Finite Group Databasebeta

← #14306007

Subgroups — #14306007

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#11754581C211yesmaximal, characteristic
2C2×#91089C211yesmaximal, characteristic, complemented
3C42×C5×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3937C211yesmaximal, characteristic, complemented
5#11748440C211yesmaximal, characteristic, complemented
6C2×#90980C211yesmaximal, characteristic, complemented
7C2×#90977C211yesmaximal, characteristic, complemented
8#10600188—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3—22no—
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C4×C5×Dic3C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17#91569—22no—
18#91458C411yescharacteristic, complemented
19C4×#1202—22no—
20#91389C411yescharacteristic, complemented
21#91374C411yescharacteristic, complemented
22#91364C411yescharacteristic, complemented
23#91089—22no—
24C3×#3937—22no—
25Dic3⋊#168—22no—
26#90980—22no—
27#90977—22no—
28#70356—51nomaximal, Hall
29#65811—31no—
30C43×C5—31no—
31C2×#3937—31no—
32C2×#3937—31no—
33C2×#3937—31no—
34C2×#3937D311yescharacteristic, complemented
35#65213—31no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C2×#1220—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42#6817—21no—
43C2×D60C2×C411yescharacteristic
44C2×C4×D15—21no—
45C2×Dic30C2×C411yescharacteristic
46C22×C5×Dic3—21no—
47C22×C5×Dic3D412yes—
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×#1202—22no—
57C3×C42×C5—21no—
58C3×C42×C5D412yescomplemented
59#6710—21no—
60#6710—21no—
61C2×C3⋊#168—21no—
62C4×C5×Dic3—28no—
63C4×C5×Dic3—21no—
64C4×C5×Dic3C2×C411yescharacteristic
65C4×C5×Dic3C2×C411yescharacteristic
66C4×C5×Dic3D412yes—
67C4×C5×Dic3D412yescomplemented
68C3×#681—21no—
69C3×#681—21no—
70C2×C3×#168—21no—
71Dic3⋊Dic5—22no—
72#6667—22no—
73#6665—22no—
74#5832—51no—
75C2×C3×#104—51no—
76C2×#837—51no—
77C2×#825—51no—
78#5118—51no—
79C42×Dic3D511yescharacteristic, complemented
80C2×#805—51no—
81C2×#709—31no—
82C2×#709C2×D311yescharacteristic
83C2×C42×C5—31no—
84C2×C42×C5—31no—
85C2×C42×C5—62no—
86C2×C42×C5C2×D311yescharacteristic
87#4076—62no—
88C2×#681—31no—
89C2×#681C2×D311yescharacteristic
90#4021—31no—
91#4021—31no—
92#4020—31no—
93C4×D20—62no—
94#4015—31no—
95#3937—22no—
96#3937—62no—
97#3937—62no—
98#3937—62no—
99C4⋊#168—62no—
100#1962—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×D15—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
112#1220—41no—
113#1220—41no—
114C2×Dic15—21no—
115D60—21no—
116D60—21no—
117C4×D15—41no—
118C4×D15—41no—
119Dic30—21no—
120Dic30—21no—
121C2×C5×Dic3—24no—
122C2×C5×Dic3—22no—
123C2×C5×Dic3—24no—
124C2×C5×Dic3C22⋊C412yes—
125C2×C5×Dic3C22⋊C412yes—
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#1202—44no—
136C3⋊#168—41no—
137C3×#168—41no—
138C2×C3×#41—51no—
139C2×#235—51no—
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—22no—
149C2×C4×Dic3C2×D511yescharacteristic
150C2×C4×Dic3C2×D511yescharacteristic
151C2×#208—51no—
152C3×#104—102no—
153#837—102no—
154#825—102no—
155Dic3⋊C8—102no—
156#805—102no—
157C22×C4×C5—21no—
158C22×C4×C5—32no—
159C22×C4×C5—61no—
160C22×C4×C5C22×D311yescharacteristic
161C2×C22⋊D5—21no—
162C2×C22⋊D5—61no—
163#709—21no—
164#709—21no—
165#709—21no—
166#709—21no—
167#709—61no—
168#709—61no—
169#709—61no—
170#709—61no—
171C2×D20—31no—
172C2×D20—62no—
173C2×D20C4×D311yescharacteristic
174C2×C4×D5—21no—
175C2×C4×D5—61no—
176C2×Dic10—31no—
177C2×Dic10C4×D311yescharacteristic
178C42×C5—21no—
179C42×C5—31no—
180C42×C5—31no—
181C42×C5—32no—
182C42×C5—32no—
183C42×C5—68no—
184C42×C5—61no—
185C42×C5#6712yescomplemented
186#685—62no—
187#685—62no—
188C4⋊Dic5—62no—
189#681—21no—
190#681—21no—
191#681—61no—
192#681—61no—
193C2×#168—21no—
194C2×#168—61no—
195#532—151no—
196#423—151no—
197C2×#104—51no—
198C2×#104—151no—
199C2×#104—151no—
200C2×#104—151no—
201C43—31no—
202C22×C3×C5—22no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5#12111yescharacteristic
205C22×C3×C5#12111yescharacteristic
206C2×D15—41no—
207C2×D15—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—41no—
216C3×Dic5—41no—
217C5×Dic3—24no—
218C5×Dic3—22no—
219C5×Dic3#10414yescomplemented
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—101no—
231C2×#67—102no—
232C22×Dic3—21no—
233C22×Dic3C22⋊D512yes—
234#235—101no—
235#235—101no—
236#235—101no—
237#235—101no—
238C2×D12—51no—
239C2×C4×D3—101no—
240C2×Dic6—51no—
241C3×M16—101no—
242C3×M16—101no—
243C2×C3×C8—101no—
244C3×C42—21no—
245C3×C42D2012yescomplemented
246#217—102no—
247(C2×D3)⋊C4—102no—
248Dic3⋊C4—102no—
249C4×Dic3—28no—
250C4×Dic3—21no—
251C4×Dic3C4×D511yescharacteristic
252C4×Dic3C4×D511yescharacteristic
253C4×Dic3D2012yes—
254C4×Dic3D2012yescomplemented
255#208—101no—
256#208—101no—
257C2×C3⋊C8—101no—
258C23×C5D3×D411yescharacteristic
259C22×D5—21no—
260C22×D5—61no—
261C2×C4×C5—24no—
262C2×C4×C5—22no—
263C2×C4×C5—32no—
264C2×C4×C5—32no—
265C2×C4×C5—64no—
266C2×C4×C5—64no—
267C2×C4×C5—62no—
268C2×C4×C5C2×C4×D311yescharacteristic
269C2×C4×C5#23511yescharacteristic
270C2×C4×C5D3×D411yescharacteristic
271C2×C4×C5D3×D411yescharacteristic
272C2×C4×C5#23711yescharacteristic
273C2×C4×C5C2×#6711yescharacteristic
274C22⋊D5—41no—
275C22⋊D5—41no—
276C22⋊D5—121no—
277C22⋊D5—121no—
278C2×Dic5—21no—
279C2×Dic5—61no—
280D20—21no—
281D20—21no—
282D20—61no—
283D20—61no—
284D20—124no—
285C4×D5—41no—
286C4×D5—41no—
287C4×D5—121no—
288C4×D5—121no—
289Dic10—21no—
290Dic10—21no—
291Dic10—61no—
292Dic10—61no—
293#168—41no—
294#168—121no—
295C2×#41—51no—
296C2×#41—151no—
297C2×M16—51no—
298C2×M16—151no—
299C4⋊Q8—151no—
300#127—151no—
301#124—151no—
302#124—151no—
303#121—302no—
304C4×D4—302no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—62no—
308C2×C42D3×D511yescharacteristic
309C4⋊C8—302no—
310#104—102no—
311#104—302no—
312#104—302no—
313#104—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#39011yescharacteristic
317C2×C3×C5C2×#10412yes—
318D15—81no—
319C3×D5—81no—
320C23×C3—101no—
321C23×C3C2×C22⋊D511yescharacteristic
322C22×D3—101no—
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×C4C2×D2011yescharacteristic
336C2×C3×C4C2×D2011yescharacteristic
337C2×C3×C4C2×D2011yescharacteristic
338C2×C3×C4#70911yescharacteristic
339C2×C3×C4#70911yescharacteristic
340#67—201no—
341#67—201no—
342#67—204no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—101no—
347C2×Dic3#68512yes—
348C2×Dic3#68512yes—
349D12—101no—
350D12—101no—
351C4×D3—201no—
352C4×D3—201no—
353Dic6—101no—
354Dic6—101no—
355C3×C8—201no—
356C3⋊C8—201no—
357C22×C5—22no—
358C22×C5D3×C22⋊C411yescharacteristic
359C22×C5#88411yescharacteristic
360C22×C5#93911yescharacteristic
361C2×D5—41no—
362C2×D5—42no—
363C2×D5—121no—
364C2×D5—122no—
365C4×C5—24no—
366C4×C5—34no—
367C4×C5—64no—
368C4×C5—62no—
369C4×C5#67⋊C411yescharacteristic
370C4×C5#88511yescharacteristic
371C4×C5C4×#6711yescharacteristic
372C4×C5#94011yescharacteristic
373Dic5—41no—
374Dic5—121no—
375#41—101no—
376#41—101no—
377#41—101no—
378#41—101no—
379#41—301no—
380#41—301no—
381#41—301no—
382#41—301no—
383C2×Q8—51no—
384C2×Q8—151no—
385C2×D4—51no—
386C2×D4—101no—
387C2×D4—151no—
388C2×D4—302no—
389C2×D4—301no—
390C22×C4—21no—
391C22×C4—32no—
392C22×C4—61no—
393C22×C4—101no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—101no—
397M16—101no—
398M16—301no—
399M16—301no—
400C2×C8—101no—
401C2×C8—301no—
402C4⋊C4—302no—
403C22⋊C4—302no—
404C22⋊C4—302no—
405C42—21no—
406C42—31no—
407C42—31no—
408C42—32no—
409C42—32no—
410C42—68no—
411C42—61no—
412C42#120212yescomplemented
413C3×C5#196211yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—202no—
416C22×C3—201no—
417C22×C3C2×#68511yescharacteristic
418C22×C3#407611yescharacteristic
419C22×C3#407611yescharacteristic
420C2×D3—202no—
421C2×D3—201no—
422C3×C4—24no—
423C3×C4—201no—
424C3×C4C4×D2011yescharacteristic
425C3×C4C4×D2011yescharacteristic
426C3×C4#402011yescharacteristic
427C3×C4#402111yescharacteristic
428Dic3—24no—
429Dic3—22no—
430Dic3—201no—
431Dic3#393714yescomplemented
432C2×C5—21no—
433C2×C5—21no—
434C2×C5D3×#10412yes—
435C2×C5#495711yescharacteristic
436D5—81no—
437D5—241no—
438C23—101no—
439C23—301no—
440C23D3×C22⋊D511yescharacteristic
441Q8—101no—
442Q8—101no—
443Q8—301no—
444Q8—301no—
445D4—101no—
446D4—101no—
447D4—201no—
448D4—201no—
449D4—301no—
450D4—301no—
451D4—604no—
452D4—601no—
453D4—601no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—101no—
462C2×C4—201no—
463C2×C4—201no—
464C2×C4—301no—
465C2×C4—601no—
466C2×C4—601no—
467C2×C4#677111yescharacteristic
468C2×C4C4×D3×D511yescharacteristic
469C2×C4D3×D2011yescharacteristic
470C2×C4D3×D2011yescharacteristic
471C2×C4#677911yescharacteristic
472C2×C4C2×#120211yescharacteristic
473C8—201no—
474C8—601no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—401no—
478C2×C3C2×#393712yes—
479C2×C3#6522811yescharacteristic
480D3—401no—
481C5#7035611yescharacteristic, complemented, Sylow
482C22—22no—
483C22—202no—
484C22—201no—
485C22—602no—
486C22—601no—
487C22#9145411yescharacteristic
488C22D3×#68511yescharacteristic
489C22#9156911yescharacteristic
490C4—24no—
491C4—34no—
492C4—64no—
493C4—62no—
494C4—201no—
495C4—601no—
496C4#9135311yescharacteristic
497C4#9139511yescharacteristic
498C4C4×#120211yescharacteristic
499C4#9145811yescharacteristic
500C3#1060018811yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—401no—
504C2—1201no—
505C2D3×#393712yes—
506C2#1174891211yescharacteristic
subgroupG/H
centerC2×C4D3×D20
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754581C2
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. C2×#91089C2semidirect
3. C42×C5×Dic3C2semidirect
4. C2×C3×#3937C2semidirect
5. #11748440C2semidirect
6. C2×#90980C2semidirect
7. C2×#90977C2semidirect
18. #91458C4semidirect
20. #91389C4semidirect
21. #91374C4semidirect
22. #91364C4semidirect
34. C2×#3937D3semidirect
58. C3×C42×C5D4semidirect
67. C4×C5×Dic3D4semidirect
79. C42×Dic3D5semidirect
185. C42×C5#67semidirect
219. C5×Dic3#104semidirect
245. C3×C42D20semidirect
254. C4×Dic3D20semidirect
412. C42#1202semidirect
413. C3×C5#1962semidirect
431. Dic3#3937semidirect
481. C5#70356semidirect
500. C3#10600188semidirect

Maximal subgroups

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

subgroupindex
#117545812
C2×#910892
C42×C5×Dic32
C2×C3×#39372
#117484402
C2×#909802
C2×#909772
#106001883
#703565