Finite Group Databasebeta

← #12262406

Subgroups — #12262406

Order 1280.

1926 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C5×#2381C211yesmaximal, characteristic, complemented
3#10614962C211yesmaximal, characteristic, complemented
4#10608347C214yesmaximal
5#10608214C214yesmaximal, complemented
6#10607034C214yesmaximal
7#10607031C214yesmaximal
8#10606695C214yesmaximal, complemented
9#10606691C214yesmaximal, complemented
10D5×#383C211yesmaximal, characteristic
11#10605996C211yesmaximal, characteristic, complemented
12#10605996C211yesmaximal, characteristic, complemented
13#10605989C211yesmaximal, characteristic
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×#4092C2212yes—
16C2×Q8×Dic5C2212yes—
17#66062C4116yescomplemented
18C2×#4043C2212yes—
19C2×#4038C2212yes—
20C2×#4036C2212yes—
21C2×#4033C2212yes—
22C2×C42×D5C2211yescharacteristic
23C2×#4015C2211yescharacteristic
24C5×#391C2214yes—
25C5×#387C2214yes—
26C5×#387C2214yes—
27C5×#383C2211yescharacteristic
28C5×#383C2211yescharacteristic
29#65519C2214yes—
30#65278C2214yes—
31#65270C2214yes—
32#65270C2214yes—
33#65267C2214yes—
34C4⋊C4×Dic5C2214yes—
35#65262C2214yes—
36C4×#685C2214yes—
37#65226C2211yescharacteristic
38#65226C2211yescharacteristic
39#65221C2214yes—
40C4×#683C2214yes—
41#65217C2214yes—
42#64957C2214yes—
43#64956C2212yes—
44#64955C2212yes—
45#64955C2214yes—
46#64953C2212yes—
47D5×#95C2212yes—
48D5×#95C2212yes—
49D5×#95C2212yes—
50#64944C2214yes—
51#64943C2214yes—
52#64943C2214yes—
53#64943C2214yes—
54#64942C2214yes—
55#64942C2214yes—
56#64942C2214yes—
57#64940C2212yes—
58#64940C2214yes—
59#64940C2212yes—
60#64940C2212yes—
61#64939C2212yes—
62#64939C2214yes—
63#64938C2212yes—
64#64938C2214yes—
65#64937C2212yes—
66#64935C2214yes—
67#64935C2214yes—
68#34389—51nomaximal, Sylow
69C22×C5×Q8C2312yes—
70C22×C4×D5C2311yescharacteristic
71C22×C4×D5C2314yes—
72C22×C4×D5C2311yescharacteristic
73C22×C4×D5C2311yescharacteristic
74C22×Dic10C2312yes—
75C5×C4⋊Q8C2×C418yescomplemented
76C2×C5×C4⋊C4C2314yes—
77C2×C5×C4⋊C4C2312yes—
78C2×C5×C4⋊C4C2312yes—
79C2×C42×C5C2314yes—
80C2×C42×C5C2311yescharacteristic
81#4092C2×C4116yes—
82Q8×Dic5C2×C4116yescomplemented
83C2×#685C2314yes—
84C2×#685C2314yes—
85C2×C4⋊Dic5C2314yes—
86C2×C4⋊Dic5C2312yes—
87C2×C4⋊Dic5C2312yes—
88C2×C4⋊Dic5C2314yes—
89C2×#683C2314yes—
90C2×#683C2314yes—
91C2×#683C2314yes—
92C2×#683C2314yes—
93C2×C4×Dic5C2314yes—
94C2×C4×Dic5C2314yes—
95C2×C4×Dic5C2314yes—
96C2×C4×Dic5C2314yes—
97C2×C4×Dic5C2311yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C2×C4×Dic5C2311yescharacteristic
100C2×C4×Dic5C2314yes—
101#4043C2×C4116yes—
102#4038C2×C4116yes—
103#4036C2×C4116yescomplemented
104#4033C2×C4116yescomplemented
105C42×D5C2×C418yes—
106#4015C2×C418yescomplemented
107C5×#95C2312yes—
108C5×#95C2312yes—
109C5×#95C2312yes—
110C5×#95C2312yes—
111C5×#95C2312yes—
112C5×#95C2312yes—
113C5×#95C2314yes—
114C5×#95C2314yes—
115#3963C2312yes—
116#3963C2312yes—
117#3963C2314yes—
118#3963C2314yes—
119#3963C2314yes—
120#3963C2314yes—
121#3963C2314yes—
122#3963C2314yes—
123#3963C2312yes—
124#3963C2314yes—
125#3963C2312yes—
126#3963C2314yes—
127#3963C2314yes—
128#3963C2312yes—
129#3963C2312yes—
130#3963C2314yes—
131C2×#533—51no—
132#2381—51no—
133#2381—51no—
134#2381—51no—
135#2381D511yescharacteristic, complemented
136#2368—54no—
137#2356—54no—
138#2343—54no—
139#2336—54no—
140#2329—54no—
141#2320—54no—
142#2283—51no—
143C2×#383—51no—
144C23×D5C2411yescharacteristic
145C2×C5×Q8—42no—
146C2×C5×Q8—42no—
147C2×C5×Q8C22×C418yes—
148C22×C4×C5C2414yes—
149C22×C4×C5C2414yes—
150C22×C4×C5C2414yes—
151C22×C4×C5C2411yescharacteristic
152C22×C4×C5C2411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×Dic5C2414yes—
155C22×Dic5C2414yes—
156C22×Dic5C2414yes—
157C22×Dic5C2411yescharacteristic
158C22×Dic5C2411yescharacteristic
159C22×Dic5C2411yescharacteristic
160C2×C4×D5—24no—
161C2×C4×D5—28no—
162C2×C4×D5—42no—
163C2×C4×D5—42no—
164C2×C4×D5—44no—
165C2×C4×D5—44no—
166C2×C4×D5C22×C414yes—
167C2×C4×D5C22×C414yes—
168C2×C4×D5C22×C414yes—
169C2×Dic10—42no—
170C2×Dic10—42no—
171C2×Dic10C22×C418yes—
172C5×C4⋊C4—44no—
173C5×C4⋊C4C22×C418yes—
174C5×C4⋊C4C22×C418yes—
175C42×C5—44no—
176C42×C5C22×C414yes—
177#685—28no—
178#685C22×C4116yes—
179C4⋊Dic5—44no—
180C4⋊Dic5C22×C418yes—
181C4⋊Dic5C22×C418yes—
182C4⋊Dic5C22×C4116yes—
183#683—44no—
184#683—44no—
185#683—44no—
186#683C22×C4116yes—
187C4×Dic5—44no—
188C4×Dic5—44no—
189C4×Dic5—44no—
190C4×Dic5—44no—
191C4×Dic5C22×C414yes—
192C4×Dic5C22×C414yes—
193C4×Dic5C22×C414yes—
194C4×Dic5C22×C4116yes—
195#533—516no—
196C2×C4⋊Q8—51no—
197C2×C4⋊Q8C2×D511yescharacteristic
198C2×#125—52no—
199C2×C22⋊Q8—52no—
200C2×C22⋊Q8—52no—
201C2×C4×Q8—52no—
202C2×C4×Q8—52no—
203C2×#117—52no—
204C22×C42—51no—
205#391—54no—
206#391—54no—
207#391—54no—
208#391C2×D514yes—
209#389—54no—
210#389—54no—
211#387—54no—
212#387—54no—
213#387—54no—
214#387—54no—
215#387—54no—
216#387—54no—
217#387—54no—
218#387—54no—
219#387—54no—
220#387—54no—
221#387C2×D514yes—
222#387C2×D514yes—
223#385—54no—
224#385—54no—
225#383—51no—
226#383—51no—
227#383—52no—
228#383—54no—
229#383—52no—
230#383—52no—
231#383C2×D511yescharacteristic
232#383C2×D511yescharacteristic
233#382—52no—
234#382—54no—
235#381—52no—
236#380—52no—
237C4×C4⋊C4—54no—
238C4×C4⋊C4—54no—
239C4×C4⋊C4—52no—
240C4×C4⋊C4—54no—
241C4×C22⋊C4—54no—
242C4×C22⋊C4—54no—
243C4×C22⋊C4—52no—
244#376—54no—
245#376—52no—
246C2×#95—52no—
247C2×#95—52no—
248C2×#95—52no—
249C23×C5C2511yescharacteristic
250C22×D5—22no—
251C22×D5—22no—
252C22×D5C23×C412yes—
253C22×D52−1+414yes—
254C5×Q8—48no—
255C2×C4×C5—22no—
256C2×C4×C5—24no—
257C2×C4×C5—24no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5C23×C418yes—
267C2×C4×C5C23×C412yes—
268C2×C4×C5C23×C412yes—
269C2×C4×C5C23×C412yes—
270C2×Dic5—22no—
271C2×Dic5—24no—
272C2×Dic5—22no—
273C2×Dic5—24no—
274C2×Dic5—22no—
275C2×Dic5—24no—
276C2×Dic5—24no—
277C2×Dic5—24no—
278C2×Dic5—24no—
279C2×Dic5—24no—
280C2×Dic5—24no—
281C2×Dic5C23×C412yes—
282C2×Dic5C23×C412yes—
283C2×Dic5C23×C412yes—
284C2×Dic5C23×C418yes—
285C4×D5—28no—
286C4×D5—44no—
287C4×D5—44no—
288C4×D5—88no—
289Dic10—48no—
290C22×Q8—52no—
291C22×Q8C22×D512yes—
292C23×C4—51no—
293C23×C4—51no—
294C23×C4—51no—
295C23×C4—54no—
296C4⋊Q8—58no—
297C4⋊Q8C4×D518yescomplemented
298#125—516no—
299C22⋊Q8—516no—
300C22⋊Q8—516no—
301C4×Q8—516no—
302C4×Q8—516no—
303#117—516no—
304C2×C4⋊C4—52no—
305C2×C4⋊C4—54no—
306C2×C4⋊C4—54no—
307C2×C4⋊C4—52no—
308C2×C4⋊C4—54no—
309C2×C4⋊C4—54no—
310C2×C4⋊C4—54no—
311C2×C4⋊C4—54no—
312C2×C4⋊C4C22×D514yes—
313C2×C4⋊C4C22×D512yes—
314C2×C4⋊C4C22×D512yes—
315C2×C22⋊C4—54no—
316C2×C22⋊C4—54no—
317C2×C42—54no—
318C2×C42—51no—
319C2×C42—51no—
320C2×C42—51no—
321C2×C42—58no—
322C2×C42—54no—
323C2×C42—54no—
324C2×C42—54no—
325C2×C42—54no—
326C2×C42C22×D514yes—
327C2×C42C22×D511yescharacteristic
328#95—54no—
329#95—54no—
330#95—52no—
331#95—54no—
332#95—52no—
333#95—52no—
334#95—54no—
335#95—52no—
336#95—54no—
337#95—54no—
338#95—52no—
339#95—54no—
340#95—54no—
341#95—54no—
342#95—52no—
343#95—54no—
344#95C22×D512yes—
345#95C22×D512yes—
346#95C22×D512yes—
347#95C22×D512yes—
348#95C22×D512yes—
349#95C22×D514yes—
350#95C22×D512yes—
351#95C22×D514yes—
352C22×C5C24×C411yescharacteristic
353C22×C5C2×2−1+412yes—
354C22×C5#58512yes—
355C22×C5#58512yes—
356C2×D5—22no—
357C2×D5—22no—
358C2×D5—28no—
359C2×D5#51914yes—
360C4×C5—22no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—28no—
364C4×C5—44no—
365C4×C5—44no—
366Dic5—22no—
367Dic5—28no—
368Dic5—22no—
369Dic5—22no—
370Dic5—44no—
371Dic5—44no—
372C24—51no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—58no—
376C2×Q8—202no—
377C2×Q8—202no—
378C2×Q8C2×C4×D518yes—
379C22×C4—51no—
380C22×C4—51no—
381C22×C4—54no—
382C22×C4—51no—
383C22×C4—54no—
384C22×C4—54no—
385C22×C4—54no—
386C22×C4—54no—
387C22×C4—54no—
388C22×C4—104no—
389C22×C4—108no—
390C22×C4—202no—
391C22×C4—202no—
392C22×C4—204no—
393C22×C4—204no—
394C22×C4C23×D514yes—
395C22×C4C23×D514yes—
396C22×C4C23×D511yescharacteristic
397C22×C4C23×D511yescharacteristic
398C22×C4C23×D511yescharacteristic
399C22×C4C23×D514yes—
400C4⋊C4—44no—
401C4⋊C4—58no—
402C4⋊C4—516no—
403C4⋊C4—516no—
404C4⋊C4—58no—
405C4⋊C4—204no—
406C4⋊C4—204no—
407C4⋊C4—204no—
408C4⋊C4—204no—
409C4⋊C4C2×C4×D518yes—
410C4⋊C4C2×C4×D518yes—
411C22⋊C4—516no—
412C22⋊C4—108no—
413C42—44no—
414C42—54no—
415C42—54no—
416C42—54no—
417C42—516no—
418C42—204no—
419C42—204no—
420C42—204no—
421C42—204no—
422C42C2×C4×D514yes—
423C2×C5C2×#51911yescharacteristic
424C2×C5#342811yescharacteristic
425C2×C5#342811yescharacteristic
426C2×C5#356414yes—
427D5—24no—
428C23—52no—
429C23—54no—
430C23—102no—
431C23—102no—
432C23C24×D511yescharacteristic
433Q8—48no—
434Q8—208no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—52no—
447C2×C4—52no—
448C2×C4—52no—
449C2×C4—58no—
450C2×C4—108no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—104no—
454C2×C4—102no—
455C2×C4—104no—
456C2×C4—104no—
457C2×C4—104no—
458C2×C4—104no—
459C2×C4—104no—
460C2×C4—104no—
461C2×C4—104no—
462C2×C4—204no—
463C2×C4—204no—
464C2×C4—408no—
465C2×C4C22×C4×D512yes—
466C2×C4C22×C4×D512yes—
467C2×C4C22×C4×D512yes—
468C2×C4C22×C4×D518yes—
469C5#3438911yescharacteristic, complemented, Sylow
470C22—54no—
471C22—102no—
472C22—108no—
473C22—102no—
474C22C23×C4×D511yescharacteristic
475C22#6628512yes—
476C22D5×2−1+412yes—
477C22#6628712yes—
478C4—22no—
479C4—28no—
480C4—22no—
481C4—22no—
482C4—44no—
483C4—44no—
484C4—102no—
485C4—102no—
486C4—108no—
487C4—102no—
488C4—204no—
489C4—204no—
490C2—104no—
491C2#1061775711yescharacteristic
492C2#1061777311yescharacteristic
493C2D5×#51911yescharacteristic
494C2#1061903914yes—
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2381C2
socle, cosocleC23×C5C25
prime radicalC23×D5C24
hypercenter#2381D5
hypocenterC5#34389
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. C5×#2381C2semidirect
3. #10614962C2semidirect
5. #10608214C2semidirect
8. #10606695C2semidirect
9. #10606691C2semidirect
11. #10605996C2semidirect
12. #10605996C2semidirect
17. #66062C4semidirect
75. C5×C4⋊Q8C2×C4semidirect
82. Q8×Dic5C2×C4semidirect
103. #4036C2×C4semidirect
104. #4033C2×C4semidirect
106. #4015C2×C4semidirect
135. #2381D5semidirect
297. C4⋊Q8C4×D5semidirect
469. C5#34389semidirect

Maximal subgroups

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

subgroupindex
C2×#660622
C5×#23812
#106149622
#106083472
#106082142
#106070342
#106070312
#106066952
#106066912
D5×#3832
#106059962
#106059892
#343895