Finite Group Databasebeta

← #14302022

Subgroups — #14302022

Order 1920.

716 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756283C211yesmaximal, characteristic
2C2×#91386C211yesmaximal, characteristic, complemented
3C2×#90981C211yesmaximal, characteristic, complemented
4C2×#90980C211yesmaximal, characteristic, complemented
5#11748351C211yesmaximal, characteristic, complemented
6C5×#4532C211yesmaximal, characteristic, complemented
7C3×#64752C211yesmaximal, characteristic, complemented
8#10599550—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C4×D15C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91786C412yescomplemented
13#91784C412yescomplemented
14#91775—21no—
15#91771—21no—
16C4×D60—22no—
17C2×C4×C5×Dic3C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21C2×#6669—21no—
22C2×#6667—21no—
23#91386—22no—
24#91386—22no—
25#90981—22no—
26#90980—22no—
27#90957—22no—
28#68757—51nomaximal, Hall
29#65858—31no—
30C2×#4018—31no—
31C2×#3957—31no—
32#65415—31no—
33C2×#3937—31no—
34C5×#339—31no—
35#64752D311yescharacteristic, complemented
36C23×D15D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C2×#1220—21no—
39C22×Dic15D411yescharacteristic
40#6817—21no—
41#6817—21no—
42#6817—21no—
43#6817—21no—
44C2×D60—22no—
45C2×D60C2×C411yescharacteristic
46C2×C4×D15—21no—
47C2×C4×D15—21no—
48C2×C4×D15—21no—
49C2×C4×D15—21no—
50C2×C4×D15—21no—
51C2×C4×D15C2×C412yes—
52C2×C4×D15D412yes—
53C2×Dic30C2×C411yescharacteristic
54C22×C5×Dic3—21no—
55C22×C3×Dic5—21no—
56C3×C5×C4⋊C4C2×C412yes—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59#6714—22no—
60#6714—22no—
61C4⋊Dic15—22no—
62C4×C5×Dic3—21no—
63C4×C5×Dic3D412yescomplemented
64C5×#208—21no—
65C5×#208—21no—
66C2×C5×C3⋊C8—21no—
67C3×C4×Dic5—21no—
68C3×C4×Dic5D412yescomplemented
69C3×#681—21no—
70C3×#681—21no—
71C2×C3×#168—21no—
72#6669—24no—
73#6667—24no—
74#5524—51no—
75C2×#872—51no—
76C2×#825—51no—
77#5126—51no—
78C2×#805—51no—
79C3×#339—51no—
80#4532D511yescharacteristic, complemented
81C2×#709—31no—
82C22×C4×D5—31no—
83C2×C5×M16—31no—
84C5×#117C2×D311yescharacteristic
85C2×C42×C5—31no—
86C2×#685—61no—
87C2×#683—61no—
88C2×C4×Dic5C2×D311yescharacteristic
89C2×#681C2×D311yescharacteristic
90#4043—32no—
91#4041—32no—
92#4032—61no—
93#4028—61no—
94C4×D20—62no—
95#4018—62no—
96#4018—62no—
97#3957—62no—
98#3952—62no—
99#3937—62no—
100#1897—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×D15—21no—
103C22×D15—21no—
104C22×D15—21no—
105C22×D15—42no—
106C22×D15C22⋊C412yes—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5C22×C411yescharacteristic
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—41no—
115#1220—41no—
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—22no—
126C4×D15—22no—
127C4×D15—41no—
128C4×D15—41no—
129Dic30—21no—
130Dic30—21no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3—42no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—42no—
135C3×#168—41no—
136C5×C3⋊C8—41no—
137C2×#235—51no—
138C22×C4×D3—51no—
139C2×C3×M16—51no—
140C3×#117C2×D511yescharacteristic
141C2×C3×C42—51no—
142C2×(C2×D3)⋊C4—101no—
143C2×Dic3⋊C4—101no—
144C2×C4×Dic3C2×D511yescharacteristic
145C2×#208C2×D511yescharacteristic
146#897—52no—
147#895—52no—
148#886—101no—
149#882—101no—
150C4×D12—102no—
151#872—102no—
152#872—102no—
153#825—102no—
154(C2×D3)⋊C8—102no—
155#805—102no—
156C23×D5—31no—
157C22×C4×C5—61no—
158C22×C4×C5C22×D311yescharacteristic
159C2×C22⋊D5—61no—
160C22×Dic5—21no—
161C22×Dic5—31no—
162#709—61no—
163#709—61no—
164#709—61no—
165#709—61no—
166C2×D20—31no—
167C2×D20—62no—
168C2×C4×D5—32no—
169C2×C4×D5—32no—
170C2×C4×D5—61no—
171C2×C4×D5—61no—
172C2×C4×D5—61no—
173C2×C4×D5—61no—
174C2×C4×D5—61no—
175C2×Dic10—31no—
176C5×M16—61no—
177C5×M16—61no—
178C2×C5×C8—61no—
179C5×C4⋊C4C4×D312yes—
180C5×C22⋊C4—21no—
181C42×C5—21no—
182C42×C5—32no—
183C42×C5—61no—
184#685—64no—
185#685—62no—
186#685—62no—
187C4⋊Dic5—62no—
188#683—64no—
189C4×Dic5—21no—
190C4×Dic5D1212yescomplemented
191#681—21no—
192#681—21no—
193C2×#168—21no—
194#525—151no—
195C2×#117—151no—
196C2×#104—151no—
197C2×#104—151no—
198#407—151no—
199#339—31no—
200#339—51no—
201C22×C3×C5—22no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C22×C3×C5#12011yescharacteristic
204C22×C3×C5#12311yescharacteristic
205C2×D15—22no—
206C2×D15—22no—
207C2×D15—22no—
208C2×D15—42no—
209C2×D15—42no—
210C2×D15—41no—
211C2×D15—42no—
212C3×C4×C5—42no—
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5C4×D411yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12111yescharacteristic
217Dic15—22no—
218Dic15—41no—
219C3×Dic5—42no—
220C5×Dic3—42no—
221C23×D3—51no—
222C22×C3×C4—101no—
223C22×C3×C4C22×D511yescharacteristic
224C2×#67—101no—
225C22×Dic3—21no—
226C22×Dic3—51no—
227#235—101no—
228#235—101no—
229#235—101no—
230#235—101no—
231C2×D12—51no—
232C2×D12—102no—
233C2×C4×D3—52no—
234C2×C4×D3—52no—
235C2×C4×D3—101no—
236C2×C4×D3—101no—
237C2×C4×D3—101no—
238C2×C4×D3—101no—
239C2×C4×D3—101no—
240C2×Dic6—51no—
241C3×M16—101no—
242C3×M16—101no—
243C2×C3×C8—101no—
244C3×C4⋊C4C4×D512yes—
245C3×C22⋊C4—21no—
246C3×C42—21no—
247C3×C42—52no—
248C3×C42—101no—
249(C2×D3)⋊C4—104no—
250(C2×D3)⋊C4—102no—
251(C2×D3)⋊C4—102no—
252C4⋊Dic3—102no—
253Dic3⋊C4—104no—
254C4×Dic3—21no—
255C4×Dic3D2012yescomplemented
256#208—21no—
257#208—21no—
258C2×C3⋊C8—21no—
259C23×C5D3×D411yescharacteristic
260C22×D5—32no—
261C22×D5—61no—
262C22×D5—61no—
263C22×D5—61no—
264C22×D5—122no—
265C2×C4×C5—22no—
266C2×C4×C5—62no—
267C2×C4×C5—122no—
268C2×C4×C5C2×C4×D311yescharacteristic
269C2×C4×C5C2×D1211yescharacteristic
270C2×C4×C5#23511yescharacteristic
271C2×C4×C5D3×D411yescharacteristic
272C2×C4×C5D3×D411yescharacteristic
273C2×C4×C5#23911yescharacteristic
274C22⋊D5—121no—
275C22⋊D5—121no—
276C2×Dic5—22no—
277C2×Dic5—32no—
278C2×Dic5—42no—
279C2×Dic5—61no—
280C2×Dic5—61no—
281C2×Dic5—61no—
282D20—61no—
283D20—61no—
284D20—124no—
285C4×D5—62no—
286C4×D5—62no—
287C4×D5—62no—
288C4×D5—62no—
289C4×D5—121no—
290C4×D5—121no—
291Dic10—61no—
292Dic10—61no—
293C5×C8—121no—
294#168—41no—
295C2×#41—151no—
296C23×C4—151no—
297C2×M16—31no—
298C2×M16—51no—
299#123—301no—
300C22⋊Q8—152no—
301#121—152no—
302#120—301no—
303C4×D4—302no—
304#117—302no—
305#117—302no—
306#117D3×D511yescharacteristic
307C2×C4⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—31no—
310C2×C42—51no—
311#104—302no—
312#104—302no—
313C22⋊C8—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317C2×C3×C5#42112yes—
318D15—42no—
319D15—81no—
320C23×C3D4×D511yescharacteristic
321C22×D3—52no—
322C22×D3—101no—
323C22×D3—101no—
324C22×D3—101no—
325C22×D3—202no—
326C2×C3×C4—22no—
327C2×C3×C4—102no—
328C2×C3×C4—202no—
329C2×C3×C4C2×C4×D511yescharacteristic
330C2×C3×C4C2×D2011yescharacteristic
331C2×C3×C4#70911yescharacteristic
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4#71311yescharacteristic
335#67—201no—
336#67—201no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—52no—
340C2×Dic3—101no—
341C2×Dic3—101no—
342C2×Dic3—101no—
343D12—101no—
344D12—101no—
345D12—204no—
346C4×D3—102no—
347C4×D3—102no—
348C4×D3—102no—
349C4×D3—102no—
350C4×D3—201no—
351C4×D3—201no—
352Dic6—101no—
353Dic6—101no—
354C3×C8—201no—
355C3⋊C8—41no—
356C22×C5—22no—
357C22×C5D3×C22⋊C411yescharacteristic
358C22×C5#88211yescharacteristic
359C22×C5#89411yescharacteristic
360C2×D5—62no—
361C2×D5—62no—
362C2×D5—62no—
363C2×D5—122no—
364C2×D5—122no—
365C2×D5—121no—
366C2×D5—122no—
367C4×C5—42no—
368C4×C5—122no—
369C4×C5C4×D1211yescharacteristic
370C4×C5#88411yescharacteristic
371C4×C5#89311yescharacteristic
372C4×C5#89511yescharacteristic
373Dic5—42no—
374Dic5—62no—
375Dic5—121no—
376C24—151no—
377#41—301no—
378#41—301no—
379#41—301no—
380#41—301no—
381C2×Q8—151no—
382C2×D4—151no—
383C2×D4—302no—
384C2×D4—301no—
385C22×C4—61no—
386C22×C4—101no—
387C22×C4—151no—
388C22×C4—152no—
389C22×C4—152no—
390C22×C4—301no—
391C22×C4—301no—
392C22×C4—301no—
393C22×C4—301no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—61no—
397M16—61no—
398M16—101no—
399M16—101no—
400C2×C8—61no—
401C2×C8—101no—
402C4⋊C4—302no—
403C4⋊C4—304no—
404C4⋊C4#120012yes—
405C22⋊C4—21no—
406C22⋊C4—302no—
407C22⋊C4—302no—
408C22⋊C4—304no—
409C42—21no—
410C42—32no—
411C42—52no—
412C42—61no—
413C42—101no—
414C3×C5#189711yescharacteristic, complemented, Hall
415C22×C3—22no—
416C22×C3D5×C22⋊C411yescharacteristic
417C22×C3#402811yescharacteristic
418C22×C3#404011yescharacteristic
419C2×D3—102no—
420C2×D3—102no—
421C2×D3—102no—
422C2×D3—202no—
423C2×D3—202no—
424C2×D3—202no—
425C2×D3—201no—
426C3×C4—42no—
427C3×C4—202no—
428C3×C4C4×D2011yescharacteristic
429C3×C4#403011yescharacteristic
430C3×C4#403911yescharacteristic
431C3×C4#404111yescharacteristic
432Dic3—42no—
433Dic3—102no—
434Dic3—201no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5#466511yescharacteristic
438C2×C5#482112yes—
439D5—122no—
440D5—241no—
441C23—152no—
442C23—301no—
443C23—301no—
444C23—301no—
445C23—602no—
446C23#679011yescharacteristic
447Q8—301no—
448Q8—301no—
449D4—301no—
450D4—301no—
451D4—604no—
452D4—601no—
453D4—601no—
454C2×C4—22no—
455C2×C4—62no—
456C2×C4—102no—
457C2×C4—122no—
458C2×C4—152no—
459C2×C4—202no—
460C2×C4—301no—
461C2×C4—301no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—301no—
467C2×C4—601no—
468C2×C4—601no—
469C2×C4#676911yescharacteristic
470C2×C4#677311yescharacteristic
471C2×C4D5×D1211yescharacteristic
472C2×C4D3×D2011yescharacteristic
473C2×C4C2×#120011yescharacteristic
474C2×C4#679011yescharacteristic
475C8—121no—
476C8—201no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3#6495411yescharacteristic
480C2×C3#6511012yes—
481D3—202no—
482D3—401no—
483C5#6875711yescharacteristic, complemented, Sylow
484C22—22no—
485C22—302no—
486C22—302no—
487C22—302no—
488C22—602no—
489C22—602no—
490C22—601no—
491C22—602no—
492C22#9143311yescharacteristic
493C22#9147611yescharacteristic
494C22#9156111yescharacteristic
495C4—42no—
496C4—122no—
497C4—202no—
498C4—302no—
499C4—601no—
500C4#9141811yescharacteristic
501C4#9142011yescharacteristic
502C4#9142811yescharacteristic
503C4#9145911yescharacteristic
504C3#1059955011yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—602no—
508C2—1201no—
509C2#1174890411yescharacteristic
510C2#1174943112yes—
subgroupG/H
centerC2×C4#6790
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11756283C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1897
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#91386C2semidirect
3. C2×#90981C2semidirect
4. C2×#90980C2semidirect
5. #11748351C2semidirect
6. C5×#4532C2semidirect
7. C3×#64752C2semidirect
12. #91786C4semidirect
13. #91784C4semidirect
35. #64752D3semidirect
63. C4×C5×Dic3D4semidirect
68. C3×C4×Dic5D4semidirect
80. #4532D5semidirect
190. C4×Dic5D12semidirect
255. C4×Dic3D20semidirect
414. C3×C5#1897semidirect
483. C5#68757semidirect
504. C3#10599550semidirect

Maximal subgroups

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

subgroupindex
#117562832
C2×#913862
C2×#909812
C2×#909802
#117483512
C5×#45322
C3×#647522
#105995503
#687575