Finite Group Databasebeta

← #14302020

Subgroups — #14302020

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
1C5×#5828C211yesmaximal, characteristic
2C2×#91371C211yesmaximal, characteristic, complemented
3C2×#91121C211yesmaximal, characteristic, complemented
4C2×C3×#3969C211yesmaximal, characteristic, complemented
5#11748346C211yesmaximal, characteristic, complemented
6#11745977C211yesmaximal, characteristic, complemented
7#11745975C211yesmaximal, characteristic, complemented
8#10602505—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C4×C5×D3C2211yescharacteristic
11C2×C4×Dic15C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#946C412yescomplemented
14C5×#938C412yescomplemented
15C5×#937—21no—
16C5×#935—21no—
17C4×C5×#67—22no—
18C5×#926C2211yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21C2×Dic3⋊Dic5—21no—
22C2×(C2×D3)⋊Dic5—21no—
23#91371—22no—
24#91371—22no—
25#91121—22no—
26C3×#3969—22no—
27(C2×D3)⋊#168—22no—
28#70383—51nomaximal, Hall
29C5×#525—31no—
30C2×#4072—31no—
31C2×#3969—31no—
32C2×#3969D311yescharacteristic, complemented
33#65496—31no—
34#64752—31no—
35#64752—31no—
36C23×C5×D3D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C2×C3×C5×Q8C2×C411yescharacteristic
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4C2×C411yescharacteristic
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5C2311yescharacteristic
46C22×Dic15—21no—
47C2×C5×#67—22no—
48C22×C5×Dic3D411yescharacteristic
49C2×C4×C5×D3—21no—
50C2×C4×C5×D3—21no—
51C2×C4×C5×D3—21no—
52C2×C4×C5×D3—21no—
53C2×C4×C5×D3C2×C412yes—
54C2×C4×C5×D3D412yes—
55C22×C3×Dic5—21no—
56C4×Dic15—21no—
57C4×Dic15D412yescomplemented
58#6710—21no—
59#6710—21no—
60C2×C3⋊#168—21no—
61C5×#217—21no—
62C5×#217—22no—
63C5×(C2×D3)⋊C4—22no—
64C5×C4⋊Dic3C2×C412yes—
65C5×Dic3⋊C4—22no—
66C4×C5×Dic3—21no—
67C3×C4×Dic5—21no—
68C3×C4×Dic5D412yescomplemented
69C3×#681—21no—
70C3×#681—21no—
71C2×C3×#168—21no—
72Dic3⋊Dic5—24no—
73(C2×D3)⋊Dic5—24no—
74#5828D511yescharacteristic
75C2×#872—51no—
76C2×C3×#104—51no—
77C2×#837—51no—
78#5126—51no—
79#4555—51no—
80#4523—51no—
81C2×C5×#41C2×D311yescharacteristic
82C23×C4×C5—31no—
83C5×#123—61no—
84C5×C22⋊Q8—32no—
85C5×#121—32no—
86C5×#120—61no—
87C4×C5×D4—62no—
88C5×#117—31no—
89C2×C22⋊Dic5—61no—
90#4072—62no—
91#4072—62no—
92C2×C4⋊Dic5—61no—
93C2×C4×Dic5—31no—
94C2×C4×Dic5C2×D311yescharacteristic
95C2×#681—31no—
96C2×#681C2×D311yescharacteristic
97#3969—22no—
98#3969—62no—
99C22⋊#168—62no—
100#1897—151noSylow
101C23×C3×C5—21no—
102C23×C3×C5C2×D411yescharacteristic
103C22×C5×D3—21no—
104C22×C5×D3—21no—
105C22×C5×D3—42no—
106C22×C5×D3C22⋊C412yes—
107C3×C5×Q8—21no—
108C3×C5×Q8—21no—
109C3×C5×D4—21no—
110C3×C5×D4—21no—
111C3×C5×D4—41no—
112C3×C5×D4—41no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5C22×C411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5#4111yescharacteristic
121C2×C3×C4×C5#4111yescharacteristic
122C2×Dic15—22no—
123C2×Dic15—42no—
124C5×#67—44no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3C22⋊C412yes—
129C4×C5×D3—22no—
130C4×C5×D3—22no—
131C4×C5×D3—22no—
132C4×C5×D3—22no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—42no—
135C3⋊#168—41no—
136C3×#168—41no—
137C2×C3×#41C2×D511yescharacteristic
138C22×C4×D3C2×D511yescharacteristic
139C2×C3×M16—51no—
140C2×C3×C42—51no—
141#946Dic512yescomplemented
142#938Dic512yescomplemented
143#937—21no—
144#935—21no—
145C4×#67—22no—
146C2×(C2×D3)⋊C4—101no—
147#926C2×D511yescharacteristic
148C2×Dic3⋊C4—101no—
149C2×C4×Dic3—51no—
150C2×#208—51no—
151#872—102no—
152#872—102no—
153C3×#104—102no—
154#837—102no—
155(C2×D3)⋊C8—102no—
156C24×C5—31no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41—21no—
160C5×#41—21no—
161C2×C5×Q8C4×D311yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4—62no—
164C2×C5×D4C4×D311yescharacteristic
165C22×C4×C5—21no—
166C22×C4×C5—31no—
167C22×C4×C5—32no—
168C22×C4×C5—32no—
169C22×C4×C5—61no—
170C22×C4×C5—61no—
171C22×C4×C5—61no—
172C22×C4×C5—61no—
173C22×C4×C5C22×D311yescharacteristic
174C22×Dic5—21no—
175C22×Dic5—61no—
176C5×C4⋊C4—32no—
177C5×C4⋊C4—62no—
178C5×C22⋊C4—62no—
179C5×C22⋊C4—62no—
180C5×C22⋊C4—61no—
181C42×C5—61no—
182C22⋊Dic5—64no—
183C4⋊Dic5—64no—
184C4×Dic5—21no—
185C4×Dic5—32no—
186C4×Dic5—61no—
187C4×Dic5#6712yescomplemented
188#681—21no—
189#681—21no—
190#681—61no—
191#681—61no—
192C2×#168—21no—
193C2×#168—61no—
194#525—31no—
195C2×#117—151no—
196C2×#104—51no—
197C2×#104—151no—
198#407—151no—
199#339—151no—
200#339—151no—
201C22×C3×C5—22no—
202C22×C3×C5—41no—
203C22×C3×C5—42no—
204C22×C3×C5C2×C22⋊C411yescharacteristic
205C22×C3×C5#12011yescharacteristic
206C22×C3×C5#12311yescharacteristic
207C2×C5×D3—22no—
208C2×C5×D3—22no—
209C2×C5×D3—22no—
210C2×C5×D3—42no—
211C2×C5×D3—42no—
212C3×C4×C5—41no—
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5C4×D411yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12111yescharacteristic
217Dic15—42no—
218C3×Dic5—42no—
219C5×Dic3—22no—
220C5×Dic3—42no—
221C23×D3C22⋊D511yescharacteristic
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C2×C3×Q8C2×Dic511yescharacteristic
227C2×C3×D4—21no—
228C2×C3×D4C2×Dic511yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—101no—
231C22×C3×C4C22×D511yescharacteristic
232C2×#67—22no—
233C22×Dic3—101no—
234C22×Dic3C22⋊D511yescharacteristic
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3C2×Dic512yes—
240C2×C4×D3C22⋊D512yes—
241C3×M16—101no—
242C3×M16—101no—
243C2×C3×C8—101no—
244C3×C42—52no—
245C3×C42—101no—
246#217—21no—
247#217—22no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—104no—
250C4⋊Dic3C2×Dic512yes—
251Dic3⋊C4—22no—
252Dic3⋊C4—104no—
253C4×Dic3—21no—
254C4×Dic3—52no—
255C4×Dic3—101no—
256#208—101no—
257#208—101no—
258C2×C3⋊C8—101no—
259C23×C5—21no—
260C23×C5—32no—
261C23×C5—61no—
262C23×C5—61no—
263C23×C5—122no—
264C23×C5D3×D411yescharacteristic
265C5×Q8—21no—
266C5×Q8—21no—
267C5×D4—21no—
268C5×D4—21no—
269C5×D4—41no—
270C5×D4—41no—
271C5×D4—124no—
272C2×C4×C5—21no—
273C2×C4×C5—32no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C2×C4×C5—62no—
279C2×C4×C5—62no—
280C2×C4×C5—62no—
281C2×C4×C5—62no—
282C2×C4×C5—62no—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×C4×C5#23511yescharacteristic
285C2×C4×C5D3×D411yescharacteristic
286C2×C4×C5D3×D411yescharacteristic
287C2×C4×C5#23711yescharacteristic
288C2×C4×C5C2×#6711yescharacteristic
289C2×Dic5—22no—
290C2×Dic5—42no—
291C2×Dic5—62no—
292C2×Dic5—122no—
293#168—41no—
294#168—121no—
295C2×#41D3×D511yescharacteristic
296C23×C4—31no—
297C2×M16—51no—
298C2×M16—151no—
299#123—61no—
300C22⋊Q8—32no—
301#121—32no—
302#120—61no—
303C4×D4—62no—
304#117—31no—
305#117—302no—
306#117—302no—
307C2×C4⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—51no—
310C2×C42—151no—
311#104—102no—
312#104—302no—
313C22⋊C8—302no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5—81no—
317C2×C3×C5#38611yescharacteristic
318C2×C3×C5#42112yes—
319C5×D3—42no—
320C23×C3—21no—
321C23×C3C2×C22⋊D511yescharacteristic
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—42no—
325C22×D3C22⋊Dic512yes—
326C3×Q8—21no—
327C3×Q8—21no—
328C3×D4—21no—
329C3×D4—21no—
330C3×D4—41no—
331C3×D4—41no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—102no—
336C2×C3×C4—202no—
337C2×C3×C4D4×D511yescharacteristic
338C2×C3×C4D4×D511yescharacteristic
339C2×C3×C4#71111yescharacteristic
340C2×C3×C4#71111yescharacteristic
341C2×C3×C4C22×Dic511yescharacteristic
342C2×C3×C4C2×C22⋊D511yescharacteristic
343#67—44no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—22no—
347C2×Dic3—102no—
348C2×Dic3—202no—
349C2×Dic3C22⋊Dic512yes—
350C4×D3—22no—
351C4×D3—22no—
352C4×D3—22no—
353C4×D3—22no—
354C3×C8—201no—
355C3⋊C8—201no—
356C22×C5—22no—
357C22×C5—41no—
358C22×C5—42no—
359C22×C5—62no—
360C22×C5—62no—
361C22×C5—62no—
362C22×C5—122no—
363C22×C5—122no—
364C22×C5D3×C22⋊C411yescharacteristic
365C22×C5#88311yescharacteristic
366C22×C5#93711yescharacteristic
367C4×C5—41no—
368C4×C5—62no—
369C4×C5—122no—
370C4×C5#67⋊C411yescharacteristic
371C4×C5#88411yescharacteristic
372C4×C5C4×#6711yescharacteristic
373C4×C5#93811yescharacteristic
374Dic5—42no—
375Dic5—122no—
376C24—31no—
377#41—21no—
378#41—21no—
379#41—21no—
380#41—21no—
381C2×Q8D3×Dic511yescharacteristic
382C2×D4—21no—
383C2×D4—62no—
384C2×D4D3×Dic511yescharacteristic
385C22×C4—21no—
386C22×C4—31no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
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—32no—
403C4⋊C4—62no—
404C4⋊C4—304no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—61no—
408C22⋊C4—304no—
409C42—52no—
410C42—61no—
411C42—101no—
412C42—152no—
413C42—301no—
414C3×C5#189711yescharacteristic, complemented, Hall
415C22×C3—22no—
416C22×C3—41no—
417C22×C3—42no—
418C22×C3#408111yescharacteristic
419C22×C3#408311yescharacteristic
420C22×C3C2×C22⋊Dic511yescharacteristic
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—42no—
425C2×D3—42no—
426C3×C4—41no—
427C3×C4—202no—
428C3×C4D4×Dic511yescharacteristic
429C3×C4D4×Dic511yescharacteristic
430C3×C4#408511yescharacteristic
431C3×C4#408511yescharacteristic
432Dic3—22no—
433Dic3—42no—
434Dic3—202no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5—81no—
438C2×C5—122no—
439C2×C5#482112yes—
440C2×C5#495611yescharacteristic
441C23—21no—
442C23—32no—
443C23—61no—
444C23—61no—
445C23—122no—
446C23D3×C22⋊D511yescharacteristic
447Q8—21no—
448Q8—21no—
449D4—21no—
450D4—21no—
451D4—41no—
452D4—41no—
453D4—124no—
454C2×C4—21no—
455C2×C4—32no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—102no—
466C2×C4—202no—
467C2×C4—302no—
468C2×C4—602no—
469C2×C4#678011yescharacteristic
470C2×C4C2×D3×Dic511yescharacteristic
471C2×C4#678211yescharacteristic
472C2×C4D5×#6711yescharacteristic
473C2×C4D3×C22⋊D511yescharacteristic
474C2×C4#679011yescharacteristic
475C8—201no—
476C8—601no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—81no—
480C2×C3#6550911yescharacteristic
481C2×C3#6552512yes—
482D3—42no—
483C5#7038311yescharacteristic, complemented, Sylow
484C22—22no—
485C22—42no—
486C22—41no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—122no—
491C22—122no—
492C22D3×C22⋊Dic511yescharacteristic
493C22#9155511yescharacteristic
494C22#9157711yescharacteristic
495C4—41no—
496C4—62no—
497C4—122no—
498C4—202no—
499C4—602no—
500C4Dic5×#6711yescharacteristic
501C4#9155811yescharacteristic
502C4#67⋊Dic511yescharacteristic
503C4#9156511yescharacteristic
504C3#1060250511yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—81no—
508C2—122no—
509C2#1174942912yes—
510C2#1174951911yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊D5
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC5×#5828C2
cosocleC22×C3×C4×C5C23
hypercenterC2×#41D3×D5
hypocenterC3×C5#1897
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×#91371C2semidirect
3. C2×#91121C2semidirect
4. C2×C3×#3969C2semidirect
5. #11748346C2semidirect
6. #11745977C2semidirect
7. #11745975C2semidirect
13. C5×#946C4semidirect
14. C5×#938C4semidirect
32. C2×#3969D3semidirect
57. C4×Dic15D4semidirect
68. C3×C4×Dic5D4semidirect
141. #946Dic5semidirect
142. #938Dic5semidirect
187. C4×Dic5#67semidirect
414. C3×C5#1897semidirect
483. C5#70383semidirect
504. C3#10602505semidirect

Maximal subgroups

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

subgroupindex
C5×#58282
C2×#913712
C2×#911212
C2×C3×#39692
#117483462
#117459772
#117459752
#106025053
#703835