Finite Group Databasebeta

← #12260405

Subgroups — #12260405

Order 1280.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65906C211yesmaximal, characteristic
2C5×#2276C211yesmaximal, characteristic, complemented
3Q8×#685C211yesmaximal, characteristic
4Q8×#685C211yesmaximal, characteristic
5#10614192C211yesmaximal, characteristic, complemented
6Q8×#683C211yesmaximal, characteristic, complemented
7C4×Q8×Dic5C211yesmaximal, characteristic, complemented
8#10605491C213yesmaximal
9#10605375C213yesmaximal, complemented
10#10605372C213yesmaximal, complemented
11#10605365C213yesmaximal
12#10605364C213yesmaximal
13#10605349C213yesmaximal, complemented
14#10605117C213yesmaximal, complemented
15#10605116C213yesmaximal
16C22×Q8×D5C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×Q8C2211yescharacteristic
20C2×Q8×Dic5C2211yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23#65906C4116yescomplemented
24C2×#4038C2213yes—
25C2×#4033C2213yes—
26C2×#4018C2213yes—
27C2×C4×Dic10C2213yes—
28C4×C5×C4⋊C4C2213yes—
29C4×C5×C4⋊C4C2213yes—
30C4×C5×C4⋊C4C2213yes—
31C5×#376C2213yes—
32#65276C2213yes—
33#65276C2213yes—
34#65266C2213yes—
35#65265C2213yes—
36C4⋊C4×Dic5C2213yes—
37C4⋊C4×Dic5C2213yes—
38#65263C2213yes—
39#65263C2213yes—
40#65262C2213yes—
41#65262C2213yes—
42#65229C2216yes—
43C4×#685C2213yes—
44C4×#685C2213yes—
45C4×C4⋊Dic5C2213yes—
46#65222C2216yes—
47#65221C2216yes—
48C4×#683C2216yes—
49C4×#683C2213yes—
50C4×#683C2213yes—
51C42×Dic5C2213yes—
52#65218C2213yes—
53#65218C2213yes—
54#65217C2213yes—
55#65217C2213yes—
56#64958C2213yes—
57#64958C2213yes—
58#64958C2211yescharacteristic
59#64958C2213yes—
60#64941C2213yes—
61#64941C2216yes—
62#64941C2213yes—
63#64941C2213yes—
64#64941C2213yes—
65#64940C2216yes—
66#64939C2216yes—
67#34262—51nomaximal, Sylow
68C22×C5×Q8C2311yescharacteristic
69C2×Q8×D5—24no—
70C2×Q8×D5—24no—
71C2×Q8×D5C2×C418yes—
72C22×C4×D5C2313yes—
73C22×Dic10C2313yes—
74C4×C5×Q8—24no—
75C4×C5×Q8—24no—
76C4×C5×Q8C2×C418yes—
77C2×C5×C4⋊C4C2313yes—
78C2×C5×C4⋊C4C2313yes—
79C2×C5×C4⋊C4C2313yes—
80C2×C42×C5C2313yes—
81C2×C42×C5C2313yes—
82C2×C42×C5C2316yes—
83C2×C42×C5C2313yes—
84Q8×Dic5—24no—
85Q8×Dic5—24no—
86Q8×Dic5C2×C418yes—
87C2×#685C2311yescharacteristic
88C2×#685C2311yescharacteristic
89C2×#685C2313yes—
90C2×#685C2313yes—
91C2×#685C2311yescharacteristic
92C2×#685C2313yes—
93C2×C4⋊Dic5C2313yes—
94C2×C4⋊Dic5C2313yes—
95C2×C4⋊Dic5C2313yes—
96C2×#683C2311yescharacteristic
97C2×#683C2313yes—
98C2×#683C2313yes—
99C2×#683C2316yes—
100C2×#683C2313yes—
101C2×#683C2313yes—
102C2×#683C2313yes—
103C2×C4×Dic5C2313yes—
104C2×C4×Dic5C2313yes—
105C2×C4×Dic5C2311yescharacteristic
106C2×C4×Dic5C2313yes—
107C2×C4×Dic5C2313yes—
108C2×C4×Dic5C2313yes—
109C2×C4×Dic5C2316yes—
110C2×C4×Dic5C2313yes—
111#4038C2×C4124yes—
112#4033C2×C4124yes—
113#4018C2×C4124yes—
114C4×Dic10C2×C4124yes—
115C5×#95C2311yescharacteristic
116C5×#95C2313yes—
117C5×#95C2313yes—
118C5×#95C2313yes—
119#3963C2313yes—
120#3963C2316yes—
121#3963C2316yes—
122#3963C2313yes—
123#3963C2313yes—
124#3963C2316yes—
125#3963C2316yes—
126#3963C2311yescharacteristic
127#3963C2313yes—
128#3963C2313yes—
129#3963C2313yes—
130#3963C2313yes—
131C2×#519—51no—
132#2356—53no—
133#2355—53no—
134#2351—51no—
135Q8×C4⋊C4—51no—
136#2347—53no—
137#2346—53no—
138Q8×C22⋊C4—51no—
139Q8×C22⋊C4—51no—
140#2320—53no—
141#2314—53no—
142#2298—53no—
143#2295—53no—
144#2276D511yescharacteristic, complemented
145C42×Q8—51no—
146C23×D5C2411yescharacteristic
147C2×C5×Q8C22×C414yes—
148C2×C5×Q8#4114yes—
149C2×C5×Q8#4114yes—
150C22×C4×C5C2413yes—
151C22×C4×C5C2413yes—
152C22×C4×C5C2411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×C4×C5C2413yes—
155C22×C4×C5C2411yescharacteristic
156C22×C4×C5C2413yes—
157C22×Dic5C2413yes—
158C22×Dic5C2413yes—
159C22×Dic5C2413yes—
160C22×Dic5C2411yescharacteristic
161C22×Dic5C2413yes—
162C22×Dic5C2411yescharacteristic
163C22×Dic5C2411yescharacteristic
164Q8×D5—232no—
165C2×C4×D5—26no—
166C2×C4×D5—26no—
167C2×C4×D5C22×C4112yes—
168C2×Dic10—26no—
169C2×Dic10—26no—
170C2×Dic10C22×C4112yes—
171C5×C4⋊C4—26no—
172C5×C4⋊C4—26no—
173C5×C4⋊C4C22×C4112yes—
174C42×C5—26no—
175C42×C5—26no—
176C42×C5—46no—
177C42×C5C22×C4112yes—
178#685—22no—
179#685—22no—
180#685—43no—
181#685—43no—
182#685C22×C414yes—
183#685C22×C4112yes—
184C4⋊Dic5—26no—
185C4⋊Dic5—26no—
186C4⋊Dic5C22×C4112yes—
187#683—22no—
188#683—26no—
189#683—46no—
190#683—43no—
191#683—43no—
192#683C22×C4112yes—
193#683C22×C4112yes—
194C4×Dic5—26no—
195C4×Dic5—26no—
196C4×Dic5—22no—
197C4×Dic5—26no—
198C4×Dic5—43no—
199C4×Dic5—43no—
200C4×Dic5C22×C4124yes—
201C4×Dic5C22×C4112yes—
202C23×Q8—51no—
203#519—516no—
204C2×C4×Q8—51no—
205C2×C4×Q8—51no—
206C2×C4×Q8—51no—
207C2×C4×Q8—53no—
208C2×C4×Q8—53no—
209C2×C4×Q8C2×D511yescharacteristic
210C2×C4×Q8C2×D511yescharacteristic
211C2×C4×Q8C2×D511yescharacteristic
212C2×#117—53no—
213C2×#117—53no—
214#391—53no—
215#391—53no—
216#391—53no—
217#391—53no—
218#389—53no—
219#389—53no—
220#389—53no—
221#389—53no—
222#388—56no—
223#388—53no—
224#388—53no—
225#388—53no—
226#388—51no—
227#387—56no—
228#387—56no—
229#387—53no—
230#387—53no—
231#387—53no—
232#387—53no—
233#384—53no—
234#383—56no—
235#382—53no—
236#382—56no—
237#380—53no—
238#380—53no—
239C4×C4⋊C4—53no—
240C4×C4⋊C4—53no—
241C4×C4⋊C4—53no—
242C4×C4⋊C4—56no—
243C4×C4⋊C4—53no—
244C4×C4⋊C4—53no—
245C4×C4⋊C4C2×D513yes—
246C4×C4⋊C4C2×D513yes—
247C4×C4⋊C4C2×D513yes—
248C4×C22⋊C4—53no—
249C4×C22⋊C4—53no—
250#376—56no—
251#376C2×D513yes—
252C43—53no—
253C23×C5C2511yescharacteristic
254C22×D5—21no—
255C22×D5—21no—
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—21no—
259C22×D5—21no—
260C22×D5C23×C412yes—
261C5×Q8#117116yes—
262C2×C4×C5—21no—
263C2×C4×C5—23no—
264C2×C4×C5—21no—
265C2×C4×C5—23no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—23no—
269C2×C4×C5—23no—
270C2×C4×C5—23no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—23no—
276C2×C4×C5—23no—
277C2×C4×C5—23no—
278C2×C4×C5C23×C416yes—
279C2×C4×C5C23×C412yes—
280C2×C4×C5C23×C416yes—
281C2×C4×C5C2×#4116yes—
282C2×C4×C5C2×#4116yes—
283C2×Dic5—21no—
284C2×Dic5—21no—
285C2×Dic5—23no—
286C2×Dic5—23no—
287C2×Dic5—21no—
288C2×Dic5—23no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—23no—
293C2×Dic5—23no—
294C2×Dic5—23no—
295C2×Dic5—23no—
296C2×Dic5—23no—
297C2×Dic5—23no—
298C2×Dic5—23no—
299C2×Dic5C23×C412yes—
300C2×Dic5C23×C416yes—
301C2×Dic5C23×C416yes—
302C2×Dic52−1+412yes—
303C2×Dic52−1+412yes—
304C4×D5—224no—
305Dic10—224no—
306C22×Q8—58no—
307C22×Q8—53no—
308C22×Q8—104no—
309C22×Q8—104no—
310C22×Q8C22×D511yescharacteristic
311C23×C4—53no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—58no—
315C4×Q8—524no—
316C4×Q8—524no—
317C4×Q8—104no—
318C4×Q8—104no—
319C4×Q8C4×D518yes—
320#117—524no—
321#117—524no—
322C2×C4⋊C4—53no—
323C2×C4⋊C4—53no—
324C2×C4⋊C4—53no—
325C2×C4⋊C4—53no—
326C2×C4⋊C4—53no—
327C2×C4⋊C4—51no—
328C2×C4⋊C4—53no—
329C2×C4⋊C4—53no—
330C2×C4⋊C4—56no—
331C2×C4⋊C4—53no—
332C2×C4⋊C4C22×D513yes—
333C2×C4⋊C4C22×D513yes—
334C2×C4⋊C4C22×D513yes—
335C2×C22⋊C4—53no—
336C2×C22⋊C4—51no—
337C2×C22⋊C4—51no—
338C2×C22⋊C4—51no—
339C2×C22⋊C4—53no—
340C2×C22⋊C4—53no—
341C2×C42—56no—
342C2×C42—53no—
343C2×C42—53no—
344C2×C42—53no—
345C2×C42—51no—
346C2×C42—53no—
347C2×C42—53no—
348C2×C42—53no—
349C2×C42C22×D513yes—
350C2×C42C22×D513yes—
351C2×C42C22×D516yes—
352C2×C42C22×D513yes—
353#95—53no—
354#95—53no—
355#95—56no—
356#95—56no—
357#95—53no—
358#95—53no—
359#95—53no—
360#95—56no—
361#95—51no—
362#95—56no—
363#95—53no—
364#95—53no—
365#95C22×D511yescharacteristic
366#95C22×D513yes—
367#95C22×D513yes—
368#95C22×D513yes—
369C22×C5C24×C411yescharacteristic
370C22×C5C22×#4111yescharacteristic
371C22×C5C22×#4111yescharacteristic
372C22×C5C2×2−1+411yescharacteristic
373C22×C5C2×2−1+411yescharacteristic
374C22×C5#58511yescharacteristic
375C22×C5#58511yescharacteristic
376C2×D5—22no—
377C2×D5—22no—
378C2×D5—22no—
379C2×D5—41no—
380C2×D5—41no—
381C2×D5—41no—
382C2×D5—41no—
383C4×C5—22no—
384C4×C5—26no—
385C4×C5—43no—
386C4×C5—41no—
387C4×C5—41no—
388C4×C5—43no—
389C4×C5C2×#117112yes—
390Dic5—26no—
391Dic5—26no—
392Dic5—22no—
393Dic5—22no—
394Dic5—22no—
395Dic5—43no—
396Dic5—43no—
397C24—51no—
398C2×Q8—512no—
399C2×Q8—1032no—
400C2×Q8—106no—
401C2×Q8—106no—
402C2×Q8C2×C4×D514yes—
403C2×Q8#70914yes—
404C2×Q8#70914yes—
405C22×C4—53no—
406C22×C4—51no—
407C22×C4—51no—
408C22×C4—51no—
409C22×C4—53no—
410C22×C4—512no—
411C22×C4—53no—
412C22×C4—53no—
413C22×C4—106no—
414C22×C4—106no—
415C22×C4C23×D511yescharacteristic
416C22×C4C23×D513yes—
417C22×C4C23×D511yescharacteristic
418C22×C4C23×D513yes—
419C22×C4C23×D513yes—
420C22×C4C23×D511yescharacteristic
421C22×C4C23×D513yes—
422C4⋊C4—26no—
423C4⋊C4—26no—
424C4⋊C4—512no—
425C4⋊C4—512no—
426C4⋊C4—512no—
427C4⋊C4—102no—
428C4⋊C4—106no—
429C4⋊C4—106no—
430C4⋊C4—106no—
431C4⋊C4—203no—
432C4⋊C4—206no—
433C4⋊C4—203no—
434C4⋊C4C2×C4×D5112yes—
435C22⋊C4—512no—
436C22⋊C4—54no—
437C22⋊C4—102no—
438C22⋊C4—102no—
439C22⋊C4—203no—
440C22⋊C4—203no—
441C42—26no—
442C42—26no—
443C42—46no—
444C42—512no—
445C42—524no—
446C42—106no—
447C42—102no—
448C42—106no—
449C42—106no—
450C42—203no—
451C42—203no—
452C42C2×C4×D5112yes—
453C2×C5C22×#11711yescharacteristic
454C2×C5C4×2−1+411yescharacteristic
455C2×C5C4×2−1+411yescharacteristic
456C2×C5#351611yescharacteristic
457C2×C5#351811yescharacteristic
458C2×C5#352311yescharacteristic
459C2×C5#352411yescharacteristic
460D5—42no—
461C23—52no—
462C23—101no—
463C23—101no—
464C23—101no—
465C23—101no—
466C23—101no—
467C23—101no—
468C23C24×D511yescharacteristic
469Q8—1024no—
470Q8#4018116yes—
471C2×C4—23no—
472C2×C4—21no—
473C2×C4—23no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—21no—
487C2×C4—56no—
488C2×C4—52no—
489C2×C4—56no—
490C2×C4—52no—
491C2×C4—52no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—103no—
495C2×C4—103no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—103no—
500C2×C4—101no—
501C2×C4—103no—
502C2×C4—1024no—
503C2×C4—103no—
504C2×C4—103no—
505C2×C4—103no—
506C2×C4—103no—
507C2×C4—103no—
508C2×C4—103no—
509C2×C4C22×C4×D516yes—
510C2×C4C22×C4×D512yes—
511C2×C4C22×C4×D516yes—
512C2×C4C2×#70916yes—
513C2×C4C2×#70916yes—
514C5#3426211yescharacteristic, complemented, Sylow
515C22—102no—
516C22—102no—
517C22—102no—
518C22—201no—
519C22—201no—
520C22—201no—
521C22—201no—
522C22C23×C4×D511yescharacteristic
523C22C22×#70911yescharacteristic
524C22C22×#70911yescharacteristic
525C22#6628511yescharacteristic
526C22D5×2−1+411yescharacteristic
527C22D5×2−1+411yescharacteristic
528C22#6628711yescharacteristic
529C4—22no—
530C4—26no—
531C4—41no—
532C4—41no—
533C4—43no—
534C4—43no—
535C4—106no—
536C4—102no—
537C4—102no—
538C4—102no—
539C4—106no—
540C4—203no—
541C4—203no—
542C4C2×#4018112yes—
543C2—202no—
544C2C22×#401811yescharacteristic
545C2#1061777411yescharacteristic
546C2#1061777911yescharacteristic
547C2#1061778511yescharacteristic
548C2#1061778811yescharacteristic
549C2#1061779911yescharacteristic
550C2#1061780711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2276C2
socle, cosocleC23×C5C25
prime radicalC23×D5C24
hypercenter#2276D5
hypocenterC5#34262
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C5×#2276C2semidirect
5. #10614192C2semidirect
6. Q8×#683C2semidirect
7. C4×Q8×Dic5C2semidirect
9. #10605375C2semidirect
10. #10605372C2semidirect
13. #10605349C2semidirect
14. #10605117C2semidirect
23. #65906C4semidirect
144. #2276D5semidirect
514. C5#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#659062
C5×#22762
Q8×#6852
#106141922
Q8×#6832
C4×Q8×Dic52
#106054912
#106053752
#106053722
#106053652
#106053642
#106053492
#106051172
#106051162
#342625