Finite Group Databasebeta

← #12108822

Subgroups — #12108822

Order 1280.

1054 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#65124C211yesmaximal, characteristic
3#10605484C211yesmaximal, characteristic
4C5×#1988C211yesmaximal, characteristic, complemented
5#10602573C211yesmaximal, characteristic, complemented
6#10601252C211yesmaximal, characteristic, complemented
7#10600282C211yesmaximal, characteristic, complemented
8#10599606C214yesmaximal
9#10599598C214yesmaximal, complemented
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×#4092—21no—
12C2×Q8×Dic5—21no—
13#66062—24no—
14#66062C418yescomplemented
15C2×#4043—21no—
16C2×#4038C2212yes—
17C2×#4036C2212yes—
18C2×#4033—21no—
19C2×#4018—21no—
20C2×C42×D5C2211yescharacteristic
21C2×#4015C2211yescharacteristic
22C2×C5×C4⋊C8C2211yescharacteristic
23C4×C5×C4⋊C4C2211yescharacteristic
24C2×#3952C2212yes—
25C2×#3947C2212yes—
26#65276C2212yes—
27#65265C2212yes—
28C4×C4⋊Dic5C2211yescharacteristic
29C4×#683—21no—
30C2×C4⋊#168C2211yescharacteristic
31#65124—24no—
32#64956—21no—
33#64939—21no—
34C5×#340C2214yes—
35#64771C2214yes—
36#64749C2214yes—
37#64749C2214yes—
38#21839—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⋊C4C2312yes—
49C2×C5×C4⋊C4C2312yes—
50C2×C42×C5—21no—
51C2×C42×C5C2311yescharacteristic
52#4092—28no—
53Q8×Dic5—28no—
54C2×#685—22no—
55C2×#685—22no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5—22no—
58C2×C4⋊Dic5C2312yes—
59C2×C4⋊Dic5C2312yes—
60C2×#683—22no—
61C2×#683—22no—
62C2×#683—22no—
63C2×C4×Dic5—22no—
64C2×C4×Dic5—21no—
65C2×C4×Dic5D411yescharacteristic
66C2×C4×Dic5D411yescharacteristic
67C2×C4×Dic5C2311yescharacteristic
68C22×#168C2312yes—
69#4043—28no—
70#4038—24no—
71#4038C2×C418yes—
72#4036—24no—
73#4036C2×C418yes—
74#4033—28no—
75#4018—44no—
76C42×D5C2×C414yes—
77C42×D5D414yes—
78#4015—22no—
79#4015C2×C414yes—
80C5×C4⋊C8—22no—
81C5×#95—21no—
82#3963—21no—
83#3963—22no—
84#3952—24no—
85#3947—24no—
86C4⋊#168—22no—
87C2×#533—51no—
88C2×#432—51no—
89#2285—51no—
90#1992—54no—
91#1988—51no—
92#1988D511yescharacteristic, complemented
93#1987—51no—
94#1987—51no—
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—42no—
115C2×C4×D5C22⋊C414yes—
116C2×C4×D5C22⋊C414yes—
117C2×C4×D5C22×C412yes—
118C2×C4×D5C2×D412yes—
119C2×Dic10—24no—
120C2×Dic10—44no—
121C2×C5×C8—22no—
122C2×C5×C8—24no—
123C5×C4⋊C4—24no—
124C5×C4⋊C4—22no—
125C5×C4⋊C4—22no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4C22×C414yes—
128C42×C5—42no—
129C42×C5C22×C412yes—
130C42×C5C2×D412yes—
131#685—28no—
132#685—44no—
133C4⋊Dic5—24no—
134C4⋊Dic5—22no—
135C4⋊Dic5—28no—
136C4⋊Dic5—22no—
137C4⋊Dic5—22no—
138C4⋊Dic5C22×C414yes—
139#683—28no—
140#683—44no—
141#683—44no—
142C4×Dic5—28no—
143C4×Dic5—42no—
144C4×Dic5C22⋊C414yes—
145C4×Dic5C22⋊C414yes—
146C4×Dic5C22×C412yes—
147C4×Dic5C2×D412yes—
148C2×#168—22no—
149C2×#168—24no—
150#533—58no—
151#533—104no—
152C2×C4⋊Q8—51no—
153C2×C4⋊Q8C2×D511yescharacteristic
154C2×#125—52no—
155C2×C22⋊Q8—101no—
156C2×C22⋊Q8—101no—
157C2×C4×Q8—101no—
158C2×C4×Q8—101no—
159C2×#117—52no—
160C2×#117—101no—
161C22×C42—51no—
162#432—104no—
163C2×C4⋊C8—51no—
164C2×C4⋊C8C2×D511yescharacteristic
165C2×C22⋊C8—52no—
166C2×#97—52no—
167#384—52no—
168#382—101no—
169#380—52no—
170#380—101no—
171C4×C4⋊C4—51no—
172C4×C4⋊C4—101no—
173C4×C4⋊C4C2×D511yescharacteristic
174#340—54no—
175#340—54no—
176#340—54no—
177#340C2×D514yes—
178C23×C5C22×D411yescharacteristic
179C22×D5—22no—
180C22×D5—21no—
181C22×D5—21no—
182C22×D5—22no—
183C22×D5C2×C22⋊C412yes—
184C5×Q8—48no—
185C2×C4×C5—22no—
186C2×C4×C5—24no—
187C2×C4×C5—42no—
188C2×C4×C5—42no—
189C2×C4×C5—42no—
190C2×C4×C5—42no—
191C2×C4×C5—42no—
192C2×C4×C5C2×C22⋊C412yes—
193C2×C4×C5C2×C22⋊C412yes—
194C2×C4×C5AGL(1,ℤ/8)12yes—
195C2×C4×C5#13712yes—
196C2×C4×C5C23×C411yescharacteristic
197C2×C4×C5C22×D411yescharacteristic
198C2×C4×C52−1+412yes—
199C2×C4×C52−1+412yes—
200C2×Dic5—22no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—22no—
204C2×Dic5—24no—
205C2×Dic5—44no—
206C2×Dic5—42no—
207C2×Dic5—42no—
208C2×Dic5—42no—
209C2×Dic5C2×C22⋊C412yes—
210C2×Dic5C2×C22⋊C412yes—
211C2×Dic5C2×C22⋊C412yes—
212C4×D5—24no—
213C4×D5—22no—
214C4×D5—22no—
215C4×D5—28no—
216C4×D5—44no—
217Dic10—48no—
218C5×C8—42no—
219#168—42no—
220C22×Q8—21no—
221C22×Q8—101no—
222C23×C4—51no—
223C23×C4—51no—
224C23×C4—51no—
225C22×C8—52no—
226C22×C8C22×D512yes—
227C4⋊Q8—22no—
228C4⋊Q8—54no—
229C4⋊Q8—102no—
230C4⋊Q8C4×D514yes—
231#125—58no—
232#125—104no—
233C22⋊Q8—108no—
234C22⋊Q8—108no—
235C4×Q8—108no—
236C4×Q8—108no—
237#117—58no—
238#117—104no—
239#117—204no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—52no—
242C2×C4⋊C4—52no—
243C2×C4⋊C4—102no—
244C2×C4⋊C4—102no—
245C2×C4⋊C4—101no—
246C2×C4⋊C4—102no—
247C2×C4⋊C4—102no—
248C2×C4⋊C4C22×D512yes—
249C2×C4⋊C4C22×D512yes—
250C2×C22⋊C4—102no—
251C2×C22⋊C4—102no—
252C2×C42—21no—
253C2×C42—51no—
254C2×C42—51no—
255C2×C42—51no—
256C2×C42—54no—
257C2×C42—54no—
258C2×C42—102no—
259C2×C42—101no—
260C2×C42C22×D511yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—102no—
263C22⋊C8—104no—
264#97—104no—
265#95—21no—
266#95—102no—
267#95—101no—
268C22×C5C22×C22⋊C411yescharacteristic
269C22×C5#53612yes—
270C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C5C2×#13711yescharacteristic
272C22×C5#57512yes—
273C2×D5—22no—
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—42no—
277C2×D5—42no—
278C4×C5—22no—
279C4×C5—44no—
280C4×C5—82no—
281C4×C5#41714yes—
282C4×C5#51912yes—
283C4×C5#53611yescharacteristic
284C4×C5#53611yescharacteristic
285Dic5—22no—
286Dic5—22no—
287Dic5—22no—
288Dic5—44no—
289Dic5—82no—
290C24—51no—
291C2×Q8—24no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—104no—
295C2×Q8—204no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—51no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—52no—
302C22×C4—52no—
303C22×C4—54no—
304C22×C4—54no—
305C22×C4—102no—
306C22×C4—102no—
307C22×C4—102no—
308C22×C4—102no—
309C22×C4—102no—
310C22×C4—102no—
311C22×C4—202no—
312C22×C4D4×D511yescharacteristic
313C22×C4D4×D511yescharacteristic
314C22×C4C23×D511yescharacteristic
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—102no—
318C2×C8—104no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—54no—
324C4⋊C4—108no—
325C4⋊C4—108no—
326C4⋊C4—102no—
327C4⋊C4—104no—
328C4⋊C4—102no—
329C4⋊C4—102no—
330C4⋊C4—204no—
331C4⋊C4—204no—
332C4⋊C4C2×C4×D514yes—
333C22⋊C4—108no—
334C22⋊C4—204no—
335C42—42no—
336C42—52no—
337C42—52no—
338C42—54no—
339C42—54no—
340C42—108no—
341C42—202no—
342C42C2×C4×D512yes—
343C42D4×D512yes—
344C2×C5#231011yescharacteristic
345C2×C5C2×#41711yescharacteristic
346C2×C5#289611yescharacteristic
347C2×C5#323512yes—
348C2×C5#323712yes—
349D5—42no—
350C23—52no—
351C23—101no—
352C23—101no—
353C23—102no—
354C23—102no—
355C23C2×D4×D511yescharacteristic
356Q8—48no—
357Q8—208no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—52no—
366C2×C4—52no—
367C2×C4—52no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—102no—
371C2×C4—102no—
372C2×C4—102no—
373C2×C4—108no—
374C2×C4—104no—
375C2×C4—102no—
376C2×C4—104no—
377C2×C4—204no—
378C2×C4—204no—
379C2×C4—202no—
380C2×C4—202no—
381C2×C4—202no—
382C2×C4D5×C22⋊C412yes—
383C2×C4D5×C22⋊C412yes—
384C2×C4#405712yes—
385C2×C4#406112yes—
386C2×C4C22×C4×D511yescharacteristic
387C2×C4C2×D4×D511yescharacteristic
388C2×C4#414712yes—
389C2×C4#415012yes—
390C8—42no—
391C8—202no—
392C5#2183911yescharacteristic, complemented, Sylow
393C22—102no—
394C22—102no—
395C22—102no—
396C22—202no—
397C22—202no—
398C22C2×D5×C22⋊C411yescharacteristic
399C22#6596212yes—
400C22C2×#405711yescharacteristic
401C22C2×#406111yescharacteristic
402C22#6611112yes—
403C4—22no—
404C4—44no—
405C4—82no—
406C4—102no—
407C4—102no—
408C4—102no—
409C4—204no—
410C4—402no—
411C4#6506014yes—
412C4#6590612yes—
413C4#6600911yescharacteristic
414C4#6605811yescharacteristic
415C2—202no—
416C2#1060834111yescharacteristic
417C2C2×#6506011yescharacteristic
418C2#1061068711yescharacteristic
419C2#1061270212yes—
420C2#1061274812yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1988C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1988D5
hypocenterC5#21839
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. C5×#1988C2semidirect
5. #10602573C2semidirect
6. #10601252C2semidirect
7. #10600282C2semidirect
9. #10599598C2semidirect
14. #66062C4semidirect
92. #1988D5semidirect
392. C5#21839semidirect

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#651242
#106054842
C5×#19882
#106025732
#106012522
#106002822
#105996062
#105995982
#218395