Finite Group Databasebeta

← #12109869

Subgroups — #12109869

Order 1280.

830 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64994C211yesmaximal, characteristic
2#10606049C211yesmaximal, characteristic
3#10605879C211yesmaximal, characteristic
4C5×#1847C211yesmaximal, characteristic, complemented
5#10601426C212yesmaximal
6#10601408C212yesmaximal, complemented
7#10601373C212yesmaximal
8#10601356C212yesmaximal, complemented
9#10600608C211yesmaximal, characteristic, complemented
10#10600371C211yesmaximal, characteristic, complemented
11#10600341C211yesmaximal, characteristic, complemented
12C2×#4038—21no—
13C2×#4038C2211yescharacteristic
14C2×#4038C2211yescharacteristic
15C2×C42×D5C2211yescharacteristic
16C2×C5×#107C2212yes—
17C2×C5×#106C2212yes—
18C2×C5×#97C2211yescharacteristic
19C5×#384C2211yescharacteristic
20C5×#382C2211yescharacteristic
21C2×#3952C2212yes—
22C2×#3946C2212yes—
23#65276—21no—
24#65276C2212yes—
25#65266—21no—
26#65265C2211yescharacteristic
27#65265C2211yescharacteristic
28C4⋊C4×Dic5—21no—
29C4⋊C4×Dic5C2212yes—
30C2×#3940C2212yes—
31C2×#3939C2212yes—
32#65225C2211yescharacteristic
33#65224C2211yescharacteristic
34C2×#3935C2211yescharacteristic
35#64994C418yescomplemented
36#64956—21no—
37#64956—21no—
38#64938—21no—
39#64938—21no—
40#64771C2212yes—
41#64771C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#21988—51nomaximal, Sylow
45C22×C4×D5D411yescharacteristic
46C22×C4×D5D411yescharacteristic
47C22×C4×D5C2311yescharacteristic
48C22×C5×C8C2312yes—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4C2311yescharacteristic
52C2×C5×C4⋊C4C2311yescharacteristic
53C2×C5×C4⋊C4C2312yes—
54C2×C42×C5C2311yescharacteristic
55C2×#685—22no—
56C2×#685—21no—
57C2×#685—21no—
58C2×C4⋊Dic5—21no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5C2311yescharacteristic
61C2×C4⋊Dic5C2311yescharacteristic
62C2×C4⋊Dic5C2312yes—
63C2×#683—21no—
64C2×#683—21no—
65C2×#683—21no—
66C2×#683—21no—
67C2×C4×Dic5—22no—
68C2×C4×Dic5—21no—
69C2×C4×Dic5—21no—
70C2×C4×Dic5D411yescharacteristic
71C2×C4×Dic5D411yescharacteristic
72C2×C4×Dic5C2311yescharacteristic
73C22×#168C2312yes—
74#4038—42no—
75#4038—42no—
76#4038—44no—
77C42×D5C2×C414yes—
78C42×D5Q814yes—
79C5×#107—42no—
80C5×#106—42no—
81C5×#97C2×C414yes—
82C5×#95—21no—
83C5×#95—21no—
84#3963—21no—
85#3963—21no—
86#3963—22no—
87#3963—22no—
88#3952C2×C418yes—
89#3946C2×C418yes—
90#3940—42no—
91#3939—42no—
92#3935C2×C414yes—
93C2×#432—51no—
94#2285—51no—
95#2285—51no—
96#1948—52no—
97#1947—52no—
98#1854—52no—
99#1854—52no—
100#1847—51no—
101#1847D511yescharacteristic, complemented
102#1836—51no—
103#1836—51no—
104C23×D5C2×D411yescharacteristic
105C22×C4×C5—22no—
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5C2×D411yescharacteristic
110C22×C4×C5C2411yescharacteristic
111C22×Dic5—21no—
112C22×Dic5—21no—
113C22×Dic5—22no—
114C22×Dic5C2×D411yescharacteristic
115C22×Dic5C2×D411yescharacteristic
116C22×Dic5C2×D411yescharacteristic
117C2×C4×D5—22no—
118C2×C4×D5—22no—
119C2×C4×D5—22no—
120C2×C4×D5—22no—
121C2×C4×D5—41no—
122C2×C4×D5—41no—
123C2×C4×D5C4⋊C414yes—
124C2×C4×D5C4⋊C414yes—
125C2×C4×D5C22×C412yes—
126C2×C4×D5C2×Q812yes—
127C2×C5×C8—22no—
128C2×C5×C8—22no—
129C2×C5×C8C22×C414yes—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—22no—
132C5×C4⋊C4—22no—
133C5×C4⋊C4—22no—
134C5×C4⋊C4—42no—
135C5×C4⋊C4—42no—
136C42×C5C22×C412yes—
137C42×C5C2×Q812yes—
138#685—42no—
139#685—42no—
140#685—44no—
141C4⋊Dic5—22no—
142C4⋊Dic5—22no—
143C4⋊Dic5—24no—
144C4⋊Dic5—42no—
145C4⋊Dic5—42no—
146#683—42no—
147#683—42no—
148#683—42no—
149#683—42no—
150C4×Dic5—42no—
151C4×Dic5—42no—
152C4×Dic5—44no—
153C4×Dic5C4⋊C414yes—
154C4×Dic5C4⋊C414yes—
155C4×Dic5C22×C412yes—
156C4×Dic5C2×Q812yes—
157C2×#168—22no—
158C2×#168—22no—
159C2×#168C22×C414yes—
160C2×#117—51no—
161C2×#117—51no—
162C2×#117—101no—
163C22×C42—51no—
164#432—58no—
165C2×#107—52no—
166C2×#107C2×D512yes—
167C2×#106—52no—
168C2×#106C2×D512yes—
169C2×C4⋊C8—52no—
170C2×C22⋊C8—52no—
171C2×#97—51no—
172C2×#97C2×D511yescharacteristic
173#384—51no—
174#384—51no—
175#384—51no—
176#384C2×D511yescharacteristic
177#382—51no—
178#382—101no—
179#382C2×D511yescharacteristic
180#380—52no—
181#380—101no—
182#380—101no—
183#380—101no—
184C4×C4⋊C4—52no—
185C4×C4⋊C4—101no—
186C4×C4⋊C4—101no—
187C4×C4⋊C4—101no—
188#340—52no—
189#340—52no—
190#340—52no—
191#340—52no—
192C23×C5C22×D411yescharacteristic
193C22×D5—21no—
194C22×D5—21no—
195C22×D5—21no—
196C22×D5—21no—
197C22×D5—21no—
198C22×D5—21no—
199C22×D5C2×C4⋊C412yes—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5—41no—
205C2×C4×C5—41no—
206C2×C4×C5—42no—
207C2×C4×C5—42no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5—42no—
213C2×C4×C5C2×C4⋊C412yes—
214C2×C4×C5C2×C4⋊C412yes—
215C2×C4×C5C23×C411yescharacteristic
216C2×C4×C5C22×Q811yescharacteristic
217C2×C4×C5H(5,𝔽2)12yes—
218C2×C4×C52−1+412yes—
219C2×Dic5—21no—
220C2×Dic5—21no—
221C2×Dic5—21no—
222C2×Dic5—21no—
223C2×Dic5—41no—
224C2×Dic5—41no—
225C2×Dic5—42no—
226C2×Dic5—42no—
227C2×Dic5—41no—
228C2×Dic5—41no—
229C2×Dic5—41no—
230C2×Dic5—41no—
231C2×Dic5—42no—
232C2×Dic5C2×C4⋊C412yes—
233C2×Dic5C2×C4⋊C412yes—
234C2×Dic5C2×C4⋊C412yes—
235C2×Dic5AGL(1,ℤ/8)12yes—
236C2×Dic5#13712yes—
237C4×D5—24no—
238C4×D5—24no—
239C4×D5—28no—
240C4×D5—44no—
241C5×C8—24no—
242#168—24no—
243C23×C4—51no—
244C23×C4—51no—
245C23×C4—51no—
246C22×C8—52no—
247C22×C8C22×D512yes—
248#117—202no—
249#117—202no—
250#117—204no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—51no—
254C2×C4⋊C4—51no—
255C2×C4⋊C4—52no—
256C2×C4⋊C4—101no—
257C2×C4⋊C4—101no—
258C2×C4⋊C4—101no—
259C2×C4⋊C4—101no—
260C2×C4⋊C4—101no—
261C2×C4⋊C4—101no—
262C2×C4⋊C4C22×D511yescharacteristic
263C2×C4⋊C4C22×D511yescharacteristic
264C2×C4⋊C4C22×D512yes—
265C2×C22⋊C4—101no—
266C2×C22⋊C4—101no—
267C2×C22⋊C4—102no—
268C2×C42—51no—
269C2×C42—51no—
270C2×C42—51no—
271C2×C42—54no—
272C2×C42—54no—
273C2×C42—101no—
274C2×C42—101no—
275C2×C42—102no—
276C2×C42C22×D511yescharacteristic
277#107—42no—
278#107—202no—
279#106—42no—
280#106—202no—
281C4⋊C8—58no—
282C22⋊C8—58no—
283#97—54no—
284#97C4×D514yes—
285#95—21no—
286#95—21no—
287#95—102no—
288#95—102no—
289#95—101no—
290#95—101no—
291C22×C5C22×C4⋊C411yescharacteristic
292C22×C5#53511yescharacteristic
293C22×C5#53611yescharacteristic
294C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C5C2×#13711yescharacteristic
296C22×C5#57511yescharacteristic
297C22×C5#57511yescharacteristic
298C2×D5—22no—
299C2×D5—22no—
300C2×D5—22no—
301C2×D5—41no—
302C2×D5—41no—
303C2×D5—41no—
304C2×D5—41no—
305C4×C5—22no—
306C4×C5—22no—
307C4×C5—81no—
308C4×C5—82no—
309C4×C5—81no—
310C4×C5#52012yes—
311C4×C5#54412yes—
312Dic5—22no—
313Dic5—22no—
314Dic5—81no—
315Dic5—81no—
316Dic5—82no—
317Dic5#42814yes—
318C24—51no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—52no—
326C22×C4—52no—
327C22×C4—54no—
328C22×C4—54no—
329C22×C4—102no—
330C22×C4—102no—
331C22×C4—102no—
332C22×C4—102no—
333C22×C4—101no—
334C22×C4—101no—
335C22×C4—102no—
336C22×C4—201no—
337C22×C4—201no—
338C22×C4D4×D511yescharacteristic
339C22×C4D4×D511yescharacteristic
340C22×C4C23×D511yescharacteristic
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—54no—
344C2×C8—102no—
345C2×C8—102no—
346C2×C8C2×C4×D514yes—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—102no—
354C4⋊C4—102no—
355C4⋊C4—104no—
356C4⋊C4—202no—
357C4⋊C4—202no—
358C4⋊C4—202no—
359C4⋊C4—202no—
360C4⋊C4—202no—
361C4⋊C4—202no—
362C22⋊C4—202no—
363C22⋊C4—202no—
364C22⋊C4—204no—
365C42—52no—
366C42—52no—
367C42—54no—
368C42—54no—
369C42—204no—
370C42—202no—
371C42—202no—
372C42C2×C4×D512yes—
373C42Q8×D512yes—
374C2×C5#231711yescharacteristic
375C2×C5C2×#42811yescharacteristic
376C2×C5#291111yescharacteristic
377C2×C5#321411yescharacteristic
378C2×C5#321611yescharacteristic
379C2×C5#323511yescharacteristic
380C2×C5#323711yescharacteristic
381D5—42no—
382C23—52no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23C2×D4×D511yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—52no—
404C2×C4—52no—
405C2×C4—52no—
406C2×C4—52no—
407C2×C4—52no—
408C2×C4—108no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—104no—
414C2×C4—104no—
415C2×C4—204no—
416C2×C4—201no—
417C2×C4—201no—
418C2×C4—202no—
419C2×C4—202no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—201no—
423C2×C4—201no—
424C2×C4—202no—
425C2×C4D5×C4⋊C412yes—
426C2×C4D5×C4⋊C412yes—
427C2×C4C22×C4×D511yescharacteristic
428C2×C4#414412yes—
429C2×C4C2×Q8×D511yescharacteristic
430C2×C4#414712yes—
431C8—24no—
432C8—104no—
433C5#2198811yescharacteristic, complemented, Sylow
434C22—102no—
435C22—102no—
436C22—102no—
437C22—201no—
438C22—201no—
439C22—201no—
440C22—201no—
441C22C2×D5×C4⋊C411yescharacteristic
442C22#6604911yescharacteristic
443C22#6605811yescharacteristic
444C22D5×AGL(1,ℤ/8)11yescharacteristic
445C22#6610711yescharacteristic
446C22D5×#13711yescharacteristic
447C22#6611111yescharacteristic
448C4—22no—
449C4—22no—
450C4—54no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4—102no—
455C4—102no—
456C4—401no—
457C4—401no—
458C4—402no—
459C4#6583812yes—
460C4#6606312yes—
461C2—202no—
462C2#1060609011yescharacteristic
463C2D5×#42811yescharacteristic
464C2#1061093911yescharacteristic
465C2#1061358111yescharacteristic
466C2#1061360611yescharacteristic
467C2#1061371011yescharacteristic
468C2#1061371211yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1847C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1847D5
hypocenterC5#21988
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. C5×#1847C2semidirect
6. #10601408C2semidirect
8. #10601356C2semidirect
9. #10600608C2semidirect
10. #10600371C2semidirect
11. #10600341C2semidirect
35. #64994C4semidirect
101. #1847D5semidirect
433. C5#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#649942
#106060492
#106058792
C5×#18472
#106014262
#106014082
#106013732
#106013562
#106006082
#106003712
#106003412
#219885