Finite Group Databasebeta

← #27299561

Subgroups — #27299561

Order 13568.

1160 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810413C212yesmaximal, complemented
2C2×#16810410C212yesmaximal, complemented
3#20274643C211yesmaximal, characteristic, complemented
4#20274633C211yesmaximal, characteristic, complemented
5C2×C53×#525C211yesmaximal, characteristic
6C2×#13133136—21no—
7C2×#13133134C2212yes—
8C2×#13133132—21no—
9C2×#13133132C2212yes—
10#16810587—21no—
11#16810486—21no—
12#16810413—24no—
13#16810410—24no—
14C22×#41×C53C2211yescharacteristic
15C24×C4×C53C411yescharacteristic, complemented
16C53×#525—24no—
17C53×#525—24no—
18C2×#123×C53C411yescharacteristic, complemented
19C2×#123×C53C2211yescharacteristic
20C2×C22⋊Q8×C53—21no—
21C2×#121×C53—21no—
22C2×#120×C53C411yescharacteristic, complemented
23C2×#120×C53C2211yescharacteristic
24C2×C4×D4×C53—21no—
25C2×C4×D4×C53—21no—
26C2×#117×C53C411yescharacteristic, complemented
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29#13133136—42no—
30#13133134—22no—
31#13133134—22no—
32#13133132—22no—
33#13133132—22no—
34#13133132—42no—
35C22⋊#88322—42no—
36#13133124—42no—
37C25×C53C2×C411yescharacteristic
38C2×#41×C53—22no—
39C2×#41×C53—22no—
40C2×#41×C53—42no—
41C2×#41×C53—42no—
42C22×Q8×C53D411yescharacteristic
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53C2×C411yescharacteristic
46C22×D4×C53D411yescharacteristic
47C22×D4×C53C2311yescharacteristic
48C23×C4×C53—22no—
49C23×C4×C53—21no—
50C23×C4×C53—21no—
51C23×C4×C53—22no—
52C23×C4×C53—42no—
53C23×C4×C53—42no—
54C23×C4×C53C2×C411yescharacteristic
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58#123×C53—22no—
59#123×C53—22no—
60#123×C53—22no—
61#123×C53D414yes—
62C22⋊Q8×C53—28no—
63#121×C53—28no—
64#120×C53—22no—
65#120×C53—22no—
66#120×C53—22no—
67#120×C53D414yes—
68C4×D4×C53—28no—
69C4×D4×C53—28no—
70#117×C53—22no—
71#117×C53—22no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C22⋊C4×C53—21no—
76C2×C22⋊C4×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53C2×C411yescharacteristic
80C2×C22⋊C4×C53C2×C411yescharacteristic
81C2×C42×C53—21no—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—41no—
85C24×C53—41no—
86C24×C53—44no—
87C24×C53C22⋊C411yescharacteristic
88C24×C53C22⋊C412yes—
89C24×C53C22×C411yescharacteristic
90C24×C53C2×D411yescharacteristic
91#41×C53—48no—
92#41×C53—48no—
93C2×Q8×C53—22no—
94C2×Q8×C53—22no—
95C2×Q8×C53—41no—
96C2×D4×C53—24no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—24no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—41no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—44no—
108C2×D4×C53—44no—
109C2×D4×C53C2×D412yes—
110C2×D4×C53C2×D412yes—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—24no—
116C22×C4×C53—24no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—48no—
128C22×C4×C53—48no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—41no—
134C22×C4×C53—41no—
135C22×C4×C53C22⋊C411yescharacteristic
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C412yes—
139C22×C4×C53C2×D411yescharacteristic
140C22×Dic53—42no—
141C4⋊C4×C53—24no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—24no—
144C22⋊C4×C53—24no—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—24no—
149C22⋊C4×C53—24no—
150C22⋊C4×C53C22⋊C412yes—
151C22⋊C4×C53C22⋊C412yes—
152C42×C53—24no—
153C22⋊Dic53—42no—
154C22⋊Dic53—42no—
155#11719917—42no—
156#11719917—42no—
157C2×#88322—42no—
158C23×C53—24no—
159C23×C53—22no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—42no—
168C23×C53—44no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—44no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—44no—
176C23×C53—48no—
177C23×C53—44no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53C2×C22⋊C411yescharacteristic
181C23×C53C2×C22⋊C411yescharacteristic
182C23×C53C2×C22⋊C411yescharacteristic
183C23×C53#12012yes—
184Q8×C53—44no—
185D4×C53—42no—
186D4×C53—42no—
187D4×C53—44no—
188D4×C53—48no—
189D4×C53—48no—
190D4×C53—42no—
191D4×C53—42no—
192C2×C4×C53—24no—
193C2×C4×C53—24no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—44no—
207C2×C4×C53—44no—
208C2×C4×C53—48no—
209C2×C4×C53—48no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53#9912yes—
217C2×C4×C53#9912yes—
218C2×C4×C53#12012yes—
219C2×Dic53—82no—
220C2×Dic53—84no—
221#88322—82no—
222#12689—531nomaximal, Sylow
223C22×C53—22no—
224C22×C53—24no—
225C22×C53—22no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—42no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—42no—
232C22×C53—44no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—44no—
236C22×C53—44no—
237C22×C53—48no—
238C22×C53—42no—
239C22×C53—44no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—41no—
244C22×C53—81no—
245C22×C53—81no—
246C22×C53#37911yescharacteristic
247C22×C53C2×#9911yescharacteristic
248C22×C53C2×#9911yescharacteristic
249C4×C53—22no—
250C4×C53—22no—
251C4×C53—22no—
252C4×C53—42no—
253C4×C53—42no—
254C4×C53—41no—
255C4×C53—42no—
256Dic53—162no—
257C2×#525D5311yescharacteristic
258C2×#352—532no—
259C2×AΣL(1,𝔽16)—532no—
260#1785—531no—
261#1782—531no—
262C2×C53—21no—
263C2×C53—21no—
264C2×C53—42no—
265C2×C53—44no—
266C2×C53—41no—
267C2×C53—41no—
268C2×C53—81no—
269C2×C53#178111yescharacteristic
270C2×C53#212012yes—
271C22×#41C2×D5311yescharacteristic
272C24×C4Dic5311yescharacteristic, complemented
273#525—24no—
274#525—24no—
275C2×#123Dic5311yescharacteristic, complemented
276C2×#123C2×D5311yescharacteristic
277C2×C22⋊Q8—21no—
278C2×#121—21no—
279C2×#120Dic5311yescharacteristic, complemented
280C2×#120C2×D5311yescharacteristic
281C2×C4×D4—21no—
282C2×C4×D4—21no—
283C2×#117Dic5311yescharacteristic, complemented
284C2×#101—1061no—
285C2×#100—532no—
286C2×#99—532no—
287C2×#99—1061no—
288#407—1061no—
289#381—1061no—
290#352—1064no—
291AΣL(1,𝔽16)—1064no—
292C53#1268911yescharacteristic, complemented, Sylow
293C25C2×Dic5311yescharacteristic
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—42no—
297C2×#41—42no—
298C22×Q8C22⋊D5311yescharacteristic
299C22×D4—21no—
300C22×D4—21no—
301C22×D4C2×Dic5311yescharacteristic
302C22×D4C22⋊D5311yescharacteristic
303C22×D4C22×D5311yescharacteristic
304C23×C4—22no—
305C23×C4—21no—
306C23×C4—22no—
307C23×C4—21no—
308C23×C4—42no—
309C23×C4—42no—
310C23×C4C2×Dic5311yescharacteristic
311C23×C4C2×Dic5311yescharacteristic
312C23×C4C22⋊D5311yescharacteristic
313C23×C4C22⋊D5311yescharacteristic
314C2×M16—1062no—
315#123—22no—
316#123—22no—
317#123—22no—
318#123C22⋊D5314yes—
319C22⋊Q8—28no—
320#121—28no—
321#120—22no—
322#120—22no—
323#120—22no—
324#120C22⋊D5314yes—
325C4×D4—28no—
326C4×D4—28no—
327#117—22no—
328#117—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—1062no—
337C2×C22⋊C4C2×Dic5311yescharacteristic
338C2×C22⋊C4C2×Dic5311yescharacteristic
339C2×C42—21no—
340#101—2122no—
341#100—1062no—
342#100—1062no—
343#99—1062no—
344#99—1062no—
345#99—2122no—
346C22⋊C8—2122no—
347#95—2122no—
348C24—21no—
349C24—21no—
350C24—41no—
351C24—41no—
352C24—44no—
353C24C22⋊Dic5311yescharacteristic
354C24C22⋊Dic5312yes—
355C24C22×Dic5311yescharacteristic
356C24C2×C22⋊D5311yescharacteristic
357#41—48no—
358#41—48no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4C2×C22⋊D5312yes—
376C2×D4C2×C22⋊D5312yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—2122no—
402C22×C4C22⋊Dic5311yescharacteristic
403C22×C4C22⋊Dic5311yescharacteristic
404C22×C4C22⋊Dic5312yes—
405C22×C4C22⋊Dic5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407M16—2122no—
408M16—2122no—
409C2×C8—2122no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C22⋊C4C22⋊Dic5312yes—
422C22⋊C4C22⋊Dic5312yes—
423C42—24no—
424C23—24no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—48no—
443C23—44no—
444C23—41no—
445C23—41no—
446C23C2×C22⋊Dic5311yescharacteristic
447C23C2×C22⋊Dic5311yescharacteristic
448C23C2×C22⋊Dic5311yescharacteristic
449C23#1313318612yes—
450Q8—44no—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—48no—
455D4—48no—
456D4—42no—
457D4—42no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—4242no—
483C2×C4—4244no—
484C2×C4#1313313212yes—
485C2×C4#1313313212yes—
486C2×C4#1313318612yes—
487C8—4242no—
488C22—22no—
489C22—24no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—44no—
502C22—48no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—81no—
510C22—81no—
511C22#1681047911yescharacteristic
512C22C2×#1313313211yescharacteristic
513C22C2×#1313313211yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—8482no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—44no—
526C2—41no—
527C2—41no—
528C2—81no—
529C2#2027463011yescharacteristic
530C2#2027635612yes—
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC2×C53×#525C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenterC2×#525D53
hypocenterC53#12689
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#16810413C2semidirect
2. C2×#16810410C2semidirect
3. #20274643C2semidirect
4. #20274633C2semidirect
15. C24×C4×C53C4semidirect
18. C2×#123×C53C4semidirect
22. C2×#120×C53C4semidirect
26. C2×#117×C53C4semidirect
272. C24×C4Dic53semidirect
275. C2×#123Dic53semidirect
279. C2×#120Dic53semidirect
283. C2×#117Dic53semidirect
292. C53#12689semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
C2×#168104102
#202746432
#202746332
C2×C53×#5252
#1268953