Finite Group Databasebeta

← #28330017

Subgroups — #28330017

Order 13568.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291626C213yesmaximal, complemented
2#20291624C213yesmaximal, complemented
3#20291620C213yesmaximal, complemented
4#20291619C213yesmaximal, complemented
5#20291618C211yesmaximal, characteristic, complemented
6#20291617C211yesmaximal, characteristic, complemented
7#20291609C211yesmaximal, characteristic, complemented
8#20291608C211yesmaximal, characteristic, complemented
9Q8×#11719940C211yesmaximal, characteristic, complemented
10#20291510C213yesmaximal, complemented
11#20291503C213yesmaximal, complemented
12C2×Q8×#88324C211yesmaximal, characteristic, complemented
13#20291410C213yesmaximal
14#20291398C213yesmaximal
15Q8×#11719947C211yesmaximal, characteristic
16#16811675C2216yes—
17#16811675C2216yes—
18#16811674C2213yes—
19#16811674C2213yes—
20#16811674C2213yes—
21#16811674C2213yes—
22#16811673C2213yes—
23#16811673C2213yes—
24#16811673C2216yes—
25#16811672C2213yes—
26#16811672C2213yes—
27#16811672C2216yes—
28Q8×#88324C2211yescharacteristic
29Q8×#88324C2211yescharacteristic
30Q8×#88324C2213yes—
31Q8×#88324C2213yes—
32Q8×#88324C2211yescharacteristic
33Q8×#88324C2211yescharacteristic
34Q8×#88324C2213yes—
35Q8×#88324C2213yes—
36#16811648C2213yes—
37#16811648C2216yes—
38#16811648C2213yes—
39#16811647C2213yes—
40#16811647C2216yes—
41#16811647C2213yes—
42#16811645C2213yes—
43#16811644C2213yes—
44#16811642C2213yes—
45#16811642C2213yes—
46#16811641C2213yes—
47#16811640C2213yes—
48#16811639C2213yes—
49#16811639C2213yes—
50#16811629C2213yes—
51#16811629C2213yes—
52#16811626C2213yes—
53C2×C4⋊#88324C2213yes—
54#16811596C2211yescharacteristic
55#16811596C2213yes—
56#16811587C2216yes—
57C4×#11719940C2213yes—
58C2×C4×#88324C2213yes—
59Q8×Dic106C411yescharacteristic
60#16811348C413yes—
61#16811331C413yes—
62#16811331C413yes—
63#16811330C413yes—
64Q8×C22⋊D53C411yescharacteristic
65Q8×C22⋊D53C411yescharacteristic
66#16811178C2213yes—
67#16811175C2213yes—
68#16811175C2213yes—
69C4×#11719947C2213yes—
70C2×Q8×Dic53C2211yescharacteristic
71C2×Q8×Dic53C2211yescharacteristic
72C4×Q8×D53C2211yescharacteristic
73D4×Q8×C53C411yescharacteristic
74C2×#11719940C2311yescharacteristic
75C2×#11719940C2313yes—
76C22×#88324C2311yescharacteristic
77C22×#88324C2313yes—
78#13133284C2316yes—
79#13133284C2313yes—
80#13133284C2313yes—
81#13133284C2313yes—
82#13133284C2313yes—
83C4⋊#88324C2313yes—
84C4⋊#88324C2313yes—
85C4⋊#88324C2316yes—
86C4⋊#88324C2313yes—
87C4⋊#88324C2313yes—
88C4⋊#88324C2313yes—
89C4⋊#88324C2313yes—
90C4⋊#88324C2313yes—
91C4⋊#88324C2313yes—
92#13133274C2311yescharacteristic
93#13133274C2313yes—
94#13133274C2311yescharacteristic
95#13133274C2313yes—
96C22⋊#88324C2313yes—
97C22⋊#88324C2311yescharacteristic
98C22⋊#88324C2311yescharacteristic
99C22⋊#88324C2313yes—
100#13133272C2313yes—
101#13133272C2313yes—
102#13133272C2316yes—
103#13133272C2316yes—
104C4×#88324C2313yes—
105C4×#88324C2313yes—
106C4×#88324C2313yes—
107C4×#88324C2313yes—
108C4×#88324C2313yes—
109C4×#88324C2313yes—
110C4×#88324C2316yes—
111C4×#88324C2316yes—
112C2×#11719947C2311yescharacteristic
113C2×Q8×D53C2×C411yescharacteristic
114#13133217C2×C413yes—
115#13133216C2×C413yes—
116#13133216C2×C413yes—
117#13133215C2×C413yes—
118#13133215C2×C413yes—
119#13133208C2×C416yes—
120#13133207C2×C416yes—
121#13133207C2×C416yes—
122#13133197C2×C416yes—
123#13133196C2×C413yes—
124#13133196C2×C413yes—
125#13133194C2×C416yes—
126#13133194C2×C416yes—
127#13133193C2×C413yes—
128#13133193C2×C413yes—
129Q8×Dic53C2×C411yescharacteristic
130Q8×Dic53C2×C411yescharacteristic
131Q8×Dic53C2×C413yes—
132Q8×Dic53C2×C413yes—
133Q8×Dic53C2311yescharacteristic
134Q8×Dic53C2313yes—
135Q8×Dic53C2313yes—
136C4×Dic106C2×C413yes—
137C4×Dic106C2×C413yes—
138#13133182C2×C413yes—
139#13133182C2313yes—
140#13133182C2316yes—
141D4×Dic53C2×C413yes—
142C4×C22⋊D53C2×C413yes—
143C4×C22⋊D53C2×C413yes—
144#13133177C2×C413yes—
145#13133177C2×C413yes—
146#13133175C2313yes—
147#13133173C2313yes—
148#13133173C2313yes—
149C2×C4⋊Dic53C2×C413yes—
150C2×C4⋊Dic53C2×C413yes—
151C4⋊C4×D53C2×C413yes—
152C2×C4×Dic53C2313yes—
153C2×C4×Dic53C2313yes—
154C42×D53C2313yes—
155C22×Q8×C53C2×C411yescharacteristic
156C22×Q8×C53C2×C411yescharacteristic
157C4⋊Q8×C53C2×C413yes—
158C22⋊Q8×C53C2×C413yes—
159C22⋊Q8×C53C2×C413yes—
160C4×Q8×C53C2×C411yescharacteristic
161C4×D4×C53C2×C413yes—
162C2×Q8×C53—24no—
163C2×Q8×C53—24no—
164C2×Q8×C53C22×C413yes—
165C2×Q8×C53C22×C413yes—
166C2×Q8×C53C22×C411yescharacteristic
167C2×D4×C53C22×C411yescharacteristic
168C22×C4×C53C22×C413yes—
169C22×C4×C53C22×C413yes—
170C2×C22⋊D53C22×C411yescharacteristic
171C2×C22⋊D53C22×C411yescharacteristic
172C22×Dic53C22×C413yes—
173C22×Dic53C22×C413yes—
174C22×Dic53C2411yescharacteristic
175C22×Dic53C2411yescharacteristic
176Q8×D53—24no—
177#11719947—22no—
178#11719947—22no—
179C2×C4×D53C22×C413yes—
180C2×C4×D53C22×C413yes—
181C2×C4×D53C2411yescharacteristic
182C2×Dic106C22×C411yescharacteristic
183C2×Dic106C22×C413yes—
184C2×Dic106C22×C416yes—
185#11719940—22no—
186#11719940—26no—
187C2×#88324—26no—
188C2×#88324—22no—
189C2×#88324C2413yes—
190C2×#88324C2411yescharacteristic
191C2×#88324C2411yescharacteristic
192C2×#88324C2411yescharacteristic
193C2×#88324C2413yes—
194C2×#88324C2413yes—
195C2×#88324C2411yescharacteristic
196C2×#88324C2413yes—
197C4⋊C4×C53C22×C413yes—
198C4⋊C4×C53C22×C413yes—
199C4⋊C4×C53C22×C416yes—
200C22⋊C4×C53C22×C413yes—
201C22⋊C4×C53C22×C413yes—
202C42×C53C22×C413yes—
203C22⋊Dic53C22×C413yes—
204C22⋊Dic53C22×C413yes—
205#11719921C22×C413yes—
206#11719921C22×C413yes—
207C4⋊Dic53C22×C413yes—
208C4⋊Dic53C22×C413yes—
209C4⋊Dic53C22×C413yes—
210C4⋊Dic53C22×C416yes—
211C4⋊Dic53C22×C416yes—
212C4⋊Dic53C22×C413yes—
213#11719919C22×C416yes—
214#11719919C22×C413yes—
215#11719919C22×C413yes—
216#11719919C22×C413yes—
217#11719919C22×C413yes—
218#11719919C22×C416yes—
219#11719919C22×C416yes—
220C4×Dic53C22×C413yes—
221C4×Dic53C22×C413yes—
222C4×Dic53C22×C413yes—
223C4×Dic53C22×C413yes—
224C4×Dic53C22×C413yes—
225C4×Dic53C2413yes—
226C4×Dic53C2413yes—
227C4×Dic53C2413yes—
228C4×Dic53C2413yes—
229C23×C53C23×C411yescharacteristic
230C23×C53C23×C411yescharacteristic
231C22×D53C23×C411yescharacteristic
232Q8×C53—26no—
233Q8×C53—26no—
234Q8×C53#13114yes—
235D4×C53—22no—
236C2×C4×C53—26no—
237C2×C4×C53—26no—
238C2×C4×C53C23×C413yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C23×C413yes—
241C2×C4×C53C23×C413yes—
242C2×C4×C53C23×C413yes—
243C22⋊D53—22no—
244C22⋊D53—22no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—43no—
248C2×Dic53—43no—
249C2×Dic53C23×C413yes—
250C2×Dic53C23×C413yes—
251C2×Dic53C23×C413yes—
252C2×Dic53C23×C411yescharacteristic
253C2×Dic53C23×C413yes—
254C2×Dic53C23×C411yescharacteristic
255C2×Dic532−1+412yes—
256C2×Dic532−1+412yes—
257C2×Dic53C2511yescharacteristic
258C4×D53—26no—
259C4×D53—21no—
260C4×D53—21no—
261C4×D53—43no—
262Dic106—26no—
263Dic106—22no—
264Dic106—46no—
265#88324—23no—
266#88324—21no—
267#88324—21no—
268#88324—23no—
269#88324—21no—
270#88324—23no—
271#88324—23no—
272#88324—21no—
273#60253—531nomaximal, Sylow
274C22×C53—21no—
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53#51912yes—
280C22×C53C24×C411yescharacteristic
281C2×D53—21no—
282C2×D53—21no—
283C2×D53—21no—
284C4×C53—21no—
285C4×C53—23no—
286C4×C53—23no—
287C4×C53—23no—
288C4×C53C2×#13116yes—
289Dic53—21no—
290Dic53—23no—
291Dic53—23no—
292Dic53—23no—
293Dic53—23no—
294Dic53#51912yes—
295Dic53C2×2−1+411yescharacteristic
296Dic53C2×2−1+411yescharacteristic
297Q8×#41—531no—
298#2986—533no—
299#2986—533no—
300#2985—533no—
301#2985—533no—
302#2984—531no—
303#2984—531no—
304#2982—531no—
305#2982—531no—
306Q8×M16—531no—
307#2962—533no—
308#2960—533no—
309C2×C8×Q8—531no—
310#2925—533no—
311#2922—533no—
312C2×C53—22no—
313C2×C53—41no—
314C2×C53C2×#51911yescharacteristic
315C2×C53C22×#13111yescharacteristic
316C2×C53#357311yescharacteristic
317D53—41no—
318Q82—531no—
319#557—533no—
320#554—533no—
321#554—533no—
322#554—533no—
323D4×Q8—531no—
324D4×Q8—531no—
325D4×Q8#615711yescharacteristic
326#533—533no—
327#533—533no—
328#533—533no—
329C4×#41—533no—
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8—531no—
333#446—533no—
334#446—533no—
335#446—536no—
336#446—533no—
337#446—536no—
338#446—533no—
339#446—536no—
340#446—533no—
341#446—533no—
342C8×Q8—531no—
343C8×Q8—531no—
344C8×Q8—533no—
345C8×Q8—533no—
346C8×Q8—533no—
347C8×Q8—533no—
348C8×Q8—536no—
349C8×Q8—533no—
350C8×Q8—533no—
351C8×Q8—531no—
352C8×Q8—531no—
353#433—533no—
354#433—533no—
355#433—533no—
356#433—533no—
357#433—536no—
358#433—536no—
359#432—533no—
360#432—533no—
361#431—533no—
362#431—533no—
363#431—533no—
364#431—533no—
365#431—533no—
366#431—533no—
367#424—533no—
368#424—533no—
369#423—533no—
370C2×C4⋊C8—533no—
371#408—531no—
372#408—533no—
373#405—536no—
374C4×M16—533no—
375C2×C4×C8—533no—
376C53#6025311yescharacteristic, complemented, Sylow
377C2×#41—531no—
378C22×Q8—531no—
379C22×Q8C2×#615711yescharacteristic
380C22×Q8C2×#615711yescharacteristic
381C2×M16—531no—
382C2×M16—533no—
383C22×C8—531no—
384C22×C8—533no—
385C4⋊Q8—533no—
386C4⋊Q8—533no—
387C4⋊Q8—533no—
388C4⋊Q8—533no—
389C4⋊Q8—533no—
390C4⋊Q8C2×#615713yes—
391#125—536no—
392#125—536no—
393#125—536no—
394C22⋊Q8—533no—
395C22⋊Q8—533no—
396C22⋊Q8—533no—
397C22⋊Q8—533no—
398C22⋊Q8—536no—
399C22⋊Q8—536no—
400C22⋊Q8—536no—
401C22⋊Q8C2×#615713yes—
402C22⋊Q8C2×#615713yes—
403C4×Q8—531no—
404C4×Q8—533no—
405C4×Q8—533no—
406C4×Q8—531no—
407C4×Q8—531no—
408C4×Q8—533no—
409C4×Q8—533no—
410C4×Q8—533no—
411C4×Q8—533no—
412C4×Q8—536no—
413C4×Q8—533no—
414C4×Q8—533no—
415C4×Q8C2×#615711yescharacteristic
416C4×D4—533no—
417C4×D4—533no—
418C4×D4—533no—
419C4×D4—533no—
420C4×D4—533no—
421C4×D4C2×#615713yes—
422#117—533no—
423#117—533no—
424#117—533no—
425C2×C4⋊C4—533no—
426C2×C4⋊C4—533no—
427C2×C4⋊C4—533no—
428C2×C42—533no—
429C2×C42—533no—
430C2×C42—533no—
431C4⋊C8—533no—
432C4⋊C8—533no—
433C4⋊C8—536no—
434C4⋊C8—533no—
435C4⋊C8—533no—
436C4⋊C8—536no—
437C4⋊C8—533no—
438C4⋊C8—533no—
439C4⋊C8—533no—
440C4⋊C8—533no—
441C4⋊C8—533no—
442C4⋊C8—533no—
443C4⋊C8—533no—
444C4⋊C8—533no—
445C22⋊C8—531no—
446C22⋊C8—531no—
447C22⋊C8—533no—
448C22⋊C8—531no—
449C22⋊C8—531no—
450C22⋊C8—533no—
451C22⋊C8—533no—
452C22⋊C8—533no—
453#97—533no—
454#97—533no—
455#97—536no—
456#97—536no—
457C4×C8—533no—
458C4×C8—533no—
459C4×C8—533no—
460C4×C8—533no—
461C4×C8—533no—
462C4×C8—533no—
463C4×C8—536no—
464C4×C8—536no—
465#41—1062no—
466#41—1062no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—531no—
470C2×Q8—533no—
471C2×Q8—536no—
472C2×Q8—1064no—
473C2×Q8C22×#615713yes—
474C2×Q8C22×#615713yes—
475C2×Q8C22×#615711yescharacteristic
476C2×D4—531no—
477C2×D4—531no—
478C2×D4C22×#615711yescharacteristic
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—533no—
482C22×C4—533no—
483C22×C4—533no—
484C22×C4—531no—
485C22×C4—533no—
486C22×C4C22×#615713yes—
487C22×C4C22×#615713yes—
488M16—1062no—
489M16—1066no—
490C2×C8—533no—
491C2×C8—531no—
492C2×C8—531no—
493C2×C8—531no—
494C2×C8—533no—
495C2×C8—533no—
496C2×C8—531no—
497C2×C8—533no—
498C2×C8—1066no—
499C2×C8—1062no—
500C4⋊C4—533no—
501C4⋊C4—536no—
502C4⋊C4—536no—
503C4⋊C4—536no—
504C4⋊C4—533no—
505C4⋊C4—533no—
506C4⋊C4—533no—
507C4⋊C4—533no—
508C4⋊C4—533no—
509C4⋊C4—533no—
510C4⋊C4—533no—
511C4⋊C4—536no—
512C4⋊C4—536no—
513C4⋊C4C22×#615713yes—
514C4⋊C4C22×#615713yes—
515C4⋊C4C22×#615716yes—
516C22⋊C4—533no—
517C22⋊C4—533no—
518C22⋊C4—533no—
519C22⋊C4—533no—
520C22⋊C4C22×#615713yes—
521C22⋊C4C22×#615713yes—
522C42—533no—
523C42—533no—
524C42—533no—
525C42—533no—
526C42—533no—
527C42—533no—
528C42—533no—
529C42—533no—
530C42—533no—
531C42C22×#615713yes—
532C23—531no—
533C23C23×#615711yescharacteristic
534C23C23×#615711yescharacteristic
535Q8—26no—
536Q8—26no—
537Q8—1062no—
538Q8—1066no—
539Q8—2126no—
540Q8#1313330414yes—
541D4—22no—
542D4—1062no—
543D4—1062no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—531no—
549C2×C4—531no—
550C2×C4—531no—
551C2×C4—533no—
552C2×C4—533no—
553C2×C4—533no—
554C2×C4—533no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1066no—
560C2×C4—2123no—
561C2×C4—2123no—
562C2×C4—2123no—
563C2×C4C23×#615711yescharacteristic
564C2×C4C23×#615713yes—
565C2×C4C23×#615713yes—
566C2×C4C23×#615713yes—
567C2×C4C23×#615713yes—
568C8—1063no—
569C8—1061no—
570C8—1061no—
571C8—1063no—
572C8—1061no—
573C8—1061no—
574C8—1063no—
575C8—1063no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—1061no—
582C22—1061no—
583C22—1061no—
584C22#1681169112yes—
585C22C24×#615711yescharacteristic
586C4—21no—
587C4—23no—
588C4—23no—
589C4—23no—
590C4—531no—
591C4—531no—
592C4—532no—
593C4—1061no—
594C4—1063no—
595C4—1063no—
596C4—1063no—
597C4—1063no—
598C4C2×#1313330416yes—
599C2—22no—
600C2—41no—
601C2—2121no—
602C2C2×#1681169111yescharacteristic
603C2C22×#1313330411yescharacteristic
604C2#2029171111yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingD4×Q8×C53C4
prime radicalC2×#11719947C23
cosocleC2×Dic53C25
hypercenterD4×Q8#6157
hypocenterC53#60253
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20291626C2semidirect
2. #20291624C2semidirect
3. #20291620C2semidirect
4. #20291619C2semidirect
5. #20291618C2semidirect
6. #20291617C2semidirect
7. #20291609C2semidirect
8. #20291608C2semidirect
9. Q8×#11719940C2semidirect
10. #20291510C2semidirect
11. #20291503C2semidirect
12. C2×Q8×#88324C2semidirect
376. C53#60253semidirect

Maximal subgroups

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

subgroupindex
#202916262
#202916242
#202916202
#202916192
#202916182
#202916172
#202916092
#202916082
Q8×#117199402
#202915102
#202915032
C2×Q8×#883242
#202914102
#202913982
Q8×#117199472
#6025353