Finite Group Databasebeta

← #16752886

Subgroups — #16752886

Order 3328.

1694 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128701C211yesmaximal, characteristic
2#13127723C214yesmaximal, complemented
3#13127707C214yesmaximal, complemented
4#13127532C214yesmaximal, complemented
5#13127521C214yesmaximal, complemented
6#13126004C214yesmaximal, complemented
7#13125980C211yesmaximal, characteristic, complemented
8#13125957C211yesmaximal, characteristic
9#13124916C214yesmaximal, complemented
10#13124875C211yesmaximal, characteristic, complemented
11#13124867C211yesmaximal, characteristic
12#13123917C211yesmaximal, characteristic, complemented
13C2×C13×#557C211yesmaximal, characteristic, complemented
14C13×#557—22no—
15C13×#557—22no—
16C13×#557C2214yes—
17C13×#557C2214yes—
18C2×C13×C4⋊Q8—21no—
19C2×C13×C4⋊Q8C2211yescharacteristic
20C2×C13×#125—22no—
21C2×C13×#125C2212yes—
22C2×C4×Q8×C13—21no—
23C2×C4×Q8×C13C2211yescharacteristic
24C2×C4×Q8×C13C2212yes—
25C2×C4×Q8×C13C2211yescharacteristic
26#11719521—21no—
27#11719290C2212yes—
28#11719289C2212yes—
29#11719286—22no—
30#11719284C2212yes—
31#11719273—22no—
32#11719272C2212yes—
33#11719220—22no—
34#11719209—21no—
35#11719184C2214yes—
36#11719182C2211yescharacteristic
37#11719175C2211yescharacteristic
38#11718889C2212yes—
39#11718888C2212yes—
40#11718755C2214yes—
41#11718754C2212yes—
42#11718749C2212yes—
43#11718748C2214yes—
44#11718740C2212yes—
45#11718739C2212yes—
46#11718729C2218yes—
47#11718727C2218yes—
48#11718725C2218yes—
49#11718724C2218yes—
50#11718692C2212yes—
51C4×#6107C2214yes—
52C4×#6107C2218yes—
53#11718689C2218yes—
54#11718688C2212yes—
55C4×#6106C2214yes—
56C4×#6106C2218yes—
57Q8×#1055C2214yes—
58Q8×#1055C2214yes—
59#11718683C2218yes—
60#11718682C2212yes—
61#11718681C2212yes—
62#11718632C2214yes—
63#11718631C2212yes—
64#11718629C2212yes—
65#11718628C2214yes—
66#11718586C2211yescharacteristic
67#11718586C2211yescharacteristic
68#11718586C2214yes—
69C22×Q8×C13—21no—
70C22×Q8×C13D411yescharacteristic
71C22×Q8×C13C2311yescharacteristic
72C2×#6127—21no—
73C13×C4⋊Q8—24no—
74C13×C4⋊Q8—24no—
75C13×C4⋊Q8—21no—
76C13×C4⋊Q8—21no—
77C13×C4⋊Q8C2312yes—
78C13×C4⋊Q8C2312yes—
79C13×#125—28no—
80C13×#125—28no—
81C13×#125—24no—
82C13×#125C2318yes—
83C4×Q8×C13—24no—
84C4×Q8×C13—24no—
85C4×Q8×C13—24no—
86C4×Q8×C13—21no—
87C4×Q8×C13—21no—
88C4×Q8×C13D412yes—
89C4×Q8×C13D412yes—
90C4×Q8×C13C2312yes—
91C4×Q8×C13C2312yes—
92C4×Q8×C13C2318yes—
93C4×Q8×C13C2312yes—
94C4×Q8×C13C2312yes—
95C2×C13×C4⋊C4—22no—
96C2×C13×C4⋊C4—21no—
97C2×C13×C4⋊C4—22no—
98C2×C13×C4⋊C4—21no—
99C2×C13×C4⋊C4—22no—
100C2×C13×C4⋊C4D411yescharacteristic
101C2×C13×C4⋊C4D412yes—
102C2×C13×C4⋊C4C2311yescharacteristic
103C2×C13×C4⋊C4C2312yes—
104C2×C42×C13—22no—
105C2×C42×C13—21no—
106C2×C42×C13C2311yescharacteristic
107C2×C42×C13C2312yes—
108#88201—22no—
109#88200—22no—
110#88200—22no—
111#88198—22no—
112C2×#6107C2314yes—
113C2×#6106C2314yes—
114#88184—21no—
115#88184C2312yes—
116#88183—24no—
117C4×C22⋊D13—24no—
118#88180—21no—
119#88178—21no—
120#88178C2312yes—
121#88152—24no—
122#88150—24no—
123#88148—24no—
124#88147—22no—
125#88146—22no—
126C4⋊C4×D13—22no—
127#88143—24no—
128#88141—22no—
129#88130—24no—
130#88129—21no—
131#88129C2314yes—
132#88128C2311yescharacteristic
133C4×D52—21no—
134C4×D52C2312yes—
135C4×D52C2314yes—
136#88126—24no—
137#88124—21no—
138#88124C2314yes—
139#88123C2311yescharacteristic
140#88123C2311yescharacteristic
141#88123C2311yescharacteristic
142C4×Dic26—21no—
143C4×Dic26C2312yes—
144C4×Dic26C2314yes—
145#88075C2314yes—
146#88075C2314yes—
147C22⋊#1055C2312yes—
148C22⋊#1055C2312yes—
149#88050C2318yes—
150#88050C2314yes—
151#88049C2314yes—
152#88049C2318yes—
153#88048C2314yes—
154#88048C2318yes—
155#88047C2314yes—
156#88047C2318yes—
157C4⋊#1055C2312yes—
158C4⋊#1055C2312yes—
159C4⋊#1055C2318yes—
160C4×#1055C2312yes—
161C4×#1055C2318yes—
162C4×#1055C2312yes—
163#61822—131nomaximal, Sylow
164C2×Q8×C13—24no—
165C2×Q8×C13—24no—
166C2×Q8×C13—21no—
167C2×Q8×C13—22no—
168C2×Q8×C13—22no—
169C2×Q8×C13—44no—
170C2×Q8×C13C2×D412yes—
171C2×Q8×C13C2×D412yes—
172C2×Q8×C13C2412yes—
173C22×C4×C13—21no—
174C22×C4×C13—21no—
175C22×C4×C13—22no—
176C22×C4×C13C2×D412yes—
177C22×C4×C13C2×D411yescharacteristic
178C22×C4×C13C2×D412yes—
179C22×C4×C13C2×D411yescharacteristic
180C22×C4×C13C2411yescharacteristic
181C2×C22⋊D13—22no—
182#6127—44no—
183C2×D52—21no—
184C2×D52—21no—
185C2×D52C2412yes—
186C2×C4×D13—22no—
187C2×C4×D13—24no—
188C2×Dic26—21no—
189C2×Dic26—21no—
190C2×Dic26C2412yes—
191C13×C4⋊C4—28no—
192C13×C4⋊C4—22no—
193C13×C4⋊C4—22no—
194C13×C4⋊C4—28no—
195C13×C4⋊C4—22no—
196C13×C4⋊C4—22no—
197C13×C4⋊C4—28no—
198C13×C4⋊C4—21no—
199C13×C4⋊C4—22no—
200C13×C4⋊C4C2×D412yes—
201C13×C4⋊C4C2×D414yes—
202C13×C4⋊C4C2×D412yes—
203C13×C4⋊C4C2×D414yes—
204C13×C4⋊C4C2412yes—
205C13×C4⋊C4C2414yes—
206C42×C13—28no—
207C42×C13—22no—
208C42×C13—22no—
209C42×C13C2×D411yescharacteristic
210C42×C13C2×D411yescharacteristic
211C42×C13C2×D414yes—
212C42×C13C2411yescharacteristic
213C42×C13C2414yes—
214C42×C13C2411yescharacteristic
215C22⋊Dic13—22no—
216#6107—28no—
217#6106—28no—
218#6103—22no—
219#6103—24no—
220#6103—22no—
221#6103—24no—
222#6103—24no—
223C4⋊Dic13—21no—
224C4⋊Dic13—21no—
225C4⋊Dic13—21no—
226C4⋊Dic13—21no—
227C4⋊Dic13C2412yes—
228C4⋊Dic13C2412yes—
229#6101—22no—
230#6101—22no—
231#6101—24no—
232#6101—24no—
233#6101—24no—
234C4×Dic13—22no—
235C4×Dic13—24no—
236C2×#1055C2414yes—
237C2×#1055C2414yes—
238#3507—131no—
239C2×#557D1311yescharacteristic, complemented
240#3383—134no—
241#3381—134no—
242#3365—134no—
243#3364—134no—
244#3232—134no—
245#3229—131no—
246#3226—131no—
247#3108—134no—
248#3104—131no—
249#3103—131no—
250#2982—131no—
251C23×C13C22×D411yescharacteristic
252C22×D13—22no—
253Q8×C13—24no—
254Q8×C13—24no—
255Q8×C13—24no—
256Q8×C13—22no—
257Q8×C13—44no—
258Q8×C13—44no—
259Q8×C132−1+414yes—
260C2×C4×C13—24no—
261C2×C4×C13—24no—
262C2×C4×C13—22no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—24no—
266C2×C4×C13—41no—
267C2×C4×C13—42no—
268C2×C4×C13—42no—
269C2×C4×C13—44no—
270C2×C4×C13#13514yes—
271C2×C4×C13C22×D414yes—
272C2×C4×C13C22×D411yescharacteristic
273C2×C4×C13C22×D411yescharacteristic
274C2×C4×C13C22×D414yes—
275C2×C4×C13C22×D411yescharacteristic
276C2×C4×C13C22×D411yescharacteristic
277C2×C4×C13C22×D411yescharacteristic
278C2×C4×C13C2511yescharacteristic
279C22⋊D13—44no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283D52—24no—
284D52—42no—
285D52—81no—
286C4×D13—44no—
287C4×D13—84no—
288Dic26—24no—
289Dic26—42no—
290Dic26—81no—
291#1055—24no—
292#1055—24no—
293#559—132no—
294#557—22no—
295#557—22no—
296#557—132no—
297#557C2×D1314yes—
298#557C2×D1314yes—
299#556—132no—
300#556—262no—
301#554—132no—
302#552—262no—
303#539—262no—
304#536—261no—
305#533—131no—
306#532—131no—
307C2×C4⋊Q8—21no—
308C2×C4⋊Q8C2×D1311yescharacteristic
309#529—134no—
310C2×#125—22no—
311C2×#125C2×D1312yes—
312#520—261no—
313C2×C4×Q8—21no—
314C2×C4×Q8C2×D1311yescharacteristic
315C2×C4×Q8C2×D1312yes—
316C2×C4×Q8C2×D1311yescharacteristic
317#500—132no—
318#499—138no—
319#498—132no—
320#488—138no—
321#487—132no—
322#486—132no—
323#484—132no—
324#483—134no—
325#481—132no—
326#480—134no—
327#479—138no—
328#479—138no—
329#478—138no—
330#477—132no—
331#475—134no—
332#475—132no—
333#474—134no—
334#464—138no—
335#462—132no—
336#460—132no—
337#451—132no—
338#448—132no—
339C8×Q8—134no—
340C8×Q8—134no—
341C4×Q16—134no—
342C4×Q16—138no—
343C4×SD16—134no—
344C4×SD16—138no—
345#431—131no—
346#431—131no—
347#431—134no—
348C22×C13—21no—
349C22×C13—21no—
350C22×C13C2×#13512yes—
351C22×C13C23×D411yescharacteristic
352C2×D13—42no—
353C2×D13—42no—
354C2×D13—42no—
355C4×C13—21no—
356C4×C13—24no—
357C4×C13—21no—
358C4×C13—24no—
359C4×C13—21no—
360C4×C13—24no—
361C4×C13—42no—
362C4×C13—44no—
363C4×C13C2×#13512yes—
364C4×C13C2×#13512yes—
365C4×C13C2×2−1+412yes—
366Dic13—42no—
367Dic13—42no—
368Dic13—42no—
369C2×#41—261no—
370C22×Q8—21no—
371C22×Q8C22⋊D1311yescharacteristic
372C22×Q8C22×D1311yescharacteristic
373C2×Q16—134no—
374C2×SD16—134no—
375C4⋊Q8—24no—
376C4⋊Q8—24no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—131no—
380C4⋊Q8—131no—
381C4⋊Q8—131no—
382C4⋊Q8—262no—
383C4⋊Q8C22×D1312yes—
384C4⋊Q8C22×D1312yes—
385#127—131no—
386#126—264no—
387#126—264no—
388#125—28no—
389#125—28no—
390#125—24no—
391#125—134no—
392#125—264no—
393#125—261no—
394#125C22×D1318yes—
395#124—134no—
396#124—262no—
397#124—261no—
398#123—264no—
399#123—264no—
400C22⋊Q8—132no—
401C22⋊Q8—262no—
402C22⋊Q8—262no—
403C22⋊Q8—261no—
404C22⋊Q8—264no—
405#121—132no—
406#121—261no—
407C4×Q8—24no—
408C4×Q8—24no—
409C4×Q8—24no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—134no—
413C4×Q8—132no—
414C4×Q8—262no—
415C4×Q8—261no—
416C4×Q8C22⋊D1312yes—
417C4×Q8C22⋊D1312yes—
418C4×Q8C22×D1312yes—
419C4×Q8C22×D1318yes—
420C4×Q8C22×D1312yes—
421C4×Q8C22×D1312yes—
422C4×Q8C22×D1312yes—
423C4×D4—134no—
424C4×D4—132no—
425C4×D4—264no—
426C4×D4—262no—
427C4×D4—261no—
428#117—261no—
429#117—262no—
430#117—264no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4—262no—
437C2×C4⋊C4C22⋊D1311yescharacteristic
438C2×C4⋊C4C22⋊D1312yes—
439C2×C4⋊C4C22×D1311yescharacteristic
440C2×C4⋊C4C22×D1312yes—
441C2×C42—22no—
442C2×C42—21no—
443C2×C42C22×D1312yes—
444C2×C42C22×D1311yescharacteristic
445#107—138no—
446#107—134no—
447#106—138no—
448#106—134no—
449C4⋊C8—132no—
450C4⋊C8—132no—
451C4⋊C8—138no—
452#103—134no—
453#103—134no—
454#103—134no—
455#103—138no—
456#102—134no—
457#102—138no—
458C22⋊C8—132no—
459C22⋊C8—132no—
460C4×C8—132no—
461C4×C8—138no—
462C4×C8—132no—
463C2×C13—22no—
464C2×C13#348611yescharacteristic
465C2×C13C22×#13511yescharacteristic
466C2×C13#358611yescharacteristic
467D13—82no—
468#41—524no—
469C2×Q8—24no—
470C2×Q8—24no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—44no—
475C2×Q8—132no—
476C2×Q8—261no—
477C2×Q8—261no—
478C2×Q8C2×C22⋊D1312yes—
479C2×Q8C2×C22⋊D1312yes—
480C2×Q8C23×D1312yes—
481C2×D4—132no—
482C2×D4—262no—
483C2×D4—261no—
484C2×D4—261no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—262no—
489C22×C4—264no—
490C22×C4C2×C22⋊D1312yes—
491C22×C4C2×C22⋊D1312yes—
492C22×C4C2×C22⋊D1311yescharacteristic
493C22×C4C2×C22⋊D1311yescharacteristic
494C22×C4C23×D1311yescharacteristic
495Q16—268no—
496SD16—268no—
497C2×C8—134no—
498C2×C8—134no—
499C4⋊C4—28no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—28no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—132no—
509C4⋊C4—132no—
510C4⋊C4—261no—
511C4⋊C4—264no—
512C4⋊C4—261no—
513C4⋊C4—262no—
514C4⋊C4—264no—
515C4⋊C4—261no—
516C4⋊C4—262no—
517C4⋊C4—264no—
518C4⋊C4—261no—
519C4⋊C4C2×C22⋊D1312yes—
520C4⋊C4C2×C22⋊D1312yes—
521C4⋊C4C2×C22⋊D1314yes—
522C4⋊C4C2×C22⋊D1314yes—
523C4⋊C4C23×D1312yes—
524C4⋊C4C23×D1314yes—
525C22⋊C4—262no—
526C22⋊C4—262no—
527C22⋊C4—264no—
528C22⋊C4—264no—
529C22⋊C4—262no—
530C22⋊C4—264no—
531C42—28no—
532C42—22no—
533C42—22no—
534C42—264no—
535C42—262no—
536C42C2×C22⋊D1314yes—
537C42C2×C22⋊D1311yescharacteristic
538C42C2×C22⋊D1311yescharacteristic
539C42C23×D1311yescharacteristic
540C42C23×D1314yes—
541C42C23×D1311yescharacteristic
542C13#6182211yescharacteristic, complemented, Sylow
543C23—262no—
544C23C22×C22⋊D1311yescharacteristic
545Q8—24no—
546Q8—24no—
547Q8—24no—
548Q8—22no—
549Q8—44no—
550Q8—44no—
551Q8—264no—
552Q8—522no—
553Q8—1041no—
554Q8#8825414yes—
555D4—264no—
556D4—524no—
557D4—522no—
558D4—1041no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—262no—
570C2×C4—262no—
571C2×C4—262no—
572C2×C4—524no—
573C2×C4—1044no—
574C2×C4#8820414yes—
575C2×C4C22×C22⋊D1314yes—
576C2×C4C22×C22⋊D1311yescharacteristic
577C2×C4C22×C22⋊D1314yes—
578C2×C4C22×C22⋊D1311yescharacteristic
579C2×C4C22×C22⋊D1311yescharacteristic
580C2×C4C22×C22⋊D1311yescharacteristic
581C2×C4C22×C22⋊D1311yescharacteristic
582C2×C4C24×D1311yescharacteristic
583C8—264no—
584C8—264no—
585C22—21no—
586C22—21no—
587C22—522no—
588C22—522no—
589C22—522no—
590C22C2×#8820412yes—
591C22C23×C22⋊D1311yescharacteristic
592C4—24no—
593C4—21no—
594C4—24no—
595C4—24no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—42no—
600C4—522no—
601C4—522no—
602C4—522no—
603C4C2×#8820412yes—
604C4C2×#8820412yes—
605C4C2×#8825412yes—
606C2—22no—
607C2—1042no—
608C2#1312848711yescharacteristic
609C2C22×#8820411yescharacteristic
610C2#1312962311yescharacteristic
subgroupG/H
centerC22C23×C22⋊D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×C13×#557C2
socleC22×C13C23×D4
prime radical#11719182C22
hypercenterC2×#557D13
hypocenterC13#61822
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #13127723C2semidirect
3. #13127707C2semidirect
4. #13127532C2semidirect
5. #13127521C2semidirect
6. #13126004C2semidirect
7. #13125980C2semidirect
9. #13124916C2semidirect
10. #13124875C2semidirect
12. #13123917C2semidirect
13. C2×C13×#557C2semidirect
239. C2×#557D13semidirect
542. C13#61822semidirect

Maximal subgroups

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

subgroupindex
#131287012
#131277232
#131277072
#131275322
#131275212
#131260042
#131259802
#131259572
#131249162
#131248752
#131248672
#131239172
C2×C13×#5572
#6182213