Finite Group Databasebeta

← #12260420

Subgroups — #12260420

Order 1280.

2390 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65908C211yesmaximal, characteristic
2C5×#2276C211yesmaximal, characteristic, complemented
3#10614192C211yesmaximal, characteristic, complemented
4Q8×#683C211yesmaximal, characteristic, complemented
5#10614188C211yesmaximal, characteristic
6#10614188C211yesmaximal, characteristic
7C4×Q8×Dic5C211yesmaximal, characteristic, complemented
8#10605499C213yesmaximal, complemented
9#10605498C213yesmaximal, complemented
10#10605491C213yesmaximal, complemented
11#10605377C213yesmaximal, complemented
12#10605370C213yesmaximal, complemented
13#10605348C213yesmaximal, complemented
14#10605146C213yesmaximal, complemented
15#10605116C213yesmaximal, complemented
16C22×#713C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×Q8C2211yescharacteristic
20C2×Q8×Dic5C2211yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23#65908C4116yescomplemented
24C2×#4039C2213yes—
25C2×D5×C4⋊C4C2213yes—
26C2×C4×D20C2213yes—
27C2×#4018C2216yes—
28C2×#4018C2213yes—
29C4×C5×C4⋊C4C2213yes—
30C4×C5×C4⋊C4C2213yes—
31C4×C5×C4⋊C4C2213yes—
32C5×#376C2213yes—
33#65277C2213yes—
34#65277C2213yes—
35#65276C2213yes—
36#65276C2213yes—
37#65266C2213yes—
38#65265C2213yes—
39C4⋊C4×Dic5C2213yes—
40C4⋊C4×Dic5C2213yes—
41#65262C2213yes—
42#65262C2213yes—
43#65229C2216yes—
44#65228C2213yes—
45#65228C2213yes—
46C4×#685C2216yes—
47C4×#685C2213yes—
48C4×#685C2213yes—
49C4×C4⋊Dic5C2213yes—
50#65222C2216yes—
51C4×#683C2213yes—
52C4×#683C2213yes—
53C42×Dic5C2213yes—
54#65217C2213yes—
55#65217C2213yes—
56#64958C2213yes—
57#64958C2216yes—
58#64958C2213yes—
59#64958C2213yes—
60#64958C2213yes—
61#64956C2216yes—
62#64953C2216yes—
63#64941C2211yescharacteristic
64#64941C2213yes—
65#64941C2213yes—
66#64941C2213yes—
67#34263—51nomaximal, Sylow
68C22×C5×Q8C2311yescharacteristic
69C2×#713—24no—
70C2×#713—24no—
71C2×#713C2×C418yes—
72C22×D20C2313yes—
73C22×C4×D5C2313yes—
74C22×C4×D5C2316yes—
75C4×C5×Q8—24no—
76C4×C5×Q8—24no—
77C4×C5×Q8C2×C418yes—
78C2×C5×C4⋊C4C2313yes—
79C2×C5×C4⋊C4C2313yes—
80C2×C5×C4⋊C4C2313yes—
81C2×C42×C5C2313yes—
82C2×C42×C5C2313yes—
83C2×C42×C5C2316yes—
84C2×C42×C5C2313yes—
85Q8×Dic5—24no—
86Q8×Dic5—24no—
87Q8×Dic5C2×C418yes—
88C2×#685C2313yes—
89C2×#685C2313yes—
90C2×#685C2316yes—
91C2×#685C2313yes—
92C2×#685C2313yes—
93C2×#685C2316yes—
94C2×#685C2313yes—
95C2×#685C2313yes—
96C2×C4⋊Dic5C2313yes—
97C2×C4⋊Dic5C2313yes—
98C2×C4⋊Dic5C2313yes—
99C2×#683C2313yes—
100C2×#683C2311yescharacteristic
101C2×#683C2316yes—
102C2×#683C2313yes—
103C2×#683C2313yes—
104C2×#683C2311yescharacteristic
105C2×#683C2313yes—
106C2×C4×Dic5C2313yes—
107C2×C4×Dic5C2313yes—
108C2×C4×Dic5C2313yes—
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2313yes—
111C2×C4×Dic5C2313yes—
112C2×C4×Dic5C2313yes—
113#4039C2×C4124yes—
114D5×C4⋊C4C2×C4124yes—
115C4×D20C2×C4124yes—
116#4018—412no—
117#4018C2×C4124yes—
118C5×#95C2311yescharacteristic
119C5×#95C2313yes—
120C5×#95C2313yes—
121C5×#95C2313yes—
122#3963C2311yescharacteristic
123#3963C2311yescharacteristic
124#3963C2313yes—
125#3963C2313yes—
126#3963C2313yes—
127#3963C2311yescharacteristic
128#3963C2313yes—
129#3963C2313yes—
130#3963C2316yes—
131#3963C2316yes—
132C2×#520—51no—
133#2357—53no—
134#2354—53no—
135#2351—51no—
136Q8×C4⋊C4—51no—
137#2345—53no—
138#2344—53no—
139#2341—51no—
140#2341—51no—
141#2319—53no—
142#2314—53no—
143#2298—53no—
144#2293—53no—
145#2276D511yescharacteristic, complemented
146C42×Q8—51no—
147C23×D5C2413yes—
148C2×C5×Q8C22×C414yes—
149C2×C5×Q8#4114yes—
150C2×C5×Q8#4114yes—
151C22×C4×C5C2413yes—
152C22×C4×C5C2413yes—
153C22×C4×C5C2411yescharacteristic
154C22×C4×C5C2411yescharacteristic
155C22×C4×C5C2413yes—
156C22×C4×C5C2411yescharacteristic
157C22×C4×C5C2413yes—
158C22×Dic5C2413yes—
159C22×Dic5C2413yes—
160C22×Dic5C2411yescharacteristic
161C22×Dic5C2413yes—
162C22×Dic5C2411yescharacteristic
163C22×Dic5C2411yescharacteristic
164C22×Dic5C2411yescharacteristic
165#713—232no—
166C2×D20—26no—
167C2×D20—26no—
168C2×D20C22×C4112yes—
169C2×C4×D5—26no—
170C2×C4×D5—26no—
171C2×C4×D5—46no—
172C2×C4×D5—46no—
173C2×C4×D5C22×C4112yes—
174C2×C4×D5C22×C4124yes—
175C5×C4⋊C4—26no—
176C5×C4⋊C4—26no—
177C5×C4⋊C4C22×C4112yes—
178C42×C5—26no—
179C42×C5—26no—
180C42×C5—46no—
181C42×C5C22×C4112yes—
182#685—26no—
183#685—26no—
184#685—46no—
185#685—43no—
186#685—43no—
187#685—46no—
188#685C22×C4112yes—
189#685C22×C4112yes—
190C4⋊Dic5—26no—
191C4⋊Dic5—26no—
192C4⋊Dic5C22×C4112yes—
193#683—26no—
194#683—22no—
195#683—46no—
196#683—43no—
197#683—43no—
198#683C22×C414yes—
199#683C22×C4112yes—
200C4×Dic5—26no—
201C4×Dic5—26no—
202C4×Dic5—26no—
203C4×Dic5—22no—
204C4×Dic5—43no—
205C4×Dic5—43no—
206C4×Dic5C22×C4112yes—
207C22×#41—51no—
208#520—516no—
209C2×C4×Q8—51no—
210C2×C4×Q8—51no—
211C2×C4×Q8—51no—
212C2×C4×Q8C2×D511yescharacteristic
213C2×C4×Q8C2×D511yescharacteristic
214C2×C4×Q8C2×D511yescharacteristic
215C2×C4×D4—53no—
216C2×C4×D4—53no—
217C2×#117—53no—
218C2×#117—56no—
219C22×C4⋊C4—53no—
220#390—53no—
221#390—53no—
222#390—53no—
223#390—53no—
224#389—53no—
225#389—53no—
226#389—53no—
227#389—53no—
228#388—56no—
229#388—56no—
230#388—53no—
231#388—53no—
232#388—53no—
233#388—53no—
234#387—56no—
235#387—53no—
236#387—53no—
237#387—53no—
238#387—51no—
239#384—53no—
240#382—53no—
241#381—56no—
242#380—53no—
243#380—53no—
244#380—56no—
245C4×C4⋊C4—53no—
246C4×C4⋊C4—53no—
247C4×C4⋊C4—53no—
248C4×C4⋊C4—53no—
249C4×C4⋊C4—53no—
250C4×C4⋊C4C2×D513yes—
251C4×C4⋊C4C2×D513yes—
252C4×C4⋊C4C2×D513yes—
253C4×C22⋊C4—53no—
254C4×C22⋊C4—53no—
255C4×C22⋊C4—56no—
256#376C2×D513yes—
257C43—53no—
258C23×C5C2511yescharacteristic
259C22×D5—23no—
260C22×D5—23no—
261C22×D5—23no—
262C22×D5—23no—
263C22×D5—23no—
264C22×D5—23no—
265C22×D5C23×C416yes—
266C5×Q8#117116yescomplemented
267C2×C4×C5—23no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—23no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—23no—
274C2×C4×C5—23no—
275C2×C4×C5—23no—
276C2×C4×C5—23no—
277C2×C4×C5—21no—
278C2×C4×C5—23no—
279C2×C4×C5—21no—
280C2×C4×C5—23no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5C23×C412yes—
284C2×C4×C5C23×C416yes—
285C2×C4×C5C23×C416yes—
286C2×C4×C5C2×#4116yes—
287C2×C4×C5C2×#4116yes—
288C2×Dic5—21no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—23no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—23no—
298C2×Dic5—23no—
299C2×Dic5—23no—
300C2×Dic5—23no—
301C2×Dic5—23no—
302C2×Dic5—23no—
303C2×Dic5—23no—
304C2×Dic5C23×C412yes—
305C2×Dic5C23×C416yes—
306C2×Dic5C23×C412yes—
307C2×Dic5H(5,𝔽2)12yes—
308C2×Dic52−1+412yes—
309D20—224no—
310C4×D5—224no—
311C4×D5—424no—
312C2×#41—58no—
313C2×#41—104no—
314C2×#41—104no—
315C22×Q8C22×D511yescharacteristic
316C22×D4—53no—
317C23×C4—53no—
318C23×C4—56no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—58no—
322C4×Q8—104no—
323C4×Q8—104no—
324C4×Q8C4×D518yes—
325C4×D4—524no—
326C4×D4—524no—
327#117—524no—
328#117—2012no—
329C2×C4⋊C4—53no—
330C2×C4⋊C4—53no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—524no—
333C2×C4⋊C4—53no—
334C2×C4⋊C4—53no—
335C2×C4⋊C4—53no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—56no—
338C2×C4⋊C4—53no—
339C2×C4⋊C4—53no—
340C2×C4⋊C4C22×D513yes—
341C2×C4⋊C4C22×D513yes—
342C2×C4⋊C4C22×D513yes—
343C2×C22⋊C4—53no—
344C2×C22⋊C4—53no—
345C2×C22⋊C4—53no—
346C2×C22⋊C4—53no—
347C2×C22⋊C4—56no—
348C2×C22⋊C4—56no—
349C2×C22⋊C4—53no—
350C2×C22⋊C4—53no—
351C2×C42—53no—
352C2×C42—53no—
353C2×C42—53no—
354C2×C42—53no—
355C2×C42—51no—
356C2×C42—53no—
357C2×C42—53no—
358C2×C42C22×D513yes—
359C2×C42C22×D513yes—
360C2×C42C22×D516yes—
361C2×C42C22×D513yes—
362#95—51no—
363#95—51no—
364#95—53no—
365#95—53no—
366#95—53no—
367#95—56no—
368#95—56no—
369#95—51no—
370#95—53no—
371#95—53no—
372#95C22×D511yescharacteristic
373#95C22×D513yes—
374#95C22×D513yes—
375#95C22×D513yes—
376C22×C5C24×C411yescharacteristic
377C22×C5C22×#4111yescharacteristic
378C22×C5C22×#4111yescharacteristic
379C22×C5C2×H(5,𝔽2)11yescharacteristic
380C22×C5C2×2−1+411yescharacteristic
381C22×C5#58511yescharacteristic
382C22×C5#58511yescharacteristic
383C2×D5—26no—
384C2×D5—26no—
385C2×D5—26no—
386C2×D5—43no—
387C2×D5—43no—
388C2×D5—43no—
389C2×D5—43no—
390C4×C5—26no—
391C4×C5—22no—
392C4×C5—41no—
393C4×C5—43no—
394C4×C5—43no—
395C4×C5—41no—
396C4×C5C2×#117112yes—
397Dic5—22no—
398Dic5—26no—
399Dic5—22no—
400Dic5—22no—
401Dic5—22no—
402Dic5—43no—
403Dic5—43no—
404C24—53no—
405#41—1032no—
406C2×Q8C2×C4×D514yes—
407C2×Q8#70914yes—
408C2×Q8#70914yes—
409C2×D4—512no—
410C2×D4—106no—
411C2×D4—106no—
412C22×C4—51no—
413C22×C4—51no—
414C22×C4—51no—
415C22×C4—53no—
416C22×C4—51no—
417C22×C4—512no—
418C22×C4—524no—
419C22×C4—53no—
420C22×C4—53no—
421C22×C4—106no—
422C22×C4—106no—
423C22×C4—206no—
424C22×C4—206no—
425C22×C4C23×D511yescharacteristic
426C22×C4C23×D513yes—
427C22×C4C23×D511yescharacteristic
428C22×C4C23×D513yes—
429C22×C4C23×D513yes—
430C22×C4C23×D511yescharacteristic
431C22×C4C23×D513yes—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—512no—
435C4⋊C4—512no—
436C4⋊C4—54no—
437C4⋊C4—106no—
438C4⋊C4—102no—
439C4⋊C4—106no—
440C4⋊C4—106no—
441C4⋊C4—206no—
442C4⋊C4—203no—
443C4⋊C4—203no—
444C4⋊C4C2×C4×D5112yes—
445C22⋊C4—512no—
446C22⋊C4—512no—
447C22⋊C4—106no—
448C22⋊C4—106no—
449C22⋊C4—206no—
450C22⋊C4—203no—
451C22⋊C4—203no—
452C22⋊C4—206no—
453C42—26no—
454C42—26no—
455C42—46no—
456C42—512no—
457C42—106no—
458C42—102no—
459C42—106no—
460C42—106no—
461C42—203no—
462C42—203no—
463C42C2×C4×D5112yes—
464C2×C5C22×#11711yescharacteristic
465C2×C5C4×H(5,𝔽2)11yescharacteristic
466C2×C5C4×2−1+411yescharacteristic
467C2×C5#351611yescharacteristic
468C2×C5#351711yescharacteristic
469C2×C5#352211yescharacteristic
470C2×C5#352411yescharacteristic
471D5—46no—
472C23—56no—
473C23—103no—
474C23—103no—
475C23—103no—
476C23—103no—
477C23—103no—
478C23—103no—
479C23C24×D511yescharacteristic
480Q8#4018116yescomplemented
481D4—1024no—
482C2×C4—23no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—23no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—23no—
489C2×C4—23no—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—23no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—23no—
497C2×C4—21no—
498C2×C4—52no—
499C2×C4—56no—
500C2×C4—52no—
501C2×C4—52no—
502C2×C4—52no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—101no—
506C2×C4—101no—
507C2×C4—103no—
508C2×C4—101no—
509C2×C4—101no—
510C2×C4—101no—
511C2×C4—101no—
512C2×C4—103no—
513C2×C4—1024no—
514C2×C4—103no—
515C2×C4—103no—
516C2×C4—103no—
517C2×C4—103no—
518C2×C4—103no—
519C2×C4—103no—
520C2×C4—2024no—
521C2×C4C22×C4×D512yes—
522C2×C4C22×C4×D516yes—
523C2×C4C22×C4×D516yes—
524C2×C4C2×#70916yes—
525C2×C4C2×#70916yes—
526C5#3426311yescharacteristic, complemented, Sylow
527C22—106no—
528C22—106no—
529C22—106no—
530C22—203no—
531C22—203no—
532C22—203no—
533C22—203no—
534C22C23×C4×D511yescharacteristic
535C22C22×#70911yescharacteristic
536C22C22×#70911yescharacteristic
537C22D5×H(5,𝔽2)11yescharacteristic
538C22D5×2−1+411yescharacteristic
539C22#6628711yescharacteristic
540C22#6628711yescharacteristic
541C4—22no—
542C4—26no—
543C4—41no—
544C4—41no—
545C4—43no—
546C4—43no—
547C4—102no—
548C4—102no—
549C4—106no—
550C4—102no—
551C4—102no—
552C4—203no—
553C4—203no—
554C4C2×#4018112yes—
555C2—206no—
556C2C22×#401811yescharacteristic
557C2#1061777411yescharacteristic
558C2#1061778011yescharacteristic
559C2#1061778411yescharacteristic
560C2#1061778711yescharacteristic
561C2#1061779711yescharacteristic
562C2#1061780911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2276C2
socle, cosocleC23×C5C25
prime radicalC22×#713C22
hypercenter#2276D5
hypocenterC5#34263
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C5×#2276C2semidirect
3. #10614192C2semidirect
4. Q8×#683C2semidirect
7. C4×Q8×Dic5C2semidirect
8. #10605499C2semidirect
9. #10605498C2semidirect
10. #10605491C2semidirect
11. #10605377C2semidirect
12. #10605370C2semidirect
13. #10605348C2semidirect
14. #10605146C2semidirect
15. #10605116C2semidirect
23. #65908C4semidirect
145. #2276D5semidirect
266. C5×Q8#117semidirect
480. Q8#4018semidirect
526. C5#34263semidirect

Maximal subgroups

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

subgroupindex
C2×#659082
C5×#22762
#106141922
Q8×#6832
#106141882
C4×Q8×Dic52
#106054992
#106054982
#106054912
#106053772
#106053702
#106053482
#106051462
#106051162
#342635