Finite Group Databasebeta

← #12108792

Subgroups — #12108792

Order 1280.

1074 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#64974C211yesmaximal, characteristic
3#10606049C211yesmaximal, characteristic
4C5×#1990C211yesmaximal, characteristic, complemented
5#10602580C211yesmaximal, characteristic, complemented
6#10600570C211yesmaximal, characteristic, complemented
7#10600375C211yesmaximal, characteristic, complemented
8#10599606C214yesmaximal
9#10599594C214yesmaximal, complemented
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×#4092—21no—
12C2×Q8×Dic5—21no—
13#66062—24no—
14#66062C418yescomplemented
15C2×#4043—21no—
16C2×#4038—21no—
17C2×#4038C2212yes—
18C2×#4036C2212yes—
19C2×#4033—21no—
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#21835—51nomaximal, Sylow
39C22×C5×Q8—21no—
40C22×C4×D5D411yescharacteristic
41C22×C4×D5D411yescharacteristic
42C22×C4×D5C2311yescharacteristic
43C22×Dic10—21no—
44C22×C5×C8C2312yes—
45C5×C4⋊Q8—22no—
46C5×C4⋊Q8C2×C414yes—
47C2×C5×C4⋊C4—21no—
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4C2312yes—
51C2×C5×C4⋊C4C2312yes—
52C2×C42×C5C2311yescharacteristic
53#4092—28no—
54Q8×Dic5—28no—
55C2×#685—22no—
56C2×#685—22no—
57C2×C4⋊Dic5—21no—
58C2×C4⋊Dic5—22no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5C2312yes—
62C2×C4⋊Dic5C2312yes—
63C2×#683—22no—
64C2×C4×Dic5—22no—
65C2×C4×Dic5—22no—
66C2×C4×Dic5D411yescharacteristic
67C2×C4×Dic5D411yescharacteristic
68C2×C4×Dic5C2311yescharacteristic
69C22×#168C2312yes—
70#4043—28no—
71#4038—24no—
72#4038—44no—
73#4038C2×C418yes—
74#4036—24no—
75#4036C2×C418yes—
76#4033—28no—
77C42×D5C2×C414yes—
78C42×D5D414yes—
79#4015—22no—
80#4015C2×C414yes—
81C4×C5×C8—22no—
82#3963—22no—
83#3963—22no—
84#3952—24no—
85#3946—24no—
86C4×#168—22no—
87C2×#533—51no—
88C2×#431—51no—
89#2285—51no—
90#1990—51no—
91#1990—51no—
92#1990—51no—
93#1990D511yescharacteristic, complemented
94#1989—54no—
95#1825—54no—
96C23×D5C2×D411yescharacteristic
97C2×C5×Q8—24no—
98C2×C5×Q8—42no—
99C2×C5×Q8—42no—
100C22×C4×C5—22no—
101C22×C4×C5—22no—
102C22×C4×C5C2×D411yescharacteristic
103C22×C4×C5C2×D411yescharacteristic
104C22×C4×C5C2411yescharacteristic
105C22×Dic5—22no—
106C22×Dic5—22no—
107C22×Dic5C2×D411yescharacteristic
108C22×Dic5C2×D411yescharacteristic
109C22×Dic5C2×D411yescharacteristic
110C2×C4×D5—22no—
111C2×C4×D5—22no—
112C2×C4×D5—22no—
113C2×C4×D5—22no—
114C2×C4×D5—22no—
115C2×C4×D5—22no—
116C2×C4×D5C22⋊C414yes—
117C2×C4×D5C22⋊C414yes—
118C2×C4×D5C22×C412yes—
119C2×C4×D5C2×D412yes—
120C2×Dic10—24no—
121C2×Dic10—42no—
122C2×Dic10—42no—
123C2×C5×C8—22no—
124C2×C5×C8—22no—
125C2×C5×C8—22no—
126C5×C4⋊C4—24no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4—24no—
129C5×C4⋊C4—42no—
130C5×C4⋊C4—42no—
131C5×C4⋊C4C22×C414yes—
132C42×C5C22×C412yes—
133C42×C5C2×D412yes—
134#685—28no—
135#685—44no—
136C4⋊Dic5—24no—
137C4⋊Dic5—22no—
138C4⋊Dic5—28no—
139C4⋊Dic5—24no—
140C4⋊Dic5—42no—
141C4⋊Dic5—42no—
142C4⋊Dic5C22×C414yes—
143#683—28no—
144C4×Dic5—28no—
145C4×Dic5—44no—
146C4×Dic5C22⋊C414yes—
147C4×Dic5C22⋊C414yes—
148C4×Dic5C22×C412yes—
149C4×Dic5C2×D412yes—
150C2×#168—22no—
151C2×#168—22no—
152C2×#168—22no—
153#533—58no—
154#533—104no—
155C2×C4⋊Q8—51no—
156C2×C4⋊Q8C2×D511yescharacteristic
157C2×#125—52no—
158C2×C22⋊Q8—101no—
159C2×C22⋊Q8—101no—
160C2×C4×Q8—101no—
161C2×C4×Q8—101no—
162C2×#117—52no—
163C2×#117—101no—
164C22×C42—51no—
165#431—104no—
166C2×C4⋊C8—52no—
167C2×C22⋊C8—52no—
168C2×C4×C8—51no—
169C2×C4×C8C2×D511yescharacteristic
170#384—51no—
171#384C2×D511yescharacteristic
172#382—101no—
173#380—52no—
174#380—101no—
175C4×C4⋊C4—52no—
176C4×C4⋊C4—101no—
177#340—54no—
178#340—54no—
179#340—54no—
180#340C2×D514yes—
181C23×C5C22×D411yescharacteristic
182C22×D5—21no—
183C22×D5—21no—
184C22×D5—21no—
185C22×D5—21no—
186C22×D5—21no—
187C22×D5—21no—
188C22×D5C2×C22⋊C412yes—
189C5×Q8—48no—
190C2×C4×C5—24no—
191C2×C4×C5—42no—
192C2×C4×C5—42no—
193C2×C4×C5—42no—
194C2×C4×C5—42no—
195C2×C4×C5—42no—
196C2×C4×C5C2×C22⋊C412yes—
197C2×C4×C5C2×C22⋊C412yes—
198C2×C4×C5#13512yes—
199C2×C4×C5#13512yes—
200C2×C4×C5AGL(1,ℤ/8)12yes—
201C2×C4×C5#13712yes—
202C2×C4×C5C23×C411yescharacteristic
203C2×C4×C5C22×D411yescharacteristic
204C2×C4×C52−1+412yes—
205C2×C4×C52−1+412yes—
206C2×Dic5—21no—
207C2×Dic5—21no—
208C2×Dic5—21no—
209C2×Dic5—21no—
210C2×Dic5—21no—
211C2×Dic5—21no—
212C2×Dic5—24no—
213C2×Dic5—42no—
214C2×Dic5—42no—
215C2×Dic5—42no—
216C2×Dic5—42no—
217C2×Dic5—42no—
218C2×Dic5C2×C22⋊C412yes—
219C2×Dic5C2×C22⋊C412yes—
220C2×Dic5C2×C22⋊C412yes—
221C4×D5—24no—
222C4×D5—24no—
223C4×D5—28no—
224C4×D5—28no—
225Dic10—48no—
226C5×C8—42no—
227#168—42no—
228C22×Q8—21no—
229C22×Q8—101no—
230C23×C4—51no—
231C23×C4—51no—
232C23×C4—51no—
233C22×C8—52no—
234C22×C8C22×D512yes—
235C4⋊Q8—22no—
236C4⋊Q8—54no—
237C4⋊Q8—102no—
238C4⋊Q8C4×D514yes—
239#125—58no—
240#125—104no—
241C22⋊Q8—108no—
242C22⋊Q8—108no—
243C4×Q8—108no—
244C4×Q8—108no—
245#117—58no—
246#117—104no—
247#117—204no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—52no—
252C2×C4⋊C4—52no—
253C2×C4⋊C4—102no—
254C2×C4⋊C4—102no—
255C2×C4⋊C4—101no—
256C2×C4⋊C4—101no—
257C2×C4⋊C4—101no—
258C2×C4⋊C4C22×D512yes—
259C2×C4⋊C4C22×D512yes—
260C2×C22⋊C4—102no—
261C2×C22⋊C4—102no—
262C2×C42—51no—
263C2×C42—51no—
264C2×C42—51no—
265C2×C42—54no—
266C2×C42—54no—
267C2×C42—102no—
268C2×C42—102no—
269C2×C42C22×D511yescharacteristic
270C4⋊C8—104no—
271C22⋊C8—104no—
272C4×C8—22no—
273C4×C8—102no—
274#95—102no—
275#95—102no—
276C22×C5C22×C22⋊C411yescharacteristic
277C22×C5#53611yescharacteristic
278C22×C5#53611yescharacteristic
279C22×C5C2×#13511yescharacteristic
280C22×C5C2×#13511yescharacteristic
281C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C5C2×#13711yescharacteristic
283C2×D5—22no—
284C2×D5—22no—
285C2×D5—22no—
286C2×D5—41no—
287C2×D5—41no—
288C2×D5—41no—
289C2×D5—41no—
290C4×C5—44no—
291C4×C5—82no—
292C4×C5#41614yes—
293C4×C5#41714yes—
294C4×C5#51912yes—
295C4×C5#53612yes—
296Dic5—22no—
297Dic5—22no—
298Dic5—22no—
299Dic5—44no—
300Dic5—82no—
301C24—51no—
302C2×Q8—24no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—104no—
306C2×Q8—202no—
307C2×Q8—202no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—51no—
311C22×C4—51no—
312C22×C4—51no—
313C22×C4—52no—
314C22×C4—52no—
315C22×C4—54no—
316C22×C4—54no—
317C22×C4—102no—
318C22×C4—102no—
319C22×C4—102no—
320C22×C4—102no—
321C22×C4—102no—
322C22×C4—102no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4D4×D511yescharacteristic
326C22×C4D4×D511yescharacteristic
327C22×C4C23×D511yescharacteristic
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—102no—
332C2×C8—102no—
333C2×C8—102no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—54no—
340C4⋊C4—108no—
341C4⋊C4—108no—
342C4⋊C4—102no—
343C4⋊C4—104no—
344C4⋊C4—104no—
345C4⋊C4—202no—
346C4⋊C4—202no—
347C4⋊C4C2×C4×D514yes—
348C22⋊C4—108no—
349C22⋊C4—204no—
350C42—52no—
351C42—52no—
352C42—54no—
353C42—54no—
354C42—108no—
355C42—204no—
356C42C2×C4×D512yes—
357C42D4×D512yes—
358C2×C5#231011yescharacteristic
359C2×C5C2×#41611yescharacteristic
360C2×C5C2×#41711yescharacteristic
361C2×C5#322811yescharacteristic
362C2×C5#322811yescharacteristic
363C2×C5#323111yescharacteristic
364C2×C5#323111yescharacteristic
365D5—42no—
366C23—52no—
367C23—101no—
368C23—101no—
369C23—101no—
370C23—101no—
371C23—101no—
372C23—101no—
373C23C2×D4×D511yescharacteristic
374Q8—48no—
375Q8—208no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—52no—
383C2×C4—52no—
384C2×C4—52no—
385C2×C4—101no—
386C2×C4—101no—
387C2×C4—101no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—108no—
392C2×C4—108no—
393C2×C4—104no—
394C2×C4—104no—
395C2×C4—104no—
396C2×C4—202no—
397C2×C4—202no—
398C2×C4—202no—
399C2×C4—202no—
400C2×C4—202no—
401C2×C4D5×C22⋊C412yes—
402C2×C4D5×C22⋊C412yes—
403C2×C4#405712yes—
404C2×C4#406112yes—
405C2×C4#406212yes—
406C2×C4#406512yes—
407C2×C4C22×C4×D511yescharacteristic
408C2×C4C2×D4×D511yescharacteristic
409C2×C4#414712yes—
410C2×C4#415012yes—
411C8—42no—
412C8—202no—
413C5#2183511yescharacteristic, complemented, Sylow
414C22—102no—
415C22—102no—
416C22—102no—
417C22—201no—
418C22—201no—
419C22—201no—
420C22—201no—
421C22C2×D5×C22⋊C411yescharacteristic
422C22#6600911yescharacteristic
423C22#6605811yescharacteristic
424C22C2×#405711yescharacteristic
425C22C2×#406111yescharacteristic
426C22C2×#406211yescharacteristic
427C22C2×#406511yescharacteristic
428C4—44no—
429C4—82no—
430C4—102no—
431C4—102no—
432C4—102no—
433C4—204no—
434C4—402no—
435C4#6506014yes—
436C4#6509314yes—
437C4#6590612yes—
438C4#6596212yes—
439C2—202no—
440C2#1060834111yescharacteristic
441C2C2×#6506011yescharacteristic
442C2C2×#6509311yescharacteristic
443C2#1061330511yescharacteristic
444C2#1061334111yescharacteristic
445C2#1061362811yescharacteristic
446C2#1061367311yescharacteristic
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#21835
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. #10599594C2semidirect
14. #66062C4semidirect
93. #1990D5semidirect
413. C5#21835semidirect

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#649742
#106060492
C5×#19902
#106025802
#106005702
#106003752
#105996062
#105995942
#218355