Finite Group Databasebeta

← #14313202

Subgroups — #14313202

Order 1920.

560 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
1C3×C5×#534C211yesmaximal, characteristic
2C2×#91118C211yesmaximal, characteristic, complemented
3C2×#91117C211yesmaximal, characteristic, complemented
4#11746360C212yesmaximal, complemented
5#11746357C212yesmaximal, complemented
6#10602640—31nomaximal, Hall
7C22×C3×C5×D4C411yescharacteristic, complemented
8C22×C3×C5×D4C2211yescharacteristic
9C3×C5×#123—21no—
10C3×C5×#123C411yescharacteristic, complemented
11C3×C5×#123C2211yescharacteristic
12C3×C5×#121—21no—
13C3×C5×#121—21no—
14C3×C5×#120—21no—
15C3×C5×#120C411yescharacteristic, complemented
16C3×C5×#120C2211yescharacteristic
17C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
18C2×#6719—21no—
19C2×#6710—21no—
20#91118—21no—
21#91118C2212yes—
22#91117—21no—
23#91117C2212yes—
24#71847—51nomaximal, Hall
25C5×#534D311yescharacteristic
26C2×#3966—31no—
27C2×#3965—31no—
28#64759—32no—
29#64756—32no—
30C24×C3×C5C2×C411yescharacteristic
31C24×C3×C5C2×C411yescharacteristic, complemented
32C24×C3×C5D411yescharacteristic
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—41no—
38C2×C3×C5×D4—41no—
39C2×C3×C5×D4—41no—
40C2×C3×C5×D4—41no—
41C2×C3×C5×D4C2×C411yescharacteristic
42C2×C3×C5×D4D411yescharacteristic
43C2×C3×C5×D4D411yescharacteristic
44C2×C3×C5×D4C2311yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5C2×C411yescharacteristic, complemented
47C22×C3×C4×C5D411yescharacteristic
48C22×Dic15—41no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
57C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
58#6719—22no—
59#6719—41no—
60#6710—22no—
61#6710—41no—
62C2×C3⋊#168—41no—
63C3×#534D511yescharacteristic
64C2×#834—51no—
65C2×#833—51no—
66#4539—52no—
67#4536—52no—
68C22×C5×D4Dic311yescharacteristic, complemented
69C22×C5×D4C2×D311yescharacteristic
70C5×#123—21no—
71C5×#123Dic311yescharacteristic, complemented
72C5×#123C2×D311yescharacteristic
73C5×#121—21no—
74C5×#121—21no—
75C5×#120—21no—
76C5×#120Dic311yescharacteristic, complemented
77C5×#120C2×D311yescharacteristic
78C2×C5×C22⋊C4Dic311yescharacteristic, complemented
79C2×C22⋊Dic5—61no—
80C2×#681—61no—
81#3966—32no—
82#3966—61no—
83#3965—32no—
84#3965—61no—
85#2121—151noSylow
86C23×C3×C5—21no—
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5—21no—
90C23×C3×C5—21no—
91C23×C3×C5—41no—
92C23×C3×C5—81no—
93C23×C3×C5—81no—
94C23×C3×C5—81no—
95C23×C3×C5C22⋊C411yescharacteristic
96C23×C3×C5C22⋊C411yescharacteristic
97C23×C3×C5C22⋊C411yescharacteristic
98C23×C3×C5C22×C411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C3×C5×D4—41no—
101C3×C5×D4—41no—
102C3×C5×D4—41no—
103C3×C5×D4—41no—
104C3×C5×D4—42no—
105C3×C5×D4—41no—
106C3×C5×D4—41no—
107C3×C5×D4—81no—
108C3×C5×D4—81no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—81no—
116C2×C3×C4×C5C22⋊C411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×Dic15—42no—
119C2×Dic15—82no—
120C3⋊#168—42no—
121C22×C3×D4Dic511yescharacteristic, complemented
122C22×C3×D4C2×D511yescharacteristic
123C3×#123—21no—
124C3×#123Dic511yescharacteristic, complemented
125C3×#123C2×D511yescharacteristic
126C3×#121—21no—
127C3×#121—21no—
128C3×#120—21no—
129C3×#120Dic511yescharacteristic, complemented
130C3×#120C2×D511yescharacteristic
131C2×C3×C22⋊C4Dic511yescharacteristic, complemented
132C2×#217—101no—
133C2×#208—101no—
134#834—52no—
135#834—101no—
136#833—52no—
137#833—101no—
138C24×C5C2×Dic311yescharacteristic
139C24×C5C2×Dic311yescharacteristic, complemented
140C24×C5#6711yescharacteristic
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—41no—
146C2×C5×D4—41no—
147C2×C5×D4—41no—
148C2×C5×D4—41no—
149C2×C5×D4C2×Dic311yescharacteristic
150C2×C5×D4#6711yescharacteristic
151C2×C5×D4#6711yescharacteristic
152C2×C5×D4C22×D311yescharacteristic
153C22×C4×C5—21no—
154C22×C4×C5C2×Dic311yescharacteristic, complemented
155C22×C4×C5#6711yescharacteristic
156C22×Dic5—121no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4C2×Dic311yescharacteristic, complemented
165C5×C22⋊C4C2×Dic311yescharacteristic, complemented
166C22⋊Dic5—62no—
167C22⋊Dic5—121no—
168#681—62no—
169#681—121no—
170C2×#168—121no—
171#534D1511yescharacteristic
172C2×#100—151no—
173C2×#99—151no—
174#352—152no—
175AΣL(1,𝔽16)—152no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—21no—
179C22×C3×C5—21no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5—42no—
186C22×C3×C5—41no—
187C22×C3×C5—41no—
188C22×C3×C5—41no—
189C22×C3×C5—81no—
190C22×C3×C5—81no—
191C22×C3×C5—81no—
192C22×C3×C5—82no—
193C22×C3×C5#9911yescharacteristic
194C22×C3×C5#9911yescharacteristic
195C22×C3×C5C2×C22⋊C411yescharacteristic
196C3×C4×C5—21no—
197C3×C4×C5—21no—
198C3×C4×C5—41no—
199C3×C4×C5—41no—
200C3×C4×C5—41no—
201Dic15—82no—
202C24×C3C2×Dic511yescharacteristic
203C24×C3C2×Dic511yescharacteristic, complemented
204C24×C3C22⋊D511yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—41no—
210C2×C3×D4—41no—
211C2×C3×D4—41no—
212C2×C3×D4—41no—
213C2×C3×D4C2×Dic511yescharacteristic
214C2×C3×D4C22⋊D511yescharacteristic
215C2×C3×D4C22⋊D511yescharacteristic
216C2×C3×D4C22×D511yescharacteristic
217C22×C3×C4—21no—
218C22×C3×C4C2×Dic511yescharacteristic, complemented
219C22×C3×C4C22⋊D511yescharacteristic
220C22×Dic3—201no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C22⋊C4—21no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4C2×Dic511yescharacteristic, complemented
229C3×C22⋊C4C2×Dic511yescharacteristic, complemented
230#217—102no—
231#217—201no—
232#208—102no—
233#208—201no—
234C2×C3⋊C8—201no—
235C23×C5—21no—
236C23×C5—21no—
237C23×C5—21no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—41no—
241C23×C5—81no—
242C23×C5—81no—
243C23×C5—81no—
244C23×C5#21711yescharacteristic
245C23×C5#21711yescharacteristic
246C23×C5#21711yescharacteristic
247C23×C5C22×Dic311yescharacteristic
248C23×C5C2×#6711yescharacteristic
249C5×D4—41no—
250C5×D4—41no—
251C5×D4—41no—
252C5×D4—41no—
253C5×D4—42no—
254C5×D4—41no—
255C5×D4—41no—
256C5×D4—81no—
257C5×D4—81no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—41no—
264C2×C4×C5—81no—
265C2×C4×C5#21711yescharacteristic
266C2×C4×C5C2×#6711yescharacteristic
267C2×Dic5—122no—
268C2×Dic5—242no—
269#168—122no—
270C22×D4Dic1511yescharacteristic, complemented
271C22×D4C2×D1511yescharacteristic
272C2×M16—301no—
273#123—21no—
274#123Dic1511yescharacteristic, complemented
275#123C2×D1511yescharacteristic
276#121—21no—
277#121—21no—
278#120—21no—
279#120Dic1511yescharacteristic, complemented
280#120C2×D1511yescharacteristic
281C2×C22⋊C4—301no—
282C2×C22⋊C4Dic1511yescharacteristic, complemented
283#100—152no—
284#100—301no—
285#99—152no—
286#99—301no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—41no—
291C2×C3×C5—41no—
292C2×C3×C5—41no—
293C2×C3×C5—41no—
294C2×C3×C5—81no—
295C2×C3×C5C2×#9911yescharacteristic
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—41no—
302C23×C3—81no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3C22⋊Dic511yescharacteristic
306C23×C3C22⋊Dic511yescharacteristic
307C23×C3C22⋊Dic511yescharacteristic
308C23×C3C22×Dic511yescharacteristic
309C23×C3C2×C22⋊D511yescharacteristic
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—81no—
318C3×D4—81no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—81no—
326C2×C3×C4C22⋊Dic511yescharacteristic
327C2×C3×C4C2×C22⋊D511yescharacteristic
328C2×Dic3—202no—
329C2×Dic3—402no—
330C3⋊C8—202no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—21no—
334C22×C5—21no—
335C22×C5—41no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—41no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5—41no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—81no—
345C22×C5—81no—
346C22×C5—81no—
347C22×C5—82no—
348C22×C5#83411yescharacteristic
349C22×C5#83411yescharacteristic
350C22×C5C2×#21711yescharacteristic
351C4×C5—21no—
352C4×C5—21no—
353C4×C5—41no—
354C4×C5—41no—
355C4×C5—41no—
356Dic5—242no—
357C24C2×Dic1511yescharacteristic
358C24C2×Dic1511yescharacteristic, complemented
359C24#122011yescharacteristic
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4C2×Dic1511yescharacteristic
369C2×D4#122011yescharacteristic
370C2×D4#122011yescharacteristic
371C2×D4C22×D1511yescharacteristic
372C22×C4—21no—
373C22×C4—601no—
374C22×C4C2×Dic1511yescharacteristic, complemented
375C22×C4#122011yescharacteristic
376M16—302no—
377M16—601no—
378C2×C8—601no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—302no—
387C22⋊C4—601no—
388C22⋊C4C2×Dic1511yescharacteristic, complemented
389C22⋊C4C2×Dic1511yescharacteristic, complemented
390C3×C5#212111yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—82no—
408C22×C3#396611yescharacteristic
409C22×C3#396611yescharacteristic
410C22×C3C2×C22⋊Dic511yescharacteristic
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—41no—
416Dic3—402no—
417C2×C5—21no—
418C2×C5—21no—
419C2×C5—21no—
420C2×C5—41no—
421C2×C5—41no—
422C2×C5—41no—
423C2×C5—41no—
424C2×C5—81no—
425C2×C5C2×#83411yescharacteristic
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23#671911yescharacteristic
436C23#671911yescharacteristic
437C23#671911yescharacteristic
438C23C22×Dic1511yescharacteristic
439C23C2×#122011yescharacteristic
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—602no—
457C2×C4—1202no—
458C2×C4#671911yescharacteristic
459C2×C4C2×#122011yescharacteristic
460C8—602no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—81no—
469C2×C3C2×#396611yescharacteristic
470C5#7184711yescharacteristic, complemented, Sylow
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22#9111811yescharacteristic
489C22#9111811yescharacteristic
490C22C2×#671911yescharacteristic
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—1202no—
497C3#1060264011yescharacteristic, complemented, Sylow
498C2—21no—
499C2—21no—
500C2—21no—
501C2—41no—
502C2—41no—
503C2—41no—
504C2—41no—
505C2—81no—
506C2C2×#9111811yescharacteristic
subgroupG/H
centerC2C2×#91118
commutatorC23×C3×C5C22×C4
FrattiniC2×D4C22×D15
Fitting, prime radicalC3×C5×#534C2
socleC2×C3×C5C2×#99
cosocleC2×C3×C5×D4C23
hypercenter#534D15
hypocenterC3×C5#2121
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #534

Complements

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

NH ≅ G/Ntype
2. C2×#91118C2semidirect
3. C2×#91117C2semidirect
4. #11746360C2semidirect
5. #11746357C2semidirect
7. C22×C3×C5×D4C4semidirect
10. C3×C5×#123C4semidirect
15. C3×C5×#120C4semidirect
17. C2×C3×C5×C22⋊C4C4semidirect
31. C24×C3×C5C2×C4semidirect
46. C22×C3×C4×C5C2×C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
57. C3×C5×C22⋊C4C2×C4semidirect
68. C22×C5×D4Dic3semidirect
71. C5×#123Dic3semidirect
76. C5×#120Dic3semidirect
78. C2×C5×C22⋊C4Dic3semidirect
121. C22×C3×D4Dic5semidirect
124. C3×#123Dic5semidirect
129. C3×#120Dic5semidirect
131. C2×C3×C22⋊C4Dic5semidirect
139. C24×C5C2×Dic3semidirect
154. C22×C4×C5C2×Dic3semidirect
164. C5×C22⋊C4C2×Dic3semidirect
165. C5×C22⋊C4C2×Dic3semidirect
203. C24×C3C2×Dic5semidirect
218. C22×C3×C4C2×Dic5semidirect
228. C3×C22⋊C4C2×Dic5semidirect
229. C3×C22⋊C4C2×Dic5semidirect
270. C22×D4Dic15semidirect
274. #123Dic15semidirect
279. #120Dic15semidirect
282. C2×C22⋊C4Dic15semidirect
358. C24C2×Dic15semidirect
374. C22×C4C2×Dic15semidirect
388. C22⋊C4C2×Dic15semidirect
389. C22⋊C4C2×Dic15semidirect
390. C3×C5#2121semidirect
470. C5#71847semidirect
497. C3#10602640semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5342
C2×#911182
C2×#911172
#117463602
#117463572
#106026403
#718475