Finite Group Databasebeta

← #15870918

Subgroups — #15870918

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117578C211yesmaximal, characteristic, complemented
2#13117549C211yesmaximal, characteristic
3#13115806C211yesmaximal, characteristic, complemented
4#13115653C211yesmaximal, characteristic
5#13115535C211yesmaximal, characteristic
6#13115528C211yesmaximal, characteristic, complemented
7#13115521C211yesmaximal, characteristic
8#13115514C211yesmaximal, characteristic, complemented
9#13114964C211yesmaximal, characteristic, complemented
10#13114668C211yesmaximal, characteristic, complemented
11#13114663C211yesmaximal, characteristic
12#103×Dic13C211yesmaximal, characteristic, complemented
13#102×Dic13C211yesmaximal, characteristic
14#13114522C211yesmaximal, characteristic, complemented
15C2×C13×#488C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C2×C13×#124C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20C13×#488C418yescomplemented
21C2×C13×#103C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C2×C4×C8×C13C2211yescharacteristic
26#11718891C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29#11718882C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88055C2211yescharacteristic
33C2×#88055C2211yescharacteristic
34C2×C8×Dic13C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36#11718646—21no—
37#11718646—21no—
38#11718646C2211yescharacteristic
39#11718646C2211yescharacteristic
40#11718642—21no—
41#11718642—21no—
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44#11718612—21no—
45#11718612—21no—
46#11718611—21no—
47#11718611—21no—
48#11718596C2211yescharacteristic
49#11718596C2211yescharacteristic
50#11718593C2211yescharacteristic
51C2×#88043C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×Dic13—21no—
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C13×#125—22no—
66C13×#125C2×C414yescomplemented
67C13×#124—22no—
68C13×#124C2×C414yes—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C22⋊C4—21no—
74C2×C13×C22⋊C4—21no—
75C2×C42×C13C2311yescharacteristic
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13—21no—
79Q8×Dic13—22no—
80Q8×Dic13—22no—
81D4×Dic13—22no—
82D4×Dic13—22no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13—21no—
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C13×#103C2×C414yescomplemented
102C13×#103C2×C414yescomplemented
103C13×#102C2×C414yes—
104C13×#102C2×C414yes—
105C4×C8×C13C2×C414yescomplemented
106#88075—22no—
107#88075—22no—
108#88072—22no—
109#88072—22no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071D411yescharacteristic
117#88071D411yescharacteristic
118#88071D411yescharacteristic
119#88071D411yescharacteristic
120#88055—22no—
121#88055—22no—
122C8×Dic13—22no—
123C8×Dic13—22no—
124#88043—22no—
125#21643—131nomaximal, Sylow
126C24×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13C22×C412yes—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13C22×C412yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—42no—
150C22×Dic13—81no—
151C22×Dic13—81no—
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13C22×C412yes—
161C2×C8×C13C22×C412yes—
162C13×C4⋊C4—24no—
163C13×C4⋊C4—24no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4C22×C412yes—
167C13×C4⋊C4C22×C412yes—
168C13×C22⋊C4—24no—
169C13×C22⋊C4—24no—
170C42×C13C22×C412yes—
171C42×C13#4112yes—
172C22⋊Dic13—42no—
173C22⋊Dic13—81no—
174C22⋊Dic13—81no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—42no—
184C4⋊Dic13—81no—
185C4⋊Dic13—81no—
186C4⋊Dic13—81no—
187C4⋊Dic13—81no—
188C4⋊Dic13—81no—
189C4⋊Dic13—81no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—42no—
193C4×Dic13—81no—
194C4×Dic13—81no—
195C4×Dic13#4112yes—
196C4×Dic13#4112yes—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055#4112yes—
202C2×#1055#4112yes—
203C2×#488D1311yescharacteristic
204#2330—131no—
205#2326—131no—
206#1992—131no—
207#1959—131no—
208#1936—131no—
209#1935—131no—
210#1934—131no—
211#1933—131no—
212#1834—131no—
213#1763—131no—
214#1762—131no—
215C4×#103—131no—
216C4×#102—131no—
217#1726—131no—
218C23×C13—22no—
219C23×C13—41no—
220C23×C13—41no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13C22×D411yescharacteristic
226Q8×C13—22no—
227Q8×C13—22no—
228Q8×C13—41no—
229Q8×C13—41no—
230D4×C13—22no—
231D4×C13—22no—
232D4×C13—41no—
233D4×C13—41no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13C4×D412yes—
248C2×C4×C13C4×D412yes—
249C2×C4×C13C23×C411yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—84no—
278C2×Dic13#12412yes—
279C2×Dic13#12412yes—
280C8×C13—22no—
281C8×C13—22no—
282#1055—22no—
283#1055—22no—
284C2×#125C2×D1311yescharacteristic
285C2×#124C2×D1311yescharacteristic
286C2×C4×Q8—131no—
287C2×C4×D4—131no—
288#488Dic1318yescomplemented
289C2×#103—131no—
290C2×#103—131no—
291C2×#103C2×D1311yescharacteristic
292C2×#103C2×D1311yescharacteristic
293C2×#102—131no—
294C2×#102—131no—
295C2×#102C2×D1311yescharacteristic
296C2×#102C2×D1311yescharacteristic
297C2×#97—131no—
298C2×#97—131no—
299C2×#97—131no—
300C2×C4×C8—131no—
301C2×C4×C8—131no—
302C2×C4×C8C2×D1311yescharacteristic
303#391—131no—
304#390—131no—
305#389—131no—
306#389—131no—
307#389—261no—
308#389—261no—
309#388—261no—
310#388—261no—
311#387—261no—
312#387—261no—
313#385—261no—
314#385—261no—
315#384—131no—
316#384—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—131no—
319#376—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328C22×C13—42no—
329C22×C13—42no—
330C22×C13—42no—
331C22×C13—81no—
332C22×C13—81no—
333C22×C13—81no—
334C22×C13—81no—
335C22×C13C2×C4×D411yescharacteristic
336C22×C13C2×#12411yescharacteristic
337C22×C13#53211yescharacteristic
338C22×C13#57611yescharacteristic
339C22×C13#57711yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57811yescharacteristic
342C4×C13—22no—
343C4×C13—22no—
344C4×C13—42no—
345C4×C13—42no—
346C4×C13—42no—
347C4×C13C4×#4111yescharacteristic
348C4×C13C4×#4111yescharacteristic
349C4×C13#52911yescharacteristic
350C4×C13#52911yescharacteristic
351Dic13—22no—
352Dic13—22no—
353Dic13—41no—
354Dic13—41no—
355Dic13—81no—
356Dic13—81no—
357Dic13—81no—
358Dic13—81no—
359C22×Q8C22×D1311yescharacteristic
360C22×D4C22×D1311yescharacteristic
361C23×C4—261no—
362C22×C8—131no—
363C22×C8—131no—
364C22×C8C22×D1311yescharacteristic
365C22×C8C22×D1311yescharacteristic
366#125—22no—
367#125C2×Dic1314yescomplemented
368#124—22no—
369#124C2×Dic1314yes—
370C4×Q8—262no—
371C4×Q8—262no—
372C4×D4—262no—
373C4×D4—262no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4C22×D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—261no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—261no—
394C2×C42—131no—
395C2×C42—131no—
396C2×C42—261no—
397C2×C42—261no—
398C2×C42—261no—
399C2×C42C22×D1311yescharacteristic
400#103—262no—
401#103—262no—
402#103C2×Dic1314yescomplemented
403#103C2×Dic1314yescomplemented
404#102—262no—
405#102—262no—
406#102C2×Dic1314yes—
407#102C2×Dic1314yes—
408#97—262no—
409#97—262no—
410#97—262no—
411C4×C8—262no—
412C4×C8—262no—
413C4×C8C2×Dic1314yescomplemented
414#95—131no—
415#95—131no—
416#95—131no—
417#95—131no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95—261no—
423#95—261no—
424C2×C13—82no—
425C2×C13C4×#12411yescharacteristic
426C2×C13#295111yescharacteristic
427C2×C13#295211yescharacteristic
428C2×C13#313611yescharacteristic
429C2×C13#313711yescharacteristic
430C2×C13#317811yescharacteristic
431C2×C13#317911yescharacteristic
432C24—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C22×Dic1312yes—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C22×Dic1312yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—131no—
449C22×C4—131no—
450C22×C4—131no—
451C22×C4—131no—
452C22×C4—261no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—261no—
456C22×C4—522no—
457C22×C4—1041no—
458C22×C4—1041no—
459C22×C4D4×D1311yescharacteristic
460C22×C4D4×D1311yescharacteristic
461C22×C4C23×D1311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—132no—
467C2×C8—132no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8C22×Dic1312yes—
473C2×C8C22×Dic1312yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—522no—
487C4⋊C4—1041no—
488C4⋊C4—1041no—
489C4⋊C4—1041no—
490C4⋊C4—1041no—
491C4⋊C4—1041no—
492C4⋊C4—1041no—
493C4⋊C4C22×Dic1312yes—
494C4⋊C4C22×Dic1312yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—522no—
498C22⋊C4—1041no—
499C22⋊C4—1041no—
500C42—132no—
501C42—132no—
502C42—261no—
503C42—261no—
504C42—522no—
505C42—1041no—
506C42—1041no—
507C42#612912yes—
508C42C22×Dic1312yes—
509C13#2164311yescharacteristic, complemented, Sylow
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D1311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—132no—
540C2×C4—132no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—1044no—
564C2×C4D4×Dic1312yes—
565C2×C4D4×Dic1312yes—
566C2×C4C2×#612911yescharacteristic
567C2×C4#41×D1311yescharacteristic
568C2×C4#41×D1311yescharacteristic
569C2×C4#41×D1311yescharacteristic
570C2×C4#41×D1311yescharacteristic
571C2×C4C23×Dic1311yescharacteristic
572C8—22no—
573C8—22no—
574C8—262no—
575C8—262no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22#1171937311yescharacteristic
584C22D13×#12411yescharacteristic
585C22#1171947411yescharacteristic
586C22#1171947511yescharacteristic
587C22#1171947511yescharacteristic
588C22#1171947611yescharacteristic
589C22C2×D4×Dic1311yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—262no—
596C4—262no—
597C4—521no—
598C4—521no—
599C4—1041no—
600C4—1041no—
601C4—1041no—
602C4—1041no—
603C4#1171938511yescharacteristic
604C4#1171938511yescharacteristic
605C4#41×Dic1311yescharacteristic
606C4#41×Dic1311yescharacteristic
607C2—82no—
608C2#124×Dic1311yescharacteristic
609C2#1312371211yescharacteristic
610C2#1312372011yescharacteristic
611C2#1312518911yescharacteristic
612C2#1312520011yescharacteristic
613C2#1312550311yescharacteristic
614C2#1312551811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#488C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#488D13
hypocenterC13#21643
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #13117578C2semidirect
3. #13115806C2semidirect
6. #13115528C2semidirect
8. #13115514C2semidirect
9. #13114964C2semidirect
10. #13114668C2semidirect
12. #103×Dic13C2semidirect
14. #13114522C2semidirect
20. C13×#488C4semidirect
66. C13×#125C2×C4semidirect
101. C13×#103C2×C4semidirect
102. C13×#103C2×C4semidirect
105. C4×C8×C13C2×C4semidirect
288. #488Dic13semidirect
367. #125C2×Dic13semidirect
402. #103C2×Dic13semidirect
403. #103C2×Dic13semidirect
413. C4×C8C2×Dic13semidirect
509. C13#21643semidirect

Maximal subgroups

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

subgroupindex
#131175782
#131175492
#131158062
#131156532
#131155352
#131155282
#131155212
#131155142
#131149642
#131146682
#131146632
#103×Dic132
#102×Dic132
#131145222
C2×C13×#4882
#2164313