Finite Group Databasebeta

← #15785680

Subgroups — #15785680

Order 3328.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718328C211yesmaximal, characteristic
2C2×C4×#6103C211yesmaximal, characteristic
3C2×D13×#95C211yesmaximal, characteristic
4#13113094C212yesmaximal, complemented
5#13113091C212yesmaximal, complemented
6#13113063C211yesmaximal, characteristic, complemented
7#13113062C211yesmaximal, characteristic, complemented
8#13113062C211yesmaximal, characteristic, complemented
9#13113057C212yesmaximal
10#13113057C212yesmaximal
11C13×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D13C2212yes—
13C23×C4×D13C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C22×#6103C2212yes—
16C22×#6103C2211yescharacteristic
17C22×#6103C2211yescharacteristic
18C22×#6101C2212yes—
19C22×C4×Dic13C2211yescharacteristic
20C2×C13×#95C2212yes—
21C2×C13×#95C2211yescharacteristic
22C2×C13×#95C2211yescharacteristic
23C2×C13×#95C2212yes—
24C2×#88071C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27C2×#88071C2212yes—
28C2×#88071C2211yescharacteristic
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2211yescharacteristic
34C2×#88071C2211yescharacteristic
35C4×#6103C418yescomplemented
36#11718328—24no—
37D13×#95—24no—
38C24×D13C2311yescharacteristic
39C23×C4×C13C2312yes—
40C23×C4×C13C2311yescharacteristic
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2312yes—
43C23×C4×C13C2311yescharacteristic
44C23×Dic13C2312yes—
45C23×Dic13C2311yescharacteristic
46C23×Dic13C2311yescharacteristic
47C23×Dic13C2312yes—
48C23×Dic13C2311yescharacteristic
49C22×C4×D13—24no—
50C22×C4×D13—24no—
51C22×C4×D13—24no—
52C22×C4×D13—24no—
53C22×C4×D13—24no—
54C22×C4×D13—42no—
55C22×C4×D13—42no—
56C22×C4×D13—42no—
57C22×C4×D13—42no—
58C22×C4×D13C2×C414yes—
59C22×C4×D13D414yes—
60C22×C4×D13Q814yes—
61C2×C42×C13—42no—
62C2×C42×C13C2×C414yescomplemented
63C2×#6103—22no—
64C2×#6103—22no—
65C2×#6103—22no—
66C2×#6103—22no—
67C2×#6103—24no—
68C2×#6103—24no—
69C2×#6103—24no—
70C2×#6103C2×C414yes—
71C2×#6103C2×C414yes—
72C2×#6101—24no—
73C2×#6101—42no—
74C2×#6101—42no—
75C2×C4×Dic13—42no—
76C2×C4×Dic13C2×C414yescomplemented
77C13×#95—22no—
78C13×#95—22no—
79C13×#95—24no—
80C13×#95—42no—
81#88071—22no—
82#88071—22no—
83#88071—24no—
84#88071—24no—
85#88071—24no—
86#88071—24no—
87#88071—42no—
88#88071—42no—
89#88071—42no—
90#88071C2×C414yescomplemented
91#88071C2×C414yescomplemented
92#13966—131nomaximal, Sylow
93C24×C13C2411yescharacteristic
94C23×D13—22no—
95C23×D13—22no—
96C23×D13—22no—
97C23×D13—22no—
98C23×D13C22×C412yes—
99C23×D13C2×D412yes—
100C23×D13C2×Q812yes—
101C23×D13#4112yes—
102C23×D13#4112yes—
103C23×D13#4112yes—
104C23×D13#4112yes—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—21no—
111C22×C4×C13—21no—
112C22×C4×C13—21no—
113C22×C4×C13—21no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C2×D412yes—
132C22×C4×C13C2×Q812yes—
133C22×C4×C13#4114yes—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13C22×C412yes—
158C22×Dic13C22×C412yes—
159C22×Dic13C22×C412yes—
160C22×Dic13C2×D412yes—
161C22×Dic13C2×Q812yes—
162C22×Dic13#4114yes—
163C2×C4×D13—216no—
164C2×C4×D13—44no—
165C2×C4×D13—44no—
166C2×C4×D13—48no—
167C2×C4×D13—48no—
168C2×C4×D13—48no—
169C2×C4×D13—48no—
170C2×C4×D13—48no—
171C2×C4×D13—48no—
172C2×C4×D13—48no—
173C2×C4×D13—48no—
174C2×C4×D13C4⋊C4116yes—
175C42×C13—44no—
176#6103—28no—
177#6103—28no—
178#6103—48no—
179#6101—48no—
180C4×Dic13—44no—
181C2×#380—131no—
182C2×C4×C22⋊C4—131no—
183C22×#95—131no—
184#1442—132no—
185#1442—132no—
186#1434—131no—
187#1434—131no—
188#1434—131no—
189#1434D1311yescharacteristic, complemented
190C22⋊#95—132no—
191C22⋊#95—132no—
192C23×C13C23×C411yescharacteristic
193C23×C13C22×D411yescharacteristic
194C23×C13C22×Q811yescharacteristic
195C23×C13C2×#4111yescharacteristic
196C23×C13C2×#4111yescharacteristic
197C23×C13C2×#4111yescharacteristic
198C23×C13C2×#4111yescharacteristic
199C23×C13C2×#4112yes—
200C23×C13C2×#4112yes—
201C23×C13H(5,𝔽2)12yes—
202C23×C13H(5,𝔽2)12yes—
203C22×D13—22no—
204C22×D13—22no—
205C22×D13—22no—
206C22×D13—22no—
207C22×D13—24no—
208C22×D13—24no—
209C22×D13—24no—
210C22×D13—24no—
211C22×D13—24no—
212C22×D13—24no—
213C22×D13—24no—
214C22×D13—24no—
215C22×D13—24no—
216C22×D13—24no—
217C22×D13—24no—
218C22×D13—24no—
219C22×D13C2×C4⋊C414yes—
220C22×D13#11714yes—
221C22×D13#11714yes—
222C22×D13#12414yes—
223C22×D13#12414yes—
224C22×D13#12514yes—
225C22×D13#12514yes—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—24no—
237C2×C4×C13—24no—
238C2×C4×C13—24no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13—42no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13—42no—
252C2×C4×C13—42no—
253C2×C4×C13—42no—
254C2×C4×C13C2×C4⋊C414yes—
255C2×Dic13—22no—
256C2×Dic13—22no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—22no—
260C2×Dic13—22no—
261C2×Dic13—22no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—24no—
266C2×Dic13—24no—
267C2×Dic13—24no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13—42no—
276C2×Dic13—42no—
277C2×Dic13—42no—
278C2×Dic13—42no—
279C2×Dic13—42no—
280C2×Dic13—42no—
281C2×Dic13—42no—
282C2×Dic13—42no—
283C2×Dic13C2×C4⋊C414yes—
284C4×D13—416no—
285C4×D13—432no—
286C24×C4—131no—
287C24×C4—132no—
288C22×C4⋊C4—132no—
289C22×C22⋊C4—131no—
290C22×C22⋊C4—131no—
291C22×C22⋊C4—132no—
292C22×C42—131no—
293C22×C42C2×D1311yescharacteristic
294#380—264no—
295C4×C22⋊C4—138no—
296C2×#95—131no—
297C2×#95—131no—
298C2×#95—132no—
299C2×#95—132no—
300C2×#95—132no—
301C2×#95—131no—
302C2×#95—131no—
303C2×#95—132no—
304C2×#95—132no—
305C2×#95—132no—
306C2×#95—132no—
307C2×#95—264no—
308C2×#95C2×D1312yes—
309C2×#95C2×D1311yescharacteristic
310C2×#95C2×D1311yescharacteristic
311C2×#95C2×D1312yes—
312C22×C13C22×C4⋊C411yescharacteristic
313C22×C13C2×#11711yescharacteristic
314C22×C13C2×#11711yescharacteristic
315C22×C13C4×#4111yescharacteristic
316C22×C13C4×#4111yescharacteristic
317C22×C13#51811yescharacteristic
318C22×C13#51811yescharacteristic
319C22×C13#52512yes—
320C22×C13C2×#12411yescharacteristic
321C22×C13C2×#12411yescharacteristic
322C22×C13C2×#12511yescharacteristic
323C22×C13C2×#12511yescharacteristic
324C22×C13#53312yes—
325C22×C13#53412yes—
326C22×C13#54312yes—
327C22×C13#55112yes—
328C22×C13#55112yes—
329C22×C13#55112yes—
330C22×C13#55112yes—
331C22×C13#55312yes—
332C22×C13#55312yes—
333C22×C13#55312yes—
334C22×C13#55312yes—
335C2×D13—28no—
336C2×D13—28no—
337C2×D13—28no—
338C2×D13—24no—
339C2×D13—28no—
340C2×D13—24no—
341C2×D13—24no—
342C2×D13—24no—
343C2×D13—24no—
344C2×D13—24no—
345C2×D13#38218yes—
346C4×C13—24no—
347C4×C13—44no—
348C4×C13—42no—
349C4×C13—42no—
350C4×C13—82no—
351Dic13—24no—
352Dic13—44no—
353Dic13—42no—
354Dic13—42no—
355Dic13—82no—
356C25—131no—
357C23×C4—131no—
358C23×C4—134no—
359C23×C4—134no—
360C23×C4—134no—
361C23×C4—131no—
362C23×C4—131no—
363C23×C4—132no—
364C23×C4—132no—
365C23×C4—264no—
366C23×C4—264no—
367C23×C4—264no—
368C23×C4—264no—
369C23×C4—264no—
370C23×C4—522no—
371C23×C4—522no—
372C23×C4—522no—
373C23×C4—522no—
374C23×C4C22×D1312yes—
375C23×C4C22×D1311yescharacteristic
376C23×C4C22×D1311yescharacteristic
377C23×C4C22×D1312yes—
378C23×C4C22×D1311yescharacteristic
379C2×C4⋊C4—264no—
380C2×C4⋊C4—522no—
381C2×C4⋊C4—522no—
382C2×C22⋊C4—134no—
383C2×C22⋊C4—134no—
384C2×C22⋊C4—262no—
385C2×C22⋊C4—262no—
386C2×C22⋊C4—262no—
387C2×C22⋊C4—262no—
388C2×C22⋊C4—264no—
389C2×C22⋊C4—264no—
390C2×C22⋊C4—264no—
391C2×C42—42no—
392C2×C42—134no—
393C2×C42—522no—
394C2×C42C4×D1314yescomplemented
395#95—24no—
396#95—22no—
397#95—22no—
398#95—42no—
399#95—134no—
400#95—134no—
401#95—264no—
402#95—264no—
403#95—264no—
404#95—264no—
405#95—262no—
406#95—262no—
407#95—522no—
408#95—522no—
409#95—522no—
410C2×C13C2×#38211yescharacteristic
411C2×C13#229611yescharacteristic
412C2×C13#231511yescharacteristic
413C2×C13#235311yescharacteristic
414C2×C13#235311yescharacteristic
415C2×C13#235311yescharacteristic
416C2×C13#235311yescharacteristic
417C2×C13#256112yes—
418C2×C13#256912yes—
419C2×C13#256912yes—
420C2×C13#259112yes—
421D13—28no—
422C24—132no—
423C24—132no—
424C24—132no—
425C24—132no—
426C24—132no—
427C24—132no—
428C24—132no—
429C24—262no—
430C24—262no—
431C24—262no—
432C24—262no—
433C24C23×D1311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—132no—
458C22×C4—132no—
459C22×C4—132no—
460C22×C4—1316no—
461C22×C4—132no—
462C22×C4—132no—
463C22×C4—134no—
464C22×C4—2616no—
465C22×C4—262no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4—262no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—262no—
484C22×C4—262no—
485C22×C4—262no—
486C22×C4—262no—
487C22×C4—262no—
488C22×C4—524no—
489C22×C4—524no—
490C22×C4—528no—
491C22×C4—528no—
492C22×C4—528no—
493C22×C4—528no—
494C22×C4—528no—
495C22×C4—528no—
496C22×C4—528no—
497C22×C4—528no—
498C22×C4C2×C4×D1312yes—
499C22×C4C2×C4×D1312yes—
500C22×C4C2×C4×D1312yes—
501C22×C4#612714yes—
502C22×C4D4×D1312yes—
503C22×C4Q8×D1312yes—
504C4⋊C4—528no—
505C22⋊C4—268no—
506C22⋊C4—268no—
507C22⋊C4—528no—
508C42—44no—
509C42—524no—
510C13#1396611yescharacteristic, complemented, Sylow
511C23—134no—
512C23—134no—
513C23—134no—
514C23—134no—
515C23—134no—
516C23—134no—
517C23—134no—
518C23—262no—
519C23—262no—
520C23—262no—
521C23—262no—
522C23—264no—
523C23—264no—
524C23—264no—
525C23—264no—
526C23—264no—
527C23—264no—
528C23—264no—
529C23—264no—
530C23—264no—
531C23—264no—
532C23—264no—
533C23—264no—
534C23C22×C4×D1311yescharacteristic
535C23C2×#612712yes—
536C23C2×D4×D1311yescharacteristic
537C23#8825112yes—
538C23C2×Q8×D1311yescharacteristic
539C23#41×D1311yescharacteristic
540C23#41×D1311yescharacteristic
541C23#41×D1311yescharacteristic
542C23#41×D1311yescharacteristic
543C23#41×D1312yes—
544C23#8825612yes—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—134no—
574C2×C4—262no—
575C2×C4—262no—
576C2×C4—264no—
577C2×C4—264no—
578C2×C4—262no—
579C2×C4—262no—
580C2×C4—262no—
581C2×C4—262no—
582C2×C4—262no—
583C2×C4—262no—
584C2×C4—264no—
585C2×C4—264no—
586C2×C4—264no—
587C2×C4—5216no—
588C2×C4—5232no—
589C2×C4—522no—
590C2×C4—522no—
591C2×C4—522no—
592C2×C4—522no—
593C2×C4—522no—
594C2×C4—522no—
595C2×C4—522no—
596C2×C4—522no—
597C2×C4—522no—
598C2×C4—522no—
599C2×C4—522no—
600C2×C4—522no—
601C2×C4—522no—
602C2×C4—522no—
603C2×C4—522no—
604C2×C4C4⋊C4×D1314yes—
605C22—138no—
606C22—264no—
607C22—268no—
608C22—268no—
609C22—268no—
610C22—264no—
611C22—264no—
612C22—264no—
613C22—264no—
614C22—264no—
615C22—268no—
616C22C4×#612711yescharacteristic
617C22#1171919111yescharacteristic
618C22C2×C4⋊C4×D1311yescharacteristic
619C22D13×#11711yescharacteristic
620C22D13×#11711yescharacteristic
621C22#1171922611yescharacteristic
622C22#1171922711yescharacteristic
623C22#1171923212yes—
624C22#1171923512yes—
625C22#1171925212yes—
626C22#1171926112yes—
627C22#1171926312yes—
628C22#1171928312yes—
629C22#1171931012yes—
630C22#1171933912yes—
631C22#1171936412yes—
632C22#1171936612yes—
633C22D13×#12411yescharacteristic
634C22D13×#12411yescharacteristic
635C22D13×#12511yescharacteristic
636C22D13×#12511yescharacteristic
637C22#1171941012yes—
638C22#1171941412yes—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—264no—
645C4—522no—
646C4—522no—
647C4—524no—
648C4—1042no—
649C2—268no—
650C2D13×#38211yescharacteristic
651C2#1311731611yescharacteristic
652C2#1311744911yescharacteristic
653C2#1311776911yescharacteristic
654C2#1311777311yescharacteristic
655C2#1311777711yescharacteristic
656C2#1311777911yescharacteristic
657C2#1311999212yes—
658C2#1312005812yes—
659C2#1312005912yes—
660C2#1312031412yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1434C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenter#1434D13
hypocenterC13#13966
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #13113094C2semidirect
5. #13113091C2semidirect
6. #13113063C2semidirect
7. #13113062C2semidirect
8. #13113062C2semidirect
11. C13×#1434C2semidirect
35. C4×#6103C4semidirect
62. C2×C42×C13C2×C4semidirect
76. C2×C4×Dic13C2×C4semidirect
90. #88071C2×C4semidirect
91. #88071C2×C4semidirect
189. #1434D13semidirect
394. C2×C42C4×D13semidirect
510. C13#13966semidirect

Maximal subgroups

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

subgroupindex
C2×#117183282
C2×C4×#61032
C2×D13×#952
#131130942
#131130912
#131130632
#131130622
#131130572
C13×#14342
#1396613