Finite Group Databasebeta

← #12260417

Subgroups — #12260417

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×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#10614191C211yesmaximal, characteristic, complemented
4#10614188C211yesmaximal, characteristic
5#10614188C211yesmaximal, characteristic
6#10614181C211yesmaximal, characteristic, complemented
7C4×Q8×Dic5C211yesmaximal, characteristic, complemented
8#10607932C213yesmaximal, complemented
9#10607931C213yesmaximal, complemented
10#10607700C213yesmaximal, complemented
11#10607692C213yesmaximal, complemented
12#10606089C213yesmaximal, complemented
13#10606055C213yesmaximal, complemented
14#10605376C213yesmaximal, complemented
15#10605351C213yesmaximal, 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×#4038C2216yes—
26C2×D5×C4⋊C4C2213yes—
27C2×C4×D20C2213yes—
28C2×#4018C2213yes—
29C5×#389C2213yes—
30C5×#389C2213yes—
31C5×#384C2213yes—
32C4×C5×C4⋊C4C2213yes—
33#65279C2216yes—
34#65279C2216yes—
35#65279C2213yes—
36#65279C2213yes—
37#65277C2213yes—
38#65277C2213yes—
39#65276C2213yes—
40#65276C2213yes—
41#65276C2216yes—
42#65274C2213yes—
43#65274C2213yes—
44#65267C2213yes—
45#65267C2216yes—
46#65267C2211yescharacteristic
47C4⋊C4×Dic5C2213yes—
48C4⋊C4×Dic5C2213yes—
49C4⋊C4×Dic5C2213yes—
50C4⋊C4×Dic5C2213yes—
51#65228C2213yes—
52#65228C2213yes—
53C4×#685C2213yes—
54C4×#685C2213yes—
55#65225C2213yes—
56C4×C4⋊Dic5C2213yes—
57C4×C4⋊Dic5C2213yes—
58C4×C4⋊Dic5C2213yes—
59#65221C2213yes—
60C42×Dic5C2213yes—
61#64959C2213yes—
62#64959C2213yes—
63#64955C2216yes—
64#64953C2216yes—
65#64944C2213yes—
66#64944C2213yes—
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⋊C4C2311yescharacteristic
79C2×C5×C4⋊C4C2313yes—
80C2×C5×C4⋊C4C2313yes—
81C2×C5×C4⋊C4C2313yes—
82C2×C5×C4⋊C4C2316yes—
83C2×C5×C4⋊C4C2313yes—
84C2×C42×C5C2313yes—
85C2×C42×C5C2313yes—
86C2×C42×C5C2313yes—
87Q8×Dic5—24no—
88Q8×Dic5—24no—
89Q8×Dic5C2×C418yes—
90C2×#685C2313yes—
91C2×#685C2313yes—
92C2×#685C2316yes—
93C2×#685C2313yes—
94C2×#685C2316yes—
95C2×#685C2313yes—
96C2×#685C2313yes—
97C2×#685C2313yes—
98C2×C4⋊Dic5C2313yes—
99C2×C4⋊Dic5C2313yes—
100C2×C4⋊Dic5C2313yes—
101C2×C4⋊Dic5C2313yes—
102C2×C4⋊Dic5C2313yes—
103C2×#683C2311yescharacteristic
104C2×#683C2313yes—
105C2×C4×Dic5C2313yes—
106C2×C4×Dic5C2313yes—
107C2×C4×Dic5C2313yes—
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2316yes—
110C2×C4×Dic5C2313yes—
111C2×C4×Dic5C2313yes—
112C2×C4×Dic5C2313yes—
113#4039C2×C4124yes—
114#4038—412no—
115D5×C4⋊C4C2×C4124yes—
116C4×D20C2×C4124yes—
117#4018C2×C4124yes—
118C5×#95C2313yes—
119C5×#95C2313yes—
120#3963C2311yescharacteristic
121#3963C2311yescharacteristic
122#3963C2313yes—
123#3963C2313yes—
124#3963C2313yes—
125#3963C2313yes—
126#3963C2316yes—
127#3963C2316yes—
128#3963C2311yescharacteristic
129#3963C2313yes—
130#3963C2311yescharacteristic
131#3963C2313yes—
132C2×#520—51no—
133#2357—53no—
134#2354—53no—
135#2351—51no—
136Q8×C4⋊C4D511yescharacteristic, complemented
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#2276—51no—
146C42×Q8—51no—
147C23×D5C2413yes—
148C2×C5×Q8C22×C414yes—
149C2×C5×Q8#4114yes—
150C2×C5×Q8#4114yes—
151C22×C4×C5C2411yescharacteristic
152C22×C4×C5C2413yes—
153C22×C4×C5C2413yes—
154C22×C4×C5C2411yescharacteristic
155C22×C4×C5C2413yes—
156C22×C4×C5C2411yescharacteristic
157C22×C4×C5C2413yes—
158C22×Dic5C2411yescharacteristic
159C22×Dic5C2413yes—
160C22×Dic5C2413yes—
161C22×Dic5C2411yescharacteristic
162C22×Dic5C2413yes—
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⋊C4—22no—
178C5×C4⋊C4—43no—
179C5×C4⋊C4—46no—
180C5×C4⋊C4C22×C4112yes—
181C42×C5—26no—
182C42×C5—26no—
183C42×C5C22×C4112yes—
184#685—26no—
185#685—26no—
186#685—46no—
187#685—43no—
188#685—46no—
189#685—43no—
190#685C22×C4112yes—
191#685C22×C4112yes—
192C4⋊Dic5—26no—
193C4⋊Dic5—26no—
194C4⋊Dic5—26no—
195C4⋊Dic5—43no—
196C4⋊Dic5C22×C4112yes—
197#683C22×C414yes—
198#683C22×C4112yes—
199C4×Dic5—26no—
200C4×Dic5—26no—
201C4×Dic5—22no—
202C4×Dic5—26no—
203C4×Dic5—46no—
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—56no—
218C2×#117—53no—
219C22×C4⋊C4—53no—
220#390—53no—
221#390—53no—
222#390—53no—
223#390—53no—
224#389—53no—
225#389—53no—
226#389C2×D513yes—
227#389C2×D513yes—
228#388—56no—
229#388—56no—
230#388—53no—
231#388—53no—
232#388—53no—
233#388—53no—
234#387—53no—
235#387—56no—
236#387—51no—
237#387—53no—
238#387—53no—
239#384C2×D513yes—
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⋊C4—53no—
251C4×C4⋊C4—53no—
252C4×C4⋊C4C2×D513yes—
253C4×C22⋊C4—53no—
254C4×C22⋊C4—53no—
255C4×C22⋊C4—56no—
256#376—53no—
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—21no—
268C2×C4×C5—23no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—23no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—23no—
275C2×C4×C5—23no—
276C2×C4×C5—23no—
277C2×C4×C5—23no—
278C2×C4×C5—21no—
279C2×C4×C5—23no—
280C2×C4×C5—21no—
281C2×C4×C5—23no—
282C2×C4×C5C23×C412yes—
283C2×C4×C5C23×C416yes—
284C2×C4×C5C23×C416yes—
285C2×C4×C5C2×#4116yes—
286C2×C4×C5C2×#4116yes—
287C2×C4×C52−1+412yes—
288C2×Dic5—21no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—23no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—23no—
296C2×Dic5—21no—
297C2×Dic5—23no—
298C2×Dic5—23no—
299C2×Dic5—23no—
300C2×Dic5—23no—
301C2×Dic5—23no—
302C2×Dic5—23no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5C23×C412yes—
306C2×Dic5C23×C416yes—
307C2×Dic5C23×C412yes—
308C2×Dic5H(5,𝔽2)12yes—
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—53no—
337C2×C4⋊C4C22×D513yes—
338C2×C4⋊C4C22×D513yes—
339C2×C4⋊C4C22×D511yescharacteristic
340C2×C4⋊C4C22×D516yes—
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—53no—
349C2×C22⋊C4—56no—
350C2×C22⋊C4—53no—
351C2×C42—53no—
352C2×C42—53no—
353C2×C42—53no—
354C2×C42—53no—
355C2×C42—51no—
356C2×C42—56no—
357C2×C42—53no—
358C2×C42—53no—
359C2×C42C22×D513yes—
360C2×C42C22×D513yes—
361C2×C42C22×D513yes—
362#95—51no—
363#95—53no—
364#95—53no—
365#95—53no—
366#95—51no—
367#95—51no—
368#95—53no—
369#95—53no—
370#95—56no—
371#95—56no—
372#95—51no—
373#95—53no—
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—22no—
393C4×C5—41no—
394C4×C5—43no—
395C4×C5—43no—
396C4×C5C2×#117112yes—
397Dic5—22no—
398Dic5—26no—
399Dic5—22no—
400Dic5—22no—
401Dic5—43no—
402Dic5—41no—
403Dic5—43no—
404C24—53no—
405#41—1032no—
406C2×Q8C2×C4×D514yes—
407C2×Q8#71114yes—
408C2×Q8#71314yes—
409C2×D4—512no—
410C2×D4—106no—
411C2×D4—106no—
412C22×C4—51no—
413C22×C4—51no—
414C22×C4—53no—
415C22×C4—51no—
416C22×C4—512no—
417C22×C4—524no—
418C22×C4—51no—
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×D513yes—
428C22×C4C23×D511yescharacteristic
429C22×C4C23×D513yes—
430C22×C4C23×D511yescharacteristic
431C22×C4C23×D513yes—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—22no—
435C4⋊C4—46no—
436C4⋊C4—43no—
437C4⋊C4—512no—
438C4⋊C4—512no—
439C4⋊C4—54no—
440C4⋊C4—106no—
441C4⋊C4—106no—
442C4⋊C4—106no—
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—206no—
452C22⋊C4—203no—
453C42—26no—
454C42—26no—
455C42—512no—
456C42—106no—
457C42—106no—
458C42—106no—
459C42—102no—
460C42—206no—
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#4038116yescomplemented
481D4—1024no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—23no—
486C2×C4—21no—
487C2×C4—23no—
488C2×C4—21no—
489C2×C4—23no—
490C2×C4—23no—
491C2×C4—21no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—23no—
495C2×C4—23no—
496C2×C4—21no—
497C2×C4—52no—
498C2×C4—56no—
499C2×C4—52no—
500C2×C4—52no—
501C2×C4—101no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—101no—
506C2×C4—101no—
507C2×C4—103no—
508C2×C4—103no—
509C2×C4—1024no—
510C2×C4—101no—
511C2×C4—103no—
512C2×C4—103no—
513C2×C4—103no—
514C2×C4—103no—
515C2×C4—101no—
516C2×C4—103no—
517C2×C4—103no—
518C2×C4—101no—
519C2×C4—2024no—
520C2×C4C22×C4×D512yes—
521C2×C4C22×C4×D516yes—
522C2×C4C22×C4×D516yes—
523C2×C4C2×#71116yes—
524C2×C4C2×#71316yes—
525C2×C4#414712yes—
526C5#3426311yescharacteristic, complemented, Sylow
527C22—106no—
528C22—106no—
529C22—106no—
530C22—203no—
531C22—203no—
532C22—203no—
533C22—203no—
534C22C23×C4×D511yescharacteristic
535C22C22×#71111yescharacteristic
536C22C22×#71311yescharacteristic
537C22C2×#414711yescharacteristic
538C22#6628311yescharacteristic
539C22D5×H(5,𝔽2)11yescharacteristic
540C22#6628711yescharacteristic
541C4—22no—
542C4—22no—
543C4—26no—
544C4—41no—
545C4—43no—
546C4—43no—
547C4—102no—
548C4—102no—
549C4—106no—
550C4—102no—
551C4—201no—
552C4—203no—
553C4—203no—
554C4C2×#4038112yes—
555C2—206no—
556C2C22×#403811yescharacteristic
557C2C4×#414711yescharacteristic
558C2#1061778711yescharacteristic
559C2#1061865011yescharacteristic
560C2#1061867311yescharacteristic
561C2#1061892711yescharacteristic
562C2#1061894411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×Q8×C4⋊C4C2
socle, cosocleC23×C5C25
prime radicalC22×#713C22
hypercenterQ8×C4⋊C4D5
hypocenterC5#34263
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×Q8×C4⋊C4C2semidirect
3. #10614191C2semidirect
6. #10614181C2semidirect
7. C4×Q8×Dic5C2semidirect
8. #10607932C2semidirect
9. #10607931C2semidirect
10. #10607700C2semidirect
11. #10607692C2semidirect
12. #10606089C2semidirect
13. #10606055C2semidirect
14. #10605376C2semidirect
15. #10605351C2semidirect
23. #65908C4semidirect
136. Q8×C4⋊C4D5semidirect
266. C5×Q8#117semidirect
480. Q8#4038semidirect
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×Q8×C4⋊C42
#106141912
#106141882
#106141812
C4×Q8×Dic52
#106079322
#106079312
#106077002
#106076922
#106060892
#106060552
#106053762
#106053512
#342635