Finite Group Databasebeta

← #555×#6157 #28328800

Subgroups — #555×#6157 #28328800

Order 13568.

2564 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291251C214yesmaximal, complemented
2#20291248C211yesmaximal, characteristic, complemented
3#20291242C214yesmaximal, complemented
4#20291233C212yesmaximal, complemented
5#20291225C214yesmaximal, complemented
6C4⋊Q8×#6157C211yesmaximal, characteristic, complemented
7#124×#6157C214yesmaximal, complemented
8#121×#6157C214yesmaximal, complemented
9C4×D4×#6157C212yesmaximal, complemented
10#117×#6157C214yesmaximal, complemented
11#555×D53C211yesmaximal, characteristic
12C2×Q8×#6157C2212yes—
13C2×D4×#6157C2212yes—
14C2×D4×#6157C2212yes—
15C2×D4×#6157C2214yes—
16C2×#13133296C2214yes—
17C22×C4×#6157C2214yes—
18(C2×#6157)⋊Q8C2214yes—
19(C2×#6157)⋊Q8C2212yes—
20#16811575C2214yes—
21#16811575C2214yes—
22#16811574C2214yes—
23#16811574C2214yes—
24#16811573C2218yes—
25#16811573C2218yes—
26#16811573C2218yes—
27#16811571C2218yes—
28#16811569C2211yescharacteristic
29#16811567C2214yes—
30#16811565C2214yes—
31C4⋊C4×#6157C2212yes—
32C4⋊C4×#6157C2214yes—
33C4⋊C4×#6157C2214yes—
34C4⋊C4×#6157C2214yes—
35C4⋊C4×#6157C2212yes—
36C4×C4⋊#6157C2212yes—
37C22⋊C4×#6157C2218yes—
38C22⋊C4×#6157C2218yes—
39C22⋊C4×#6157C2214yes—
40C22⋊C4×#6157C2214yes—
41C4×C22⋊#6157C2214yes—
42#16811559C2218yes—
43C42×#6157C2214yes—
44C42×#6157C2211yescharacteristic
45#16811339C411yescharacteristic, complemented
46#16811338C412yescomplemented
47#16811337C414yescomplemented
48#16811336C414yescomplemented
49#16811335C414yescomplemented
50C4⋊Q8×D53C2211yescharacteristic
51#124×D53C2214yes—
52#121×D53C2214yes—
53C4×D4×D53C2212yes—
54#117×D53C2214yes—
55C53×#555C411yescharacteristic, complemented
56C23×#6157C2314yes—
57C23×#6157C2314yes—
58Q8×#6157—44no—
59D4×#6157—28no—
60D4×#6157—28no—
61D4×#6157—48no—
62#13133296—216no—
63C2×C4⋊#6157C2312yes—
64C2×C4⋊#6157C2314yes—
65C2×C4⋊#6157C2314yes—
66C2×C4⋊#6157C2314yes—
67C2×C4⋊#6157C2312yes—
68C2×C4⋊#6157C2312yes—
69C2×C22⋊#6157C2314yes—
70C2×C22⋊#6157C2314yes—
71C2×C22⋊#6157C2318yes—
72C2×C22⋊#6157C2318yes—
73C2×C4×#6157—216no—
74C2×C4×#6157C2314yes—
75C2×C4×#6157C2318yes—
76C2×C4×#6157C2312yes—
77C2×C4×#6157C2311yescharacteristic
78C2×C4×#6157C2314yes—
79C2×C4×#6157C2314yes—
80C2×C4×#6157C2314yes—
81C2×C4×#6157C2311yescharacteristic
82C2×C4×#6157C2314yes—
83C2×C4×#6157C2312yes—
84(C2×#6157)⋊C4C2314yes—
85(C2×#6157)⋊C4C2318yes—
86(C2×#6157)⋊C4C2318yes—
87(C2×#6157)⋊C4C2318yes—
88(C2×#6157)⋊C4C2314yes—
89(C2×#6157)⋊C4C2314yes—
90C2×Q8×D53C2312yes—
91C2×D4×D53C2312yes—
92C2×D4×D53C2314yes—
93C22×C4×D53C2314yes—
94#13133217C2×C411yescharacteristic, complemented
95#13133216C2×C414yescomplemented
96#13133215C2×C412yescomplemented
97#13133206C2×C414yescomplemented
98#13133205C2×C414yescomplemented
99#13133204C2×C414yescomplemented
100#13133203C2×C418yescomplemented
101#13133203C2×C418yescomplemented
102#13133192C2×C414yescomplemented
103#13133191C2×C414yescomplemented
104#13133190C2×C418yescomplemented
105#13133189C2×C418yescomplemented
106#13133188C2×C414yescomplemented
107C4×D212C2×C412yescomplemented
108D4×Dic53C2×C412yescomplemented
109#13133179C2×C412yescomplemented
110C4×C22⋊D53C2×C414yescomplemented
111#13133177C2×C414yescomplemented
112#13133176C2×C414yescomplemented
113#13133175C2×C414yescomplemented
114#13133175C2×C414yescomplemented
115#13133174C2×C418yescomplemented
116#13133173C2×C418yescomplemented
117C4⋊C4×D53C2312yes—
118C4⋊C4×D53C2314yes—
119C22⋊C4×D53C2318yes—
120C22⋊C4×D53C2314yes—
121C42×D53C2314yes—
122C42×D53C2311yescharacteristic
123C4⋊Q8×C53C2×C411yescharacteristic, complemented
124#124×C53C2×C414yescomplemented
125#121×C53C2×C414yescomplemented
126C4×D4×C53C2×C412yescomplemented
127#117×C53C2×C414yescomplemented
128C23×D53C2414yes—
129C22×#6157—28no—
130C22×#6157—28no—
131C22×#6157—28no—
132C22×#6157—28no—
133C22×#6157—28no—
134C22×#6157—44no—
135C22×#6157C2414yes—
136C22×#6157C2414yes—
137C22×#6157C2411yescharacteristic
138C22×#6157C2412yes—
139C22×#6157C2414yes—
140C22×#6157C2411yescharacteristic
141C2×Q8×C53C22×C412yes—
142C2×D4×C53C22×C412yes—
143C2×D4×C53C22×C414yes—
144C22×C4×C53C22×C414yes—
145C2×C22⋊D53C22×C414yes—
146C2×C22⋊D53C22×C418yes—
147C22×Dic53C22×C414yes—
148Q8×D53—42no—
149Q8×D53—42no—
150D4×D53—24no—
151D4×D53—24no—
152D4×D53—44no—
153D4×D53—44no—
154C2×D212C22×C412yes—
155C2×D212C22×C414yes—
156C2×C4×D53—28no—
157C2×C4×D53—28no—
158C2×C4×D53C22×C414yes—
159C2×C4×D53C22×C414yes—
160C2×C4×D53C2414yes—
161C2×C4×D53C2411yescharacteristic
162C2×C4×D53C2414yes—
163C2×C4×D53C2412yes—
164C2×Dic106C22×C412yes—
165C2×Dic106C22×C414yes—
166C22⋊#6157—216no—
167C22⋊#6157—28no—
168C22⋊#6157—28no—
169C22⋊#6157—48no—
170C4⋊#6157—28no—
171C4⋊#6157—24no—
172C4⋊#6157—44no—
173C4⋊#6157—44no—
174C4⋊#6157—42no—
175C4⋊#6157#4118yes—
176C4×#6157—28no—
177C4×#6157—28no—
178C4×#6157—216no—
179C4×#6157—28no—
180C4×#6157—22no—
181C4×#6157—28no—
182C4×#6157—24no—
183C4×#6157—44no—
184C4×#6157—41no—
185C4×#6157#4118yes—
186C4⋊C4×C53C22×C412yes—
187C4⋊C4×C53C22×C414yes—
188C22⋊C4×C53C22×C418yes—
189C22⋊C4×C53C22×C414yes—
190C42×C53C22×C414yes—
191C42×C53C22×C411yescharacteristic
192C22⋊Dic53C22×C418yes—
193C22⋊Dic53C22×C414yes—
194#11719921C22×C418yes—
195#11719921C22×C418yes—
196#11719921C22×C414yes—
197#11719921C22×C414yes—
198C4⋊Dic53C22×C412yes—
199C4⋊Dic53C22×C414yes—
200#11719919C22×C414yes—
201#11719919C22×C418yes—
202C4×Dic53C22×C414yes—
203C4×Dic53C22×C418yes—
204C4×Dic53C22×C411yescharacteristic
205C4×Dic53C22×C412yes—
206C23×C53C23×C414yes—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53C23×C414yes—
214C22×D53C2511yescharacteristic
215C2×#6157—24no—
216C2×#6157—22no—
217C2×#6157—24no—
218C2×#6157—22no—
219C2×#6157—24no—
220C2×#6157—28no—
221C2×#6157—22no—
222C2×#6157—24no—
223C2×#6157—21no—
224C2×#6157—24no—
225C2×#6157—24no—
226C2×#6157—416no—
227C2×#6157—416no—
228C2×#6157C2×#4114yes—
229C2×#6157C2×#4118yes—
230C2×#6157C2×#4114yes—
231C2×#6157H(5,𝔽2)12yes—
232Q8×C53—42no—
233D4×C53—24no—
234D4×C53—44no—
235C2×C4×C53—28no—
236C2×C4×C53C23×C411yescharacteristic
237C2×C4×C53C23×C414yes—
238C2×C4×C53C23×C414yes—
239C2×C4×C53C23×C412yes—
240C22⋊D53—28no—
241C22⋊D53—48no—
242C2×Dic53—28no—
243C2×Dic53C23×C411yescharacteristic
244C2×Dic53C23×C414yes—
245C2×Dic53C23×C414yes—
246C2×Dic53C23×C412yes—
247D212—24no—
248D212—44no—
249C4×D53—24no—
250C4×D53—24no—
251C4×D53—24no—
252C4×D53—21no—
253C4×D53—24no—
254C4×D53—21no—
255C4×D53—28no—
256C4×D53—28no—
257C4×D53C2×#4114yes—
258C4×D53C2×#4114yes—
259Dic106—44no—
260Dic106—42no—
261C4×#555—531nomaximal, Sylow
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—24no—
265C22×C53C24×C411yescharacteristic
266C2×D53—24no—
267C2×D53—24no—
268C2×D53—24no—
269C2×D53—44no—
270C2×D53—44no—
271C2×D53—44no—
272C2×D53—44no—
273C2×D53C22×#4112yes—
274C2×D53C22×#4112yes—
275C2×D53C2×H(5,𝔽2)11yescharacteristic
276C2×D53#58511yescharacteristic
277#6157—24no—
278#6157—28no—
279#6157—44no—
280#6157—44no—
281#6157—41no—
282#6157#55514yesdirect product factor
283C4×C53—24no—
284C4×C53—21no—
285C4×C53—24no—
286C4×C53C4×#4114yes—
287Dic53—24no—
288Dic53—21no—
289Dic53—24no—
290Dic53C4×#4114yes—
291C2×#555—531no—
292#2372—534no—
293#2369—531no—
294#2363—534no—
295#2354—532no—
296#2346—534no—
297C4×C4⋊Q8—531no—
298C4×#124—534no—
299C4×#121—534no—
300C42×D4—532no—
301C4×#117—534no—
302C2×C53—44no—
303C2×C53C2×C4×#4112yes—
304C2×C53C4×H(5,𝔽2)11yescharacteristic
305D53—44no—
306D53C2×#55511yescharacteristic
307D53#347512yes—
308D53#348311yescharacteristic
309#555—531no—
310#555—534no—
311#555—534no—
312#555—532no—
313#555—534no—
314#555#615711yescharacteristic, direct product factor
315C2×C4⋊Q8—531no—
316C2×#124—534no—
317C2×#121—534no—
318C2×C4×Q8—532no—
319C2×C4×D4—532no—
320C2×C4×D4—532no—
321C2×C4×D4—532no—
322C2×C4×D4—534no—
323C2×#117—534no—
324C2×#117—534no—
325C22×C42—534no—
326#391—532no—
327#391—534no—
328#390—534no—
329#390—534no—
330#389—534no—
331#389—534no—
332#388—538no—
333#388—538no—
334#388—538no—
335#386—538no—
336#384—531no—
337#382—534no—
338#380—534no—
339C4×C4⋊C4—534no—
340C4×C4⋊C4—532no—
341C4×C4⋊C4—534no—
342C4×C4⋊C4—532no—
343C4×C4⋊C4—534no—
344C4×C4⋊C4—532no—
345C4×C22⋊C4—534no—
346C4×C22⋊C4—538no—
347C4×C22⋊C4—534no—
348C4×C22⋊C4—538no—
349C4×C22⋊C4—534no—
350#376—538no—
351C43—531no—
352C43—534no—
353C53C4×#55511yescharacteristic, complemented, Sylow
354C22×Q8—532no—
355C22×D4—532no—
356C22×D4—534no—
357C23×C4—534no—
358C23×C4—534no—
359C23×C4—534no—
360C4⋊Q8—532no—
361C4⋊Q8—534no—
362C4⋊Q8—531no—
363C4⋊Q8C2×#615711yescharacteristic, complemented
364#124—534no—
365#124—534no—
366#124—538no—
367#124—538no—
368#124—534no—
369#124C2×#615714yescomplemented
370#121—534no—
371#121—538no—
372#121—534no—
373#121—534no—
374#121—538no—
375#121C2×#615714yescomplemented
376C4×Q8—2124no—
377C4×D4—532no—
378C4×D4—534no—
379C4×D4—532no—
380C4×D4—534no—
381C4×D4—532no—
382C4×D4—1068no—
383C4×D4—1068no—
384C4×D4—2128no—
385C4×D4C2×#615712yescomplemented
386#117—534no—
387#117—534no—
388#117—534no—
389#117—538no—
390#117—538no—
391#117—10616no—
392#117C2×#615714yescomplemented
393C2×C4⋊C4—532no—
394C2×C4⋊C4—532no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—532no—
398C2×C4⋊C4—534no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—532no—
401C2×C22⋊C4—538no—
402C2×C22⋊C4—534no—
403C2×C22⋊C4—538no—
404C2×C22⋊C4—534no—
405C2×C22⋊C4—534no—
406C2×C22⋊C4—538no—
407C2×C42—532no—
408C2×C42—534no—
409C2×C42—531no—
410C2×C42—538no—
411C2×C42—534no—
412C2×C42—532no—
413C2×C42—531no—
414C2×C42—534no—
415C2×C42—534no—
416C2×C42—534no—
417C2×C42—531no—
418C2×C42—534no—
419C2×C42—10616no—
420#95—534no—
421#95—538no—
422#95—538no—
423#95—538no—
424#95—534no—
425#95—534no—
426C24—534no—
427C2×Q8—534no—
428C2×Q8—532no—
429C2×Q8—2122no—
430C2×Q8—2122no—
431C2×Q8C22×#615712yes—
432C2×D4—534no—
433C2×D4—538no—
434C2×D4—532no—
435C2×D4—534no—
436C2×D4—1064no—
437C2×D4—1064no—
438C2×D4—2124no—
439C2×D4—2124no—
440C2×D4C22×#615712yes—
441C2×D4C22×#615714yes—
442C22×C4—531no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—534no—
446C22×C4—534no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—534no—
450C22×C4—531no—
451C22×C4—534no—
452C22×C4—534no—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—1068no—
456C22×C4—1068no—
457C22×C4—1068no—
458C22×C4—1068no—
459C22×C4—1068no—
460C22×C4—1068no—
461C22×C4—1068no—
462C22×C4—2124no—
463C22×C4C22×#615714yes—
464C4⋊C4—538no—
465C4⋊C4—534no—
466C4⋊C4—532no—
467C4⋊C4—538no—
468C4⋊C4—534no—
469C4⋊C4—1068no—
470C4⋊C4—1064no—
471C4⋊C4—2124no—
472C4⋊C4—2124no—
473C4⋊C4—2122no—
474C4⋊C4C22×#615712yes—
475C4⋊C4C22×#615714yes—
476C22⋊C4—538no—
477C22⋊C4—534no—
478C22⋊C4—538no—
479C22⋊C4—538no—
480C22⋊C4—534no—
481C22⋊C4—534no—
482C22⋊C4—10616no—
483C22⋊C4—1068no—
484C22⋊C4—1068no—
485C22⋊C4—2128no—
486C22⋊C4C22×#615718yes—
487C22⋊C4C22×#615714yes—
488C42—538no—
489C42—534no—
490C42—538no—
491C42—531no—
492C42—532no—
493C42—1068no—
494C42—1068no—
495C42—1068no—
496C42—10616no—
497C42—1064no—
498C42—1068no—
499C42—1062no—
500C42—2124no—
501C42—2121no—
502C42C22×#615714yes—
503C42C22×#615711yescharacteristic
504C23—531no—
505C23—534no—
506C23—1064no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—1064no—
511C23—1064no—
512C23C23×#615714yes—
513Q8—42no—
514Q8—2124no—
515Q8—2122no—
516D4—24no—
517D4—44no—
518D4—1068no—
519D4—1064no—
520D4—2124no—
521D4—2128no—
522C2×C4—28no—
523C2×C4—534no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—534no—
527C2×C4—534no—
528C2×C4—534no—
529C2×C4—538no—
530C2×C4—534no—
531C2×C4—531no—
532C2×C4—534no—
533C2×C4—1068no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1061no—
538C2×C4—1064no—
539C2×C4—1061no—
540C2×C4—1068no—
541C2×C4—1064no—
542C2×C4—1062no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—1068no—
546C2×C4—1064no—
547C2×C4—1062no—
548C2×C4—1068no—
549C2×C4—1064no—
550C2×C4—1061no—
551C2×C4—1064no—
552C2×C4—1064no—
553C2×C4—21216no—
554C2×C4—21216no—
555C2×C4C23×#615714yes—
556C2×C4C23×#615711yescharacteristic
557C2×C4C23×#615714yes—
558C2×C4C23×#615712yes—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—532no—
563C22—532no—
564C22—531no—
565C22—531no—
566C22—1064no—
567C22—1064no—
568C22—1064no—
569C22—2124no—
570C22—2124no—
571C22—2124no—
572C22—2124no—
573C22C24×#615711yescharacteristic
574C4—24no—
575C4—24no—
576C4—21no—
577C4—534no—
578C4—534no—
579C4—1064no—
580C4—1068no—
581C4—1064no—
582C4—1064no—
583C4—1061no—
584C4—2124no—
585C4—2121no—
586C4—2124no—
587C4#41×#615714yes—
588C2—44no—
589C2—531no—
590C2—532no—
591C2—531no—
592C2—2124no—
593C2C2×#41×#615712yes—
594C2H(5,𝔽2)×#615711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#555C4
prime radical#555×D53C2
cosocleC22×D53C25
hypercenter#555#6157
hypocenterC53C4×#555
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20291251C2semidirect
2. #20291248C2semidirect
3. #20291242C2semidirect
4. #20291233C2semidirect
5. #20291225C2semidirect
6. C4⋊Q8×#6157C2semidirect
7. #124×#6157C2semidirect
8. #121×#6157C2semidirect
9. C4×D4×#6157C2semidirect
10. #117×#6157C2semidirect
45. #16811339C4semidirect
46. #16811338C4semidirect
47. #16811337C4semidirect
48. #16811336C4semidirect
49. #16811335C4semidirect
55. C53×#555C4semidirect
94. #13133217C2×C4semidirect
95. #13133216C2×C4semidirect
96. #13133215C2×C4semidirect
97. #13133206C2×C4semidirect
98. #13133205C2×C4semidirect
99. #13133204C2×C4semidirect
100. #13133203C2×C4semidirect
101. #13133203C2×C4semidirect
102. #13133192C2×C4semidirect
103. #13133191C2×C4semidirect
104. #13133190C2×C4semidirect
105. #13133189C2×C4semidirect
106. #13133188C2×C4semidirect
107. C4×D212C2×C4semidirect
108. D4×Dic53C2×C4semidirect
109. #13133179C2×C4semidirect
110. C4×C22⋊D53C2×C4semidirect
111. #13133177C2×C4semidirect
112. #13133176C2×C4semidirect
113. #13133175C2×C4semidirect
114. #13133175C2×C4semidirect
115. #13133174C2×C4semidirect
116. #13133173C2×C4semidirect
123. C4⋊Q8×C53C2×C4semidirect
124. #124×C53C2×C4semidirect
125. #121×C53C2×C4semidirect
126. C4×D4×C53C2×C4semidirect
127. #117×C53C2×C4semidirect
282. #6157#555direct (N × H)
314. #555#6157direct (N × H)
353. C53C4×#555semidirect
363. C4⋊Q8C2×#6157semidirect
369. #124C2×#6157semidirect
375. #121C2×#6157semidirect
385. C4×D4C2×#6157semidirect
392. #117C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202912512
#202912482
#202912422
#202912332
#202912252
C4⋊Q8×#61572
#124×#61572
#121×#61572
C4×D4×#61572
#117×#61572
#555×D532
C4×#55553