Finite Group Databasebeta

← #28314115

Subgroups — #28314115

Order 13568.

1174 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291614C212yesmaximal, complemented
2#20289976C212yesmaximal, complemented
3#20289974C212yesmaximal, complemented
4#20289969C212yesmaximal, complemented
5#20289966C212yesmaximal, complemented
6#20289954C212yesmaximal, complemented
7#20289949C212yesmaximal, complemented
8C2×#16811104C211yesmaximal, characteristic
9#16811675—28no—
10#16811644C2214yes—
11#16811643C2214yes—
12#16811640C2214yes—
13#16811468C2214yes—
14#16811467C2214yes—
15#16811467C2214yes—
16#16811466C2214yes—
17C22×#11719945C411yescharacteristic
18#16811104—28no—
19C2×#13133182C411yescharacteristic
20C2×#13133182C2211yescharacteristic
21C2×#13133179C411yescharacteristic
22C2×#13133179C2211yescharacteristic
23C2×C4×C22⋊D53C411yescharacteristic
24C2×C4×C22⋊D53C411yescharacteristic
25C2×C4×C22⋊D53C2211yescharacteristic
26C2×C4×C22⋊D53C2211yescharacteristic
27C2×#13133176C2211yescharacteristic
28C2×#13133175C2211yescharacteristic
29C2×#13133175C2211yescharacteristic
30C2×C4⋊C4×D53C411yescharacteristic
31C2×C4⋊C4×D53C411yescharacteristic
32C22×C4⋊C4×C53C411yescharacteristic
33#13133284—24no—
34#13133284—24no—
35C4⋊#88324—24no—
36C22⋊#88324C2318yes—
37#13133272—24no—
38#13133272—24no—
39C4×#88324—24no—
40C2×#11719945—24no—
41C2×#11719945—24no—
42C2×#11719945—24no—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53C2×C411yescharacteristic
46C22×C22⋊D53C2×C411yescharacteristic
47C22×C4×D53C2×C411yescharacteristic
48C22×C4×D53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53#13133182—24no—
54#13133182D418yes—
55#13133179—24no—
56#13133179—24no—
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60C4×C22⋊D53—24no—
61#13133176—24no—
62#13133175—24no—
63#13133175—24no—
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×#11719919C2×C411yescharacteristic
67C2×#11719919C2×C411yescharacteristic
68C2×#11719919C2×C411yescharacteristic
69C2×#11719919C2×C411yescharacteristic
70C4⋊C4×D53—24no—
71C4⋊C4×D53—24no—
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2311yescharacteristic
74C2×#11719921C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×#11719921C2311yescharacteristic
77C2×#11719921C2311yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53C2×C411yescharacteristic
89C2×C4⋊C4×C53C2×C411yescharacteristic
90C2×C4⋊C4×C53C2×C411yescharacteristic
91C2×C4⋊C4×C53C2×C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C23×D53C22×C411yescharacteristic
94C23×D53C22×C411yescharacteristic
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—81no—
101C22×C4×C53—81no—
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C22×C411yescharacteristic
108C2×C22⋊D53—22no—
109C2×C22⋊D53—22no—
110C2×C22⋊D53—22no—
111C2×C22⋊D53—22no—
112C2×C22⋊D53—22no—
113C2×C22⋊D53M1614yes—
114C22×Dic53C22×C411yescharacteristic
115C22×Dic53C22×C411yescharacteristic
116C22×Dic53C22×C411yescharacteristic
117C22×Dic53C22×C411yescharacteristic
118C22×Dic53C22×C411yescharacteristic
119C22×Dic53C2411yescharacteristic
120#11719945—232no—
121C2×D212—22no—
122C2×D212—22no—
123C2×D212—22no—
124C2×C4×D53—22no—
125C2×C4×D53—22no—
126C2×C4×D53—22no—
127C2×C4×D53—22no—
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—81no—
134C2×C4×D53—81no—
135C2×C4×D53—81no—
136C2×C4×D53—81no—
137C2×C4×D53M1614yes—
138C2×Dic106—22no—
139C2×Dic106—22no—
140C2×Dic106C22⋊C414yes—
141C2×#88324—28no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53C22⋊C414yes—
146C22⋊Dic53—22no—
147C22⋊Dic53—22no—
148#11719921—22no—
149#11719921—22no—
150#11719921—22no—
151#11719921—22no—
152C4⋊Dic53—22no—
153C4⋊Dic53—22no—
154#11719919—22no—
155#11719919—22no—
156#11719919C22⋊C414yes—
157#11719919C22⋊C414yes—
158C4×Dic53—22no—
159C4×Dic53—22no—
160C4×Dic53—22no—
161C4×Dic53C2×D414yes—
162C4×Dic53C2×D414yes—
163C4×Dic53C2×D414yes—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53C2×M1612yes—
169C23×C53C23×C411yescharacteristic
170C22×D53—21no—
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—41no—
178C22×D53C2×M1612yes—
179C22×D53C2×M1612yes—
180C2×C4×C53—28no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—82no—
192C2×C4×C53—82no—
193C2×C4×C53C2×C22⋊C412yes—
194C2×C4×C53C2×C22⋊C412yes—
195C2×C4×C53C2×C22⋊C412yes—
196C2×C4×C53C2×M1612yes—
197C22⋊D53—28no—
198C22⋊D53—28no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53C2×C22⋊C412yes—
208C2×Dic53C2×C22⋊C412yes—
209C2×Dic53C2×C22⋊C412yes—
210C2×Dic53C2×M1612yes—
211C2×Dic53C2×M1612yes—
212C2×Dic53C22×D412yes—
213D212—28no—
214C4×D53—28no—
215C4×D53—28no—
216C4×D53—82no—
217C4×D53—82no—
218C4×D53—82no—
219C4×D53—82no—
220Dic106—28no—
221#88324—48no—
222#17372—531nomaximal, Sylow
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—22no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53C22×C22⋊C411yescharacteristic
231C22×C53C22×M1611yescharacteristic
232C22×C53C22×M1611yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—42no—
236C2×D53—42no—
237C2×D53—41no—
238C2×D53—41no—
239C2×D53—41no—
240C2×D53—41no—
241C2×D53—42no—
242C2×D53—42no—
243C4×C53—22no—
244C4×C53—22no—
245C4×C53—22no—
246C4×C53—22no—
247C4×C53—41no—
248C4×C53—41no—
249Dic53—22no—
250Dic53—22no—
251Dic53—22no—
252Dic53—41no—
253Dic53—41no—
254Dic53#40714yes—
255C2×#520—531no—
256#2983—532no—
257#1466—532no—
258#1466—532no—
259#1464—532no—
260#1464—532no—
261#1460—532no—
262#1458—532no—
263C2×C53—21no—
264C2×C53—21no—
265C2×C53—42no—
266C2×C53C2×#40711yescharacteristic
267C2×C53#288411yescharacteristic
268C2×C53#288811yescharacteristic
269D53—42no—
270D53—42no—
271C22×#41—531no—
272#520—1068no—
273C2×C4×Q8—531no—
274C2×C4×Q8—531no—
275C2×C4×D4—531no—
276C2×C4×D4—531no—
277C2×C4×D4—531no—
278C2×C4×D4—531no—
279C2×C4×D4—531no—
280C2×C4×D4—531no—
281C2×#117—531no—
282C2×#117—531no—
283C2×#117—531no—
284C22×C4⋊C4—531no—
285C22×C4⋊C4—531no—
286C22×C4⋊C4#615711yescharacteristic
287#446—1068no—
288#432—534no—
289#432—534no—
290#431—534no—
291#324—534no—
292#323—534no—
293#323—534no—
294#323—534no—
295C53#1737211yescharacteristic, complemented, Sylow
296C2×#41—1064no—
297C2×#41—1064no—
298C2×#41—1064no—
299C22×Q8—531no—
300C22×D4—531no—
301C22×D4—531no—
302C22×D4—531no—
303C23×C4—531no—
304C23×C4—531no—
305C23×C4—531no—
306C23×C4—531no—
307C23×C4—531no—
308C23×C4—531no—
309C23×C4C2×#615711yescharacteristic
310C23×C4C2×#615711yescharacteristic
311C23×C4C2×#615711yescharacteristic
312C4×Q8—538no—
313C4×Q8—1064no—
314C4×D4—1064no—
315C4×D4—1064no—
316C4×D4—1064no—
317C4×D4—1064no—
318C4×D4—1064no—
319C4×D4—1064no—
320#117—1064no—
321#117—1064no—
322#117—1064no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—531no—
325C2×C4⋊C4—531no—
326C2×C4⋊C4—531no—
327C2×C4⋊C4—531no—
328C2×C4⋊C4—531no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—1064no—
331C2×C4⋊C4—1064no—
332C2×C4⋊C4C2×#615711yescharacteristic
333C2×C4⋊C4C2×#615711yescharacteristic
334C2×C4⋊C4C2×#615711yescharacteristic
335C2×C4⋊C4C2×#615711yescharacteristic
336C2×C22⋊C4—531no—
337C2×C22⋊C4—531no—
338C2×C22⋊C4—531no—
339C2×C22⋊C4—531no—
340C2×C22⋊C4—531no—
341C2×C22⋊C4—531no—
342C2×C42—531no—
343C2×C42—531no—
344C2×C42—531no—
345C2×C42—531no—
346C2×C42—531no—
347C2×C42—531no—
348C4⋊C8—1064no—
349C4⋊C8—1064no—
350C4⋊C8—1064no—
351C22⋊C8—538no—
352#97—1064no—
353#97—1064no—
354C4×C8—1064no—
355C24—531no—
356C24—531no—
357C24C22×#615711yescharacteristic
358#41—10632no—
359C2×Q8—534no—
360C2×Q8—1062no—
361C2×Q8—1062no—
362C2×D4—534no—
363C2×D4—1062no—
364C2×D4—1062no—
365C2×D4—1062no—
366C2×D4—1062no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—1062no—
370C2×D4—1062no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—534no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4—1062no—
390C22×C4—1062no—
391C22×C4—1062no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—4241no—
395C22×C4—4241no—
396C22×C4—4241no—
397C22×C4—4241no—
398C22×C4C22×#615711yescharacteristic
399C22×C4C22×#615711yescharacteristic
400C22×C4C22×#615711yescharacteristic
401C22×C4C22×#615711yescharacteristic
402C22×C4C22×#615711yescharacteristic
403C22×C4C22×#615711yescharacteristic
404C2×C8—1068no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—534no—
409C4⋊C4—534no—
410C4⋊C4—1062no—
411C4⋊C4—1062no—
412C4⋊C4—1062no—
413C4⋊C4—1062no—
414C4⋊C4C22⋊#615714yes—
415C22⋊C4—1062no—
416C22⋊C4—1062no—
417C22⋊C4—1062no—
418C22⋊C4—1062no—
419C22⋊C4—1062no—
420C22⋊C4—1062no—
421C42—534no—
422C42—534no—
423C42—534no—
424C42—1062no—
425C42—1062no—
426C42—1062no—
427C23—21no—
428C23—21no—
429C23—41no—
430C23—41no—
431C23—532no—
432C23—532no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—2121no—
438C23—2121no—
439C23—2121no—
440C23—2121no—
441C23D53⋊M1612yes—
442C23C23×#615711yescharacteristic
443Q8—1068no—
444D4—1068no—
445D4—1068no—
446D4—1068no—
447C2×C4—28no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—532no—
461C2×C4—532no—
462C2×C4—532no—
463C2×C4—532no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—1068no—
467C2×C4—1068no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—4242no—
473C2×C4—4242no—
474C2×C4—4242no—
475C2×C4—4242no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4C2×C22⋊#615712yes—
481C2×C4C2×C22⋊#615712yes—
482C2×C4C2×C22⋊#615712yes—
483C2×C4D53⋊M1612yes—
484C8—2128no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—41no—
491C22—41no—
492C22—1062no—
493C22—1062no—
494C22—2122no—
495C22—2122no—
496C22—2122no—
497C22—2122no—
498C22—2121no—
499C22—2121no—
500C22—2121no—
501C22—2121no—
502C22C22×C22⋊#615711yescharacteristic
503C22C2×D53⋊M1611yescharacteristic
504C22C2×D53⋊M1611yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—534no—
512C4—1062no—
513C4—1062no—
514C4—1062no—
515C4—2121no—
516C4—2121no—
517C2—21no—
518C2—21no—
519C2—42no—
520C2—2122no—
521C2—2122no—
522C2D53⋊#40711yescharacteristic
523C2#2029131611yescharacteristic
524C2#2029132511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×#11719945C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#17372
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291614C2semidirect
2. #20289976C2semidirect
3. #20289974C2semidirect
4. #20289969C2semidirect
5. #20289966C2semidirect
6. #20289954C2semidirect
7. #20289949C2semidirect
295. C53#17372semidirect

Maximal subgroups

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

subgroupindex
#202916142
#202899762
#202899742
#202899692
#202899662
#202899542
#202899492
C2×#168111042
#1737253