Finite Group Databasebeta

← #14302000

Subgroups — #14302000

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#11754563C211yesmaximal, characteristic
2C2×#91357C211yesmaximal, characteristic, complemented
3C2×C5×#825C211yesmaximal, characteristic, complemented
4C3×#65415C211yesmaximal, characteristic, complemented
5C2×#90978C211yesmaximal, characteristic, complemented
6#11746376C211yesmaximal, characteristic, complemented
7#11745976C211yesmaximal, characteristic, complemented
8#10601176—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×C4×Dic15C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91563—21no—
17C2×#6668—21no—
18#91539—21no—
19C2×#6665—21no—
20#91449C412yescomplemented
21C4×#1201—22no—
22#91359C412yescomplemented
23#91357—22no—
24#91357—22no—
25C5×#825—22no—
26C3×#3952—22no—
27#90978—22no—
28#69966—51nomaximal, Hall
29#66162—31no—
30C2×#4018—31no—
31C2×C5×#104—31no—
32C2×#3969—31no—
33#65415D311yescharacteristic, complemented
34#64775—31no—
35#64743—31no—
36C23×C3×D5D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C2×C5×#67—21no—
40C22×C5×Dic3—21no—
41C5×#235—21no—
42C5×#235—21no—
43C5×#235—21no—
44C5×#235—21no—
45C2×C5×D12C2×C411yescharacteristic
46C2×C4×C5×D3—21no—
47C2×C5×Dic6C2×C411yescharacteristic
48C22×C3×Dic5D411yescharacteristic
49C2×C3×C4×D5—21no—
50C2×C3×C4×D5—21no—
51C2×C3×C4×D5—21no—
52C2×C3×C4×D5—21no—
53C2×C3×C4×D5C2×C412yes—
54C2×C3×C4×D5D412yes—
55C2×#1201—22no—
56C3×C5×M16—21no—
57C3×C5×M16—21no—
58C2×C3×C5×C8—21no—
59#6719—21no—
60C4⋊Dic15C2×C412yes—
61C4×Dic15—21no—
62C4×Dic15—21no—
63C4×Dic15D412yescomplemented
64C4×C5×Dic3—21no—
65C4×C5×Dic3D412yescomplemented
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168—21no—
69#6669—22no—
70#6668—24no—
71(C2×D3)⋊Dic5—22no—
72#6665—24no—
73#6665—22no—
74#5791—51no—
75C2×#926—51no—
76C3×#407—51no—
77C2×#825—51no—
78C2×#825D511yescharacteristic, complemented
79#4555—51no—
80#4555—51no—
81C2×C5×#41—31no—
82C22×C4×D5C2×D311yescharacteristic
83C2×C5×M16C2×D311yescharacteristic
84C2×C42×C5—31no—
85#4092—32no—
86#4084—32no—
87#4083—61no—
88#4081—61no—
89C4×C22⋊D5—62no—
90C2×#685—61no—
91#4072—31no—
92C2×#683—61no—
93C2×C4×Dic5—31no—
94C2×#681C2×D311yescharacteristic
95#4018—62no—
96#4018—62no—
97C5×#104—62no—
98#3969—62no—
99#3952—22no—
100#1897—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C22×C3×D5—21no—
104C22×C3×D5—21no—
105C22×C3×D5—42no—
106C22×C3×D5C22⋊C412yes—
107C2×C3×C4×C5C22×C411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×Dic15—22no—
114C2×Dic15—22no—
115C2×Dic15—42no—
116C5×#67—41no—
117C5×#67—41no—
118C2×C5×Dic3—21no—
119C2×C5×Dic3—22no—
120C2×C5×Dic3—42no—
121C5×D12—21no—
122C5×D12—21no—
123C4×C5×D3—41no—
124C4×C5×D3—41no—
125C5×Dic6—21no—
126C5×Dic6—21no—
127C2×C3×Dic5—21no—
128C2×C3×Dic5—21no—
129C2×C3×Dic5C22⋊C412yes—
130C3×C4×D5—22no—
131C3×C4×D5—22no—
132C3×C4×D5—22no—
133C3×C4×D5—22no—
134#1201—44no—
135C3×C5×C8—41no—
136C3×#168—41no—
137C23×C3×C4—51no—
138C2×#235C2×D511yescharacteristic
139C2×C3×M16—51no—
140C2×C3×M16C2×D511yescharacteristic
141#953—101no—
142C2×#217—101no—
143#930—52no—
144#929—101no—
145C4×#67—102no—
146#926—51no—
147#926—102no—
148#926—102no—
149C2×C4⋊Dic3—101no—
150#924—52no—
151C2×C4×Dic3—51no—
152C2×C4×Dic3C2×D511yescharacteristic
153C3×C22⋊C8—102no—
154#825—22no—
155#825—102no—
156C23×D5D1211yescharacteristic
157C5×#41—61no—
158C5×#41—61no—
159C5×#41—61no—
160C5×#41—61no—
161C2×C5×Q8—31no—
162C2×C5×D4—31no—
163C2×C5×D4—61no—
164C22×C4×C5—61no—
165C22×C4×C5—61no—
166C22×C4×C5C22×D311yescharacteristic
167C2×C22⋊D5—62no—
168C22×Dic5—61no—
169C22×Dic5D1211yescharacteristic
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C2×C4×D5—21no—
173C2×C4×D5—21no—
174C2×C4×D5C4×D312yes—
175C2×C4×D5#6712yes—
176C5×M16—21no—
177C5×M16—21no—
178C2×C5×C8—21no—
179C42×C5—32no—
180C42×C5—61no—
181C22⋊Dic5—61no—
182C22⋊Dic5—62no—
183#685—64no—
184#685—62no—
185C4⋊Dic5—32no—
186#683—64no—
187#683—62no—
188C4×Dic5—32no—
189C4×Dic5—61no—
190C4×Dic5—61no—
191#681—21no—
192#681—21no—
193C2×#168—21no—
194#525—151no—
195C2×#117—151no—
196C2×#104—31no—
197C2×#104—151no—
198#407—51no—
199#339—151no—
200#339—151no—
201C22×C3×C5—22no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C22×C3×C5#12011yescharacteristic
204C22×C3×C5#12311yescharacteristic
205C2×C5×D3—41no—
206C2×C5×D3—42no—
207C2×C3×D5—22no—
208C2×C3×D5—22no—
209C2×C3×D5—22no—
210C2×C3×D5—42no—
211C2×C3×D5—42no—
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12111yescharacteristic
216Dic15—42no—
217Dic15—42no—
218C3×Dic5—22no—
219C5×Dic3—42no—
220C5×Dic3—41no—
221C24×C3—51no—
222C22×C3×C4—51no—
223C22×C3×C4—52no—
224C22×C3×C4—52no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×C3×C4—101no—
228C22×C3×C4—101no—
229C22×C3×C4C22×D511yescharacteristic
230C2×#67—21no—
231C2×#67—102no—
232C22×Dic3—21no—
233C22×Dic3—101no—
234#235—21no—
235#235—21no—
236#235—21no—
237#235—21no—
238C2×D12C4×D511yescharacteristic
239C2×C4×D3—21no—
240C2×Dic6C4×D511yescharacteristic
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—101no—
244C3×M16—101no—
245C2×C3×C8—21no—
246C2×C3×C8—101no—
247#217—102no—
248#217—104no—
249#217—101no—
250(C2×D3)⋊C4—102no—
251C4⋊Dic3—52no—
252C4⋊Dic3—104no—
253Dic3⋊C4—102no—
254C4×Dic3—21no—
255C4×Dic3—52no—
256C4×Dic3—101no—
257C4×Dic3—101no—
258C4×Dic3C22⋊D512yescomplemented
259C23×C5—61no—
260C23×C5C2×D1211yescharacteristic
261C22×D5—21no—
262C22×D5—21no—
263C22×D5—42no—
264C22×D5(C2×D3)⋊C412yes—
265C5×Q8—61no—
266C5×Q8—61no—
267C5×D4—61no—
268C5×D4—61no—
269C5×D4—121no—
270C5×D4—121no—
271C2×C4×C5—62no—
272C2×C4×C5—61no—
273C2×C4×C5—122no—
274C2×C4×C5—121no—
275C2×C4×C5—121no—
276C2×C4×C5C2×C4×D311yescharacteristic
277C2×C4×C5D3×D411yescharacteristic
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5#23711yescharacteristic
280C2×C4×C5#23711yescharacteristic
281C2×C4×C5C2×#6711yescharacteristic
282C22⋊D5—124no—
283C2×Dic5—21no—
284C2×Dic5—21no—
285C2×Dic5—62no—
286C2×Dic5—62no—
287C2×Dic5—122no—
288C2×Dic5(C2×D3)⋊C412yes—
289C4×D5—22no—
290C4×D5—22no—
291C4×D5—22no—
292C4×D5—22no—
293C5×C8—41no—
294#168—41no—
295C2×#41—31no—
296C23×C4—51no—
297C2×M16—51no—
298C2×M16D3×D511yescharacteristic
299#123—301no—
300C22⋊Q8—152no—
301#121—152no—
302#120—301no—
303C4×D4—302no—
304#117—151no—
305#117—302no—
306#117—302no—
307C2×C4⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—31no—
310C2×C42—151no—
311#104—62no—
312#104—302no—
313C22⋊C8—102no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317C2×C3×C5#42112yes—
318C3×D5—42no—
319C5×D3—81no—
320C23×C3—52no—
321C23×C3—101no—
322C23×C3—101no—
323C23×C3—202no—
324C23×C3D4×D511yescharacteristic
325C22×D3—21no—
326C2×C3×C4—52no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—102no—
333C2×C3×C4C2×C4×D511yescharacteristic
334C2×C3×C4#70911yescharacteristic
335C2×C3×C4D4×D511yescharacteristic
336C2×C3×C4D4×D511yescharacteristic
337C2×C3×C4#71111yescharacteristic
338C2×C3×C4C2×C22⋊D511yescharacteristic
339#67—41no—
340#67—41no—
341#67—204no—
342C2×Dic3—21no—
343C2×Dic3—22no—
344C2×Dic3—42no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—202no—
348D12—21no—
349D12—21no—
350C4×D3—41no—
351C4×D3—41no—
352Dic6—21no—
353Dic6—21no—
354C3×C8—41no—
355C3×C8—201no—
356C22×C5—22no—
357C22×C5—122no—
358C22×C5—121no—
359C22×C5#88211yescharacteristic
360C22×C5#88611yescharacteristic
361C22×C5C2×(C2×D3)⋊C411yescharacteristic
362C2×D5—22no—
363C2×D5—22no—
364C2×D5—22no—
365C2×D5—42no—
366C2×D5—42no—
367C4×C5—122no—
368C4×C5—121no—
369C4×C5#67⋊C411yescharacteristic
370C4×C5#67⋊C411yescharacteristic
371C4×C5#93911yescharacteristic
372C4×C5#93911yescharacteristic
373Dic5—22no—
374Dic5—122no—
375Dic5—122no—
376C24—51no—
377#41—61no—
378#41—61no—
379#41—61no—
380#41—61no—
381C2×Q8—31no—
382C2×D4—31no—
383C2×D4—61no—
384C2×D4—302no—
385C22×C4—51no—
386C22×C4—52no—
387C22×C4—52no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—101no—
391C22×C4—101no—
392C22×C4—101no—
393C22×C4—101no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—21no—
397M16—21no—
398M16—101no—
399M16—101no—
400C2×C8—21no—
401C2×C8—101no—
402C4⋊C4—152no—
403C4⋊C4—302no—
404C4⋊C4—304no—
405C22⋊C4—302no—
406C22⋊C4—302no—
407C22⋊C4—304no—
408C22⋊C4—301no—
409C42—32no—
410C42—61no—
411C42—152no—
412C42—301no—
413C42—301no—
414C3×C5#189711yescharacteristic, complemented, Hall
415C22×C3—22no—
416C22×C3—102no—
417C22×C3—102no—
418C22×C3—102no—
419C22×C3—202no—
420C22×C3—202no—
421C22×C3D5×C22⋊C411yescharacteristic
422C22×C3#402911yescharacteristic
423C22×C3#408311yescharacteristic
424C2×D3—42no—
425C2×D3—41no—
426C3×C4—102no—
427C3×C4#402711yescharacteristic
428C3×C4#403011yescharacteristic
429C3×C4C4×C22⋊D511yescharacteristic
430C3×C4#408411yescharacteristic
431Dic3—42no—
432Dic3—41no—
433Dic3—202no—
434Dic3—202no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5—241no—
438C2×C5#495811yescharacteristic
439C2×C5#513912yes—
440D5—42no—
441C23—52no—
442C23—61no—
443C23—101no—
444C23—101no—
445C23—202no—
446C23D5×D1211yescharacteristic
447Q8—61no—
448Q8—61no—
449D4—61no—
450D4—61no—
451D4—121no—
452D4—121no—
453D4—604no—
454C2×C4—52no—
455C2×C4—61no—
456C2×C4—62no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—602no—
469C2×C4#676511yescharacteristic
470C2×C4C4×D3×D511yescharacteristic
471C2×C4#677711yescharacteristic
472C2×C4#678011yescharacteristic
473C2×C4D5×#6711yescharacteristic
474C2×C4D3×C22⋊D511yescharacteristic
475C8—41no—
476C8—201no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—202no—
480C2×C3#6511012yes—
481C2×C3#6524511yescharacteristic
482D3—81no—
483C5#6996611yescharacteristic, complemented, Sylow
484C22—22no—
485C22—102no—
486C22—102no—
487C22—102no—
488C22—121no—
489C22—122no—
490C22—202no—
491C22—202no—
492C22#9141411yescharacteristic
493C22D5×(C2×D3)⋊C411yescharacteristic
494C22#9147511yescharacteristic
495C4—102no—
496C4—122no—
497C4—121no—
498C4—602no—
499C4—602no—
500C4#9140811yescharacteristic
501C4#9144111yescharacteristic
502C4#9144211yescharacteristic
503C4#9154911yescharacteristic
504C3#1060117611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—202no—
508C2—241no—
509C2#1174822912yes—
510C2#1174890611yescharacteristic
subgroupG/H
centerC2×C4D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754563C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1897
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91357C2semidirect
3. C2×C5×#825C2semidirect
4. C3×#65415C2semidirect
5. C2×#90978C2semidirect
6. #11746376C2semidirect
7. #11745976C2semidirect
20. #91449C4semidirect
22. #91359C4semidirect
33. #65415D3semidirect
63. C4×Dic15D4semidirect
65. C4×C5×Dic3D4semidirect
78. C2×#825D5semidirect
258. C4×Dic3C22⋊D5semidirect
414. C3×C5#1897semidirect
483. C5#69966semidirect
504. C3#10601176semidirect

Maximal subgroups

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

subgroupindex
#117545632
C2×#913572
C2×C5×#8252
C3×#654152
C2×#909782
#117463762
#117459762
#106011763
#699665