Finite Group Databasebeta

← #27301587

Subgroups — #27301587

Order 13568.

818 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810607C211yesmaximal, characteristic, complemented
2C2×#16810410C212yesmaximal, complemented
3#20274633C211yesmaximal, characteristic, complemented
4#20272833C212yesmaximal, complemented
5C53×#2404C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810607—24no—
8C2×#13133136—21no—
9C2×#13133134C2212yes—
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810486—21no—
13#16810410—24no—
14C22×#41×C53C2211yescharacteristic
15C2×#120×C53C411yescharacteristic, complemented
16C2×#120×C53C2211yescharacteristic
17C22×C22⋊C4×C53C411yescharacteristic, complemented
18C53×#393—21no—
19C53×#392—21no—
20C53×#386—21no—
21C53×#386—21no—
22C53×#386C411yescharacteristic, complemented
23C53×#386C2211yescharacteristic
24C53×#381C411yescharacteristic, complemented
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—42no—
28C2×#11719917—41no—
29C22×#88322—41no—
30C2×C22⋊Dic53—22no—
31#13133136—24no—
32#13133134—22no—
33#13133134D414yescomplemented
34#13133132—22no—
35#13133132—22no—
36#13133132—42no—
37#13133124—41no—
38#13133124—41no—
39C25×C53C2×C411yescharacteristic
40C2×#41×C53—22no—
41C2×#41×C53—22no—
42C2×#41×C53—41no—
43C2×#41×C53—41no—
44C2×#41×C53—42no—
45C22×Q8×C53D411yescharacteristic
46C22×D4×C53—21no—
47C22×D4×C53—21no—
48C22×D4×C53C2×C411yescharacteristic
49C22×D4×C53D411yescharacteristic
50C22×D4×C53C2311yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C23×C4×C53D411yescharacteristic
55#120×C53—22no—
56#120×C53—22no—
57#120×C53—42no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—41no—
65C2×C22⋊C4×C53—41no—
66C2×C22⋊C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53C2×C411yescharacteristic
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53—21no—
73C24×C53—22no—
74C24×C53—41no—
75C24×C53—42no—
76C24×C53—41no—
77C24×C53—81no—
78C24×C53C22⋊C411yescharacteristic
79C24×C53C22×C411yescharacteristic
80C24×C53C2×D411yescharacteristic
81#41×C53—44no—
82#41×C53—44no—
83#41×C53—44no—
84#41×C53—44no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—42no—
98C2×D4×C53—42no—
99C2×D4×C53—42no—
100C2×D4×C53—42no—
101C2×D4×C53—81no—
102C2×D4×C53—81no—
103C2×D4×C53C2×D412yes—
104C2×D4×C53#4112yes—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—81no—
123C22×C4×C53C22⋊C411yescharacteristic
124C22×C4×C53C22⋊C411yescharacteristic
125C22×C4×C53C22⋊C411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×Dic53—42no—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—42no—
130C22⋊C4×C53—42no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—81no—
137C22⋊C4×C53—81no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140#11719917—24no—
141#11719917—44no—
142#11719917—42no—
143C2×#88322—44no—
144C2×#88322—42no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—22no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—82no—
167C23×C53—84no—
168C23×C53—84no—
169C23×C53—82no—
170C23×C53#9912yes—
171C23×C53C2×C22⋊C411yescharacteristic
172C23×C53C4×D412yes—
173C23×C53#12112yes—
174Q8×C53—42no—
175Q8×C53—42no—
176D4×C53—42no—
177D4×C53—42no—
178D4×C53—44no—
179D4×C53—44no—
180D4×C53—84no—
181D4×C53—84no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—82no—
210C2×C4×C53#9911yescharacteristic
211C2×C4×C53#9911yescharacteristic
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12311yescharacteristic
214C2×Dic53—82no—
215C2×Dic53—82no—
216C2×Dic53—82no—
217#88322—44no—
218#13083—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—42no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—84no—
242C22×C53—84no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53#38611yescharacteristic
246C22×C53C2×#9911yescharacteristic
247C22×C53C2×#9911yescharacteristic
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—41no—
253C4×C53—41no—
254C4×C53—81no—
255C4×C53—81no—
256C4×C53—81no—
257Dic53—162no—
258C2×#413—531no—
259#2404D5311yescharacteristic
260C2×AΣL(1,𝔽16)—532no—
261#1782—531no—
262#1390—532no—
263C2×C53—21no—
264C2×C53—21no—
265C2×C53—42no—
266C2×C53—42no—
267C2×C53—42no—
268C2×C53—82no—
269C2×C53#189811yescharacteristic
270C2×C53#212011yescharacteristic
271C2×C53#212211yescharacteristic
272C22×#41C2×D5311yescharacteristic
273C22×M16—1061no—
274C2×#120Dic5311yescharacteristic, complemented
275C2×#120C2×D5311yescharacteristic
276C22×C22⋊C4Dic5311yescharacteristic, complemented
277#413—1064no—
278C2×#101—1061no—
279C2×#100—532no—
280C2×#99—532no—
281C2×#99—1061no—
282#393—21no—
283#392—21no—
284#386—21no—
285#386—21no—
286#386Dic5311yescharacteristic, complemented
287#386C2×D5311yescharacteristic
288#381—1061no—
289#381Dic5311yescharacteristic, complemented
290AΣL(1,𝔽16)—1064no—
291C53#1308311yescharacteristic, complemented, Sylow
292C25C2×Dic5311yescharacteristic
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—41no—
296C2×#41—41no—
297C2×#41—42no—
298C22×Q8C22⋊D5311yescharacteristic
299C22×D4—21no—
300C22×D4—21no—
301C22×D4C2×Dic5311yescharacteristic
302C22×D4C22⋊D5311yescharacteristic
303C22×D4C22×D5311yescharacteristic
304C23×C4C2×Dic5311yescharacteristic
305C23×C4C2×Dic5311yescharacteristic
306C23×C4C22⋊D5311yescharacteristic
307C23×C4C22⋊D5311yescharacteristic
308C2×M16—1064no—
309C2×M16—1062no—
310C2×M16—2122no—
311C2×M16—2121no—
312C22×C8—2121no—
313#120—22no—
314#120—22no—
315#120—42no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—41no—
323C2×C22⋊C4—41no—
324C2×C22⋊C4—1062no—
325C2×C22⋊C4C2×Dic5311yescharacteristic
326C2×C22⋊C4C2×Dic5311yescharacteristic
327#101—1064no—
328#100—534no—
329#100—1062no—
330#99—1062no—
331#99—1062no—
332#99—2122no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—2121no—
338#95—2121no—
339C24—21no—
340C24—22no—
341C24—41no—
342C24—42no—
343C24—41no—
344C24—81no—
345C24C22⋊Dic5311yescharacteristic
346C24C22×Dic5311yescharacteristic
347C24C2×C22⋊D5311yescharacteristic
348#41—44no—
349#41—44no—
350#41—44no—
351#41—44no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4D4×D5312yes—
371C2×D4#1171994712yes—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—81no—
390C22×C4—2122no—
391C22×C4C22⋊Dic5311yescharacteristic
392C22×C4C22⋊Dic5311yescharacteristic
393C22×C4C22⋊Dic5311yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395M16—1064no—
396M16—2124no—
397M16—2122no—
398C2×C8—2124no—
399C2×C8—2122no—
400C22⋊C4—22no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—81no—
409C22⋊C4—81no—
410C22⋊C4—2122no—
411C22⋊C4—2122no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—82no—
434C23—84no—
435C23—84no—
436C23—82no—
437C23#1313313212yes—
438C23C2×C22⋊Dic5311yescharacteristic
439C23D4×Dic5312yes—
440C23#1313318912yes—
441Q8—42no—
442Q8—42no—
443D4—42no—
444D4—42no—
445D4—44no—
446D4—44no—
447D4—84no—
448D4—84no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—4242no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4#1313313211yescharacteristic
481C2×C4#1313313211yescharacteristic
482C2×C4#1313318511yescharacteristic
483C2×C4#1313319811yescharacteristic
484C8—2124no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—84no—
508C22—84no—
509C22—81no—
510C22—81no—
511C22#1681050311yescharacteristic
512C22C2×#1313313211yescharacteristic
513C22C2×#1313313211yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—8482no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—82no—
530C2#2027516911yescharacteristic
531C2#2027635611yescharacteristic
532C2#2027636511yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC53×#2404C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenter#2404D53
hypocenterC53#13083
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#16810607C2semidirect
2. C2×#16810410C2semidirect
3. #20274633C2semidirect
4. #20272833C2semidirect
15. C2×#120×C53C4semidirect
17. C22×C22⋊C4×C53C4semidirect
22. C53×#386C4semidirect
24. C53×#381C4semidirect
33. #13133134D4semidirect
274. C2×#120Dic53semidirect
276. C22×C22⋊C4Dic53semidirect
286. #386Dic53semidirect
289. #381Dic53semidirect
291. C53#13083semidirect

Maximal subgroups

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

subgroupindex
C2×#168106072
C2×#168104102
#202746332
#202728332
C53×#24042
#1308353