Finite Group Databasebeta

← #15785487

Subgroups — #15785487

Order 3328.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718612C212yesmaximal
2#13113089C212yesmaximal, complemented
3#13113085C212yesmaximal
4#13113068C212yesmaximal, complemented
5#13113066C211yesmaximal, characteristic, complemented
6#13113063C211yesmaximal, characteristic, complemented
7#13113063C211yesmaximal, characteristic, complemented
8#13113059C211yesmaximal, characteristic
9#13113059C211yesmaximal, characteristic
10C4×#88071C211yesmaximal, characteristic, complemented
11C22×C13×#117C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C2×C13×#117—28no—
14C2×C13×#117C418yescomplemented
15C22×C13×C4⋊C4C2211yescharacteristic
16C22×C13×C4⋊C4C2211yescharacteristic
17C22×C13×C22⋊C4C2211yescharacteristic
18C22×C13×C22⋊C4C2211yescharacteristic
19C22×C42×C13C2211yescharacteristic
20C22×C42×C13C2211yescharacteristic
21C22×C22⋊Dic13C2212yes—
22C22×C22⋊Dic13C2212yes—
23C22×C4⋊Dic13C2212yes—
24C22×C4×Dic13C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27C2×#88071C2212yes—
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30C2×#88071C2211yescharacteristic
31C2×#88071C2211yescharacteristic
32C2×#88071C2211yescharacteristic
33C2×#88071C2211yescharacteristic
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37#11718612—28no—
38C25×C13C2311yescharacteristic
39C23×C4×C13—24no—
40C23×C4×C13—22no—
41C23×C4×C13—22no—
42C23×C4×C13—24no—
43C23×C4×C13C2×C414yes—
44C23×C4×C13C2311yescharacteristic
45C23×C4×C13C2311yescharacteristic
46C23×C4×C13C2311yescharacteristic
47C23×C4×C13C2311yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2311yescharacteristic
50C23×Dic13C2312yes—
51C23×Dic13C2312yes—
52C23×Dic13C2312yes—
53C23×Dic13C2312yes—
54C13×#117—232no—
55C2×C13×C4⋊C4—24no—
56C2×C13×C4⋊C4—24no—
57C2×C13×C4⋊C4C2×C414yes—
58C2×C13×C4⋊C4C2×C414yes—
59C2×C13×C22⋊C4—24no—
60C2×C13×C22⋊C4C2×C414yes—
61C2×C13×C22⋊C4C2×C414yes—
62C2×C13×C22⋊C4D418yes—
63C2×C42×C13—24no—
64C2×C42×C13—24no—
65C2×C42×C13C2×C414yes—
66C2×C42×C13C2×C414yes—
67C2×C22⋊Dic13—24no—
68C2×C22⋊Dic13—24no—
69C2×C22⋊Dic13—24no—
70C2×C22⋊Dic13—24no—
71C2×C22⋊Dic13—42no—
72C2×C22⋊Dic13—42no—
73C2×C4⋊Dic13—24no—
74C2×C4⋊Dic13—24no—
75C2×C4⋊Dic13—42no—
76C2×C4×Dic13—24no—
77C2×C4×Dic13—24no—
78C2×C4×Dic13—42no—
79#88071—24no—
80#88071—24no—
81#88071—22no—
82#88071—22no—
83#88071—24no—
84#88071—24no—
85#88071—22no—
86#88071—22no—
87#88071—24no—
88#88071—24no—
89#88071—42no—
90#88071—42no—
91#13943—131nomaximal, Sylow
92C24×C13—22no—
93C24×C13—22no—
94C24×C13—21no—
95C24×C13—21no—
96C24×C13—22no—
97C24×C13—21no—
98C24×C13—21no—
99C24×C13—21no—
100C24×C13—21no—
101C24×C13C22×C412yes—
102C24×C13C2×D414yes—
103C24×C13C2411yescharacteristic
104C22×C4×C13—216no—
105C22×C4×C13—216no—
106C22×C4×C13—28no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—22no—
111C22×C4×C13—21no—
112C22×C4×C13—21no—
113C22×C4×C13—22no—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—28no—
119C22×C4×C13—28no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C412yes—
131C22×C4×C13C22×C412yes—
132C22×C4×C13C22×C412yes—
133C22×C4×C13C22×C412yes—
134C22×C4×C13C2×D414yes—
135C22×C4×C13C2×D414yes—
136C22×C4×C13#4112yes—
137C22×C4×C13#4112yes—
138C22×C4×C13#4114yes—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—22no—
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×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—22no—
164C22×Dic13—22no—
165C22×Dic13#4114yes—
166C22×Dic13#4114yes—
167C13×C4⋊C4—28no—
168C13×C4⋊C4—28no—
169C13×C22⋊C4—28no—
170C13×C22⋊C4C22⋊C4116yes—
171C42×C13—28no—
172C42×C13—28no—
173C22⋊Dic13—48no—
174C22⋊Dic13—48no—
175C4⋊Dic13—48no—
176C4×Dic13—48no—
177C22×#117D1311yescharacteristic
178C2×#388—132no—
179#1441—132no—
180#1440—132no—
181#1436—132no—
182C4⋊#95—131no—
183#1434—131no—
184#1434—131no—
185C22⋊#95—131no—
186C22⋊#95—131no—
187C4×#95—131no—
188C23×C13—22no—
189C23×C13—22no—
190C23×C13—22no—
191C23×C13—22no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13—24no—
195C23×C13—24no—
196C23×C13—24no—
197C23×C13—24no—
198C23×C13—24no—
199C23×C13—24no—
200C23×C13—42no—
201C23×C13—42no—
202C23×C13—41no—
203C23×C13—41no—
204C23×C13—41no—
205C23×C13—41no—
206C23×C13—42no—
207C23×C13—42no—
208C23×C13—42no—
209C23×C13—42no—
210C23×C13C2×C22⋊C414yes—
211C23×C13C23×C411yescharacteristic
212C23×C13C22×D412yes—
213C23×C13C2×#4111yescharacteristic
214C23×C13C2×#4111yescharacteristic
215C23×C13C2×#4112yes—
216C23×C13C2×#4112yes—
217C23×C13C2×#4112yes—
218C23×C13H(5,𝔽2)11yescharacteristic
219C23×C13H(5,𝔽2)11yescharacteristic
220C23×C132−1+411yescharacteristic
221C23×C132−1+411yescharacteristic
222C2×C4×C13—232no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—24no—
232C2×C4×C13—24no—
233C2×C4×C13—24no—
234C2×C4×C13—24no—
235C2×C4×C13—24no—
236C2×C4×C13—24no—
237C2×C4×C13—24no—
238C2×C4×C13—24no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13C2×C22⋊C414yes—
250C2×C4×C13C2×C22⋊C414yes—
251C2×C4×C13#11714yes—
252C2×C4×C13#11714yes—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—24no—
256C2×Dic13—24no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
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×Dic13—42no—
279C2×Dic13—42no—
280C2×Dic13—42no—
281C24×C4C2×D1311yescharacteristic
282C2×#117—28no—
283C2×#117Dic1318yescomplemented
284C22×C4⋊C4—132no—
285C22×C4⋊C4C2×D1311yescharacteristic
286C22×C4⋊C4C2×D1311yescharacteristic
287C22×C22⋊C4—132no—
288C22×C22⋊C4—132no—
289C22×C22⋊C4C2×D1311yescharacteristic
290C22×C22⋊C4C2×D1311yescharacteristic
291C22×C42—132no—
292C22×C42C2×D1311yescharacteristic
293C22×C42C2×D1311yescharacteristic
294#388—268no—
295C2×#95—131no—
296C2×#95—131no—
297C2×#95—132no—
298C2×#95—131no—
299C2×#95—131no—
300C2×#95—132no—
301C2×#95—132no—
302C2×#95—132no—
303C2×#95—132no—
304C2×#95—132no—
305C2×#95—132no—
306C2×#95—132no—
307C22×C13—24no—
308C22×C13—24no—
309C22×C13—24no—
310C22×C13—42no—
311C22×C13—42no—
312C22×C13—42no—
313C22×C13—44no—
314C22×C13—44no—
315C22×C13—44no—
316C22×C13—44no—
317C22×C13—42no—
318C22×C13C22×C22⋊C411yescharacteristic
319C22×C13C2×#11711yescharacteristic
320C22×C13C2×#11711yescharacteristic
321C22×C13C4×#4111yescharacteristic
322C22×C13C4×#4111yescharacteristic
323C22×C13#52011yescharacteristic
324C22×C13#52011yescharacteristic
325C22×C13#52512yes—
326C22×C13#52512yes—
327C22×C13#52912yes—
328C22×C13#52912yes—
329C22×C13#54011yescharacteristic
330C22×C13#54111yescharacteristic
331C22×C13#54211yescharacteristic
332C22×C13#54211yescharacteristic
333C22×C13#54612yes—
334C22×C13#54612yes—
335C22×C13#54712yes—
336C22×C13#54712yes—
337C22×C13#55212yes—
338C22×C13#55312yes—
339C22×C13#55512yes—
340C22×C13#55612yes—
341C4×C13—24no—
342C4×C13—24no—
343C4×C13—24no—
344C4×C13—24no—
345C4×C13—42no—
346C4×C13—42no—
347Dic13—44no—
348Dic13—44no—
349Dic13—82no—
350Dic13—82no—
351C25C22×D1311yescharacteristic
352C23×C4—24no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—24no—
356C23×C4—132no—
357C23×C4—132no—
358C23×C4—132no—
359C23×C4—132no—
360C23×C4C2×Dic1314yes—
361C23×C4C22×D1311yescharacteristic
362C23×C4C22×D1311yescharacteristic
363C23×C4C22×D1311yescharacteristic
364C23×C4C22×D1311yescharacteristic
365C23×C4C22×D1311yescharacteristic
366C23×C4C22×D1311yescharacteristic
367#117—232no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—264no—
371C2×C4⋊C4—264no—
372C2×C4⋊C4—522no—
373C2×C4⋊C4C2×Dic1314yes—
374C2×C4⋊C4C2×Dic1314yes—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—264no—
377C2×C22⋊C4—264no—
378C2×C22⋊C4—264no—
379C2×C22⋊C4—264no—
380C2×C22⋊C4—522no—
381C2×C22⋊C4—522no—
382C2×C22⋊C4C2×Dic1314yes—
383C2×C22⋊C4C2×Dic1314yes—
384C2×C22⋊C4C22⋊D1318yes—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—264no—
388C2×C42—264no—
389C2×C42—522no—
390C2×C42C2×Dic1314yes—
391C2×C42C2×Dic1314yes—
392#95—264no—
393#95—264no—
394#95—262no—
395#95—262no—
396#95—264no—
397#95—264no—
398#95—262no—
399#95—262no—
400#95—264no—
401#95—264no—
402#95—522no—
403#95—522no—
404C2×C13—44no—
405C2×C13#228011yescharacteristic
406C2×C13#229811yescharacteristic
407C2×C13#231411yescharacteristic
408C2×C13#233911yescharacteristic
409C2×C13#233911yescharacteristic
410C2×C13#234611yescharacteristic
411C2×C13#234711yescharacteristic
412C2×C13#247012yes—
413C2×C13#247412yes—
414C2×C13#247612yes—
415C2×C13#247812yes—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24C22×Dic1312yes—
426C24C2×C22⋊D1314yes—
427C24C23×D1311yescharacteristic
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—134no—
453C22×C4—134no—
454C22×C4—262no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—262no—
465C22×C4—262no—
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×C4#612712yes—
481C22×C4#612712yes—
482C22×C4#612714yes—
483C22×C4C22×Dic1312yes—
484C22×C4C22×Dic1312yes—
485C22×C4C22×Dic1312yes—
486C22×C4C22×Dic1312yes—
487C22×C4C22×Dic1312yes—
488C22×C4C22×Dic1312yes—
489C22×C4C2×C22⋊D1314yes—
490C22×C4C2×C22⋊D1314yes—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—528no—
494C22⋊C4—28no—
495C22⋊C4—528no—
496C22⋊C4—528no—
497C22⋊C4C22⋊Dic13116yes—
498C42—28no—
499C42—28no—
500C42—528no—
501C13#1394311yescharacteristic, complemented, Sylow
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×C22⋊Dic1314yes—
525C23C2×#612711yescharacteristic
526C23C2×#612711yescharacteristic
527C23C2×#612712yes—
528C23#41×D1312yes—
529C23#41×D1312yes—
530C23#8825611yescharacteristic
531C23#8825611yescharacteristic
532C23#8825711yescharacteristic
533C23#8825711yescharacteristic
534C23C23×Dic1311yescharacteristic
535C23C22×C22⋊D1312yes—
536C2×C4—232no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—264no—
564C2×C4—264no—
565C2×C4—264no—
566C2×C4—264no—
567C2×C4—264no—
568C2×C4—264no—
569C2×C4—522no—
570C2×C4—522no—
571C2×C4—522no—
572C2×C4—522no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—522no—
576C2×C4—522no—
577C2×C4—522no—
578C2×C4—522no—
579C2×C4—522no—
580C2×C4—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4—522no—
584C2×C4—522no—
585C2×C4—522no—
586C2×C4—522no—
587C2×C4—522no—
588C2×C4—522no—
589C2×C4—522no—
590C2×C4—522no—
591C2×C4#8818014yes—
592C2×C4#8818014yes—
593C2×C4C2×C22⋊Dic1314yes—
594C2×C4C2×C22⋊Dic1314yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22#1171922311yescharacteristic
607C22#1171923312yes—
608C22#1171923312yes—
609C22#1171923412yes—
610C22#1171923512yes—
611C22#1171923712yes—
612C22#1171923812yes—
613C22#1171923911yescharacteristic
614C22#1171923911yescharacteristic
615C22#1171924011yescharacteristic
616C22C2×#8818011yescharacteristic
617C22C2×#8818011yescharacteristic
618C22#1171949612yes—
619C22#1171949612yes—
620C22#1171952912yes—
621C22#1171952912yes—
622C22#41×Dic1311yescharacteristic
623C22#41×Dic1311yescharacteristic
624C22#1171953211yescharacteristic
625C22#1171953211yescharacteristic
626C22#1171953412yes—
627C22#1171953412yes—
628C22C22×C22⋊Dic1311yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—524no—
636C4—524no—
637C4—1042no—
638C4—1042no—
639C2—44no—
640C2#1311721911yescharacteristic
641C2#1311733411yescharacteristic
642C2#1311744511yescharacteristic
643C2#1311766611yescharacteristic
644C2#1311766611yescharacteristic
645C2#1311772111yescharacteristic
646C2#1311772911yescharacteristic
647C2#1311892612yes—
648C2#1311898612yes—
649C2#1311901612yes—
650C2#1311904612yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC22×C13×#117C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22×#117D13
hypocenterC13#13943
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #13113089C2semidirect
4. #13113068C2semidirect
5. #13113066C2semidirect
6. #13113063C2semidirect
7. #13113063C2semidirect
10. C4×#88071C2semidirect
14. C2×C13×#117C4semidirect
283. C2×#117Dic13semidirect
501. C13#13943semidirect

Maximal subgroups

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

subgroupindex
C2×#117186122
#131130892
#131130852
#131130682
#131130662
#131130632
#131130592
C4×#880712
C22×C13×#1172
#1394313