Finite Group Databasebeta

← #28314091

Subgroups — #28314091

Order 13568.

1210 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291570C212yesmaximal, complemented
2#20289979C212yesmaximal, complemented
3#20289970C212yesmaximal, complemented
4#20289966C212yesmaximal, complemented
5#20289964C212yesmaximal, complemented
6#20289950C212yesmaximal, complemented
7D53⋊#324C212yesmaximal, complemented
8C2×#16811104C211yesmaximal, characteristic
9#16811653—28no—
10#16811643C2214yes—
11#16811594C2214yes—
12D53⋊(C22⋊C8)C2214yes—
13#16811469C2214yes—
14#16811469C2214yes—
15#16811468C2214yes—
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×#13133175C411yescharacteristic
29C2×#13133175C411yescharacteristic
30C2×C4⋊C4×D53C2211yescharacteristic
31C2×C4⋊C4×D53C2211yescharacteristic
32C22×C4⋊C4×C53C411yescharacteristic
33C2×#11719936—24no—
34C2×D53⋊C8—24no—
35#13133284—24no—
36#13133274—24no—
37#13133274—24no—
38C22⋊#88324C2318yes—
39#13133272—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×D53C2311yescharacteristic
52C22×C4×D53C2311yescharacteristic
53#13133182—24no—
54#13133182—24no—
55#13133179—24no—
56#13133179D418yes—
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×#11719919C2311yescharacteristic
69C2×#11719919C2311yescharacteristic
70C4⋊C4×D53—24no—
71C4⋊C4×D53—24no—
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2311yescharacteristic
74C2×#11719921C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2×C411yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C411yescharacteristic
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⋊D53M1614yescomplemented
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×D212C22⋊C414yes—
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—81no—
132C2×C4×D53—81no—
133C2×C4×D53—81no—
134C2×C4×D53—81no—
135C2×C4×D53M1614yescomplemented
136C2×C4×D53C2×D414yes—
137C2×C4×D53C2×D414yes—
138C2×Dic106—22no—
139C2×Dic106—22no—
140C2×Dic106—22no—
141C2×#88324—28no—
142#11719936—84no—
143D53⋊C8—84no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53C22⋊C414yes—
148C22⋊Dic53—22no—
149C22⋊Dic53—22no—
150#11719921—22no—
151#11719921—22no—
152#11719921C22⋊C414yes—
153#11719921C22⋊C414yes—
154C4⋊Dic53—22no—
155C4⋊Dic53—22no—
156#11719919—22no—
157#11719919—22no—
158#11719919—22no—
159#11719919—22no—
160C4×Dic53—22no—
161C4×Dic53—22no—
162C4×Dic53—22no—
163C4×Dic53—22no—
164C4×Dic53—22no—
165C4×Dic53C2×D414yes—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53C2×M1612yes—
171C23×C53C23×C411yescharacteristic
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—41no—
178C22×D53C2×C22⋊C412yes—
179C22×D53C2×C22⋊C412yes—
180C22×D53C2×M1612yes—
181C22×D53C2×M1612yes—
182C2×C4×C53—28no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—82no—
194C2×C4×C53—82no—
195C2×C4×C53C2×C22⋊C412yes—
196C2×C4×C53C2×C22⋊C412yes—
197C2×C4×C53C2×C22⋊C412yes—
198C2×C4×C53C2×M1612yes—
199C22⋊D53—28no—
200C22⋊D53—28no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53C2×C22⋊C412yes—
212C2×Dic53C2×M1612yes—
213C2×Dic53C2×M1612yes—
214C2×Dic53C22×D412yes—
215D212—28no—
216C4×D53—28no—
217C4×D53—28no—
218C4×D53—44no—
219C4×D53—44no—
220C4×D53—82no—
221C4×D53—82no—
222Dic106—28no—
223#88324—48no—
224#17368—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—22no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53C22×C22⋊C411yescharacteristic
233C22×C53C22×M1611yescharacteristic
234C22×C53C22×M1611yescharacteristic
235C2×D53—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53—22no—
240C2×D53—42no—
241C2×D53—42no—
242C2×D53—42no—
243C2×D53—42no—
244C2×D53#40714yes—
245C4×C53—22no—
246C4×C53—22no—
247C4×C53—22no—
248C4×C53—22no—
249C4×C53—41no—
250C4×C53—41no—
251Dic53—22no—
252Dic53—22no—
253Dic53—22no—
254Dic53—22no—
255Dic53—41no—
256Dic53—41no—
257C2×#520—531no—
258#2973—532no—
259#1467—532no—
260#1465—532no—
261#1464—532no—
262#1463—532no—
263#1459—532no—
264C2×#324—532no—
265C2×C53—21no—
266C2×C53—21no—
267C2×C53—42no—
268C2×C53C2×#40711yescharacteristic
269C2×C53#288311yescharacteristic
270C2×C53#288911yescharacteristic
271D53—24no—
272D53—42no—
273C22×#41—531no—
274#520—1068no—
275C2×C4×Q8—531no—
276C2×C4×Q8—531no—
277C2×C4×D4—531no—
278C2×C4×D4—531no—
279C2×C4×D4—531no—
280C2×C4×D4—531no—
281C2×C4×D4—531no—
282C2×C4×D4—531no—
283C2×#117—531no—
284C2×#117—531no—
285C2×#117—531no—
286C22×C4⋊C4—531no—
287C22×C4⋊C4—531no—
288C22×C4⋊C4#615711yescharacteristic
289#435—1068no—
290#432—534no—
291#407—534no—
292C2×C22⋊C8—534no—
293#324—534no—
294#324—534no—
295#324—534no—
296#323—534no—
297C53#1736811yescharacteristic, complemented, Sylow
298C2×#41—1064no—
299C2×#41—1064no—
300C2×#41—1064no—
301C22×Q8—531no—
302C22×D4—531no—
303C22×D4—531no—
304C22×D4—531no—
305C23×C4—531no—
306C23×C4—531no—
307C23×C4—531no—
308C23×C4—531no—
309C23×C4—531no—
310C23×C4—531no—
311C23×C4C2×#615711yescharacteristic
312C23×C4C2×#615711yescharacteristic
313C23×C4C2×#615711yescharacteristic
314C2×M16—1064no—
315C22×C8—1064no—
316C4×Q8—1064no—
317C4×Q8—1064no—
318C4×D4—538no—
319C4×D4—1064no—
320C4×D4—1064no—
321C4×D4—1064no—
322C4×D4—1064no—
323C4×D4—1064no—
324#117—1064no—
325#117—1064no—
326#117—1064no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—531no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—531no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—531no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—1064no—
335C2×C4⋊C4—1064no—
336C2×C4⋊C4C2×#615711yescharacteristic
337C2×C4⋊C4C2×#615711yescharacteristic
338C2×C4⋊C4C2×#615711yescharacteristic
339C2×C4⋊C4C2×#615711yescharacteristic
340C2×C22⋊C4—531no—
341C2×C22⋊C4—531no—
342C2×C22⋊C4—531no—
343C2×C22⋊C4—531no—
344C2×C22⋊C4—531no—
345C2×C22⋊C4—531no—
346C2×C42—531no—
347C2×C42—531no—
348C2×C42—531no—
349C2×C42—531no—
350C2×C42—531no—
351C2×C42—531no—
352C4⋊C8—1064no—
353C22⋊C8—538no—
354C22⋊C8—1064no—
355C22⋊C8—1064no—
356#97—1064no—
357C24—531no—
358C24—531no—
359C24C22×#615711yescharacteristic
360#41—10632no—
361C2×Q8—1062no—
362C2×Q8—1062no—
363C2×Q8—1062no—
364C2×D4—534no—
365C2×D4—534no—
366C2×D4—1062no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—1062no—
370C2×D4—1062no—
371C2×D4—1062no—
372C2×D4—1062no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—534no—
385C22×C4—534no—
386C22×C4—534no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—1062no—
390C22×C4—1062no—
391C22×C4—1062no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—4241no—
397C22×C4—4241no—
398C22×C4—4241no—
399C22×C4—4241no—
400C22×C4C22×#615711yescharacteristic
401C22×C4C22×#615711yescharacteristic
402C22×C4C22×#615711yescharacteristic
403C22×C4C22×#615711yescharacteristic
404C22×C4C22×#615711yescharacteristic
405C22×C4C22×#615711yescharacteristic
406M16—4244no—
407C2×C8—1068no—
408C2×C8—4244no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—1062no—
413C4⋊C4—1062no—
414C4⋊C4—1062no—
415C4⋊C4—1062no—
416C4⋊C4—1062no—
417C4⋊C4—1062no—
418C4⋊C4C22⋊#615714yes—
419C22⋊C4—534no—
420C22⋊C4—534no—
421C22⋊C4—1062no—
422C22⋊C4—1062no—
423C22⋊C4—1062no—
424C22⋊C4—1062no—
425C42—534no—
426C42—1062no—
427C42—1062no—
428C42—1062no—
429C42—1062no—
430C42—1062no—
431C23—21no—
432C23—21no—
433C23—41no—
434C23—41no—
435C23—532no—
436C23—532no—
437C23—532no—
438C23—532no—
439C23—1061no—
440C23—1061no—
441C23—2121no—
442C23—2121no—
443C23—2121no—
444C23—2121no—
445C23D53⋊M1612yes—
446C23C23×#615711yescharacteristic
447Q8—1068no—
448D4—1068no—
449D4—1068no—
450D4—1068no—
451C2×C4—28no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—532no—
467C2×C4—532no—
468C2×C4—1068no—
469C2×C4—1068no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—4242no—
477C2×C4—4242no—
478C2×C4—2124no—
479C2×C4—2124no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4—2121no—
484C2×C4C2×C22⋊#615712yes—
485C2×C4C2×C22⋊#615712yes—
486C2×C4C2×C22⋊#615712yes—
487C2×C4D53⋊M1612yes—
488C8—2128no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—534no—
497C22—1062no—
498C22—1062no—
499C22—1062no—
500C22—1062no—
501C22—1062no—
502C22—2122no—
503C22—2122no—
504C22—2122no—
505C22—2122no—
506C22C22×C22⋊#615711yescharacteristic
507C22C2×D53⋊M1611yescharacteristic
508C22C2×D53⋊M1611yescharacteristic
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—1062no—
516C4—1062no—
517C4—1062no—
518C4—1062no—
519C4—2121no—
520C4—2121no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2—1064no—
525C2—2122no—
526C2D53⋊#40711yescharacteristic
527C2#2029131411yescharacteristic
528C2#2029132811yescharacteristic
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#17368
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. #20291570C2semidirect
2. #20289979C2semidirect
3. #20289970C2semidirect
4. #20289966C2semidirect
5. #20289964C2semidirect
6. #20289950C2semidirect
7. D53⋊#324C2semidirect
113. C2×C22⋊D53M16semidirect
135. C2×C4×D53M16semidirect
297. C53#17368semidirect

Maximal subgroups

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

subgroupindex
#202915702
#202899792
#202899702
#202899662
#202899642
#202899502
D53⋊#3242
C2×#168111042
#1736853