Finite Group Databasebeta

← #15926549

Subgroups — #15926549

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118521C211yesmaximal, characteristic
2#13117867C211yesmaximal, characteristic
3#13116409C211yesmaximal, characteristic, complemented
4#13116331C211yesmaximal, characteristic, complemented
5#13116229C211yesmaximal, characteristic, complemented
6#13115975C211yesmaximal, characteristic, complemented
7#13115786C211yesmaximal, characteristic, complemented
8#13115420C211yesmaximal, characteristic, complemented
9#13115260C211yesmaximal, characteristic
10#13115106C211yesmaximal, characteristic, complemented
11#13115037C211yesmaximal, characteristic
12#13115013C211yesmaximal, characteristic
13#13114849C211yesmaximal, characteristic, complemented
14#13114552C211yesmaximal, characteristic
15C13×#1873C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C2×#88142C2211yescharacteristic
22C2×C13×#103C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×C22⋊C8C2211yescharacteristic
25#11718891C2211yescharacteristic
26C2×#88075C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28#11718873C2211yescharacteristic
29C4×C22⋊Dic13C2211yescharacteristic
30C2×#88058C2211yescharacteristic
31C2×#88057C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34#11718646C2211yescharacteristic
35#11718646C2211yescharacteristic
36#11718642—21no—
37#11718638C2211yescharacteristic
38C4⋊C4×Dic13—21no—
39C4⋊C4×Dic13C2211yescharacteristic
40#11718634C2211yescharacteristic
41#11718612—21no—
42#11718612—21no—
43#11718609—21no—
44#11718608—21no—
45#11718594—21no—
46#11718590—21no—
47C13×#340C2211yescharacteristic
48C13×#340C2211yescharacteristic
49#11718143C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59C22×Q8×C13C2311yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C22×Dic26—21no—
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C13×C22⋊Q8—22no—
65C13×C22⋊Q8Q814yescomplemented
66C13×#117—22no—
67C13×#117—41no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4—21no—
74C2×C42×C13—21no—
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13D411yescharacteristic
78C2×C22⋊Dic13D411yescharacteristic
79Q8×Dic13—22no—
80Q8×Dic13—41no—
81#88180—22no—
82#88180—41no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055—22no—
98C22×#1055—41no—
99C22×#1055—41no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102#88142—22no—
103#88142—41no—
104C13×#103—22no—
105C13×#103—41no—
106C13×C22⋊C8—22no—
107#88075—41no—
108#88075—41no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071D411yescharacteristic
119#88071D411yescharacteristic
120#88058—41no—
121#88057—22no—
122#88054—41no—
123#88054—41no—
124#26413—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C2×Q8×C13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13C2×Q812yes—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×Q812yes—
142C22×C4×C13#4112yes—
143C22×C4×C13C2411yescharacteristic
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C2×Dic26—42no—
153C2×Dic26—42no—
154C2×Dic26—81no—
155C2×C8×C13—21no—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C13×C4⋊C4—24no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4C2×Q812yes—
167C13×C4⋊C4#4112yes—
168C13×C22⋊C4—24no—
169C13×C22⋊C4—42no—
170C42×C13—42no—
171C22⋊Dic13—22no—
172C22⋊Dic13—41no—
173C22⋊Dic13—42no—
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—42no—
183C4⋊Dic13—42no—
184C4⋊Dic13—81no—
185C4⋊Dic13#4112yes—
186#6101—42no—
187#6101—42no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—42no—
193C4×Dic13—42no—
194C4×Dic13—42no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—44no—
202#2439—131no—
203#2362—131no—
204#2101—131no—
205#2087—131no—
206#2068—131no—
207#2024—131no—
208#1988—131no—
209#1920—131no—
210#1892—131no—
211#1873D1311yescharacteristic
212#1865—131no—
213#1854—131no—
214#1850—131no—
215#1805—131no—
216#1735—131no—
217C23×C13—21no—
218C23×C13—21no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13C22⋊Q812yes—
224C23×C13C22×D411yescharacteristic
225Q8×C13—22no—
226Q8×C13—22no—
227Q8×C13—41no—
228Q8×C13—41no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
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×C13C22⋊Q812yes—
248C2×C4×C13C22×Q811yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13H(5,𝔽2)11yescharacteristic
253C2×C4×C13H(5,𝔽2)11yescharacteristic
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—22no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
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#13512yes—
278Dic26—84no—
279C8×C13—41no—
280C8×C13—41no—
281#1055—41no—
282#1055—41no—
283C23×C8—131no—
284C2×C4⋊Q8—131no—
285C2×C22⋊Q8C2×D1311yescharacteristic
286C2×C4×Q8—131no—
287C2×#117—131no—
288C2×#117C2×D1311yescharacteristic
289C2×#107—131no—
290C2×#106—131no—
291C2×C4⋊C8—131no—
292C2×C4⋊C8—131no—
293C2×#103—131no—
294C2×#103—131no—
295C2×#103C2×D1311yescharacteristic
296C2×#103C2×D1311yescharacteristic
297C2×C22⋊C8C2×D1311yescharacteristic
298#396—261no—
299#393—261no—
300#391—131no—
301#391—261no—
302#389—131no—
303#389—131no—
304#389—131no—
305#388—261no—
306#388—261no—
307#387—261no—
308#387—261no—
309#382—131no—
310#380—131no—
311C4×C4⋊C4—131no—
312C4×C4⋊C4—261no—
313C4×C22⋊C4—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340C2×D1311yescharacteristic
323#340C2×D1311yescharacteristic
324#336—131no—
325#336—131no—
326C22×C13—22no—
327C22×C13—22no—
328C22×C13—22no—
329C22×C13—41no—
330C22×C13—41no—
331C22×C13—41no—
332C22×C13—41no—
333C22×C13C2×C22⋊Q811yescharacteristic
334C22×C13#54611yescharacteristic
335C22×C13#54611yescharacteristic
336C22×C13C2×#13511yescharacteristic
337C22×C13#57511yescharacteristic
338C22×C13#57711yescharacteristic
339C22×C13#57811yescharacteristic
340C4×C13—22no—
341C4×C13—42no—
342C4×C13—42no—
343C4×C13—42no—
344C4×C13—81no—
345C4×C13#52911yescharacteristic
346C4×C13#53811yescharacteristic
347C4×C13#55411yescharacteristic
348C4×C13#55411yescharacteristic
349Dic13—22no—
350Dic13—41no—
351Dic13—42no—
352Dic13—41no—
353Dic13—41no—
354Dic13—81no—
355Dic13—81no—
356Dic13—81no—
357C22×Q8—261no—
358C22×Q8C22×D1311yescharacteristic
359C23×C4C22×D1311yescharacteristic
360C22×C8—131no—
361C22×C8—131no—
362C22×C8—262no—
363C22×C8—521no—
364C22×C8—521no—
365C22×C8C22×D1311yescharacteristic
366C22×C8C22×D1311yescharacteristic
367C4⋊Q8—262no—
368C4⋊Q8—521no—
369C22⋊Q8—22no—
370C22⋊Q8Dic2614yescomplemented
371C4×Q8—262no—
372C4×Q8—521no—
373#117—22no—
374#117—41no—
375#117—262no—
376#117—521no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
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×C4⋊C4C22×D1311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—131no—
393C2×C22⋊C4—131no—
394C2×C22⋊C4—261no—
395C2×C22⋊C4—261no—
396C2×C42—21no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42—261no—
402#107—521no—
403#106—262no—
404C4⋊C8—521no—
405C4⋊C8—521no—
406#103—22no—
407#103—41no—
408#103—521no—
409#103—521no—
410C22⋊C8—22no—
411#95—131no—
412#95—131no—
413#95—261no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422C2×C13—42no—
423C2×C13#245011yescharacteristic
424C2×C13#307611yescharacteristic
425C2×C13#308211yescharacteristic
426C2×C13#329611yescharacteristic
427C2×C13#330211yescharacteristic
428C2×C13#331311yescharacteristic
429C2×C13#332111yescharacteristic
430C24D4×D1311yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—522no—
437C2×Q8—522no—
438C2×Q8—1041no—
439C2×Q8C2×Dic2612yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
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×C4C2×Dic2612yes—
457C22×C4D4×D1311yescharacteristic
458C22×C4#612912yes—
459C22×C4C23×D1311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8—524no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—132no—
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—522no—
488C4⋊C4—522no—
489C4⋊C4—522no—
490C4⋊C4—1041no—
491C4⋊C4C2×Dic2612yes—
492C4⋊C4#612912yes—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—262no—
496C22⋊C4—521no—
497C22⋊C4—522no—
498C22⋊C4—1041no—
499C42—42no—
500C42—261no—
501C42—261no—
502C42—261no—
503C42—261no—
504C42—522no—
505C42—522no—
506C42—522no—
507C13#2641311yescharacteristic, complemented, Sylow
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#8813212yes—
515C23C2×D4×D1311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—1044no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—132no—
540C2×C4—261no—
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—262no—
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#8813212yes—
564C2×C4C22×Dic2611yescharacteristic
565C2×C4C2×#612911yescharacteristic
566C2×C4C2×#612911yescharacteristic
567C2×C4#8825111yescharacteristic
568C2×C4#41×D1311yescharacteristic
569C2×C4#8825611yescharacteristic
570C8—41no—
571C8—41no—
572C8—521no—
573C8—521no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#8813211yescharacteristic
582C22#1171929611yescharacteristic
583C22#1171935911yescharacteristic
584C22D13×#13511yescharacteristic
585C22#1171947311yescharacteristic
586C22#1171948311yescharacteristic
587C22#1171948411yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—262no—
594C4—521no—
595C4—522no—
596C4—521no—
597C4—521no—
598C4—1041no—
599C4—1041no—
600C4—1041no—
601C4#1171924411yescharacteristic
602C4#1171924811yescharacteristic
603C4#1171930511yescharacteristic
604C4#1171932211yescharacteristic
605C2—42no—
606C2#1311866411yescharacteristic
607C2#1312462011yescharacteristic
608C2#1312468011yescharacteristic
609C2#1312666411yescharacteristic
610C2#1312675111yescharacteristic
611C2#1312690611yescharacteristic
612C2#1312702011yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1873C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1873D13
hypocenterC13#26413
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. #13116409C2semidirect
4. #13116331C2semidirect
5. #13116229C2semidirect
6. #13115975C2semidirect
7. #13115786C2semidirect
8. #13115420C2semidirect
10. #13115106C2semidirect
13. #13114849C2semidirect
65. C13×C22⋊Q8Q8semidirect
370. C22⋊Q8Dic26semidirect
507. C13#26413semidirect

Maximal subgroups

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

subgroupindex
#131185212
#131178672
#131164092
#131163312
#131162292
#131159752
#131157862
#131154202
#131152602
#131151062
#131150372
#131150132
#131148492
#131145522
C13×#18732
#2641313