—
272C2×Dic5—32no—
273C2×Dic5—61no—
274C2×Dic5—61no—
275C2×Dic5—61no—
276D20—61no—
277D20—61no—
278D20—124no—
279C4×D5—62no—
280C4×D5—62no—
281C4×D5—121no—
282C4×D5—121no—
283C4×D5—121no—
284C4×D5—121no—
285Dic10—61no—
286Dic10—61no—
287C5×C8—62no—
288C5×C8—121no—
289#168—22no—
290#168—41no—
291C2×#41—151no—
292C23×C4—151no—
293#131—302no—
294#131—302no—
295C2×M16—31no—
296C2×M16—31no—
297C2×M16—51no—
298C2×M16—51no—
299C2×M16—302no—
300#123—152no—
301C22⋊Q8—301no—
302#121—301no—
303#120—152no—
304C4×D4—302no—
305#117D3×D511yescharacteristic
306#101—301no—
307#100—301no—
308C22⋊C8—302no—
309C22⋊C8—302no—
310C2×C3×C5—21no—
311C2×C3×C5—21no—
312C2×C3×C5—21no—
313C2×C3×C5#38611yescharacteristic
314D15—42no—
315D15—42no—
316C23×C3C2×C4×D511yescharacteristic
317C22×D3—52no—
318C22×D3—101no—
319C22×D3—101no—
320C22×D3—101no—
321C22×D3—202no—
322C2×C3×C4—22no—
323C2×C3×C4C2×D2011yescharacteristic
324C2×C3×C4#70911yescharacteristic
325C2×C3×C4D4×D511yescharacteristic
326C2×C3×C4D4×D511yescharacteristic
327C2×C3×C4D4×D511yescharacteristic
328C2×C3×C4#71311yescharacteristic
329#67—102no—
330#67—102no—
331C2×Dic3—52no—
332C2×Dic3—101no—
333C2×Dic3—101no—
334C2×Dic3—101no—
335D12—101no—
336D12—101no—
337D12—204no—
338C4×D3—102no—
339C4×D3—102no—
340C4×D3—201no—
341C4×D3—201no—
342C4×D3—201no—
343C4×D3—201no—
344Dic6—101no—
345Dic6—101no—
346C3×C8—102no—
347C3×C8—201no—
348C3⋊C8—22no—
349C3⋊C8—41no—
350C22×C5—41no—
351C22×C5C4×D1211yescharacteristic
352C22×C5D3×C22⋊C411yescharacteristic
353C22×C5#89311yescharacteristic
354C2×D5—62no—
355C2×D5—62no—
356C2×D5—62no—
357C2×D5—62no—
358C2×D5—121no—
359C2×D5—121no—
360C2×D5—121no—
361C2×D5—121no—
362C2×D5—122no—
363C4×C5—42no—
364C4×C5#88211yescharacteristic
365C4×C5#88411yescharacteristic
366C4×C5#89411yescharacteristic
367C4×C5#89511yescharacteristic
368Dic5—62no—
369Dic5—121no—
370C24—151no—
371#41—301no—
372#41—301no—
373#41—301no—
374#41—301no—
375C2×Q8—151no—
376C2×D4—152no—
377C2×D4—151no—
378C2×D4—302no—
379C22×C4—152no—
380C22×C4—151no—
381C22×C4—301no—
382C22×C4—301no—
383C22×C4—301no—
384C22×C4—301no—
385C22×C4—301no—
386C22×C4—301no—
387C22×C4C2×D3×D511yescharacteristic
388M16—61no—
389M16—61no—
390M16—61no—
391M16—61no—
392M16—101no—
393M16—101no—
394M16—101no—
395M16—101no—
396M16—304no—
397C2×C8—32no—
398C2×C8—52no—
399C2×C8—61no—
400C2×C8—101no—
401C2×C8—304no—
402C4⋊C4—21no—
403C4⋊C4—302no—
404C22⋊C4—302no—
405C22⋊C4—302no—
406C22⋊C4#120012yes—
407C42—21no—
408C3×C5#189111yescharacteristic, complemented, Hall
409C22×C3—41no—
410C22×C3C4×D2011yescharacteristic
411C22×C3D5×C22⋊C411yescharacteristic
412C22×C3#403911yescharacteristic
413C2×D3—102no—
414C2×D3—102no—
415C2×D3—102no—
416C2×D3—102no—
417C2×D3—201no—
418C2×D3—201no—
419C2×D3—201no—
420C2×D3—201no—
421C2×D3—202no—
422C3×C4—42no—
423C3×C4#402811yescharacteristic
424C3×C4#403011yescharacteristic
425C3×C4#404011yescharacteristic
426C3×C4#404111yescharacteristic
427Dic3—102no—
428Dic3—201no—
429C2×C5—21no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5#466511yescharacteristic
433D5—122no—
434D5—122no—
435C23—152no—
436C23—301no—
437C23—301no—
438C23—301no—
439C23—602no—
440C23C2×#120011yescharacteristic
441Q8—301no—
442Q8—301no—
443D4—302no—
444D4—302no—
445D4—301no—
446D4—301no—
447D4—604no—
448C2×C4—22no—
449C2×C4—152no—
450C2×C4—302no—
451C2×C4—302no—
452C2×C4—301no—
453C2×C4—301no—
454C2×C4—301no—
455C2×C4—601no—
456C2×C4—601no—
457C2×C4—601no—
458C2×C4—601no—
459C2×C4#676911yescharacteristic
460C2×C4#677311yescharacteristic
461C2×C4D5×D1211yescharacteristic
462C2×C4D3×D2011yescharacteristic
463C2×C4#679011yescharacteristic
464C2×C4#679011yescharacteristic
465C8—62no—
466C8—102no—
467C8—121no—
468C8—201no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3#6495411yescharacteristic
473D3—202no—
474D3—202no—
475C5#6903611yescharacteristic, complemented, Sylow
476C22—41no—
477C22—302no—
478C22—302no—
479C22—302no—
480C22—302no—
481C22—602no—
482C22—601no—
483C22—601no—
484C22—601no—
485C22—601no—
486C22#9141811yescharacteristic
487C22#9142811yescharacteristic
488C22#9156111yescharacteristic
489C4—42no—
490C4—302no—
491C4—601no—
492C4#9142011yescharacteristic
493C4#9143311yescharacteristic
494C4#9145911yescharacteristic
495C4#9147611yescharacteristic
496C3#1059982911yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—602no—
501C2—602no—
502C2#1174890411yescharacteristic
subgroupG/H
centerC4#91476
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11756283C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1891
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11754130C2semidirect
3. #11748357C2semidirect
4. #11748351C2semidirect
5. #11748351C2semidirect
6. C5×#4533C2semidirect
7. C3×#64753C2semidirect
35. #64753D3semidirect
65. C2×C5×C3⋊C8D4semidirect
71. C2×C3×#168D4semidirect
80. #4533D5semidirect
189. C2×#168D12semidirect
254. C2×C3⋊C8D20semidirect
408. C3×C5#1891semidirect
475. C5#69036semidirect
496. C3#10599829semidirect

Maximal subgroups

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

subgroupindex
#117562832
#117541302
#117483572
#117483512
C5×#45332
C3×#647532
#105998293
#690365