Finite Group Databasebeta

← #12073155

Subgroups — #12073155

Order 1280.

1064 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#543C211yesmaximal, characteristic
2#10614472C212yesmaximal, complemented
3#10613848C211yesmaximal, characteristic
4#10602686C214yesmaximal, complemented
5#10602684C214yesmaximal, complemented
6#10602483C212yesmaximal, complemented
7C2×#64745C211yesmaximal, characteristic
8C5×#543—24no—
9C5×#543—22no—
10C5×#543C2214yes—
11C2×C5×C22⋊Q8—22no—
12C2×C5×C22⋊Q8—21no—
13C2×C5×C22⋊Q8—22no—
14C2×C5×C22⋊Q8C2212yes—
15C22×C5×C22⋊C4—21no—
16C22×C5×C22⋊C4C2211yescharacteristic
17C2×#4070C2211yescharacteristic
18#65536—21no—
19#65517—22no—
20#65501C2211yescharacteristic
21#65499—21no—
22C2×C22⋊#168C2212yes—
23#65340—24no—
24#65339C2214yescomplemented
25#65333C2214yescomplemented
26#65259—22no—
27#65238C2212yes—
28#65237—22no—
29#65236C2212yes—
30#64758C2218yescomplemented
31#64745C2214yes—
32#18287—51nomaximal, Sylow
33C25×C5D411yescharacteristic
34C22×C5×Q8—21no—
35C22×C5×Q8D412yes—
36C23×C4×C5—21no—
37C23×C4×C5—21no—
38C23×C4×C5D411yescharacteristic
39C23×C4×C5C2311yescharacteristic
40C22×Dic10C2311yescharacteristic
41C5×C22⋊Q8—28no—
42C5×C22⋊Q8—28no—
43C5×C22⋊Q8—24no—
44C5×C22⋊Q8—28no—
45C5×C22⋊Q8—24no—
46C5×C22⋊Q8—22no—
47C5×C22⋊Q8—22no—
48C5×C22⋊Q8—28no—
49C5×C22⋊Q8D414yes—
50C5×C22⋊Q8C2314yes—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—22no—
53C2×C5×C4⋊C4—22no—
54C2×C5×C4⋊C4D412yes—
55C2×C5×C22⋊C4—22no—
56C2×C5×C22⋊C4—24no—
57C2×C5×C22⋊C4—22no—
58C2×C5×C22⋊C4—24no—
59C2×C5×C22⋊C4—22no—
60C2×C5×C22⋊C4—22no—
61C2×C5×C22⋊C4D412yes—
62C2×C5×C22⋊C4C2312yes—
63C2×C22⋊Dic5—21no—
64C2×C22⋊Dic5—21no—
65C2×C22⋊Dic5—22no—
66C2×C22⋊Dic5—22no—
67C2×#689—24no—
68C2×C4⋊Dic5—21no—
69C2×C4⋊Dic5C2311yescharacteristic
70#4070—22no—
71#4070—22no—
72C2×#683—21no—
73C2×#683—22no—
74C22×#168—22no—
75#3967—24no—
76#3963—22no—
77#3963C2312yes—
78C22⋊#168—22no—
79C22⋊#168C2314yes—
80#3942—24no—
81#3939—24no—
82C2×#543D511yescharacteristic
83#3189—52no—
84#2626—51no—
85#1614—54no—
86#1602—54no—
87#1520—52no—
88C2×#323—51no—
89C24×C5—21no—
90C24×C5—21no—
91C24×C5—22no—
92C24×C5—81no—
93C24×C5—81no—
94C24×C5Q1614yescomplemented
95C24×C5C2×D412yes—
96C24×C5C2×D411yescharacteristic
97C2×C5×Q8—24no—
98C2×C5×Q8—22no—
99C2×C5×Q8—82no—
100C2×C5×Q8—82no—
101C2×C5×Q8C2×D414yes—
102C22×C4×C5—22no—
103C22×C4×C5—21no—
104C22×C4×C5—22no—
105C22×C4×C5—24no—
106C22×C4×C5—24no—
107C22×C4×C5—21no—
108C22×C4×C5—81no—
109C22×C4×C5—81no—
110C22×C4×C5—82no—
111C22×C4×C5—82no—
112C22×C4×C5C2×D412yes—
113C22×C4×C5C2×D411yescharacteristic
114C22×C4×C5C2×D412yes—
115C22×C4×C5C2×D412yes—
116C22×C4×C5C2411yescharacteristic
117C22×Dic5—21no—
118C22×Dic5—22no—
119C22×Dic5—21no—
120C2×Dic10—22no—
121C2×Dic10—81no—
122C5×C4⋊C4—24no—
123C5×C4⋊C4—28no—
124C5×C4⋊C4—24no—
125C5×C4⋊C4—22no—
126C5×C4⋊C4—24no—
127C5×C4⋊C4C2×D414yes—
128C5×C22⋊C4—28no—
129C5×C22⋊C4—24no—
130C5×C22⋊C4—24no—
131C5×C22⋊C4—24no—
132C5×C22⋊C4—24no—
133C22⋊Dic5—41no—
134C22⋊Dic5—41no—
135C22⋊Dic5—42no—
136C22⋊Dic5—82no—
137C22⋊Dic5—82no—
138#689—84no—
139C4⋊Dic5—22no—
140C4⋊Dic5—81no—
141#683—42no—
142#683—82no—
143C2×#168—24no—
144C2×#168—82no—
145#543—24no—
146#543—22no—
147#543C2×D514yes—
148C2×C22⋊Q8—22no—
149C2×C22⋊Q8—21no—
150C2×C22⋊Q8—22no—
151C2×C22⋊Q8—51no—
152C2×C22⋊Q8C2×D512yes—
153C22×C22⋊C4—21no—
154C22×C22⋊C4C2×D511yescharacteristic
155#484—54no—
156#484—102no—
157#467—104no—
158#451—54no—
159#451—102no—
160C2×C22⋊C8—52no—
161#399—52no—
162#396—102no—
163#393—52no—
164#385—101no—
165#380—51no—
166#379—101no—
167#328—58no—
168#323—54no—
169C23×C5—22no—
170C23×C5—24no—
171C23×C5—22no—
172C23×C5—24no—
173C23×C5—41no—
174C23×C5—41no—
175C23×C5—42no—
176C23×C5—42no—
177C23×C5—42no—
178C23×C5—42no—
179C23×C5—81no—
180C23×C5—81no—
181C23×C5—84no—
182C23×C5—84no—
183C23×C5#12011yescharacteristic
184C23×C5#12011yescharacteristic
185C23×C5C2×Q1614yes—
186C23×C5C2×Q1612yes—
187C23×C5#13712yes—
188C23×C5C22×D411yescharacteristic
189C5×Q8—44no—
190C5×Q8—82no—
191C5×Q8—84no—
192C2×C4×C5—24no—
193C2×C4×C5—22no—
194C2×C4×C5—24no—
195C2×C4×C5—42no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5—42no—
199C2×C4×C5—42no—
200C2×C4×C5—44no—
201C2×C4×C5—44no—
202C2×C4×C5—42no—
203C2×C4×C5—81no—
204C2×C4×C5—84no—
205C2×C4×C5—84no—
206C2×C4×C5#12012yes—
207C2×C4×C5C22×D412yes—
208C2×Dic5—42no—
209C2×Dic5—42no—
210C2×Dic5—42no—
211C2×Dic5—42no—
212C2×Dic5—41no—
213C2×Dic5—41no—
214C2×Dic5—42no—
215Dic10—44no—
216#168—44no—
217C25C22⋊D511yescharacteristic
218C22×Q8—21no—
219C22×Q8—51no—
220C22×Q8C22⋊D512yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4C22⋊D511yescharacteristic
224C23×C4C22×D511yescharacteristic
225C2×Q16—104no—
226C22×C8—102no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—28no—
235C22⋊Q8—102no—
236C22⋊Q8—102no—
237C22⋊Q8C22⋊D514yes—
238C22⋊Q8C22×D514yes—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—51no—
243C2×C4⋊C4—101no—
244C2×C4⋊C4—101no—
245C2×C4⋊C4—102no—
246C2×C4⋊C4C22⋊D512yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—101no—
254C2×C22⋊C4—101no—
255C2×C22⋊C4—102no—
256C2×C22⋊C4—102no—
257C2×C22⋊C4C22⋊D512yes—
258C2×C22⋊C4C22×D512yes—
259#107—104no—
260#103—104no—
261#103—104no—
262C22⋊C8—54no—
263C22⋊C8—102no—
264#95—52no—
265#95—102no—
266C22×C5—21no—
267C22×C5—21no—
268C22×C5—21no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—24no—
273C22×C5—42no—
274C22×C5—42no—
275C22×C5—44no—
276C22×C5—48no—
277C22×C5—42no—
278C22×C5—42no—
279C22×C5—82no—
280C22×C5—84no—
281C22×C5#45112yes—
282C22×C5#45112yes—
283C22×C5C2×#12011yescharacteristic
284C22×C5C22×Q1611yescharacteristic
285C22×C5C2×#13711yescharacteristic
286C4×C5—22no—
287C4×C5—22no—
288C4×C5—42no—
289C4×C5—44no—
290C4×C5—44no—
291Dic5—82no—
292Dic5—81no—
293Dic5—81no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—81no—
298C24—81no—
299C24#68914yescomplemented
300C24C2×C22⋊D512yes—
301C24C2×C22⋊D511yescharacteristic
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—102no—
307C2×Q8—401no—
308C2×Q8C2×C22⋊D514yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—102no—
320C22×C4—101no—
321C22×C4—101no—
322C22×C4C2×C22⋊D512yes—
323C22×C4C2×C22⋊D511yescharacteristic
324C22×C4C2×C22⋊D512yes—
325C22×C4C2×C22⋊D512yes—
326C22×C4C23×D511yescharacteristic
327Q16—404no—
328C2×C8—104no—
329C2×C8—402no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—102no—
336C4⋊C4—202no—
337C4⋊C4—402no—
338C4⋊C4—401no—
339C4⋊C4C2×C22⋊D514yes—
340C22⋊C4—28no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—201no—
346C22⋊C4—201no—
347C22⋊C4—202no—
348C22⋊C4—402no—
349C22⋊C4—402no—
350C2×C5—21no—
351C2×C5—21no—
352C2×C5—22no—
353C2×C5—22no—
354C2×C5—44no—
355C2×C5C2×#45111yescharacteristic
356C2×C5#301911yescharacteristic
357C2×C5#302611yescharacteristic
358C23—22no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—84no—
371C23—84no—
372C23#408812yes—
373C23C2×#68914yes—
374C23C2×#68912yes—
375C23#409911yescharacteristic
376C23#409911yescharacteristic
377C23C22×C22⋊D511yescharacteristic
378Q8—44no—
379Q8—82no—
380Q8—84no—
381Q8—204no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—202no—
397C2×C4—202no—
398C2×C4—202no—
399C2×C4—202no—
400C2×C4—202no—
401C2×C4—201no—
402C2×C4—201no—
403C2×C4#409912yes—
404C2×C4C22×C22⋊D512yes—
405C8—204no—
406C5#1828711yescharacteristic, complemented, Sylow
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#6551712yes—
423C22#6551712yes—
424C22C2×#408811yescharacteristic
425C22C22×#68911yescharacteristic
426C22C2×#409911yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—402no—
433C4—401no—
434C4—401no—
435C2—21no—
436C2—21no—
437C2—22no—
438C2—22no—
439C2—44no—
440C2C2×#6551711yescharacteristic
441C2#1061583511yescharacteristic
442C2#1061584111yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#543C2
socleC22×C5C2×#120
prime radicalC25×C5D4
hypercenterC2×#543D5
hypocenterC5#18287
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #10614472C2semidirect
4. #10602686C2semidirect
5. #10602684C2semidirect
6. #10602483C2semidirect
24. #65339C22semidirect
25. #65333C22semidirect
30. #64758C22semidirect
94. C24×C5Q16semidirect
299. C24#689semidirect
406. C5#18287semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5432
#106144722
#106138482
#106026862
#106026842
#106024832
C2×#647452
#182875