Finite Group Databasebeta

← #12963706

Subgroups — #12963706

Order 1280.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618175C211yesmaximal, characteristic
2C5×#2961C211yesmaximal, characteristic, complemented
3#10615059C211yesmaximal, characteristic, complemented
4#10611692C218yesmaximal, complemented
5#10611691C218yesmaximal, complemented
6D5×#424C212yesmaximal
7#10610808C212yesmaximal, complemented
8#10610779C211yesmaximal, characteristic
9#10610775C211yesmaximal, characteristic, complemented
10#10610279C212yesmaximal
11#10610273C212yesmaximal, complemented
12C2×#64997C211yesmaximal, characteristic
13#10610211C211yesmaximal, characteristic, complemented
14C5×#533C2211yescharacteristic
15C2×D5×M16C2212yes—
16C22×#676C2211yescharacteristic
17C22×C8×D5C2211yescharacteristic
18#66063C414yes—
19#66023C414yes—
20#65968C414yes—
21#65957C414yes—
22#65912C2218yes—
23D5×#117C2212yes—
24#65851C2212yes—
25C2×#4018C2211yescharacteristic
26#65804C2211yescharacteristic
27C5×#446C2218yescomplemented
28C5×#424C2212yes—
29C5×#423C2211yescharacteristic
30C5×#405C2212yes—
31C2×C5×#97C2211yescharacteristic
32#65411C2212yes—
33#65408C2212yes—
34#65407C2212yes—
35#65391C2212yes—
36#65390C2212yes—
37C2×#3947C2211yescharacteristic
38C2×C8×Dic5C2211yescharacteristic
39#65313C2214yescomplemented
40Q8×#168C2214yescomplemented
41#65288C2212yes—
42#65286C2212yes—
43#65213C2211yescharacteristic
44C2×C4×#168C2211yescharacteristic
45#65128C2218yes—
46#65128C2218yes—
47D5×C4⋊C8C2218yes—
48#65120C2214yescomplemented
49#65120C2218yescomplemented
50#65119C2214yescomplemented
51#65117C2218yescomplemented
52#64997C2214yes—
53#64997C2214yes—
54#64997C2214yes—
55#64996C2214yes—
56#64990C2218yescomplemented
57#64990C2218yescomplemented
58#64978C2214yes—
59C4×#676C2214yes—
60C8×Dic10C2218yescomplemented
61#60193—51nomaximal, Sylow
62C22×C4×D5C2311yescharacteristic
63C2×C5×M16C2312yes—
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×C4⋊Q8C2×C412yes—
67C5×#125C2×C412yes—
68C5×C22⋊Q8C2×C414yes—
69C4×C5×Q8C2314yes—
70C5×#117C2312yes—
71C2×C42×C5C2311yescharacteristic
72#4092C2×C418yes—
73#4090C2×C418yes—
74C2×#685C2×C412yes—
75#4072C2312yes—
76#4070C2×C414yes—
77C2×#683C2×C412yes—
78C2×C4×Dic5C2311yescharacteristic
79C2×#681C2312yes—
80C22×#168C2311yescharacteristic
81C22×#168C2311yescharacteristic
82D5×M16—28no—
83C2×#676Q818yescomplemented
84C2×#676C2314yes—
85C2×#676C2312yes—
86C2×#676C2312yes—
87C2×C8×D5—24no—
88C2×C8×D5C2314yes—
89C2×C8×D5C2314yes—
90#4043C2×C418yes—
91#4042C2×C418yes—
92#4042C2×C418yes—
93#4038C2×C414yes—
94D5×C4⋊C4C2×C414yes—
95#4036C2×C414yes—
96#4035C2×C418yes—
97#4035C2×C418yes—
98#4034C2×C414yes—
99#4034C2×C418yes—
100D5×C22⋊C4C2×C414yes—
101#4024C2×C418yes—
102#4023C2×C414yes—
103#4018C2×C412yes—
104#4018C2×C412yes—
105#4018C2314yes—
106#4018C2314yes—
107C42×D5C2314yes—
108#4016C2×C412yes—
109#4015C2×C412yes—
110C4×Dic10C2314yes—
111C4×Dic10C2318yes—
112C5×C4⋊C8C2314yes—
113C5×C4⋊C8C2318yes—
114C5×#97C2314yes—
115C5×#97C2314yes—
116C4×C5×C8C2314yes—
117#3947C2312yes—
118#3947C2312yes—
119#3947C2314yes—
120#3947C2314yes—
121#3946C2318yes—
122#3946C2318yes—
123#3946C2318yes—
124C8×Dic5C2314yes—
125C8×Dic5C2314yes—
126C8×Dic5C2314yes—
127C4⋊#168C2318yes—
128C4⋊#168C2314yes—
129#3935C2314yes—
130C4×#168C2314yes—
131C4×#168C2312yes—
132C4×#168C2312yes—
133#3458—51no—
134#2962—58no—
135#2962—58no—
136#2961—52no—
137#2961—51no—
138#2961D511yescharacteristic, complemented
139#2960—51no—
140#2960—52no—
141#2960—51no—
142#2905—51no—
143C2×#424—52no—
144#2873—52no—
145C2×#405—51no—
146C23×D5C22×C411yescharacteristic
147C2×C5×Q8C22×C414yes—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C2411yescharacteristic
150C22×Dic5C22×C412yes—
151C22×Dic5C22×C411yescharacteristic
152C2×C4×D5C22×C418yes—
153C2×C4×D5C22×C412yes—
154C2×C4×D5C2×Q814yes—
155C2×C4×D5#4114yes—
156C2×C4×D5C2412yes—
157C2×Dic10C22×C414yes—
158C2×Dic10C22×C418yes—
159C5×M16—24no—
160C2×C5×C8—22no—
161C2×C5×C8C2×Q814yes—
162C2×C5×C8C2412yes—
163C2×C5×C8C2414yes—
164C2×C5×C8C2412yes—
165C5×C4⋊C4C22×C414yes—
166C5×C4⋊C4C22×C418yes—
167C5×C4⋊C4C22×C414yes—
168C5×C22⋊C4C22×C414yes—
169C42×C5C22×C411yescharacteristic
170C42×C5C22×C411yescharacteristic
171C42×C5C2412yes—
172C42×C5C2414yes—
173C22⋊Dic5C22×C414yes—
174#685C22×C414yes—
175#685C22×C414yes—
176#685C22×C418yes—
177C4⋊Dic5C22×C414yes—
178C4⋊Dic5C22×C418yes—
179C4⋊Dic5C22×C414yes—
180#683C22×C418yes—
181#683C22×C418yes—
182#683C22×C418yes—
183#683C22×C414yes—
184#683C22×C414yes—
185#683C22×C418yes—
186C4×Dic5C22×C418yes—
187C4×Dic5C22×C412yes—
188C4×Dic5C2414yes—
189C4×Dic5C2412yes—
190#681—24no—
191C2×#168—22no—
192C2×#168C2×Q814yes—
193C2×#168C2412yes—
194C2×#168C2414yes—
195C2×#168C2411yescharacteristic
196C2×#168C2411yescharacteristic
197#676—28no—
198#676C2×Q818yes—
199#676C2×Q818yes—
200C8×D5—28no—
201C8×D5—28no—
202C22×M16—52no—
203C22×M16—51no—
204C23×C8—51no—
205#544—54no—
206#544—54no—
207#544—54no—
208#543—54no—
209#533—58no—
210#533—52no—
211#533—51no—
212#533C2×D511yescharacteristic
213C2×#117—52no—
214C2×#117—51no—
215#446—54no—
216#446—58no—
217#446—54no—
218#446—58no—
219#446—58no—
220#446—58no—
221#446C2×D518yescomplemented
222C8×Q8—54no—
223C8×Q8—54no—
224C8×Q8—58no—
225#424—52no—
226#424—52no—
227#424—52no—
228#424—58no—
229#424—58no—
230#424C2×D512yes—
231#423—52no—
232#423—51no—
233#423C2×D511yescharacteristic
234C2×C4⋊C8—58no—
235#405—52no—
236#405—52no—
237#405—52no—
238#405—54no—
239#405—54no—
240#405—54no—
241#405—54no—
242#405C2×D512yes—
243C4×M16—54no—
244C4×M16—54no—
245C2×#97—51no—
246C2×#97C2×D511yescharacteristic
247C2×C4×C8—51no—
248C2×C4×C8—51no—
249C23×C5C23×C411yescharacteristic
250C22×D5—42no—
251C22×D5C4×Q814yes—
252C22×D5C23×C412yes—
253C5×Q8—44no—
254C2×C4×C5—42no—
255C2×C4×C5C23×C418yes—
256C2×C4×C5C23×C412yes—
257C2×C4×C5C23×C412yes—
258C2×C4×C5C23×C411yescharacteristic
259C2×C4×C5C22×Q812yes—
260C2×C4×C5C2×#4112yes—
261C2×C4×C5C2511yescharacteristic
262C2×Dic5—42no—
263C2×Dic5C4×Q814yes—
264C2×Dic5C23×C414yes—
265C2×Dic5C23×C418yes—
266C2×Dic5C23×C412yes—
267C4×D5—48no—
268C4×D5C4×Q818yes—
269C4×D5C22×Q814yes—
270C4×D5C2×#4114yes—
271Dic10—44no—
272Dic10—48no—
273C5×C8—22no—
274C5×C8—24no—
275C5×C8C22×Q814yes—
276#168—22no—
277#168—24no—
278#168C22×Q812yes—
279#168C22×Q812yes—
280C23×C4—51no—
281C2×M16—52no—
282C2×M16—58no—
283C2×M16—52no—
284C2×M16—52no—
285C2×M16—54no—
286C2×M16—108no—
287C2×M16C22×D512yes—
288C22×C8—51no—
289C22×C8—51no—
290C22×C8—54no—
291C22×C8—54no—
292C22×C8—104no—
293C22×C8C22×D511yescharacteristic
294C22×C8C22×D511yescharacteristic
295C4⋊Q8—58no—
296C4⋊Q8—58no—
297C4⋊Q8—54no—
298C4⋊Q8—52no—
299C4⋊Q8C4×D512yes—
300#125—58no—
301#125—58no—
302#125—54no—
303#125—52no—
304#125C4×D512yes—
305C22⋊Q8—58no—
306C22⋊Q8—54no—
307C22⋊Q8—58no—
308C22⋊Q8—58no—
309C22⋊Q8—58no—
310C22⋊Q8—58no—
311C22⋊Q8C4×D514yes—
312C4×Q8—54no—
313C4×Q8—58no—
314C4×Q8C22×D514yes—
315#117—52no—
316#117—54no—
317#117—54no—
318#117—52no—
319#117—52no—
320#117—54no—
321#117—54no—
322#117C22×D512yes—
323C2×C4⋊C4—52no—
324C2×C4⋊C4—54no—
325C2×C22⋊C4—52no—
326C2×C22⋊C4—54no—
327C2×C42—51no—
328C2×C42—54no—
329C2×C42C22×D511yescharacteristic
330C4⋊C8—58no—
331C4⋊C8—58no—
332C4⋊C8—58no—
333C4⋊C8—58no—
334C4⋊C8—54no—
335C4⋊C8C22×D518yes—
336C4⋊C8C22×D514yes—
337#97—52no—
338#97—52no—
339#97—54no—
340#97—54no—
341#97—54no—
342#97C22×D514yes—
343#97C22×D514yes—
344C4×C8—54no—
345C4×C8—54no—
346C4×C8—54no—
347C4×C8—52no—
348C4×C8—52no—
349C4×C8—54no—
350C4×C8C22×D514yes—
351C22×C5—22no—
352C22×C5C2×C4×Q812yes—
353C22×C5C24×C411yescharacteristic
354C2×D5—24no—
355C2×D5—44no—
356C2×D5C2×C4×Q814yes—
357C4×C5—22no—
358C4×C5—28no—
359C4×C5—22no—
360C4×C5C2×C4×Q812yes—
361C4×C5C23×Q811yescharacteristic
362C4×C5C22×#4111yescharacteristic
363Dic5—28no—
364Dic5—24no—
365Dic5C2×C4×Q814yes—
366C24—51no—
367C2×Q8—58no—
368C2×Q8—54no—
369C2×Q8C2×C4×D514yes—
370C22×C4—51no—
371C22×C4—52no—
372C22×C4—54no—
373C22×C4—54no—
374C22×C4—52no—
375C22×C4—52no—
376C22×C4—58no—
377C22×C4C2×C4×D512yes—
378C22×C4C23×D511yescharacteristic
379M16—24no—
380M16—58no—
381M16—58no—
382M16—108no—
383M16—104no—
384C2×C8—22no—
385C2×C8—54no—
386C2×C8—51no—
387C2×C8—51no—
388C2×C8—52no—
389C2×C8—54no—
390C2×C8—108no—
391C2×C8—108no—
392C2×C8—102no—
393C2×C8Q8×D514yes—
394C2×C8C23×D514yes—
395C2×C8C23×D512yes—
396C2×C8C23×D512yes—
397C4⋊C4—58no—
398C4⋊C4—54no—
399C4⋊C4—54no—
400C4⋊C4—54no—
401C4⋊C4—58no—
402C4⋊C4—58no—
403C4⋊C4—58no—
404C4⋊C4—58no—
405C4⋊C4—54no—
406C4⋊C4C2×C4×D514yes—
407C4⋊C4C2×C4×D514yes—
408C4⋊C4C2×C4×D518yes—
409C22⋊C4—54no—
410C22⋊C4—58no—
411C22⋊C4—54no—
412C22⋊C4—54no—
413C22⋊C4C2×C4×D514yes—
414C42—58no—
415C42—52no—
416C42—52no—
417C42—54no—
418C42C2×C4×D511yescharacteristic
419C42C2×C4×D511yescharacteristic
420C42C23×D512yes—
421C42C23×D514yes—
422C2×C5—22no—
423C2×C5C22×C4×Q811yescharacteristic
424C2×C5#357312yes—
425D5—24no—
426C23—52no—
427C23—54no—
428C23—202no—
429C23C22×C4×D511yescharacteristic
430Q8—44no—
431Q8—204no—
432Q8—208no—
433C2×C4—42no—
434C2×C4—54no—
435C2×C4—52no—
436C2×C4—58no—
437C2×C4—54no—
438C2×C4—54no—
439C2×C4—58no—
440C2×C4—54no—
441C2×C4—208no—
442C2×C4—202no—
443C2×C4C22×C4×D518yes—
444C2×C4C22×C4×D512yes—
445C2×C4C22×C4×D512yes—
446C2×C4C22×C4×D511yescharacteristic
447C2×C4C2×Q8×D512yes—
448C2×C4D5×#4112yes—
449C2×C4C24×D511yescharacteristic
450C8—22no—
451C8—24no—
452C8—52no—
453C8—52no—
454C8—102no—
455C8—104no—
456C8C2×Q8×D514yes—
457C5#6019311yescharacteristic, complemented, Sylow
458C22—22no—
459C22—54no—
460C22—104no—
461C22—204no—
462C22C4×Q8×D512yes—
463C22C23×C4×D511yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—54no—
468C4—108no—
469C4—104no—
470C4C4×Q8×D512yes—
471C4C22×Q8×D511yescharacteristic
472C4C2×D5×#4111yescharacteristic
473C2—22no—
474C2—104no—
475C2C2×C4×Q8×D511yescharacteristic
476C2#1061906712yes—
subgroupG/H
center, FrattiniC2×C4C24×D5
commutator, socleC22×C5C24×C4
FittingC5×#2961C2
prime radicalC23×D5C22×C4
cosocleC2×C4×C5C25
hypercenter#2961D5
hypocenterC5#60193
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. C5×#2961C2semidirect
3. #10615059C2semidirect
4. #10611692C2semidirect
5. #10611691C2semidirect
7. #10610808C2semidirect
9. #10610775C2semidirect
11. #10610273C2semidirect
13. #10610211C2semidirect
27. C5×#446C22semidirect
39. #65313C22semidirect
40. Q8×#168C22semidirect
48. #65120C22semidirect
49. #65120C22semidirect
50. #65119C22semidirect
51. #65117C22semidirect
56. #64990C22semidirect
57. #64990C22semidirect
60. C8×Dic10C22semidirect
83. C2×#676Q8semidirect
138. #2961D5semidirect
221. #446C2×D5semidirect
457. C5#60193semidirect

Maximal subgroups

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

subgroupindex
#106181752
C5×#29612
#106150592
#106116922
#106116912
D5×#4242
#106108082
#106107792
#106107752
#106102792
#106102732
C2×#649972
#106102112
#601935