Finite Group Databasebeta

← #15755318

Subgroups — #15755318

Order 3328.

676 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13115810C211yesmaximal, characteristic, complemented
2#13113553C211yesmaximal, characteristic, complemented
3#13113456C211yesmaximal, characteristic, complemented
4#13113456C211yesmaximal, characteristic, complemented
5#13113374C211yesmaximal, characteristic, complemented
6#13113374C211yesmaximal, characteristic, complemented
7C13×#3552C211yesmaximal, characteristic
8C13×#563—22no—
9C13×#557C411yescharacteristic, complemented
10C13×#556C411yescharacteristic, complemented
11C13×#554—21no—
12C13×#553—21no—
13D4×Q8×C13—21no—
14D4×Q8×C13C411yescharacteristic, complemented
15C13×#548—21no—
16C13×#547C411yescharacteristic, complemented
17C13×#544—21no—
18C13×#542—21no—
19C13×#541—21no—
20C13×#539—22no—
21C13×#536—21no—
22C2×C13×C4⋊Q8C2211yescharacteristic
23C13×#529C2211yescharacteristic
24C13×#529C2211yescharacteristic
25#11718597C2211yescharacteristic
26#11718587C2211yescharacteristic
27#11718587C2211yescharacteristic
28C2×#88043C2211yescharacteristic
29#11718139—22no—
30#11718135—21no—
31#11718135—21no—
32#11718132—21no—
33#11718132—21no—
34#11718130—21no—
35#11718130—21no—
36#11718129—21no—
37#11718129—21no—
38#11718121—22no—
39C2×C13×#41—21no—
40C22×Q8×C13—21no—
41C22×Q8×C13—21no—
42C13×C4⋊Q8—22no—
43C13×C4⋊Q8—21no—
44C13×C4⋊Q8—21no—
45C13×C4⋊Q8—21no—
46C13×C4⋊Q8—21no—
47C13×C4⋊Q8C2×C411yescharacteristic
48C13×C4⋊Q8C2×C411yescharacteristic
49C13×C4⋊Q8D412yes—
50C13×#126—22no—
51C13×#126—22no—
52C13×#126—22no—
53C13×#126—21no—
54C13×#126—21no—
55C13×#125—22no—
56C13×#125—22no—
57C13×#125—22no—
58C13×#125—21no—
59C13×#125D411yescharacteristic
60C13×#125D411yescharacteristic
61C13×#124—22no—
62C13×#124—21no—
63C13×#124D411yescharacteristic
64C13×#124D411yescharacteristic
65C13×#123—22no—
66C13×#123—22no—
67C13×#123—21no—
68C13×#123—21no—
69C13×#123—22no—
70C13×C22⋊Q8—22no—
71C13×C22⋊Q8—22no—
72C13×C22⋊Q8—22no—
73C13×C22⋊Q8—22no—
74C13×C22⋊Q8—22no—
75C13×C22⋊Q8—21no—
76C13×C22⋊Q8—22no—
77C13×C22⋊Q8—21no—
78C13×C22⋊Q8—21no—
79C13×C22⋊Q8D411yescharacteristic
80C13×C22⋊Q8D411yescharacteristic
81C13×#121—22no—
82C13×#121—21no—
83C13×#121D411yescharacteristic
84C13×#121D411yescharacteristic
85C4×Q8×C13—21no—
86C4×Q8×C13—21no—
87C4×Q8×C13—21no—
88C4×Q8×C13C2×C411yescharacteristic
89C4×Q8×C13C2×C411yescharacteristic
90C4×D4×C13—22no—
91C4×D4×C13—22no—
92C4×D4×C13—21no—
93C4×D4×C13—21no—
94C4×D4×C13—21no—
95C4×D4×C13C2×C411yescharacteristic
96C4×D4×C13C2×C411yescharacteristic
97C13×#117—21no—
98C13×#117—21no—
99C13×#117—21no—
100C2×C13×C4⋊C4—21no—
101C2×C13×C4⋊C4—21no—
102C2×C13×C4⋊C4D412yes—
103C2×C42×C13C2311yescharacteristic
104C2×C4⋊Dic13—21no—
105C22×#1055—21no—
106#88071—21no—
107#88071—21no—
108C22⋊#1055—21no—
109C22⋊#1055—21no—
110C4⋊#1055—21no—
111C4⋊#1055—21no—
112#88043—21no—
113#88043—21no—
114#88043—21no—
115#88043—21no—
116#8499—131nomaximal, Sylow
117C13×#41—82no—
118C2×Q8×C13—22no—
119C2×Q8×C13—21no—
120C2×Q8×C13—21no—
121C2×Q8×C13—22no—
122C2×Q8×C13—82no—
123C2×Q8×C13—82no—
124C2×Q8×C13C22⋊C411yescharacteristic
125C2×Q8×C13C22⋊C411yescharacteristic
126C2×D4×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—22no—
129C2×D4×C13C22⋊C411yescharacteristic
130C2×D4×C13C22⋊C411yescharacteristic
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×Dic13—41no—
140C13×C4⋊C4—22no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—22no—
143C13×C4⋊C4—22no—
144C13×C4⋊C4—22no—
145C13×C4⋊C4—22no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—21no—
149C13×C4⋊C4—21no—
150C13×C4⋊C4—22no—
151C13×C4⋊C4—22no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4C22⋊C411yescharacteristic
159C13×C4⋊C4C22⋊C412yes—
160C13×C4⋊C4C22⋊C412yes—
161C13×C4⋊C4C22⋊C411yescharacteristic
162C13×C4⋊C4C22⋊C411yescharacteristic
163C13×C4⋊C4C22⋊C411yescharacteristic
164C13×C22⋊C4—22no—
165C13×C22⋊C4—22no—
166C13×C22⋊C4—22no—
167C13×C22⋊C4—22no—
168C13×C22⋊C4—21no—
169C13×C22⋊C4—21no—
170C13×C22⋊C4—21no—
171C13×C22⋊C4—21no—
172C13×C22⋊C4—21no—
173C13×C22⋊C4—21no—
174C13×C22⋊C4—21no—
175C13×C22⋊C4—21no—
176C13×C22⋊C4—21no—
177C13×C22⋊C4—21no—
178C42×C13—22no—
179C42×C13—22no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13C22×C411yescharacteristic
183C42×C13C2×D411yescharacteristic
184C42×C13C2×D411yescharacteristic
185C42×C13C2×D411yescharacteristic
186C4⋊Dic13—42no—
187C2×#1055—22no—
188C2×#1055—41no—
189C2×#1055—41no—
190C2×#1055—41no—
191C2×#1055—41no—
192#3552D1311yescharacteristic
193#1993—131no—
194#1556—131no—
195#1532—131no—
196#1532—131no—
197#1512—131no—
198#1512—131no—
199C23×C13—21no—
200C23×C13—21no—
201C23×C13#12011yescharacteristic
202Q8×C13—42no—
203Q8×C13—42no—
204Q8×C13—41no—
205Q8×C13—41no—
206Q8×C13—82no—
207Q8×C13—82no—
208D4×C13—41no—
209D4×C13—41no—
210D4×C13—41no—
211D4×C13—41no—
212D4×C13—41no—
213D4×C13—41no—
214D4×C13—82no—
215C2×C4×C13—21no—
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—21no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—42no—
228C2×C4×C13—42no—
229C2×C4×C13—82no—
230C2×C4×C13—82no—
231C2×C4×C13—82no—
232C2×C4×C13C2×C22⋊C411yescharacteristic
233C2×C4×C13C2×C22⋊C411yescharacteristic
234C2×C4×C13C2×C22⋊C411yescharacteristic
235C2×C4×C13#12011yescharacteristic
236C2×C4×C13#12011yescharacteristic
237C2×C4×C13#12011yescharacteristic
238C2×Dic13—81no—
239C2×Dic13—81no—
240C2×Dic13—81no—
241#1055—42no—
242#1055—81no—
243#1055—81no—
244#563—22no—
245#557Dic1311yescharacteristic, complemented
246#556Dic1311yescharacteristic, complemented
247#554—21no—
248#553—21no—
249D4×Q8—21no—
250D4×Q8Dic1311yescharacteristic, complemented
251#548—21no—
252#547Dic1311yescharacteristic, complemented
253#544—21no—
254#542—21no—
255#541—21no—
256#539—22no—
257#536—21no—
258C2×C4⋊Q8C2×D1311yescharacteristic
259#529C2×D1311yescharacteristic
260#529C2×D1311yescharacteristic
261#432—131no—
262#432—131no—
263C2×#97—131no—
264#382—131no—
265#340—262no—
266#332—262no—
267#330—261no—
268#330—261no—
269#329—261no—
270#329—261no—
271#328—261no—
272#328—261no—
273#327—261no—
274#327—261no—
275C22×C13—21no—
276C22×C13—21no—
277C22×C13—41no—
278C22×C13—41no—
279C22×C13—41no—
280C22×C13—41no—
281C22×C13—41no—
282C22×C13—41no—
283C22×C13#37911yescharacteristic
284C22×C13#45411yescharacteristic
285C22×C13#45611yescharacteristic
286C4×C13—21no—
287C4×C13—21no—
288C4×C13—21no—
289C4×C13—21no—
290C4×C13—21no—
291C4×C13—21no—
292C4×C13—41no—
293C4×C13—41no—
294C4×C13—41no—
295C4×C13—42no—
296C4×C13—42no—
297C4×C13—41no—
298C4×C13—41no—
299C4×C13—41no—
300Dic13—161no—
301C2×#41—21no—
302C22×Q8—21no—
303C22×Q8—21no—
304C22×C8—261no—
305C4⋊Q8—22no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C2×Dic1311yescharacteristic
311C4⋊Q8C2×Dic1311yescharacteristic
312C4⋊Q8C22⋊D1312yes—
313#126—22no—
314#126—22no—
315#126—22no—
316#126—21no—
317#126—21no—
318#125—22no—
319#125—22no—
320#125—22no—
321#125—21no—
322#125C22⋊D1311yescharacteristic
323#125C22⋊D1311yescharacteristic
324#124—22no—
325#124—21no—
326#124C22⋊D1311yescharacteristic
327#124C22⋊D1311yescharacteristic
328#123—22no—
329#123—22no—
330#123—21no—
331#123—21no—
332#123—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—22no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8C22⋊D1311yescharacteristic
343C22⋊Q8C22⋊D1311yescharacteristic
344#121—22no—
345#121—21no—
346#121C22⋊D1311yescharacteristic
347#121C22⋊D1311yescharacteristic
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C2×Dic1311yescharacteristic
352C4×Q8C2×Dic1311yescharacteristic
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4C2×Dic1311yescharacteristic
359C4×D4C2×Dic1311yescharacteristic
360#117—21no—
361#117—21no—
362#117—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4C22⋊D1312yes—
367C2×C42C22×D1311yescharacteristic
368C4⋊C8—261no—
369C4⋊C8—261no—
370C22⋊C8—261no—
371C22⋊C8—261no—
372#97—261no—
373#97—261no—
374#97—261no—
375#97—261no—
376#95—261no—
377#95—261no—
378C2×C13—21no—
379C2×C13—21no—
380C2×C13—81no—
381C2×C13—81no—
382C2×C13#179311yescharacteristic
383C2×C13#179511yescharacteristic
384C2×C13#179611yescharacteristic
385#41—82no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8C22⋊Dic1311yescharacteristic
393C2×Q8C22⋊Dic1311yescharacteristic
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4C22⋊Dic1311yescharacteristic
398C2×D4C22⋊Dic1311yescharacteristic
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—521no—
405C22×C4C2×C22⋊D1311yescharacteristic
406C22×C4C2×C22⋊D1311yescharacteristic
407C22×C4C2×C22⋊D1311yescharacteristic
408C2×C8—262no—
409C2×C8—521no—
410C2×C8—521no—
411C2×C8—521no—
412C2×C8—521no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—522no—
432C4⋊C4C22⋊Dic1311yescharacteristic
433C4⋊C4C22⋊Dic1312yes—
434C4⋊C4C22⋊Dic1312yes—
435C4⋊C4C22⋊Dic1311yescharacteristic
436C4⋊C4C22⋊Dic1311yescharacteristic
437C4⋊C4C22⋊Dic1311yescharacteristic
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42C22×Dic1311yescharacteristic
457C42C2×C22⋊D1311yescharacteristic
458C42C2×C22⋊D1311yescharacteristic
459C42C2×C22⋊D1311yescharacteristic
460C13#849911yescharacteristic, complemented, Sylow
461C23—21no—
462C23—21no—
463C23#8820711yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—42no—
467Q8—42no—
468Q8—82no—
469Q8—82no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—82no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—1041no—
495C2×C4—1041no—
496C2×C4—1041no—
497C2×C4C2×C22⋊Dic1311yescharacteristic
498C2×C4C2×C22⋊Dic1311yescharacteristic
499C2×C4C2×C22⋊Dic1311yescharacteristic
500C2×C4#8820711yescharacteristic
501C2×C4#8820711yescharacteristic
502C2×C4#8820711yescharacteristic
503C8—522no—
504C8—1041no—
505C8—1041no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22#1171890611yescharacteristic
515C22#1171890711yescharacteristic
516C22#1171890811yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—2081no—
532C2—21no—
533C2—21no—
534C2—81no—
535C2—81no—
536C2#1311479511yescharacteristic
537C2#1311480611yescharacteristic
538C2#1311481111yescharacteristic
subgroupG/H
centerC22#11718908
commutatorC42×C13C22×C4
FrattiniC2×C42C22×D13
Fitting, prime radicalC13×#3552C2
socleC22×C13#379
cosocleC2×C42×C13C23
hypercenter#3552D13
hypocenterC13#8499
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3552

Complements

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

NH ≅ G/Ntype
1. #13115810C2semidirect
2. #13113553C2semidirect
3. #13113456C2semidirect
4. #13113456C2semidirect
5. #13113374C2semidirect
6. #13113374C2semidirect
9. C13×#557C4semidirect
10. C13×#556C4semidirect
14. D4×Q8×C13C4semidirect
16. C13×#547C4semidirect
245. #557Dic13semidirect
246. #556Dic13semidirect
250. D4×Q8Dic13semidirect
252. #547Dic13semidirect
460. C13#8499semidirect

Maximal subgroups

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

subgroupindex
#131158102
#131135532
#131134562
#131133742
C13×#35522
#849913