Finite Group Databasebeta

← #12099108

Subgroups — #12099108

Order 1280.

1542 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4038C211yesmaximal, characteristic
2C2×#65015C211yesmaximal, characteristic
3D5×#380C211yesmaximal, characteristic
4C5×#1824C211yesmaximal, characteristic, complemented
5#10602347C211yesmaximal, characteristic, complemented
6#10600717C211yesmaximal, characteristic, complemented
7#10600294C211yesmaximal, characteristic, complemented
8#10599666C214yesmaximal, complemented
9#10599660C214yesmaximal, complemented
10C23×C4×D5C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12C22×#685—21no—
13C22×C4⋊Dic5C2211yescharacteristic
14C22×C4×Dic5—21no—
15C22×#676C2212yes—
16C2×#4038—21no—
17C2×#4038—21no—
18C2×#4038—24no—
19C2×#4038C418yescomplemented
20C2×#4038C2212yes—
21C2×#4038C2212yes—
22C2×D5×C4⋊C4C2212yes—
23C2×D5×C22⋊C4—21no—
24C2×C5×C22⋊C8C2211yescharacteristic
25C5×#380C2211yescharacteristic
26#65501C2211yescharacteristic
27C2×C22⋊#168C2211yescharacteristic
28C2×#3952C2212yes—
29#65276C2212yes—
30#65231—21no—
31#65015—24no—
32#64956—21no—
33D5×#95—21no—
34C5×#340C2214yescomplemented
35#64771C2214yescomplemented
36#64749C2214yescomplemented
37#64749C2214yescomplemented
38#20702—51nomaximal, Sylow
39C24×D5D411yescharacteristic
40C23×C4×C5—21no—
41C23×C4×C5C2311yescharacteristic
42C23×Dic5—21no—
43C23×Dic5D411yescharacteristic
44C22×C4×D5—28no—
45C22×C4×D5—22no—
46C22×C4×D5C2×C414yes—
47C22×C4×D5D411yescharacteristic
48C22×C4×D5D411yescharacteristic
49C22×C4×D5D414yes—
50C22×C4×D5C2311yescharacteristic
51C22×C4×D5C2311yescharacteristic
52C22×C5×C8C2312yes—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—22no—
55C2×C5×C4⋊C4C2×C414yescomplemented
56C2×C5×C4⋊C4C2312yes—
57C2×C5×C4⋊C4C2312yes—
58C2×C5×C22⋊C4—21no—
59C2×C22⋊Dic5—21no—
60C2×#685—28no—
61C2×#685—22no—
62C2×#685—22no—
63C2×#685—22no—
64C2×C4⋊Dic5—21no—
65C2×C4⋊Dic5—22no—
66C2×C4⋊Dic5C2×C414yescomplemented
67C2×C4⋊Dic5C2312yes—
68C2×C4⋊Dic5C2312yes—
69C2×#683—22no—
70C2×C4×Dic5—28no—
71C2×C4×Dic5—22no—
72C2×C4×Dic5—22no—
73C22×#168C2312yes—
74C2×#676—24no—
75C2×#676—28no—
76#4038—28no—
77#4038—28no—
78#4038—24no—
79#4038—24no—
80#4038C2×C418yes—
81#4038C2×C418yescomplemented
82D5×C4⋊C4—28no—
83D5×C22⋊C4—44no—
84C5×C22⋊C8—22no—
85C5×#95—21no—
86#3963—21no—
87#3963—22no—
88C22⋊#168—22no—
89#3952—24no—
90C22×#117—51no—
91C2×#407—51no—
92C2×#380—51no—
93#1825—54no—
94#1824—51no—
95#1824D511yescharacteristic, complemented
96#1823—51no—
97#1823—51no—
98#1736—54no—
99C24×C5C2×D411yescharacteristic
100C23×D5—22no—
101C23×D5—22no—
102C23×D5—24no—
103C23×D5—42no—
104C23×D5C22⋊C414yes—
105C23×D5C2×D411yescharacteristic
106C23×D5C2×D411yescharacteristic
107C22×C4×C5—24no—
108C22×C4×C5—24no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—44no—
112C22×C4×C5C22×C412yes—
113C22×C4×C5C2×D411yescharacteristic
114C22×C4×C5C2×D412yes—
115C22×C4×C5C2411yescharacteristic
116C22×Dic5—24no—
117C22×Dic5—24no—
118C22×Dic5—22no—
119C22×Dic5—22no—
120C22×Dic5—42no—
121C22×Dic5—42no—
122C22×Dic5C22⋊C414yes—
123C22×Dic5C2×D411yescharacteristic
124C22×Dic5C2×D411yescharacteristic
125C2×C4×D5—24no—
126C2×C4×D5—24no—
127C2×C4×D5—24no—
128C2×C4×D5—216no—
129C2×C4×D5—216no—
130C2×C4×D5—44no—
131C2×C4×D5—44no—
132C2×C4×D5—44no—
133C2×C4×D5C22⋊C414yes—
134C2×C4×D5C22⋊C414yes—
135C2×C4×D5C22×C412yes—
136C2×C4×D5C22×C412yes—
137C2×C4×D5C2×D412yes—
138C2×C4×D5C2×D412yes—
139C2×C4×D5#4118yes—
140C2×C5×C8—22no—
141C2×C5×C8—24no—
142C5×C4⋊C4—24no—
143C5×C4⋊C4—22no—
144C5×C4⋊C4—24no—
145C5×C4⋊C4C22×C414yes—
146C5×C22⋊C4—42no—
147C22⋊Dic5—42no—
148#685—28no—
149#685—28no—
150#685—44no—
151C4⋊Dic5—24no—
152C4⋊Dic5—22no—
153C4⋊Dic5—24no—
154C4⋊Dic5C22×C414yes—
155#683—44no—
156C4×Dic5—28no—
157C4×Dic5—28no—
158C2×#168—22no—
159C2×#168—24no—
160#676—48no—
161C24×C4—51no—
162C22×M16—52no—
163C2×#117—58no—
164C2×#117—52no—
165C2×#117—52no—
166C2×#117—101no—
167C2×#117—101no—
168C2×#117—104no—
169C22×C4⋊C4—51no—
170C22×C4⋊C4—52no—
171C22×C4⋊C4C2×D511yescharacteristic
172C22×C22⋊C4—101no—
173C22×C22⋊C4—101no—
174C22×C42—101no—
175#407—104no—
176C2×C22⋊C8—51no—
177C2×C22⋊C8—52no—
178C2×C22⋊C8C2×D511yescharacteristic
179#380—51no—
180#380—52no—
181#380—101no—
182#380—101no—
183#380C2×D511yescharacteristic
184C2×#95—101no—
185#340—54no—
186#340—54no—
187#340—54no—
188#340C2×D514yescomplemented
189C23×C5—22no—
190C23×C5—22no—
191C23×C5C2×C22⋊C412yes—
192C23×C5AGL(1,ℤ/8)12yes—
193C23×C5#13712yes—
194C23×C5C22×D411yescharacteristic
195C22×D5—21no—
196C22×D5—21no—
197C22×D5—22no—
198C22×D5—22no—
199C22×D5—21no—
200C22×D5—21no—
201C22×D5—22no—
202C22×D5—28no—
203C22×D5—28no—
204C22×D5—44no—
205C22×D5—48no—
206C22×D5—48no—
207C22×D5C2×C22⋊C412yes—
208C22×D5C2×C22⋊C412yes—
209C22×D5#12314yes—
210C2×C4×C5—22no—
211C2×C4×C5—24no—
212C2×C4×C5—24no—
213C2×C4×C5—24no—
214C2×C4×C5—48no—
215C2×C4×C5—44no—
216C2×C4×C5—42no—
217C2×C4×C5—42no—
218C2×C4×C5—42no—
219C2×C4×C5C2×C22⋊C412yes—
220C2×C4×C5C23×C411yescharacteristic
221C2×C4×C5C22×D411yescharacteristic
222C2×C4×C5C2×#4114yes—
223C2×Dic5—22no—
224C2×Dic5—28no—
225C2×Dic5—24no—
226C2×Dic5—42no—
227C2×Dic5—42no—
228C2×Dic5—48no—
229C2×Dic5—42no—
230C2×Dic5—42no—
231C2×Dic5—42no—
232C2×Dic5C2×C22⋊C412yes—
233C2×Dic5C2×C22⋊C412yes—
234C2×Dic5#12314yes—
235C4×D5—28no—
236C4×D5—24no—
237C4×D5—24no—
238C4×D5—28no—
239C4×D5—88no—
240C4×D5#11718yes—
241C4×D5#12318yes—
242C5×C8—42no—
243#168—42no—
244C25—51no—
245C23×C4—21no—
246C23×C4—51no—
247C23×C4—51no—
248C23×C4—54no—
249C23×C4—51no—
250C23×C4—51no—
251C23×C4—54no—
252C23×C4—51no—
253C23×C4—108no—
254C23×C4—102no—
255C23×C4—101no—
256C23×C4C22×D511yescharacteristic
257C2×M16—104no—
258C2×M16—108no—
259C22×C8—52no—
260C22×C8C22×D512yes—
261#117—58no—
262#117—58no—
263#117—104no—
264#117—104no—
265#117—108no—
266#117—108no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—54no—
270C2×C4⋊C4—52no—
271C2×C4⋊C4—52no—
272C2×C4⋊C4—108no—
273C2×C4⋊C4—102no—
274C2×C4⋊C4—101no—
275C2×C4⋊C4—102no—
276C2×C4⋊C4C4×D514yescomplemented
277C2×C4⋊C4C22×D512yes—
278C2×C4⋊C4C22×D512yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—102no—
281C2×C22⋊C4—102no—
282C2×C22⋊C4—108no—
283C2×C22⋊C4—102no—
284C2×C22⋊C4—101no—
285C2×C22⋊C4—204no—
286C2×C42—102no—
287C2×C42—102no—
288C2×C42—108no—
289C22⋊C8—22no—
290C22⋊C8—104no—
291C22⋊C8—102no—
292#95—21no—
293#95—102no—
294#95—101no—
295C22×C5—24no—
296C22×C5—24no—
297C22×C5—22no—
298C22×C5—22no—
299C22×C5#41714yes—
300C22×C5C22×C22⋊C411yescharacteristic
301C22×C5C2×#12312yes—
302C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C5C2×#13711yescharacteristic
304C22×C5#57512yes—
305C2×D5—24no—
306C2×D5—24no—
307C2×D5—22no—
308C2×D5—22no—
309C2×D5—22no—
310C2×D5—22no—
311C2×D5—22no—
312C2×D5—416no—
313C2×D5—42no—
314C2×D5—42no—
315C2×D5#38114yes—
316C4×C5—22no—
317C4×C5—44no—
318C4×C5—82no—
319C4×C5C2×#11712yes—
320C4×C5C2×#12312yes—
321Dic5—22no—
322Dic5—44no—
323Dic5—82no—
324Dic5#38114yes—
325C24—51no—
326C24—51no—
327C24—54no—
328C24—104no—
329C24—102no—
330C24—102no—
331C24—202no—
332C24D4×D511yescharacteristic
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—44no—
338C22×C4—51no—
339C22×C4—52no—
340C22×C4—52no—
341C22×C4—58no—
342C22×C4—51no—
343C22×C4—54no—
344C22×C4—52no—
345C22×C4—52no—
346C22×C4—54no—
347C22×C4—54no—
348C22×C4—104no—
349C22×C4—1016no—
350C22×C4—1016no—
351C22×C4—104no—
352C22×C4—104no—
353C22×C4—104no—
354C22×C4—102no—
355C22×C4—104no—
356C22×C4—102no—
357C22×C4—204no—
358C22×C4—204no—
359C22×C4—204no—
360C22×C4—202no—
361C22×C4—202no—
362C22×C4C2×C4×D512yes—
363C22×C4D4×D512yes—
364C22×C4D4×D511yescharacteristic
365C22×C4C23×D511yescharacteristic
366M16—208no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—102no—
370C2×C8—104no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—54no—
375C4⋊C4—102no—
376C4⋊C4—104no—
377C4⋊C4—104no—
378C4⋊C4—204no—
379C4⋊C4C2×C4×D514yes—
380C22⋊C4—42no—
381C22⋊C4—108no—
382C22⋊C4—108no—
383C22⋊C4—204no—
384C22⋊C4—202no—
385C42—108no—
386C42—108no—
387C2×C5—24no—
388C2×C5C2×#38111yescharacteristic
389C2×C5C2×#41711yescharacteristic
390C2×C5#289611yescharacteristic
391C2×C5#309412yes—
392C2×C5#309512yes—
393D5—24no—
394D5—42no—
395C23—22no—
396C23—22no—
397C23—52no—
398C23—54no—
399C23—52no—
400C23—108no—
401C23—101no—
402C23—101no—
403C23—102no—
404C23—101no—
405C23—101no—
406C23—102no—
407C23—108no—
408C23—102no—
409C23—204no—
410C23—208no—
411C23—208no—
412C23D5×C22⋊C412yes—
413C23#406012yes—
414C23#406412yes—
415C23C2×D4×D511yescharacteristic
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—52no—
426C2×C4—54no—
427C2×C4—58no—
428C2×C4—58no—
429C2×C4—52no—
430C2×C4—108no—
431C2×C4—108no—
432C2×C4—108no—
433C2×C4—102no—
434C2×C4—104no—
435C2×C4—104no—
436C2×C4—104no—
437C2×C4—202no—
438C2×C4—202no—
439C2×C4—208no—
440C2×C4—202no—
441C2×C4—202no—
442C2×C4—202no—
443C2×C4—408no—
444C2×C4D5×C22⋊C412yes—
445C2×C4C22×C4×D511yescharacteristic
446C2×C4C2×D4×D511yescharacteristic
447C2×C4D5×#4114yes—
448C8—42no—
449C8—202no—
450C5#2070211yescharacteristic, complemented, Sylow
451C22—22no—
452C22—24no—
453C22—24no—
454C22—22no—
455C22—54no—
456C22—104no—
457C22—102no—
458C22—104no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—102no—
463C22—2016no—
464C22—202no—
465C22—202no—
466C22#6509214yes—
467C22C2×D5×C22⋊C411yescharacteristic
468C22D5×#12312yes—
469C22C2×#406011yescharacteristic
470C22C2×#406411yescharacteristic
471C22#6610712yes—
472C4—22no—
473C4—44no—
474C4—54no—
475C4—82no—
476C4—102no—
477C4—204no—
478C4—402no—
479C4D5×#11712yes—
480C4D5×#12312yes—
481C2—24no—
482C2—104no—
483C2—202no—
484C2D5×#38111yescharacteristic
485C2C2×#6509211yescharacteristic
486C2#1061065511yescharacteristic
487C2#1061305712yes—
488C2#1061320012yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1824C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenter#1824D5
hypocenterC5#20702
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. C5×#1824C2semidirect
5. #10602347C2semidirect
6. #10600717C2semidirect
7. #10600294C2semidirect
8. #10599666C2semidirect
9. #10599660C2semidirect
19. C2×#4038C4semidirect
34. C5×#340C22semidirect
35. #64771C22semidirect
36. #64749C22semidirect
37. #64749C22semidirect
55. C2×C5×C4⋊C4C2×C4semidirect
66. C2×C4⋊Dic5C2×C4semidirect
81. #4038C2×C4semidirect
95. #1824D5semidirect
188. #340C2×D5semidirect
276. C2×C4⋊C4C4×D5semidirect
450. C5#20702semidirect

Maximal subgroups

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

subgroupindex
C22×#40382
C2×#650152
D5×#3802
C5×#18242
#106023472
#106007172
#106002942
#105996662
#105996602
#207025