Finite Group Databasebeta

← #15830701

Subgroups — #15830701

Order 3328.

1074 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128675C211yesmaximal, characteristic
2#13117746C211yesmaximal, characteristic, complemented
3#13113409C211yesmaximal, characteristic, complemented
4#13113405C211yesmaximal, characteristic, complemented
5#13113380C212yesmaximal, complemented
6#13113375C212yesmaximal, complemented
7#13113362C212yesmaximal, complemented
8#13113341C212yesmaximal, complemented
9C2×#11718059C211yesmaximal, characteristic, complemented
10C2×#11718062C211yesmaximal, characteristic, complemented
11C13×#2973C211yesmaximal, characteristic, complemented
12C2×C4×D4×C13C2211yescharacteristic
13#11719509—21no—
14C2×D4×Dic13—21no—
15#11719496C2212yes—
16C2×#88184C2211yescharacteristic
17C2×#88178C2211yescharacteristic
18#11719270C412yescomplemented
19#11719261—22no—
20#11719257C412yescomplemented
21#11719255C412yescomplemented
22#11719246C412yescomplemented
23#11719245—22no—
24#11719220—21no—
25#11719199—21no—
26#11719191—21no—
27#11719184C2212yes—
28C13×#435—22no—
29C13×#435—22no—
30C13×#432C2212yescomplemented
31C13×#407C2212yescomplemented
32C2×C13×C22⋊C8C2211yescharacteristic
33C2×C13×C22⋊C8C2211yescharacteristic
34#11718873C2211yescharacteristic
35#11718873C2211yescharacteristic, complemented
36C2×#88071C2211yescharacteristic
37C2×#88071C2211yescharacteristic, complemented
38#11718646—21no—
39#11718611—21no—
40#11718611—21no—
41#11718597C2211yescharacteristic, complemented
42C4×C4⋊Dic13C2211yescharacteristic, complemented
43#11718062C2212yes—
44#11718062C2212yes—
45#11718062C2212yes—
46#11718062C2212yes—
47#11718059C2212yescomplemented
48#11718059C2212yescomplemented
49#11718059C2212yescomplemented
50#11718059C2212yescomplemented
51C22×D4×C13D411yescharacteristic
52C23×C4×C13C2311yescharacteristic
53C23×C4×C13C2311yescharacteristic
54C22×C22⋊D13—21no—
55C23×Dic13—21no—
56C23×Dic13—21no—
57C2×#6127C2×C412yescomplemented
58C22×D52C2×C411yescharacteristic, complemented
59C22×Dic26C2×C411yescharacteristic, complemented
60C2×C13×M16—22no—
61C22×C8×C13—21no—
62C22×C8×C13—21no—
63C4×D4×C13C2×C412yes—
64C4×D4×C13C2×C412yes—
65C4×D4×C13D412yes—
66C4×D4×C13D412yes—
67C2×C13×C4⋊C4D411yescharacteristic
68C2×C13×C22⋊C4D411yescharacteristic
69C2×C13×C22⋊C4D411yescharacteristic
70C2×C42×C13C2311yescharacteristic
71#88207—22no—
72C2×C22⋊Dic13—21no—
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13—21no—
75#88193—22no—
76#88193—22no—
77#88191—22no—
78#88189—22no—
79D4×Dic13—44no—
80#88184—21no—
81#88184—21no—
82#88184C2×C412yescomplemented
83#88184C2×C412yescomplemented
84#88184C2312yes—
85#88183—22no—
86#88183—22no—
87C4×C22⋊D13—22no—
88C4×C22⋊D13—22no—
89C4×C22⋊D13—22no—
90C2×#6103—21no—
91#88180C2×C412yescomplemented
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13—21no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13C2×C411yescharacteristic, complemented
97C2×C4⋊Dic13C2×C411yescharacteristic, complemented
98#88178—21no—
99#88178—21no—
100#88178C2×C412yescomplemented
101#88178C2×C412yescomplemented
102#88178C2312yes—
103C2×#6101—21no—
104C2×C4×Dic13—21no—
105#88152—22no—
106#88150—22no—
107#88148—22no—
108#88143—22no—
109#88139—22no—
110#88138—22no—
111#88137—22no—
112#88135—22no—
113#88135—22no—
114C22⋊C4×D13—22no—
115#88133—22no—
116#88131—22no—
117#88130—22no—
118#88129C2×C412yescomplemented
119C4×D52C2×C412yescomplemented
120#88126—22no—
121#88124C2×C412yescomplemented
122C4×Dic26C2×C412yescomplemented
123C13×C4⋊C8—22no—
124C13×C22⋊C8—21no—
125C13×C22⋊C8—21no—
126C13×C22⋊C8—22no—
127C13×C22⋊C8C2312yes—
128C13×C22⋊C8C2312yes—
129C13×#97—22no—
130#88071—21no—
131#88071—21no—
132#88071—21no—
133#88071C2311yescharacteristic
134#88071C2311yescharacteristic
135#88071C2311yescharacteristic
136#88071C2311yescharacteristic
137#17622—131nomaximal, Sylow
138C24×C13C2×D411yescharacteristic
139C24×C13C2×D411yescharacteristic
140C23×D13—21no—
141C2×D4×C13—22no—
142C2×D4×C13—22no—
143C2×D4×C13C22⋊C412yes—
144C2×D4×C13C22⋊C412yes—
145C22×C4×C13—81no—
146C22×C4×C13—81no—
147C22×C4×C13C22×C411yescharacteristic
148C22×C4×C13C22×C412yes—
149C22×C4×C13C22×C411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D412yes—
157C22×C4×C13C2411yescharacteristic
158C2×C22⋊D13—22no—
159C2×C22⋊D13—22no—
160C2×C22⋊D13—22no—
161C2×C22⋊D13—22no—
162C2×C22⋊D13—42no—
163C2×C22⋊D13—42no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—21no—
168C22×Dic13—21no—
169C22×Dic13—21no—
170C22×Dic13—21no—
171C22×Dic13—42no—
172C22×Dic13—42no—
173C22×Dic13—42no—
174C22×Dic13—42no—
175C22×Dic13—81no—
176C22×Dic13—81no—
177#6127—82no—
178C2×D52—21no—
179C2×D52—81no—
180C2×D52C22×C412yes—
181C2×C4×D13—22no—
182C2×C4×D13—22no—
183C2×Dic26—21no—
184C2×Dic26—81no—
185C2×Dic26C22×C412yes—
186C13×M16—82no—
187C2×C8×C13—22no—
188C2×C8×C13—22no—
189C2×C8×C13—81no—
190C2×C8×C13—81no—
191C13×C4⋊C4C22⋊C412yes—
192C13×C4⋊C4C22⋊C412yes—
193C13×C22⋊C4C22⋊C412yes—
194C13×C22⋊C4C22⋊C412yes—
195C13×C22⋊C4C22⋊C412yes—
196C13×C22⋊C4C22⋊C412yes—
197C42×C13C22×C412yes—
198C42×C13C2×D412yes—
199C22⋊Dic13—22no—
200C22⋊Dic13—22no—
201C22⋊Dic13—22no—
202C22⋊Dic13—22no—
203C22⋊Dic13—42no—
204C22⋊Dic13—42no—
205#6103—22no—
206#6103—22no—
207#6103—22no—
208#6103—22no—
209#6103—22no—
210#6103—22no—
211C4⋊Dic13—21no—
212C4⋊Dic13—21no—
213C4⋊Dic13—42no—
214C4⋊Dic13—42no—
215C4⋊Dic13—81no—
216C4⋊Dic13—81no—
217C4⋊Dic13C22×C412yes—
218C4⋊Dic13C22×C412yes—
219#6101—22no—
220#6101—22no—
221#6101—22no—
222#6101—22no—
223#6101—22no—
224#6101—22no—
225C4×Dic13—22no—
226C4×Dic13—22no—
227C4×Dic13—42no—
228#3505—131no—
229#2973D1311yescharacteristic, complemented
230#2349—131no—
231#1520—131no—
232#1519—131no—
233#1513—132no—
234#1512—132no—
235#1509—132no—
236#1504—132no—
237C2×#328—131no—
238C2×#327—131no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—41no—
244C23×C13—41no—
245C23×C13C2×C22⋊C412yes—
246C23×C13C2×C22⋊C411yescharacteristic
247C23×C13C2×C22⋊C411yescharacteristic
248C23×C13AGL(1,ℤ/8)12yes—
249C23×C13#13712yes—
250C23×C13C22×D411yescharacteristic
251C22×D13—22no—
252C22×D13—42no—
253C22×D13—81no—
254C22×D13—81no—
255D4×C13—24no—
256D4×C13—24no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—42no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—42no—
268C2×C4×C13—41no—
269C2×C4×C13C2×C22⋊C412yes—
270C2×C4×C13C2×C22⋊C412yes—
271C2×C4×C13C2×C22⋊C411yescharacteristic
272C2×C4×C13C2×C22⋊C412yes—
273C2×C4×C13C2×C22⋊C411yescharacteristic
274C2×C4×C13C23×C411yescharacteristic
275C2×C4×C13C22×D411yescharacteristic
276C22⋊D13—42no—
277C22⋊D13—42no—
278C22⋊D13—44no—
279C22⋊D13—82no—
280C22⋊D13—82no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—42no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—81no—
298C2×Dic13—81no—
299C2×Dic13—84no—
300C2×Dic13—84no—
301D52—42no—
302D52—81no—
303C4×D13—82no—
304C4×D13—82no—
305Dic26—42no—
306Dic26—81no—
307C8×C13—42no—
308C8×C13—42no—
309#553—132no—
310#553—262no—
311#552—132no—
312#551—262no—
313#548—132no—
314#546—132no—
315#539—261no—
316#538—261no—
317#534—261no—
318#529—132no—
319#525—132no—
320C2×C22⋊Q8—131no—
321C2×#121—131no—
322#518—261no—
323C2×C4×D4—261no—
324C2×C4×D4C2×D1311yescharacteristic
325#435—22no—
326#435—22no—
327#432C2×D1312yescomplemented
328#407C2×D1312yescomplemented
329C2×C22⋊C8C2×D1311yescharacteristic
330C2×C22⋊C8C2×D1311yescharacteristic
331#389—261no—
332#385—261no—
333#385—261no—
334#382—131no—
335#380—131no—
336#380—131no—
337C4×C4⋊C4—131no—
338C2×#95—131no—
339C2×#95—131no—
340#328—132no—
341#328—132no—
342#328—132no—
343#328—132no—
344#327—132no—
345#327—132no—
346#327—132no—
347#327—132no—
348C22×C13—21no—
349C22×C13—21no—
350C22×C13—21no—
351C22×C13—21no—
352C22×C13—21no—
353C22×C13—21no—
354C22×C13—22no—
355C22×C13—22no—
356C22×C13—22no—
357C22×C13—42no—
358C22×C13—42no—
359C22×C13—42no—
360C22×C13—42no—
361C22×C13#41712yes—
362C22×C13#41712yes—
363C22×C13C22×C22⋊C411yescharacteristic
364C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C13C2×#13711yescharacteristic
366C2×D13—42no—
367C2×D13—42no—
368C2×D13—42no—
369C2×D13—162no—
370C4×C13—21no—
371C4×C13—21no—
372C4×C13—21no—
373C4×C13—22no—
374C4×C13—21no—
375C4×C13—22no—
376C4×C13—22no—
377Dic13—42no—
378Dic13—42no—
379Dic13—42no—
380Dic13—42no—
381Dic13—81no—
382Dic13—81no—
383Dic13—81no—
384C2×#41—132no—
385C22×Q8—131no—
386C22×D4—131no—
387C22×D4—261no—
388C22×D4D5211yescharacteristic
389C23×C4—261no—
390C23×C4—261no—
391C23×C4C22×D1311yescharacteristic
392C23×C4C22×D1311yescharacteristic
393C2×M16—22no—
394C22×C8—21no—
395C22×C8—21no—
396#126—262no—
397#126—262no—
398#126—262no—
399#125—132no—
400#125—262no—
401#124—132no—
402#124—262no—
403#123—262no—
404#123—262no—
405#123—262no—
406#123—262no—
407#123—262no—
408C22⋊Q8—132no—
409C22⋊Q8—132no—
410C22⋊Q8—132no—
411C22⋊Q8—261no—
412C22⋊Q8—261no—
413C22⋊Q8—262no—
414#121—132no—
415#121—132no—
416#121—132no—
417#121—262no—
418#121—262no—
419#121—261no—
420#121—261no—
421#121—262no—
422#120—262no—
423#120—262no—
424C4×Q8—132no—
425C4×D4—132no—
426C4×D4—262no—
427C4×D4—262no—
428C4×D4—262no—
429C4×D4—262no—
430C4×D4—262no—
431C4×D4—524no—
432C4×D4C4×D1312yes—
433C4×D4C4×D1312yes—
434C4×D4C22⋊D1312yes—
435C4×D4C22⋊D1312yes—
436#117—132no—
437#117—262no—
438#117—262no—
439C2×C4⋊C4—131no—
440C2×C4⋊C4—131no—
441C2×C4⋊C4—261no—
442C2×C4⋊C4—261no—
443C2×C4⋊C4—261no—
444C2×C4⋊C4—261no—
445C2×C4⋊C4—261no—
446C2×C4⋊C4D5211yescharacteristic
447C2×C22⋊C4—261no—
448C2×C22⋊C4—261no—
449C2×C22⋊C4—262no—
450C2×C22⋊C4—261no—
451C2×C22⋊C4—261no—
452C2×C22⋊C4D5211yescharacteristic
453C2×C22⋊C4D5211yescharacteristic
454C2×C42—261no—
455C2×C42C22×D1311yescharacteristic
456C4⋊C8—22no—
457C22⋊C8—21no—
458C22⋊C8—21no—
459C22⋊C8—22no—
460C22⋊C8C22×D1312yes—
461C22⋊C8C22×D1312yes—
462#97—22no—
463#95—131no—
464#95—131no—
465#95—131no—
466#95—131no—
467#95—261no—
468#95—261no—
469#95—261no—
470C2×C13—21no—
471C2×C13—22no—
472C2×C13—21no—
473C2×C13—22no—
474C2×C13—42no—
475C2×C13#288311yescharacteristic
476C2×C13C2×#41711yescharacteristic
477C2×C13#290111yescharacteristic
478D13—82no—
479C24—261no—
480C24C2×D5211yescharacteristic
481C24C2×D5211yescharacteristic
482#41—1042no—
483C2×Q8—132no—
484C2×Q8—261no—
485C2×Q8—1041no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—132no—
489C2×D4—262no—
490C2×D4—262no—
491C2×D4—262no—
492C2×D4—262no—
493C2×D4—261no—
494C2×D4—522no—
495C2×D4—522no—
496C2×D4—1041no—
497C2×D4#610312yes—
498C2×D4#610312yes—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—261no—
502C22×C4—261no—
503C22×C4—261no—
504C22×C4—261no—
505C22×C4—262no—
506C22×C4—262no—
507C22×C4—261no—
508C22×C4—261no—
509C22×C4—261no—
510C22×C4—522no—
511C22×C4—522no—
512C22×C4—522no—
513C22×C4—522no—
514C22×C4—1041no—
515C22×C4—1041no—
516C22×C4C2×C4×D1311yescharacteristic
517C22×C4C2×C4×D1311yescharacteristic
518C22×C4C2×C4×D1312yes—
519C22×C4C2×D5211yescharacteristic
520C22×C4C2×D5211yescharacteristic
521C22×C4C2×D5211yescharacteristic
522C22×C4C2×D5211yescharacteristic
523C22×C4C2×C22⋊D1311yescharacteristic
524C22×C4C2×C22⋊D1311yescharacteristic
525C22×C4C2×C22⋊D1312yes—
526C22×C4C23×D1311yescharacteristic
527M16—82no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—81no—
531C2×C8—81no—
532C4⋊C4—132no—
533C4⋊C4—132no—
534C4⋊C4—261no—
535C4⋊C4—261no—
536C4⋊C4—262no—
537C4⋊C4—262no—
538C4⋊C4—262no—
539C4⋊C4—262no—
540C4⋊C4—262no—
541C4⋊C4—262no—
542C4⋊C4—522no—
543C4⋊C4—522no—
544C4⋊C4—1041no—
545C4⋊C4—1041no—
546C4⋊C4#610312yes—
547C4⋊C4#610312yes—
548C22⋊C4—262no—
549C22⋊C4—262no—
550C22⋊C4—262no—
551C22⋊C4—262no—
552C22⋊C4—262no—
553C22⋊C4—262no—
554C22⋊C4—262no—
555C22⋊C4—262no—
556C22⋊C4—262no—
557C22⋊C4—262no—
558C22⋊C4—522no—
559C22⋊C4—522no—
560C22⋊C4#610312yes—
561C22⋊C4#610312yes—
562C22⋊C4#610312yes—
563C22⋊C4#610312yes—
564C42—262no—
565C42—262no—
566C42—522no—
567C42C2×C4×D1312yes—
568C42C2×C22⋊D1312yes—
569C13#1762211yescharacteristic, complemented, Sylow
570C23—21no—
571C23—21no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—262no—
577C23—522no—
578C23—1041no—
579C23—1041no—
580C23#8816212yes—
581C23#8816312yes—
582C23C2×#610312yes—
583C23C2×#610311yescharacteristic
584C23C2×#610311yescharacteristic
585C23C22×D5211yescharacteristic
586Q8—522no—
587Q8—1041no—
588D4—24no—
589D4—24no—
590D4—522no—
591D4—522no—
592D4—522no—
593D4—524no—
594D4—1041no—
595D4—1042no—
596D4—1042no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—262no—
610C2×C4—262no—
611C2×C4—262no—
612C2×C4—262no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—522no—
616C2×C4—521no—
617C2×C4—521no—
618C2×C4—521no—
619C2×C4—521no—
620C2×C4—521no—
621C2×C4—521no—
622C2×C4—521no—
623C2×C4—521no—
624C2×C4—521no—
625C2×C4—1042no—
626C2×C4—1042no—
627C2×C4—1041no—
628C2×C4—1041no—
629C2×C4—1044no—
630C2×C4—1044no—
631C2×C4C2×#610312yes—
632C2×C4C2×#610312yes—
633C2×C4C2×#610312yes—
634C2×C4C2×#610311yescharacteristic
635C2×C4C2×#610311yescharacteristic
636C2×C4C22×C4×D1311yescharacteristic
637C2×C4C22×C22⋊D1311yescharacteristic
638C8—42no—
639C8—42no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—522no—
654C22—522no—
655C22—522no—
656C22—2082no—
657C22#1171879412yes—
658C22#1171879412yes—
659C22C2×#8816211yescharacteristic
660C22C2×#8816311yescharacteristic
661C22C22×#610311yescharacteristic
662C4—22no—
663C4—22no—
664C4—21no—
665C4—21no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—522no—
670C4—522no—
671C4—522no—
672C4—522no—
673C4—1041no—
674C4—1041no—
675C4—1041no—
676C2—21no—
677C2—21no—
678C2—22no—
679C2—22no—
680C2—42no—
681C2—1042no—
682C2#1312329611yescharacteristic
683C2C2×#1171879411yescharacteristic
684C2#1312338111yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2973C2
socleC22×C13C22×C22⋊C4
prime radical#13128675C2
cosocleC22×C4×C13C24
hypercenter#2973D13
hypocenterC13#17622
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #13117746C2semidirect
3. #13113409C2semidirect
4. #13113405C2semidirect
5. #13113380C2semidirect
6. #13113375C2semidirect
7. #13113362C2semidirect
8. #13113341C2semidirect
9. C2×#11718059C2semidirect
10. C2×#11718062C2semidirect
11. C13×#2973C2semidirect
18. #11719270C4semidirect
20. #11719257C4semidirect
21. #11719255C4semidirect
22. #11719246C4semidirect
30. C13×#432C22semidirect
31. C13×#407C22semidirect
35. #11718873C22semidirect
37. C2×#88071C22semidirect
41. #11718597C22semidirect
42. C4×C4⋊Dic13C22semidirect
47. #11718059C22semidirect
48. #11718059C22semidirect
49. #11718059C22semidirect
50. #11718059C22semidirect
57. C2×#6127C2×C4semidirect
58. C22×D52C2×C4semidirect
59. C22×Dic26C2×C4semidirect
82. #88184C2×C4semidirect
83. #88184C2×C4semidirect
91. #88180C2×C4semidirect
96. C2×C4⋊Dic13C2×C4semidirect
97. C2×C4⋊Dic13C2×C4semidirect
100. #88178C2×C4semidirect
101. #88178C2×C4semidirect
118. #88129C2×C4semidirect
119. C4×D52C2×C4semidirect
121. #88124C2×C4semidirect
122. C4×Dic26C2×C4semidirect
229. #2973D13semidirect
327. #432C2×D13semidirect
328. #407C2×D13semidirect
569. C13#17622semidirect

Maximal subgroups

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

subgroupindex
#131286752
#131177462
#131134092
#131134052
#131133802
#131133752
#131133622
#131133412
C2×#117180592
C2×#117180622
C13×#29732
#1762213