Finite Group Databasebeta

← #12073209

Subgroups — #12073209

Order 1280.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#537C211yesmaximal, characteristic
2#10614361C212yesmaximal, complemented
3#10613914C211yesmaximal, characteristic, complemented
4#10602673C214yesmaximal, complemented
5#10602668C214yesmaximal, complemented
6#10602468C212yesmaximal, complemented
7C2×#64748C211yesmaximal, characteristic, complemented
8C22×C5×#41—21no—
9C5×#537—24no—
10C5×#537—22no—
11C5×#537C2214yes—
12C2×C5×C22⋊Q8—22no—
13C2×C5×#121—22no—
14C2×C5×#121—21no—
15C2×C5×#121C2212yes—
16C22×C5×C4⋊C4C2211yescharacteristic
17C2×#4070C2211yescharacteristic
18#65507—22no—
19#65501C2211yescharacteristic
20#65500—21no—
21#65499—21no—
22C2×C22⋊#168C2212yes—
23#65330—24no—
24#65329C2214yescomplemented
25#65323C2214yescomplemented
26#65275C2212yescomplemented
27#65273—22no—
28#65268C2212yescomplemented
29#65260—22no—
30#64757C2218yescomplemented
31#64748C2214yescomplemented
32#18293—51nomaximal, Sylow
33C2×C5×#41—24no—
34C2×C5×#41—24no—
35C2×C5×#41—44no—
36C2×C5×#41—44no—
37C22×C5×Q8—21no—
38C22×C5×D4—21no—
39C22×C5×D4—22no—
40C22×C5×D4—21no—
41C22×C5×D4D412yes—
42C23×C4×C5—22no—
43C23×C4×C5D411yescharacteristic
44C23×C4×C5D411yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C22×Dic10C2311yescharacteristic
47C5×C22⋊Q8—28no—
48C5×C22⋊Q8—28no—
49C5×#121—24no—
50C5×#121—28no—
51C5×#121—28no—
52C5×#121—24no—
53C5×#121—22no—
54C5×#121—22no—
55C5×#121D414yes—
56C5×#121C2314yes—
57C2×C5×C4⋊C4—21no—
58C2×C5×C4⋊C4—22no—
59C2×C5×C4⋊C4—22no—
60C2×C5×C4⋊C4D412yes—
61C2×C5×C4⋊C4D412yes—
62C2×C5×C4⋊C4C2312yes—
63C2×C5×C22⋊C4—22no—
64C2×C5×C22⋊C4—22no—
65C2×C22⋊Dic5—21no—
66C2×C22⋊Dic5—21no—
67C2×#687—24no—
68C2×C4⋊Dic5—21no—
69C2×C4⋊Dic5C2311yescharacteristic
70#4070—22no—
71#4070—22no—
72C2×#683—21no—
73C2×#683—22no—
74C22×#168—22no—
75#3964—24no—
76#3963—22no—
77#3963—22no—
78#3963—22no—
79#3963C2312yes—
80C22⋊#168—22no—
81C22⋊#168C2314yes—
82#3942—24no—
83#3940—24no—
84C2×#537D511yescharacteristic
85#3187—52no—
86#2627—51no—
87#1616—54no—
88#1604—54no—
89#1519—52no—
90C2×#324—51no—
91C24×C5—22no—
92C24×C5C2×D411yescharacteristic
93C5×#41—48no—
94C5×#41—416no—
95C5×#41—48no—
96C2×C5×Q8—24no—
97C2×C5×Q8—44no—
98C2×C5×D4—24no—
99C2×C5×D4—24no—
100C2×C5×D4—24no—
101C2×C5×D4—24no—
102C2×C5×D4—22no—
103C2×C5×D4—44no—
104C2×C5×D4—44no—
105C2×C5×D4—44no—
106C2×C5×D4—82no—
107C2×C5×D4—82no—
108C2×C5×D4C2×D414yes—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—22no—
112C22×C4×C5—28no—
113C22×C4×C5—21no—
114C22×C4×C5—22no—
115C22×C4×C5—21no—
116C22×C4×C5—48no—
117C22×C4×C5—81no—
118C22×C4×C5—81no—
119C22×C4×C5SD1614yescomplemented
120C22×C4×C5C2×D412yes—
121C22×C4×C5C2×D412yes—
122C22×C4×C5C2×D411yescharacteristic
123C22×C4×C5C2×D412yes—
124C22×C4×C5C2×D412yes—
125C22×C4×C5C2411yescharacteristic
126C22×Dic5—21no—
127C22×Dic5—21no—
128C22×Dic5—22no—
129C2×Dic10—22no—
130C2×Dic10—81no—
131C5×C4⋊C4—24no—
132C5×C4⋊C4—24no—
133C5×C4⋊C4—22no—
134C5×C4⋊C4—24no—
135C5×C4⋊C4C2×D414yes—
136C5×C22⋊C4—28no—
137C5×C22⋊C4—24no—
138C5×C22⋊C4—24no—
139C22⋊Dic5—41no—
140C22⋊Dic5—41no—
141C22⋊Dic5—42no—
142#687—84no—
143C4⋊Dic5—22no—
144C4⋊Dic5—81no—
145#683—42no—
146#683—82no—
147C2×#168—24no—
148C2×#168—82no—
149C22×#41—21no—
150#537—24no—
151#537—22no—
152#537C2×D514yes—
153C2×C22⋊Q8—22no—
154C2×C22⋊Q8—51no—
155C2×#121—22no—
156C2×#121—21no—
157C2×#121C2×D512yes—
158C22×C4⋊C4C2×D511yescharacteristic
159#483—102no—
160#481—54no—
161#465—104no—
162#450—102no—
163#448—54no—
164C2×C22⋊C8—52no—
165#400—52no—
166#398—102no—
167#395—52no—
168#385—101no—
169#381—101no—
170#380—51no—
171#327—58no—
172#324—54no—
173C23×C5—24no—
174C23×C5—41no—
175C23×C5—41no—
176C23×C5—44no—
177C23×C5—44no—
178C23×C5—44no—
179C23×C5#12011yescharacteristic
180C23×C5#12011yescharacteristic
181C23×C5C2×SD1612yes—
182C23×C5AGL(1,ℤ/8)12yes—
183C23×C5C22×D411yescharacteristic
184C5×Q8—48no—
185C5×D4—44no—
186C5×D4—48no—
187C5×D4—44no—
188C5×D4—44no—
189C5×D4—48no—
190C5×D4—82no—
191C5×D4—84no—
192C2×C4×C5—22no—
193C2×C4×C5—22no—
194C2×C4×C5—24no—
195C2×C4×C5—24no—
196C2×C4×C5—22no—
197C2×C4×C5—24no—
198C2×C4×C5—42no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5—42no—
202C2×C4×C5—416no—
203C2×C4×C5—42no—
204C2×C4×C5—44no—
205C2×C4×C5—42no—
206C2×C4×C5—81no—
207C2×C4×C5#12012yes—
208C2×C4×C5C2×SD1614yes—
209C2×C4×C5C22×D412yes—
210C2×Dic5—42no—
211C2×Dic5—42no—
212C2×Dic5—42no—
213C2×Dic5—42no—
214C2×Dic5—41no—
215C2×Dic5—41no—
216C2×Dic5—42no—
217Dic10—44no—
218#168—44no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8—51no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4C22⋊D512yes—
229C23×C4—22no—
230C23×C4C22⋊D511yescharacteristic
231C23×C4C22⋊D511yescharacteristic
232C23×C4C22×D511yescharacteristic
233C2×SD16—104no—
234C22×C8—102no—
235C22⋊Q8—28no—
236C22⋊Q8—28no—
237C22⋊Q8—102no—
238C22⋊Q8—102no—
239#121—24no—
240#121—28no—
241#121—28no—
242#121—24no—
243#121—22no—
244#121—22no—
245#121C22⋊D514yes—
246#121C22×D514yes—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—51no—
251C2×C4⋊C4—101no—
252C2×C4⋊C4—101no—
253C2×C4⋊C4—102no—
254C2×C4⋊C4C22⋊D512yes—
255C2×C4⋊C4C22⋊D512yes—
256C2×C4⋊C4C22×D512yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—101no—
260C2×C22⋊C4—101no—
261#106—104no—
262#103—104no—
263#102—104no—
264C22⋊C8—54no—
265C22⋊C8—102no—
266#95—52no—
267#95—102no—
268#95—102no—
269#95—102no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—21no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5—21no—
276C22×C5—42no—
277C22×C5—42no—
278C22×C5—44no—
279C22×C5—44no—
280C22×C5—44no—
281C22×C5—88no—
282C22×C5#45012yes—
283C22×C5#45012yes—
284C22×C5C2×#12011yescharacteristic
285C22×C5C22×SD1611yescharacteristic
286C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C5—24no—
288C4×C5—22no—
289C4×C5—22no—
290C4×C5—44no—
291C4×C5—42no—
292Dic5—82no—
293Dic5—81no—
294Dic5—81no—
295C24—22no—
296C24C2×C22⋊D511yescharacteristic
297#41—48no—
298#41—416no—
299#41—48no—
300C2×Q8—24no—
301C2×Q8—44no—
302C2×Q8—102no—
303C2×Q8—401no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C22⋊D514yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—48no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—102no—
326C22×C4—101no—
327C22×C4—101no—
328C22×C4#68714yescomplemented
329C22×C4C2×C22⋊D512yes—
330C22×C4C2×C22⋊D512yes—
331C22×C4C2×C22⋊D511yescharacteristic
332C22×C4C2×C22⋊D512yes—
333C22×C4C2×C22⋊D512yes—
334C22×C4C23×D511yescharacteristic
335SD16—404no—
336C2×C8—104no—
337C2×C8—402no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—102no—
343C4⋊C4—202no—
344C4⋊C4—402no—
345C4⋊C4—401no—
346C4⋊C4C2×C22⋊D514yes—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—201no—
351C22⋊C4—201no—
352C22⋊C4—202no—
353C2×C5—21no—
354C2×C5—22no—
355C2×C5—21no—
356C2×C5—22no—
357C2×C5—84no—
358C2×C5C2×#45011yescharacteristic
359C2×C5#302011yescharacteristic
360C2×C5#302611yescharacteristic
361C23—24no—
362C23—41no—
363C23—41no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23#407812yes—
368C23C2×#68712yes—
369C23#409911yescharacteristic
370C23#409911yescharacteristic
371C23C22×C22⋊D511yescharacteristic
372Q8—48no—
373Q8—204no—
374D4—44no—
375D4—48no—
376D4—44no—
377D4—44no—
378D4—48no—
379D4—82no—
380D4—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—416no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—202no—
397C2×C4—202no—
398C2×C4—202no—
399C2×C4—202no—
400C2×C4—202no—
401C2×C4—201no—
402C2×C4—201no—
403C2×C4C2×#68714yes—
404C2×C4#409912yes—
405C2×C4C22×C22⋊D512yes—
406C8—204no—
407C5#1829311yescharacteristic, complemented, Sylow
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22#6550712yes—
421C22#6550712yes—
422C22C2×#407811yescharacteristic
423C22C22×#68711yescharacteristic
424C22C2×#409911yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—402no—
431C4—401no—
432C4—401no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2—22no—
437C2—84no—
438C2C2×#6550711yescharacteristic
439C2#1061584111yescharacteristic
440C2#1061584711yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
Fitting, prime radicalC2×C5×#537C2
socleC22×C5C2×#120
hypercenterC2×#537D5
hypocenterC5#18293
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. #10614361C2semidirect
3. #10613914C2semidirect
4. #10602673C2semidirect
5. #10602668C2semidirect
6. #10602468C2semidirect
7. C2×#64748C2semidirect
24. #65329C22semidirect
25. #65323C22semidirect
26. #65275C22semidirect
28. #65268C22semidirect
30. #64757C22semidirect
31. #64748C22semidirect
119. C22×C4×C5SD16semidirect
328. C22×C4#687semidirect
407. C5#18293semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5372
#106143612
#106139142
#106026732
#106026682
#106024682
C2×#647482
#182935