Finite Group Databasebeta

← #12109015

Subgroups — #12109015

Order 1280.

1106 conjugacy classes of subgroups.

The subgroup lattice has 454 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#10600564C211yesmaximal, characteristic, complemented
7#10600372C211yesmaximal, characteristic, complemented
8#10599698C214yesmaximal, complemented
9#10599672C214yesmaximal
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×#4092—21no—
12C2×Q8×Dic5—21no—
13#66062—24no—
14#66062C418yescomplemented
15C2×#4043C2212yes—
16C2×#4038—21no—
17C2×#4038C2212yes—
18C2×#4036—21no—
19C2×#4033C2212yes—
20C2×C42×D5C2211yescharacteristic
21C2×#4015C2211yescharacteristic
22C2×C4×C5×C8C2211yescharacteristic
23C5×#384C2211yescharacteristic
24C2×#3952C2212yes—
25C2×#3948C2214yes—
26C2×#3946C2212yes—
27#65276—21no—
28#65276—21no—
29#65266C2212yes—
30C4⋊C4×Dic5—21no—
31C4⋊C4×Dic5—21no—
32C2×#3942C2214yes—
33#65224C2211yescharacteristic
34C2×C4×#168C2211yescharacteristic
35#64974—24no—
36C5×#340C2214yes—
37#64749C2214yes—
38#21864—51nomaximal, Sylow
39C22×C5×Q8—21no—
40C22×C4×D5D411yescharacteristic
41C22×C4×D5D411yescharacteristic
42C22×C4×D5C2311yescharacteristic
43C22×Dic10C2312yes—
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—21no—
59C2×C4⋊Dic5—22no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5—21no—
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—24no—
71#4043C2×C418yes—
72#4038—28no—
73#4038—44no—
74#4036—28no—
75#4033—24no—
76#4033C2×C418yes—
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#3948—44no—
86#3946—24no—
87#3942—44no—
88C4×#168—22no—
89C2×#533—51no—
90C2×#431—51no—
91#2285—51no—
92#1990—51no—
93#1990D511yescharacteristic, complemented
94#1956—51no—
95#1956—51no—
96#1873—54no—
97#1805—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
112C2×C4×D5—22no—
113C2×C4×D5—22no—
114C2×C4×D5—22no—
115C2×C4×D5—22no—
116C2×C4×D5—22no—
117C2×C4×D5—22no—
118C2×C4×D5C22⋊C414yes—
119C2×C4×D5C22⋊C414yes—
120C2×C4×D5C22×C412yes—
121C2×C4×D5C2×D412yes—
122C2×Dic10—24no—
123C2×Dic10—24no—
124C2×Dic10—22no—
125C2×Dic10C22×C414yes—
126C2×C5×C8—22no—
127C2×C5×C8—22no—
128C2×C5×C8—22no—
129C5×C4⋊C4—24no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—24no—
132C5×C4⋊C4—42no—
133C5×C4⋊C4—42no—
134C5×C4⋊C4C22×C414yes—
135C42×C5C22×C412yes—
136C42×C5C2×D412yes—
137#685—28no—
138#685—44no—
139C4⋊Dic5—24no—
140C4⋊Dic5—24no—
141C4⋊Dic5—28no—
142C4⋊Dic5—24no—
143C4⋊Dic5—42no—
144C4⋊Dic5—42no—
145#683—28no—
146C4×Dic5—28no—
147C4×Dic5—44no—
148C4×Dic5C22⋊C414yes—
149C4×Dic5C22⋊C414yes—
150C4×Dic5C22×C412yes—
151C4×Dic5C2×D412yes—
152C2×#168—22no—
153C2×#168—22no—
154C2×#168—22no—
155#533—58no—
156#533—104no—
157C2×C4⋊Q8—51no—
158C2×C4⋊Q8C2×D511yescharacteristic
159C2×#125—101no—
160C2×C22⋊Q8—52no—
161C2×C22⋊Q8—101no—
162C2×C4×Q8—52no—
163C2×C4×Q8—101no—
164C2×#117—52no—
165C2×#117—101no—
166C22×C42—51no—
167#431—104no—
168C2×C4⋊C8—52no—
169C2×#103—54no—
170C2×#103—54no—
171C2×C22⋊C8—52no—
172C2×C4×C8—51no—
173C2×C4×C8C2×D511yescharacteristic
174#384—51no—
175#384C2×D511yescharacteristic
176#382—52no—
177#380—101no—
178#380—101no—
179C4×C4⋊C4—101no—
180C4×C4⋊C4—101no—
181#340—54no—
182#340C2×D514yes—
183C23×C5C22×D411yescharacteristic
184C22×D5—21no—
185C22×D5—21no—
186C22×D5—21no—
187C22×D5—21no—
188C22×D5—21no—
189C22×D5—21no—
190C22×D5C2×C22⋊C412yes—
191C5×Q8—48no—
192C2×C4×C5—24no—
193C2×C4×C5—42no—
194C2×C4×C5—42no—
195C2×C4×C5—42no—
196C2×C4×C5—42no—
197C2×C4×C5—42no—
198C2×C4×C5C2×C22⋊C412yes—
199C2×C4×C5C2×C22⋊C412yes—
200C2×C4×C5#13512yes—
201C2×C4×C5#13512yes—
202C2×C4×C5#13712yes—
203C2×C4×C5#13712yes—
204C2×C4×C5C23×C411yescharacteristic
205C2×C4×C5C22×D411yescharacteristic
206C2×C4×C5H(5,𝔽2)12yes—
207C2×C4×C52−1+412yes—
208C2×Dic5—21no—
209C2×Dic5—21no—
210C2×Dic5—21no—
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5—21no—
214C2×Dic5—24no—
215C2×Dic5—42no—
216C2×Dic5—42no—
217C2×Dic5—42no—
218C2×Dic5—42no—
219C2×Dic5—42no—
220C2×Dic5C2×C22⋊C412yes—
221C2×Dic5C2×C22⋊C412yes—
222C2×Dic5C2×C22⋊C412yes—
223C4×D5—24no—
224C4×D5—24no—
225C4×D5—28no—
226C4×D5—28no—
227Dic10—28no—
228Dic10—44no—
229C5×C8—42no—
230#168—42no—
231C22×Q8—21no—
232C22×Q8—52no—
233C23×C4—51no—
234C23×C4—51no—
235C23×C4—51no—
236C22×C8—52no—
237C22×C8C22×D512yes—
238C4⋊Q8—22no—
239C4⋊Q8—54no—
240C4⋊Q8—102no—
241C4⋊Q8C4×D514yes—
242#125—108no—
243C22⋊Q8—58no—
244C22⋊Q8—104no—
245C22⋊Q8—108no—
246C4×Q8—58no—
247C4×Q8—104no—
248C4×Q8—108no—
249#117—108no—
250#117—204no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—52no—
255C2×C4⋊C4—102no—
256C2×C4⋊C4—102no—
257C2×C4⋊C4—101no—
258C2×C4⋊C4—101no—
259C2×C4⋊C4—101no—
260C2×C4⋊C4—101no—
261C2×C4⋊C4C22×D512yes—
262C2×C4⋊C4C22×D512yes—
263C2×C22⋊C4—102no—
264C2×C22⋊C4—102no—
265C2×C42—51no—
266C2×C42—51no—
267C2×C42—51no—
268C2×C42—54no—
269C2×C42—54no—
270C2×C42—102no—
271C2×C42—102no—
272C2×C42C22×D511yescharacteristic
273C4⋊C8—104no—
274#103—204no—
275#103—204no—
276C22⋊C8—104no—
277C4×C8—22no—
278C4×C8—102no—
279#95—102no—
280#95—102no—
281C22×C5C22×C22⋊C411yescharacteristic
282C22×C5#53511yescharacteristic
283C22×C5#53611yescharacteristic
284C22×C5C2×#13511yescharacteristic
285C22×C5C2×#13511yescharacteristic
286C22×C5C2×#13711yescharacteristic
287C22×C5C2×#13711yescharacteristic
288C2×D5—22no—
289C2×D5—22no—
290C2×D5—22no—
291C2×D5—41no—
292C2×D5—41no—
293C2×D5—41no—
294C2×D5—41no—
295C4×C5—44no—
296C4×C5—82no—
297C4×C5#41614yes—
298C4×C5#41914yes—
299C4×C5#52012yes—
300C4×C5#53712yes—
301Dic5—22no—
302Dic5—22no—
303Dic5—22no—
304Dic5—44no—
305Dic5—82no—
306C24—51no—
307C2×Q8—24no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—54no—
311C2×Q8—102no—
312C2×Q8—104no—
313C2×Q8—104no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—52no—
320C22×C4—52no—
321C22×C4—54no—
322C22×C4—54no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—102no—
329C22×C4—102no—
330C22×C4—102no—
331C22×C4D4×D511yescharacteristic
332C22×C4D4×D511yescharacteristic
333C22×C4C23×D511yescharacteristic
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—102no—
338C2×C8—102no—
339C2×C8—102no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—108no—
346C4⋊C4—108no—
347C4⋊C4—104no—
348C4⋊C4—104no—
349C4⋊C4—104no—
350C4⋊C4—202no—
351C4⋊C4—202no—
352C4⋊C4C2×C4×D514yes—
353C22⋊C4—108no—
354C22⋊C4—204no—
355C42—52no—
356C42—52no—
357C42—54no—
358C42—54no—
359C42—108no—
360C42—204no—
361C42C2×C4×D512yes—
362C42D4×D512yes—
363C2×C5#231111yescharacteristic
364C2×C5C2×#41611yescharacteristic
365C2×C5C2×#41911yescharacteristic
366C2×C5#321011yescharacteristic
367C2×C5#321011yescharacteristic
368C2×C5#323111yescharacteristic
369C2×C5#323111yescharacteristic
370D5—42no—
371C23—52no—
372C23—101no—
373C23—101no—
374C23—101no—
375C23—101no—
376C23—101no—
377C23—101no—
378C23C2×D4×D511yescharacteristic
379Q8—48no—
380Q8—108no—
381Q8—204no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—52no—
389C2×C4—52no—
390C2×C4—52no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—101no—
395C2×C4—101no—
396C2×C4—101no—
397C2×C4—108no—
398C2×C4—108no—
399C2×C4—104no—
400C2×C4—104no—
401C2×C4—104no—
402C2×C4—202no—
403C2×C4—202no—
404C2×C4—202no—
405C2×C4—202no—
406C2×C4—202no—
407C2×C4D5×C22⋊C412yes—
408C2×C4D5×C22⋊C412yes—
409C2×C4#405812yes—
410C2×C4#406112yes—
411C2×C4#406212yes—
412C2×C4#406412yes—
413C2×C4C22×C4×D511yescharacteristic
414C2×C4C2×D4×D511yescharacteristic
415C2×C4#414712yes—
416C2×C4#414912yes—
417C8—42no—
418C8—202no—
419C5#2186411yescharacteristic, complemented, Sylow
420C22—102no—
421C22—102no—
422C22—102no—
423C22—201no—
424C22—201no—
425C22—201no—
426C22—201no—
427C22C2×D5×C22⋊C411yescharacteristic
428C22#6600811yescharacteristic
429C22#6605811yescharacteristic
430C22C2×#405811yescharacteristic
431C22C2×#406111yescharacteristic
432C22C2×#406211yescharacteristic
433C22C2×#406411yescharacteristic
434C4—44no—
435C4—82no—
436C4—102no—
437C4—102no—
438C4—102no—
439C4—204no—
440C4—402no—
441C4#6506114yes—
442C4#6509114yes—
443C4#6590812yes—
444C4#6596312yes—
445C2—202no—
446C2#1060821311yescharacteristic
447C2C2×#6506111yescharacteristic
448C2C2×#6509111yescharacteristic
449C2#1061330011yescharacteristic
450C2#1061333611yescharacteristic
451C2#1061362811yescharacteristic
452C2#1061367511yescharacteristic
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#21864
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. #10600564C2semidirect
7. #10600372C2semidirect
8. #10599698C2semidirect
14. #66062C4semidirect
93. #1990D5semidirect
419. C5#21864semidirect

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
#106005642
#106003722
#105996982
#105996722
#218645