Finite Group Databasebeta

← #15789593

Subgroups — #15789593

Order 3328.

1610 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88133C211yesmaximal, characteristic
2C2×#11718311C211yesmaximal, characteristic, complemented
3C2×C4×#6101C211yesmaximal, characteristic, complemented
4#13113088C212yesmaximal, complemented
5#13113087C212yesmaximal, complemented
6#13113079C212yesmaximal
7#13113079C212yesmaximal
8#13113073C211yesmaximal, characteristic
9#13113071C211yesmaximal, characteristic, complemented
10#13113069C211yesmaximal, characteristic, complemented
11C13×#1437C211yesmaximal, characteristic
12C22×C13×C22⋊C4C2211yescharacteristic
13C22×C13×C22⋊C4C2212yes—
14C22×C42×C13C2211yescharacteristic
15C22×C22⋊Dic13C2211yescharacteristic
16C22×C22⋊Dic13C2211yescharacteristic
17C22×C22⋊Dic13C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×#6101C2212yes—
20C22×#6101C2211yescharacteristic
21C22×#6101C2211yescharacteristic
22C22×#6101C2212yes—
23C22×C4×Dic13C2211yescharacteristic
24C22×C4×Dic13C2211yescharacteristic
25C2×#88133—24no—
26C2×#88133—24no—
27C2×#88133C418yescomplemented
28C2×C13×#95C2211yescharacteristic
29C2×C13×#95C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2211yescharacteristic
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38C4×#6101—24no—
39#11718311—24no—
40C25×C13C2311yescharacteristic
41C23×C4×C13C2312yes—
42C23×C4×C13C2311yescharacteristic
43C23×C4×C13C2312yes—
44C23×C4×C13C2311yescharacteristic
45C23×Dic13C2312yes—
46C23×Dic13C2311yescharacteristic
47C23×Dic13C2311yescharacteristic
48C23×Dic13C2312yes—
49C23×Dic13C2311yescharacteristic
50C23×Dic13C2311yescharacteristic
51C2×C13×C22⋊C4—22no—
52C2×C13×C22⋊C4—22no—
53C2×C13×C22⋊C4—24no—
54C2×C13×C22⋊C4—42no—
55C2×C13×C22⋊C4—42no—
56C2×C13×C22⋊C4C2×C414yes—
57C2×C42×C13—22no—
58C2×C42×C13—42no—
59C2×C22⋊Dic13—22no—
60C2×C22⋊Dic13—22no—
61C2×C22⋊Dic13—22no—
62C2×C22⋊Dic13—22no—
63C2×C22⋊Dic13—24no—
64C2×C22⋊Dic13—42no—
65C2×C22⋊Dic13—42no—
66C2×C22⋊Dic13C2×C414yes—
67C2×C22⋊Dic13C2×C414yes—
68C2×C4⋊Dic13—22no—
69C2×C4⋊Dic13—22no—
70C2×C4⋊Dic13C2×C414yes—
71C2×#6101—22no—
72C2×#6101—22no—
73C2×#6101—24no—
74C2×#6101—22no—
75C2×#6101—22no—
76C2×#6101—24no—
77C2×#6101—24no—
78C2×#6101—24no—
79C2×#6101—42no—
80C2×#6101—42no—
81C2×#6101C2×C414yes—
82C2×#6101C2×C414yes—
83C2×C4×Dic13—22no—
84C2×C4×Dic13—42no—
85C2×C4×Dic13C2×C414yes—
86C2×C4×Dic13D414yes—
87C2×C4×Dic13D414yes—
88#88133—232no—
89C13×#95—22no—
90C13×#95—42no—
91#88071—22no—
92#88071—24no—
93#88071—24no—
94#88071—24no—
95#88071—24no—
96#88071—42no—
97#88071—42no—
98#88071—42no—
99#14291—131nomaximal, Sylow
100C24×C13—21no—
101C24×C13—21no—
102C24×C13—22no—
103C24×C13—22no—
104C24×C13—22no—
105C24×C13—22no—
106C24×C13—22no—
107C24×C13—22no—
108C24×C13C22×C412yes—
109C24×C13C2411yescharacteristic
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—21no—
114C22×C4×C13—21no—
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×C13—22no—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C2×D412yes—
132C22×C4×C13C2×D412yes—
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×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13C22×C412yes—
160C22×Dic13C22×C412yes—
161C22×Dic13C22×C412yes—
162C22×Dic13C22×C412yes—
163C22×Dic13C2×D412yes—
164C22×Dic13C2×D412yes—
165C22×Dic13C2×D412yes—
166C22×Dic13C2×D412yes—
167C22×Dic13#4114yes—
168C13×C22⋊C4—28no—
169C13×C22⋊C4—48no—
170C42×C13—44no—
171C22⋊Dic13—28no—
172C22⋊Dic13—28no—
173C22⋊Dic13—48no—
174C4⋊Dic13—28no—
175#6101—216no—
176#6101—28no—
177#6101—28no—
178#6101—48no—
179C4×Dic13—44no—
180C4×Dic13C22⋊C4116yescomplemented
181C22×#126—131no—
182C2×#382—131no—
183C2×C4×C4⋊C4—131no—
184#1441—132no—
185#1441—132no—
186#1439—132no—
187#1439—132no—
188#1437—131no—
189#1437D1311yescharacteristic
190#1436—131no—
191#1436—131no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13—22no—
195C23×C13—24no—
196C23×C13—24no—
197C23×C13—42no—
198C23×C13—42no—
199C23×C13—41no—
200C23×C13—41no—
201C23×C13—41no—
202C23×C13—41no—
203C23×C13—42no—
204C23×C13—42no—
205C23×C13—42no—
206C23×C13—42no—
207C23×C13—42no—
208C23×C13—42no—
209C23×C13—42no—
210C23×C13—42no—
211C23×C13—42no—
212C23×C13—42no—
213C23×C13C23×C411yescharacteristic
214C23×C13C22×D411yescharacteristic
215C23×C13C22×D411yescharacteristic
216C23×C13C2×#4112yes—
217C23×C13C2×#4112yes—
218C23×C13H(5,𝔽2)12yes—
219C23×C13H(5,𝔽2)12yes—
220C23×C132−1+412yes—
221C23×C132−1+412yes—
222C2×C4×C13—24no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—24no—
226C2×C4×C13—22no—
227C2×C4×C13—24no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—42no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
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×C13C2×C22⋊C414yes—
246C2×Dic13—22no—
247C2×Dic13—22no—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—24no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—24no—
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—22no—
257C2×Dic13—22no—
258C2×Dic13—24no—
259C2×Dic13—24no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—42no—
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×Dic13C2×C22⋊C414yes—
279C2×Dic13C2×C22⋊C414yes—
280C2×#126—138no—
281C2×#126—264no—
282C2×#126—264no—
283C22×C4⋊C4—131no—
284C22×C4⋊C4—131no—
285C22×C4⋊C4—131no—
286C22×C4⋊C4—132no—
287C22×C4⋊C4—132no—
288C22×C22⋊C4—131no—
289C22×C22⋊C4—131no—
290C22×C22⋊C4—132no—
291C22×C22⋊C4C2×D1311yescharacteristic
292C22×C22⋊C4C2×D1312yes—
293C22×C42—131no—
294C22×C42—131no—
295C22×C42C2×D1311yescharacteristic
296#382—264no—
297C4×C4⋊C4—264no—
298C2×#95—131no—
299C2×#95—132no—
300C2×#95—132no—
301C2×#95—132no—
302C2×#95—132no—
303C2×#95—132no—
304C2×#95—132no—
305C2×#95—132no—
306C2×#95C2×D1311yescharacteristic
307C2×#95C2×D1312yes—
308C22×C13—42no—
309C22×C13—42no—
310C22×C13—44no—
311C22×C13—44no—
312C22×C13—42no—
313C22×C13—82no—
314C22×C13—82no—
315C22×C13—82no—
316C22×C13—82no—
317C22×C13C22×C22⋊C411yescharacteristic
318C22×C13C4×#4111yescharacteristic
319C22×C13C4×#4111yescharacteristic
320C22×C13#52012yes—
321C22×C13#52012yes—
322C22×C13#52512yes—
323C22×C13#53212yes—
324C22×C13#53511yescharacteristic
325C22×C13#53511yescharacteristic
326C22×C13#53611yescharacteristic
327C22×C13#53611yescharacteristic
328C22×C13#53712yes—
329C22×C13#53712yes—
330C22×C13#53812yes—
331C22×C13#53812yes—
332C22×C13#53912yes—
333C22×C13#53912yes—
334C22×C13#54112yes—
335C22×C13#54112yes—
336C22×C13#54212yes—
337C22×C13#54212yes—
338C4×C13—24no—
339C4×C13—42no—
340C4×C13—44no—
341C4×C13—82no—
342Dic13—24no—
343Dic13—24no—
344Dic13—42no—
345Dic13—44no—
346Dic13—42no—
347Dic13—82no—
348C25C22×D1311yescharacteristic
349C23×C4—131no—
350C23×C4—131no—
351C23×C4—131no—
352C23×C4—131no—
353C23×C4—132no—
354C23×C4—132no—
355C23×C4C22×D1312yes—
356C23×C4C22×D1311yescharacteristic
357C23×C4C22×D1312yes—
358C23×C4C22×D1311yescharacteristic
359#126—2632no—
360C2×C4⋊C4—134no—
361C2×C4⋊C4—134no—
362C2×C4⋊C4—134no—
363C2×C4⋊C4—262no—
364C2×C4⋊C4—262no—
365C2×C4⋊C4—262no—
366C2×C4⋊C4—262no—
367C2×C4⋊C4—264no—
368C2×C4⋊C4—264no—
369C2×C4⋊C4—264no—
370C2×C4⋊C4—262no—
371C2×C4⋊C4—262no—
372C2×C4⋊C4—264no—
373C2×C4⋊C4—522no—
374C2×C4⋊C4—522no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—134no—
381C2×C22⋊C4—134no—
382C2×C22⋊C4—262no—
383C2×C22⋊C4—262no—
384C2×C22⋊C4—262no—
385C2×C22⋊C4—262no—
386C2×C22⋊C4—264no—
387C2×C22⋊C4—522no—
388C2×C22⋊C4—522no—
389C2×C22⋊C4C4×D1314yes—
390C2×C42—22no—
391C2×C42—42no—
392C2×C42—134no—
393C2×C42—134no—
394C2×C42—134no—
395C2×C42—262no—
396C2×C42—522no—
397#95—22no—
398#95—42no—
399#95—264no—
400#95—264no—
401#95—264no—
402#95—264no—
403#95—262no—
404#95—522no—
405#95—522no—
406#95—522no—
407C2×C13—82no—
408C2×C13#229711yescharacteristic
409C2×C13#231111yescharacteristic
410C2×C13#231111yescharacteristic
411C2×C13#233412yes—
412C2×C13#233412yes—
413C2×C13#263612yes—
414C2×C13#263712yes—
415C2×C13#266212yes—
416C2×C13#266412yes—
417C24—22no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24C2×C4×D1312yes—
426C24C23×D1311yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
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—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—132no—
447C22×C4—132no—
448C22×C4—132no—
449C22×C4—132no—
450C22×C4—132no—
451C22×C4—132no—
452C22×C4—132no—
453C22×C4—132no—
454C22×C4—134no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—262no—
474C22×C4—262no—
475C22×C4—262no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4C2×C4×D1312yes—
481C22×C4C2×C4×D1312yes—
482C22×C4#612714yes—
483C22×C4D4×D1312yes—
484C22×C4D4×D1312yes—
485C4⋊C4—268no—
486C4⋊C4—268no—
487C4⋊C4—2616no—
488C4⋊C4—268no—
489C4⋊C4—528no—
490C22⋊C4—28no—
491C22⋊C4—48no—
492C22⋊C4—268no—
493C22⋊C4—268no—
494C22⋊C4—528no—
495C42—44no—
496C42—1316no—
497C42—524no—
498C13#1429111yescharacteristic, complemented, Sylow
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23C22×C4×D1311yescharacteristic
521C23C2×#612712yes—
522C23C2×D4×D1311yescharacteristic
523C23C2×D4×D1311yescharacteristic
524C23#8825112yes—
525C23#8825112yes—
526C23#41×D1312yes—
527C23#8825712yes—
528C23#8825712yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—134no—
553C2×C4—134no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—264no—
557C2×C4—264no—
558C2×C4—264no—
559C2×C4—264no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—262no—
563C2×C4—262no—
564C2×C4—262no—
565C2×C4—262no—
566C2×C4—264no—
567C2×C4—264no—
568C2×C4—264no—
569C2×C4—264no—
570C2×C4—522no—
571C2×C4—522no—
572C2×C4—522no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—521no—
576C2×C4—522no—
577C2×C4—522no—
578C2×C4—521no—
579C2×C4—522no—
580C2×C4—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4—522no—
584C2×C4—522no—
585C2×C4—522no—
586C2×C4C22⋊C4×D1314yes—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C4×#612711yescharacteristic
597C22C2×C22⋊C4×D1311yescharacteristic
598C22#1171919912yes—
599C22#1171922312yes—
600C22#1171922711yescharacteristic
601C22#1171925112yes—
602C22#1171925112yes—
603C22#1171925212yes—
604C22#1171925312yes—
605C22#1171926612yes—
606C22#1171926912yes—
607C22#1171929711yescharacteristic
608C22#1171930212yes—
609C22#1171931312yes—
610C22#1171931812yes—
611C22#1171935311yescharacteristic
612C22#1171936512yes—
613C22#1171937412yes—
614C22#1171938011yescharacteristic
615C22#1171940612yes—
616C22#1171942011yescharacteristic
617C4—24no—
618C4—44no—
619C4—42no—
620C4—82no—
621C4—264no—
622C4—264no—
623C4—522no—
624C4—524no—
625C4—522no—
626C4—1042no—
627C2—82no—
628C2#1311732211yescharacteristic
629C2#1311742311yescharacteristic
630C2#1311742611yescharacteristic
631C2#1311760912yes—
632C2#1311761312yes—
633C2#1312073712yes—
634C2#1312074812yes—
635C2#1312097712yes—
636C2#1312100112yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1437C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenter#1437D13
hypocenterC13#14291
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#11718311C2semidirect
3. C2×C4×#6101C2semidirect
4. #13113088C2semidirect
5. #13113087C2semidirect
9. #13113071C2semidirect
10. #13113069C2semidirect
27. C2×#88133C4semidirect
180. C4×Dic13C22⋊C4semidirect
498. C13#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#881332
C2×#117183112
C2×C4×#61012
#131130882
#131130872
#131130792
#131130732
#131130712
#131130692
C13×#14372
#1429113