Finite Group Databasebeta

← #12108784

Subgroups — #12108784

Order 1280.

1302 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic
2C2×#64974C211yesmaximal, characteristic
3#10606049C211yesmaximal, characteristic
4C5×#1990C211yesmaximal, characteristic, complemented
5#10602580C211yesmaximal, characteristic, complemented
6#10600570C211yesmaximal, characteristic, complemented
7#10600375C211yesmaximal, characteristic, complemented
8#10599665C214yesmaximal
9#10599592C214yesmaximal, complemented
10C22×Q8×D5—21no—
11C2×C5×C4⋊Q8C2211yescharacteristic
12C2×#4090—21no—
13D5×C4⋊Q8—24no—
14D5×C4⋊Q8C418yescomplemented
15C2×#4038—21no—
16C2×D5×C4⋊C4—21no—
17C2×D5×C4⋊C4C2212yes—
18C2×#4034—21no—
19C2×#4034C2212yes—
20C2×C42×D5C2211yescharacteristic
21C2×#4015C2211yescharacteristic
22C2×C4×C5×C8C2211yescharacteristic
23C5×#384C2211yescharacteristic
24C2×#3952C2212yes—
25C2×#3946C2212yes—
26#65276—21no—
27#65276C2212yes—
28#65266—21no—
29C4⋊C4×Dic5—21no—
30C4⋊C4×Dic5C2212yes—
31#65224C2211yescharacteristic
32C2×C4×#168C2211yescharacteristic
33#64974—24no—
34C5×#340C2214yes—
35#64771C2214yes—
36#64749C2214yes—
37#64749C2214yes—
38#21834—51nomaximal, Sylow
39C22×C5×Q8—21no—
40C2×Q8×D5—28no—
41C2×Q8×D5—44no—
42C2×Q8×D5—44no—
43C22×C4×D5—22no—
44C22×C4×D5D411yescharacteristic
45C22×C4×D5D411yescharacteristic
46C22×C4×D5C2311yescharacteristic
47C22×Dic10—21no—
48C22×Dic10—22no—
49C22×C5×C8C2312yes—
50C5×C4⋊Q8—22no—
51C5×C4⋊Q8C2×C414yes—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4C2312yes—
56C2×C5×C4⋊C4C2312yes—
57C2×C42×C5C2311yescharacteristic
58#4090—28no—
59C2×#685—22no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5—21no—
62C2×C4⋊Dic5—21no—
63C2×C4⋊Dic5C2312yes—
64C2×C4⋊Dic5C2312yes—
65C2×#683—22no—
66C2×#683—22no—
67C2×C4×Dic5—22no—
68C2×C4×Dic5D411yescharacteristic
69C2×C4×Dic5D411yescharacteristic
70C2×C4×Dic5C2311yescharacteristic
71C22×#168C2312yes—
72#4038—44no—
73D5×C4⋊C4—28no—
74D5×C4⋊C4—24no—
75D5×C4⋊C4C2×C418yes—
76#4034—28no—
77#4034—24no—
78#4034C2×C418yes—
79C42×D5C2×C414yes—
80C42×D5D414yes—
81#4015—22no—
82#4015C2×C414yes—
83C4×C5×C8—22no—
84#3963—22no—
85#3963—22no—
86#3952—24no—
87#3946—24no—
88C4×#168—22no—
89C22×C4⋊Q8—51no—
90C2×#431—51no—
91#2285—51no—
92#1990—51no—
93#1990—51no—
94#1990—51no—
95#1990D511yescharacteristic, complemented
96#1987—54no—
97#1824—54no—
98C23×D5C2×D411yescharacteristic
99C2×C5×Q8—24no—
100C2×C5×Q8—42no—
101C2×C5×Q8—42no—
102C22×C4×C5—22no—
103C22×C4×C5—22no—
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2×D411yescharacteristic
106C22×C4×C5C2411yescharacteristic
107C22×Dic5—22no—
108C22×Dic5—22no—
109C22×Dic5C2×D411yescharacteristic
110C22×Dic5C2×D411yescharacteristic
111C22×Dic5C2×D411yescharacteristic
112Q8×D5—432no—
113C2×C4×D5—22no—
114C2×C4×D5—22no—
115C2×C4×D5—22no—
116C2×C4×D5—22no—
117C2×C4×D5—28no—
118C2×C4×D5—44no—
119C2×C4×D5—44no—
120C2×C4×D5C22⋊C414yes—
121C2×C4×D5C22⋊C414yes—
122C2×C4×D5SD1614yes—
123C2×C4×D5Q1614yes—
124C2×C4×D5C22×C412yes—
125C2×C4×D5C2×D412yes—
126C2×Dic10—24no—
127C2×Dic10—28no—
128C2×Dic10—44no—
129C2×Dic10—44no—
130C2×Dic10—42no—
131C2×Dic10—42no—
132C2×C5×C8—22no—
133C2×C5×C8—22no—
134C2×C5×C8—22no—
135C5×C4⋊C4—24no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—24no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4—42no—
140C5×C4⋊C4C22×C414yes—
141C42×C5C22×C412yes—
142C42×C5C2×D412yes—
143#685—44no—
144C4⋊Dic5—24no—
145C4⋊Dic5—22no—
146C4⋊Dic5—24no—
147C4⋊Dic5—42no—
148C4⋊Dic5—42no—
149C4⋊Dic5C22×C414yes—
150#683—28no—
151#683—28no—
152C4×Dic5—44no—
153C4×Dic5C22⋊C414yes—
154C4×Dic5C22⋊C414yes—
155C4×Dic5C22×C412yes—
156C4×Dic5C2×D412yes—
157C2×#168—22no—
158C2×#168—22no—
159C2×#168—22no—
160C23×Q8—101no—
161C2×C4⋊Q8—58no—
162C2×C4⋊Q8—52no—
163C2×C4⋊Q8—51no—
164C2×C4⋊Q8—101no—
165C2×C4⋊Q8—104no—
166C2×C4⋊Q8—101no—
167C2×C4⋊Q8C2×D511yescharacteristic
168C2×#117—101no—
169C22×C4⋊C4—52no—
170C22×C4⋊C4—101no—
171C22×C42—51no—
172#431—104no—
173C2×C4⋊C8—52no—
174C2×C22⋊C8—52no—
175C2×C4×C8—51no—
176C2×C4×C8C2×D511yescharacteristic
177#384—51no—
178#384C2×D511yescharacteristic
179#382—101no—
180#380—52no—
181#380—101no—
182C4×C4⋊C4—52no—
183C4×C4⋊C4—101no—
184#340—54no—
185#340—54no—
186#340—54no—
187#340C2×D514yes—
188C23×C5C22×D411yescharacteristic
189C22×D5—21no—
190C22×D5—21no—
191C22×D5—21no—
192C22×D5—21no—
193C22×D5C2×C22⋊C412yes—
194C22×D5C2×SD1612yes—
195C22×D5C2×Q1612yes—
196C5×Q8—48no—
197C2×C4×C5—24no—
198C2×C4×C5—42no—
199C2×C4×C5—42no—
200C2×C4×C5—42no—
201C2×C4×C5—42no—
202C2×C4×C5—42no—
203C2×C4×C5C2×C22⋊C412yes—
204C2×C4×C5C2×C22⋊C412yes—
205C2×C4×C5C2×SD1612yes—
206C2×C4×C5C2×Q1612yes—
207C2×C4×C5AGL(1,ℤ/8)12yes—
208C2×C4×C5AGL(1,ℤ/8)12yes—
209C2×C4×C5C23×C411yescharacteristic
210C2×C4×C5C22×D411yescharacteristic
211C2×C4×C52−1+412yes—
212C2×C4×C52−1+412yes—
213C2×Dic5—21no—
214C2×Dic5—21no—
215C2×Dic5—21no—
216C2×Dic5—21no—
217C2×Dic5—24no—
218C2×Dic5—42no—
219C2×Dic5—42no—
220C2×Dic5—42no—
221C2×Dic5—42no—
222C2×Dic5—42no—
223C2×Dic5C2×C22⋊C412yes—
224C2×Dic5C2×C22⋊C412yes—
225C2×Dic5C2×C22⋊C412yes—
226C2×Dic5C2×SD1612yes—
227C2×Dic5C2×Q1612yes—
228C4×D5—24no—
229C4×D5—24no—
230C4×D5—28no—
231C4×D5—416no—
232C4×D5#103116yes—
233Dic10—416no—
234Dic10—48no—
235C5×C8—42no—
236#168—42no—
237C22×Q8—21no—
238C22×Q8—102no—
239C22×Q8—108no—
240C22×Q8—101no—
241C22×Q8—204no—
242C22×Q8—204no—
243C23×C4—51no—
244C23×C4—51no—
245C23×C4—51no—
246C23×C4—102no—
247C22×C8—52no—
248C22×C8C22×D512yes—
249C4⋊Q8—22no—
250C4⋊Q8—58no—
251C4⋊Q8—54no—
252C4⋊Q8—104no—
253C4⋊Q8—108no—
254C4⋊Q8—108no—
255C4⋊Q8—102no—
256C4⋊Q8C4×D514yes—
257#117—204no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—52no—
262C2×C4⋊C4—58no—
263C2×C4⋊C4—52no—
264C2×C4⋊C4—104no—
265C2×C4⋊C4—108no—
266C2×C4⋊C4—102no—
267C2×C4⋊C4—102no—
268C2×C4⋊C4—101no—
269C2×C4⋊C4—101no—
270C2×C4⋊C4—101no—
271C2×C4⋊C4C22×D512yes—
272C2×C4⋊C4C22×D512yes—
273C2×C22⋊C4—102no—
274C2×C42—51no—
275C2×C42—51no—
276C2×C42—51no—
277C2×C42—54no—
278C2×C42—54no—
279C2×C42—102no—
280C2×C42C22×D511yescharacteristic
281C4⋊C8—104no—
282C22⋊C8—104no—
283C4×C8—22no—
284C4×C8—102no—
285#95—102no—
286#95—102no—
287C22×C5C22×C22⋊C411yescharacteristic
288C22×C5#53611yescharacteristic
289C22×C5#53611yescharacteristic
290C22×C5C22×SD1611yescharacteristic
291C22×C5C22×Q1611yescharacteristic
292C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
294C2×D5—22no—
295C2×D5—22no—
296C2×D5—22no—
297C2×D5—22no—
298C2×D5—22no—
299C2×D5—22no—
300C2×D5C2×#10314yes—
301C4×C5—44no—
302C4×C5—82no—
303C4×C5C2×#10314yes—
304C4×C5#41814yes—
305C4×C5#51912yes—
306C4×C5#53612yes—
307Dic5—22no—
308Dic5—22no—
309Dic5—44no—
310Dic5—82no—
311Dic5C2×#10314yes—
312C24—51no—
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—108no—
317C2×Q8—104no—
318C2×Q8—204no—
319C2×Q8—204no—
320C2×Q8—2032no—
321C2×Q8—202no—
322C2×Q8—202no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—51no—
326C22×C4—54no—
327C22×C4—54no—
328C22×C4—54no—
329C22×C4—51no—
330C22×C4—51no—
331C22×C4—52no—
332C22×C4—52no—
333C22×C4—54no—
334C22×C4—108no—
335C22×C4—102no—
336C22×C4—102no—
337C22×C4—102no—
338C22×C4—102no—
339C22×C4—102no—
340C22×C4—102no—
341C22×C4—204no—
342C22×C4—204no—
343C22×C4D4×D511yescharacteristic
344C22×C4D4×D511yescharacteristic
345C22×C4C23×D511yescharacteristic
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—102no—
350C2×C8—102no—
351C2×C8—102no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—54no—
358C4⋊C4—108no—
359C4⋊C4—108no—
360C4⋊C4—102no—
361C4⋊C4—104no—
362C4⋊C4—104no—
363C4⋊C4—202no—
364C4⋊C4—202no—
365C4⋊C4C2×C4×D514yes—
366C22⋊C4—204no—
367C42—52no—
368C42—52no—
369C42—54no—
370C42—54no—
371C42—204no—
372C42C2×C4×D512yes—
373C42D4×D512yes—
374C2×C5#231011yescharacteristic
375C2×C5C22×#10311yescharacteristic
376C2×C5C2×#41811yescharacteristic
377C2×C5#322711yescharacteristic
378C2×C5#322711yescharacteristic
379C2×C5#323011yescharacteristic
380C2×C5#323011yescharacteristic
381D5—24no—
382C23—52no—
383C23—52no—
384C23—52no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23C2×D4×D511yescharacteristic
390Q8—48no—
391Q8—2016no—
392Q8—208no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—52no—
400C2×C4—52no—
401C2×C4—52no—
402C2×C4—516no—
403C2×C4—52no—
404C2×C4—52no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—101no—
408C2×C4—101no—
409C2×C4—108no—
410C2×C4—104no—
411C2×C4—104no—
412C2×C4—104no—
413C2×C4—2016no—
414C2×C4—202no—
415C2×C4—202no—
416C2×C4—202no—
417C2×C4—202no—
418C2×C4—202no—
419C2×C4D5×C22⋊C412yes—
420C2×C4D5×C22⋊C412yes—
421C2×C4#405712yes—
422C2×C4D5×SD1612yes—
423C2×C4#406012yes—
424C2×C4D5×Q1612yes—
425C2×C4C22×C4×D511yescharacteristic
426C2×C4C2×D4×D511yescharacteristic
427C2×C4#414712yes—
428C2×C4#415012yes—
429C8—42no—
430C8—202no—
431C5#2183411yescharacteristic, complemented, Sylow
432C22—54no—
433C22—102no—
434C22—102no—
435C22—102no—
436C22—102no—
437C22—102no—
438C22—102no—
439C22C2×D5×C22⋊C411yescharacteristic
440C22#6600911yescharacteristic
441C22#6605811yescharacteristic
442C22C2×#405711yescharacteristic
443C22C2×D5×SD1611yescharacteristic
444C22C2×#406011yescharacteristic
445C22C2×D5×Q1611yescharacteristic
446C4—44no—
447C4—54no—
448C4—82no—
449C4—102no—
450C4—102no—
451C4—204no—
452C4—402no—
453C4#6505914yes—
454C4D5×#10314yes—
455C4#6590612yes—
456C4#6596212yes—
457C2—104no—
458C2#1060834111yescharacteristic
459C2C2×#6505911yescharacteristic
460C2C2×D5×#10311yescharacteristic
461C2#1061330311yescharacteristic
462C2#1061333911yescharacteristic
463C2#1061362511yescharacteristic
464C2#1061367411yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1990C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1990D5
hypocenterC5#21834
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C5×#1990C2semidirect
5. #10602580C2semidirect
6. #10600570C2semidirect
7. #10600375C2semidirect
9. #10599592C2semidirect
14. D5×C4⋊Q8C4semidirect
95. #1990D5semidirect
431. C5#21834semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#649742
#106060492
C5×#19902
#106025802
#106005702
#106003752
#105996652
#105995922
#218345