Finite Group Databasebeta

← #15830730

Subgroups — #15830730

Order 3328.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123865C211yesmaximal, characteristic, complemented
2#13117809C211yesmaximal, characteristic, complemented
3#13113392C212yesmaximal, complemented
4#13113374C212yesmaximal, complemented
5#13113369C211yesmaximal, characteristic, complemented
6#13113365C211yesmaximal, characteristic, complemented
7#13113361C212yesmaximal, complemented
8#13113356C211yesmaximal, characteristic, complemented
9#13113352C211yesmaximal, characteristic, complemented
10#13113327C212yesmaximal, complemented
11C13×#3505C211yesmaximal, characteristic
12C13×#553—22no—
13C13×#553C412yescomplemented
14C13×#552C412yescomplemented
15C13×#551—22no—
16C13×#548C412yescomplemented
17C13×#546C412yescomplemented
18C13×#539—21no—
19C13×#538—21no—
20C13×#534—21no—
21C13×#529C2212yes—
22C13×#525C2212yes—
23C2×C13×C22⋊Q8C2211yescharacteristic
24C2×C13×#121C2211yescharacteristic
25C13×#518—21no—
26C2×C4×D4×C13C2211yescharacteristic
27#11718882—21no—
28#11718872C2211yescharacteristic
29#11718872C2211yescharacteristic, complemented
30C4×C22⋊Dic13C2211yescharacteristic
31C4×C22⋊Dic13C2211yescharacteristic, complemented
32#11718868C2211yescharacteristic
33#11718868C2211yescharacteristic
34C2×C22⋊#1055C2212yescomplemented
35#11718672—22no—
36#11718672—22no—
37C4⋊C4×Dic13—21no—
38#11718612—21no—
39#11718612—21no—
40#11718597C2211yescharacteristic, complemented
41C4×C4⋊Dic13C2211yescharacteristic, complemented
42#11718587C2212yescomplemented
43#11718130C2212yescomplemented
44#11718130C2212yescomplemented
45#11718130C2212yescomplemented
46#11718130C2212yescomplemented
47#11718129C2212yes—
48#11718129C2212yes—
49#11718129C2212yes—
50#11718129C2212yes—
51C2×C13×#41C2×C412yes—
52C22×Q8×C13C2×C411yescharacteristic
53C22×D4×C13—21no—
54C22×D4×C13C2×C411yescharacteristic
55C22×D4×C13D411yescharacteristic
56C23×C4×C13C2311yescharacteristic
57C23×C4×C13C2311yescharacteristic
58C13×#126—22no—
59C13×#126—22no—
60C13×#126—22no—
61C13×#125—22no—
62C13×#125C2×C412yes—
63C13×#124—22no—
64C13×#124C2×C412yes—
65C13×#123—22no—
66C13×#123—22no—
67C13×#123—22no—
68C13×#123—22no—
69C13×#123—22no—
70C13×C22⋊Q8—21no—
71C13×C22⋊Q8—21no—
72C13×C22⋊Q8—22no—
73C13×C22⋊Q8C2×C412yes—
74C13×C22⋊Q8C2×C412yes—
75C13×C22⋊Q8C2312yes—
76C13×#121—22no—
77C13×#121—22no—
78C13×#121—21no—
79C13×#121—21no—
80C13×#121—22no—
81C13×#121C2×C412yes—
82C13×#121C2×C412yes—
83C13×#121C2312yes—
84C13×#120—22no—
85C13×#120—22no—
86C4×Q8×C13C2×C412yes—
87C4×D4×C13—22no—
88C4×D4×C13—22no—
89C4×D4×C13—22no—
90C4×D4×C13—22no—
91C4×D4×C13—22no—
92C4×D4×C13C2×C412yes—
93C4×D4×C13C2×C412yes—
94C4×D4×C13C2×C412yes—
95C4×D4×C13D412yes—
96C4×D4×C13D412yes—
97C13×#117—22no—
98C13×#117—22no—
99C13×#117C2×C412yes—
100C2×C13×C4⋊C4—21no—
101C2×C13×C4⋊C4C2×C411yescharacteristic
102C2×C13×C4⋊C4C2×C411yescharacteristic
103C2×C13×C4⋊C4D411yescharacteristic
104C2×C13×C22⋊C4—21no—
105C2×C13×C22⋊C4—21no—
106C2×C13×C22⋊C4—22no—
107C2×C13×C22⋊C4D411yescharacteristic
108C2×C13×C22⋊C4D411yescharacteristic
109C2×C42×C13C2311yescharacteristic
110C2×C22⋊Dic13—21no—
111C2×C22⋊Dic13—21no—
112C2×C22⋊Dic13—21no—
113C2×C22⋊Dic13—21no—
114C2×C4⋊Dic13—21no—
115C2×C4⋊Dic13—21no—
116C2×C4×Dic13—21no—
117C2×C4×Dic13—21no—
118C2×C4×Dic13—21no—
119C2×#6099—21no—
120C2×#6099—21no—
121C22×#1055—22no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071C2311yescharacteristic
126#88071C2311yescharacteristic
127#88071C2311yescharacteristic
128#88071C2311yescharacteristic
129C22⋊#1055—22no—
130C22⋊#1055—21no—
131C22⋊#1055—21no—
132C22⋊#1055C2312yes—
133C22⋊#1055C2312yes—
134C4⋊#1055—22no—
135#88043—22no—
136#17625—131nomaximal, Sylow
137C24×C13—21no—
138C24×C13C2×D411yescharacteristic
139C24×C13C2×D411yescharacteristic
140C13×#41—82no—
141C2×Q8×C13—21no—
142C2×Q8×C13—81no—
143C2×Q8×C13C22×C412yes—
144C2×D4×C13—22no—
145C2×D4×C13—22no—
146C2×D4×C13—22no—
147C2×D4×C13—22no—
148C2×D4×C13—21no—
149C2×D4×C13—41no—
150C2×D4×C13—41no—
151C2×D4×C13—42no—
152C2×D4×C13—42no—
153C2×D4×C13—81no—
154C2×D4×C13C22⋊C412yes—
155C2×D4×C13C22⋊C412yes—
156C2×D4×C13C22×C412yes—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13—22no—
161C22×C4×C13—22no—
162C22×C4×C13—81no—
163C22×C4×C13—81no—
164C22×C4×C13C22×C411yescharacteristic
165C22×C4×C13C22×C411yescharacteristic
166C22×C4×C13C22×C412yes—
167C22×C4×C13C2×D411yescharacteristic
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13C2×D411yescharacteristic
170C22×C4×C13C2×D411yescharacteristic
171C22×C4×C13C2×D412yes—
172C22×C4×C13C2×D411yescharacteristic
173C22×C4×C13C2×D411yescharacteristic
174C22×C4×C13C2411yescharacteristic
175C22×Dic13—21no—
176C22×Dic13—21no—
177C22×Dic13—21no—
178C22×Dic13—21no—
179C13×C4⋊C4—22no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—22no—
183C13×C4⋊C4—22no—
184C13×C4⋊C4—22no—
185C13×C4⋊C4—22no—
186C13×C4⋊C4—22no—
187C13×C4⋊C4C22⋊C412yes—
188C13×C4⋊C4C22⋊C412yes—
189C13×C4⋊C4C22×C412yes—
190C13×C4⋊C4C22×C412yes—
191C13×C22⋊C4—22no—
192C13×C22⋊C4—22no—
193C13×C22⋊C4—22no—
194C13×C22⋊C4—22no—
195C13×C22⋊C4—22no—
196C13×C22⋊C4—22no—
197C13×C22⋊C4—22no—
198C13×C22⋊C4—22no—
199C13×C22⋊C4—22no—
200C13×C22⋊C4—22no—
201C13×C22⋊C4C22⋊C412yes—
202C13×C22⋊C4C22⋊C412yes—
203C13×C22⋊C4C22⋊C412yes—
204C13×C22⋊C4C22⋊C412yes—
205C42×C13—22no—
206C42×C13—22no—
207C42×C13C22×C412yes—
208C42×C13C2×D412yes—
209C22⋊Dic13—42no—
210C22⋊Dic13—42no—
211C22⋊Dic13—81no—
212C22⋊Dic13—81no—
213C4⋊Dic13—42no—
214C4⋊Dic13—42no—
215C4×Dic13—42no—
216C4×Dic13—81no—
217C4×Dic13—81no—
218#6099—81no—
219#6099—81no—
220C2×#1055—22no—
221C2×#1055—22no—
222C2×#1055—82no—
223#3505D1311yescharacteristic
224#2973—131no—
225#2357—131no—
226#1516—132no—
227#1512—132no—
228#1511—131no—
229#1510—131no—
230#1509—132no—
231#1508—131no—
232#1507—131no—
233#1501—132no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13—22no—
239C23×C13—41no—
240C23×C13—41no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—42no—
244C23×C13—81no—
245C23×C13—81no—
246C23×C13C2×C22⋊C411yescharacteristic
247C23×C13C2×C22⋊C412yes—
248C23×C13C2×C22⋊C411yescharacteristic
249C23×C13C22×D411yescharacteristic
250Q8×C13—42no—
251Q8×C13—81no—
252D4×C13—42no—
253D4×C13—42no—
254D4×C13—42no—
255D4×C13—44no—
256D4×C13—42no—
257D4×C13—42no—
258D4×C13—81no—
259D4×C13—82no—
260D4×C13—82no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—22no—
266C2×C4×C13—42no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—82no—
278C2×C4×C13—82no—
279C2×C4×C13—81no—
280C2×C4×C13—81no—
281C2×C4×C13C2×C22⋊C412yes—
282C2×C4×C13C2×C22⋊C412yes—
283C2×C4×C13C2×C22⋊C412yes—
284C2×C4×C13C2×C22⋊C411yescharacteristic
285C2×C4×C13C2×C22⋊C411yescharacteristic
286C2×C4×C13C23×C411yescharacteristic
287C2×C4×C13C22×D411yescharacteristic
288C2×Dic13—22no—
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—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300#1055—42no—
301#1055—42no—
302#553—22no—
303#553Dic1312yescomplemented
304#552Dic1312yescomplemented
305#551—22no—
306#548Dic1312yescomplemented
307#546Dic1312yescomplemented
308#539—21no—
309#538—21no—
310#534—21no—
311#529C2×D1312yes—
312#525C2×D1312yes—
313C2×C22⋊Q8C2×D1311yescharacteristic
314C2×#121C2×D1311yescharacteristic
315#518—21no—
316C2×C4×D4C2×D1311yescharacteristic
317#435—262no—
318#435—262no—
319#432—132no—
320#407—131no—
321#407—131no—
322C2×C22⋊C8—132no—
323#390—261no—
324#388—261no—
325#388—261no—
326#382—131no—
327#381—131no—
328#381—131no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4—261no—
331C4×C22⋊C4—131no—
332C4×C22⋊C4—131no—
333#328—132no—
334#328—132no—
335#328—132no—
336#328—132no—
337#327—132no—
338#327—132no—
339#327—132no—
340#327—132no—
341C22×C13—21no—
342C22×C13—21no—
343C22×C13—21no—
344C22×C13—22no—
345C22×C13—21no—
346C22×C13—21no—
347C22×C13—21no—
348C22×C13—22no—
349C22×C13—21no—
350C22×C13—22no—
351C22×C13—21no—
352C22×C13—42no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—81no—
356C22×C13—81no—
357C22×C13—81no—
358C22×C13—81no—
359C22×C13—162no—
360C22×C13C22×C22⋊C411yescharacteristic
361C22×C13#57511yescharacteristic
362C22×C13#57511yescharacteristic
363C4×C13—21no—
364C4×C13—21no—
365C4×C13—22no—
366C4×C13—21no—
367C4×C13—21no—
368C4×C13—22no—
369C4×C13—22no—
370C4×C13—42no—
371C4×C13—42no—
372C4×C13—42no—
373Dic13—42no—
374Dic13—81no—
375Dic13—81no—
376Dic13—81no—
377C2×#41C2×Dic1312yes—
378C22×Q8C2×Dic1311yescharacteristic
379C22×D4—21no—
380C22×D4C2×Dic1311yescharacteristic
381C22×D4C22⋊D1311yescharacteristic
382C23×C4C22×D1311yescharacteristic
383C23×C4C22×D1311yescharacteristic
384C2×M16—261no—
385C2×M16—261no—
386C22×C8—262no—
387#126—22no—
388#126—22no—
389#126—22no—
390#125—22no—
391#125C2×Dic1312yes—
392#124—22no—
393#124C2×Dic1312yes—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8C2×Dic1312yes—
403C22⋊Q8C2×Dic1312yes—
404C22⋊Q8C22×D1312yes—
405#121—22no—
406#121—22no—
407#121—21no—
408#121—21no—
409#121—22no—
410#121C2×Dic1312yes—
411#121C2×Dic1312yes—
412#121C22×D1312yes—
413#120—22no—
414#120—22no—
415C4×Q8C2×Dic1312yes—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4C2×Dic1312yes—
422C4×D4C2×Dic1312yes—
423C4×D4C2×Dic1312yes—
424C4×D4C22⋊D1312yes—
425C4×D4C22⋊D1312yes—
426#117—22no—
427#117—22no—
428#117C2×Dic1312yes—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—261no—
431C2×C4⋊C4—261no—
432C2×C4⋊C4C2×Dic1311yescharacteristic
433C2×C4⋊C4C2×Dic1311yescharacteristic
434C2×C4⋊C4C22⋊D1311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—261no—
439C2×C22⋊C4—261no—
440C2×C22⋊C4—261no—
441C2×C22⋊C4—261no—
442C2×C22⋊C4C22⋊D1311yescharacteristic
443C2×C22⋊C4C22⋊D1311yescharacteristic
444C2×C42—261no—
445C2×C42—261no—
446C2×C42—261no—
447C2×C42C22×D1311yescharacteristic
448C4⋊C8—262no—
449C22⋊C8—132no—
450C22⋊C8—132no—
451C22⋊C8—262no—
452C22⋊C8—261no—
453C22⋊C8—261no—
454#97—262no—
455#95—131no—
456#95—131no—
457#95—131no—
458#95—131no—
459#95—261no—
460#95—261no—
461#95—261no—
462C2×C13—21no—
463C2×C13—21no—
464C2×C13—41no—
465C2×C13—42no—
466C2×C13—41no—
467C2×C13—82no—
468C2×C13#288311yescharacteristic
469C2×C13#289611yescharacteristic
470C2×C13#290111yescharacteristic
471C24—21no—
472C24C2×C22⋊D1311yescharacteristic
473C24C2×C22⋊D1311yescharacteristic
474#41—82no—
475C2×Q8—21no—
476C2×Q8—81no—
477C2×Q8C22×Dic1312yes—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—81no—
488C2×D4C22⋊Dic1312yes—
489C2×D4C22⋊Dic1312yes—
490C2×D4C22×Dic1312yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—261no—
501C22×C4—261no—
502C22×C4C22×Dic1311yescharacteristic
503C22×C4C22×Dic1311yescharacteristic
504C22×C4C22×Dic1312yes—
505C22×C4C2×C22⋊D1311yescharacteristic
506C22×C4C2×C22⋊D1311yescharacteristic
507C22×C4C2×C22⋊D1311yescharacteristic
508C22×C4C2×C22⋊D1311yescharacteristic
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C2×C22⋊D1311yescharacteristic
511C22×C4C2×C22⋊D1312yes—
512C22×C4C23×D1311yescharacteristic
513M16—1041no—
514M16—1041no—
515C2×C8—262no—
516C2×C8—262no—
517C2×C8—1042no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—522no—
527C4⋊C4—522no—
528C4⋊C4C22⋊Dic1312yes—
529C4⋊C4C22⋊Dic1312yes—
530C4⋊C4C22×Dic1312yes—
531C4⋊C4C22×Dic1312yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—522no—
543C22⋊C4—522no—
544C22⋊C4—1041no—
545C22⋊C4—1041no—
546C22⋊C4C22⋊Dic1312yes—
547C22⋊C4C22⋊Dic1312yes—
548C22⋊C4C22⋊Dic1312yes—
549C22⋊C4C22⋊Dic1312yes—
550C42—22no—
551C42—22no—
552C42—522no—
553C42—1041no—
554C42—1041no—
555C42C22×Dic1312yes—
556C42C2×C22⋊D1312yes—
557C13#1762511yescharacteristic, complemented, Sylow
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—81no—
569C23—81no—
570C23C2×C22⋊Dic1311yescharacteristic
571C23C2×C22⋊Dic1312yes—
572C23C2×C22⋊Dic1311yescharacteristic
573C23C22×C22⋊D1311yescharacteristic
574Q8—42no—
575Q8—81no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—81no—
583D4—82no—
584D4—82no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—262no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4—521no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—521no—
616C2×C4—521no—
617C2×C4C2×C22⋊Dic1312yes—
618C2×C4C2×C22⋊Dic1312yes—
619C2×C4C2×C22⋊Dic1312yes—
620C2×C4C2×C22⋊Dic1311yescharacteristic
621C2×C4C2×C22⋊Dic1311yescharacteristic
622C2×C4C23×Dic1311yescharacteristic
623C2×C4C22×C22⋊D1311yescharacteristic
624C8—522no—
625C8—522no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—162no—
645C22#1171952711yescharacteristic
646C22#1171952711yescharacteristic
647C22C22×C22⋊Dic1311yescharacteristic
648C4—22no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—522no—
659C4—1041no—
660C4—1041no—
661C4—1041no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—82no—
668C2#1312329711yescharacteristic
669C2#1312335511yescharacteristic
670C2#1312338311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
Fitting, prime radicalC13×#3505C2
socleC22×C13C22×C22⋊C4
cosocleC22×C4×C13C24
hypercenter#3505D13
hypocenterC13#17625
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3505

Complements

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

NH ≅ G/Ntype
1. #13123865C2semidirect
2. #13117809C2semidirect
3. #13113392C2semidirect
4. #13113374C2semidirect
5. #13113369C2semidirect
6. #13113365C2semidirect
7. #13113361C2semidirect
8. #13113356C2semidirect
9. #13113352C2semidirect
10. #13113327C2semidirect
13. C13×#553C4semidirect
14. C13×#552C4semidirect
16. C13×#548C4semidirect
17. C13×#546C4semidirect
29. #11718872C22semidirect
31. C4×C22⋊Dic13C22semidirect
34. C2×C22⋊#1055C22semidirect
40. #11718597C22semidirect
41. C4×C4⋊Dic13C22semidirect
42. #11718587C22semidirect
43. #11718130C22semidirect
44. #11718130C22semidirect
45. #11718130C22semidirect
46. #11718130C22semidirect
303. #553Dic13semidirect
304. #552Dic13semidirect
306. #548Dic13semidirect
307. #546Dic13semidirect
557. C13#17625semidirect

Maximal subgroups

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

subgroupindex
#131238652
#131178092
#131133922
#131133742
#131133692
#131133652
#131133612
#131133562
#131133522
#131133272
C13×#35052
#1762513