Finite Group Databasebeta

← #27280672

Subgroups — #27280672

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285468C211yesmaximal, characteristic
2#20275551C211yesmaximal, characteristic
3#20273287C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic
5#20272594C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2368C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810951C412yescomplemented
10#16810941—21no—
11#16810854—22no—
12#16810773C412yescomplemented
13#16810762—21no—
14C2×#13133131—21no—
15C2×#13133131—21no—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19#16810360—22no—
20#16810356—21no—
21#16810356—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#391—21no—
24C53×#387—21no—
25C53×#387—21no—
26C53×#385—21no—
27C4×C4⋊C4×C53—21no—
28C4×C22⋊C4×C53—21no—
29C4×C22⋊C4×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C22×D212C2×C411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719924—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133148—22no—
45#13133141—22no—
46#13133137—22no—
47#13133131—42no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322C2×C412yes—
53#13133124—21no—
54C22×Q8×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53C2311yescharacteristic
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53C2×C412yes—
60C22⋊Q8×C53D412yes—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53D411yescharacteristic
78#95×C53D411yescharacteristic
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—81no—
83C2×Q8×C53C22⋊C412yes—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—42no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—81no—
94C22×C4×C53—81no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C2×C22⋊D53—41no—
102C2×C22⋊D53—41no—
103C2×C22⋊D53—41no—
104C2×C22⋊D53—41no—
105C2×C22⋊D53—41no—
106C2×C22⋊D53—41no—
107C22×Dic53—21no—
108C22×Dic53—41no—
109C2×D212—21no—
110C2×D212—21no—
111C2×D212—81no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—42no—
116C4⋊C4×C53—81no—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53SD1612yescomplemented
124C42×C53—42no—
125C42×C53—81no—
126C42×C53—81no—
127C22⋊Dic53—41no—
128C22⋊Dic53—41no—
129#11719924—82no—
130#11719921—41no—
131#11719921—41no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4⋊Dic53—81no—
135C2×#88322—22no—
136C2×#88322—41no—
137C2×#88322—81no—
138C2×#88322—81no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53C4×D411yescharacteristic
142C23×C53#12011yescharacteristic
143C23×C53#12111yescharacteristic
144C23×C53C2×SD1611yescharacteristic
145C23×C53#13511yescharacteristic
146C23×C53AGL(1,ℤ/8)11yescharacteristic
147C23×C53#13711yescharacteristic
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—81no—
152C22×D53—81no—
153Q8×C53—42no—
154Q8×C53—81no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—82no—
180C2×C4×C53—82no—
181C2×C4×C53C2×C22⋊C411yescharacteristic
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12311yescharacteristic
185C22⋊D53—81no—
186C22⋊D53—81no—
187C22⋊D53—81no—
188C22⋊D53—81no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—81no—
193C2×Dic53—81no—
194C2×Dic53—81no—
195D212—42no—
196D212—81no—
197#88322—42no—
198#88322—81no—
199#9278—531nomaximal, Sylow
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53#38611yescharacteristic
211C22×C53C4×SD1611yescharacteristic
212C22×C53#44011yescharacteristic
213C22×C53#44811yescharacteristic
214C22×C53#44911yescharacteristic
215C22×C53#46011yescharacteristic
216C22×C53#46411yescharacteristic
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—161no—
221C2×D53—161no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—42no—
227C4×C53—42no—
228C4×C53—42no—
229C4×C53—42no—
230C4×C53—81no—
231C4×C53—81no—
232Dic53—81no—
233Dic53—161no—
234#3187—531no—
235#2368D5311yescharacteristic, complemented
236#1964—531no—
237#1520—531no—
238C2×#327—531no—
239#1294—531no—
240#1282—531no—
241C2×C53—21no—
242C2×C53—21no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—21no—
247C2×C53#188011yescharacteristic
248C2×C53#190611yescharacteristic
249C2×C53#191311yescharacteristic
250D53—161no—
251C2×C22⋊Q8C2×D5311yescharacteristic
252C2×#121—531no—
253#483—532no—
254#481—1061no—
255#465—1062no—
256#450—532no—
257#448—1061no—
258C2×#99—1061no—
259C2×#99—1061no—
260C2×C22⋊C8—531no—
261C2×C22⋊C8—531no—
262#391—21no—
263#387—21no—
264#387—21no—
265#385—21no—
266#380—531no—
267C4×C4⋊C4—21no—
268C4×C22⋊C4—21no—
269C4×C22⋊C4C2×D5311yescharacteristic
270C2×#95C2×D5311yescharacteristic
271#328—1062no—
272#327—1061no—
273#327—1061no—
274C53#927811yescharacteristic, complemented, Sylow
275C22×Q8C22⋊D5311yescharacteristic
276C22×D4—531no—
277C22×D4—1061no—
278C23×C4—21no—
279C23×C4C22×D5311yescharacteristic
280C2×SD16—1062no—
281C22×C8—1061no—
282C22×C8—1061no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8C4×D5312yes—
286C22⋊Q8D21212yes—
287#121—1061no—
288#121—1061no—
289#121—1061no—
290#121—1061no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—531no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4C22⋊D5311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—1061no—
299C2×C22⋊C4—1061no—
300C2×C22⋊C4C22⋊D5311yescharacteristic
301C2×C22⋊C4C22⋊D5311yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305#106—1062no—
306#103—1062no—
307#102—1062no—
308#99—2122no—
309#99—2122no—
310C22⋊C8—532no—
311C22⋊C8—1061no—
312C22⋊C8—1061no—
313C22⋊C8—1061no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—1061no—
321#95C22⋊D5311yescharacteristic
322#95C22⋊D5311yescharacteristic
323C24—1061no—
324C24C2×C22⋊D5311yescharacteristic
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8#1171992112yes—
328C2×D4—1061no—
329C2×D4—1061no—
330C2×D4—4241no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2121no—
335C2×D4—2121no—
336C2×D4—2121no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—1061no—
349C22×C4—2121no—
350C22×C4C2×C4×D5311yescharacteristic
351C22×C4C2×D21211yescharacteristic
352C22×C4#1171994511yescharacteristic
353C22×C4#1171994511yescharacteristic
354C22×C4C2×C22⋊D5311yescharacteristic
355C22×C4C2×C22⋊D5311yescharacteristic
356SD16—4242no—
357C2×C8—1062no—
358C2×C8—4241no—
359C2×C8—4241no—
360C2×C8—2121no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—81no—
366C4⋊C4—1061no—
367C4⋊C4—1061no—
368C4⋊C4—4241no—
369C4⋊C4#1171992112yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—2121no—
376C22⋊C4—2121no—
377C22⋊C4—2121no—
378C22⋊C4—2121no—
379C22⋊C4#1171992412yescomplemented
380C42—42no—
381C42—81no—
382C42—81no—
383C23—41no—
384C23—41no—
385C23—4241no—
386C23—4241no—
387C23—2121no—
388C23—2121no—
389C23—2121no—
390C23C4×C22⋊D5311yescharacteristic
391C23#1313318611yescharacteristic
392C23#1313319111yescharacteristic
393C23C2×#1171992411yescharacteristic
394C23#1313323711yescharacteristic
395C23#1313324211yescharacteristic
396C23#1313324811yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—4241no—
400D4—4241no—
401D4—4241no—
402D4—4241no—
403D4—4241no—
404D4—2122no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—4241no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4C2×#1171992111yescharacteristic
438C2×C4C4×C22⋊D5311yescharacteristic
439C2×C4#1313319111yescharacteristic
440C2×C4#1313320011yescharacteristic
441C8—4241no—
442C8—2122no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—8481no—
454C22—8481no—
455C22—4241no—
456C22—4241no—
457C22—4241no—
458C22#1681050211yescharacteristic
459C22C4×#1171992411yescharacteristic
460C22#1681072111yescharacteristic
461C22#1681076211yescharacteristic
462C22#1681076811yescharacteristic
463C22#1681082311yescharacteristic
464C22#1681084511yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—8481no—
476C4—4241no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—8481no—
484C2#2027507011yescharacteristic
485C2#2027521611yescharacteristic
486C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2368C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2368D53
hypocenterC53#9278
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2368

Complements

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

NH ≅ G/Ntype
3. #20273287C2semidirect
5. #20272594C2semidirect
6. #20272563C2semidirect
7. C53×#2368C2semidirect
9. #16810951C4semidirect
12. #16810773C4semidirect
123. C22⋊C4×C53SD16semidirect
235. #2368D53semidirect
274. C53#9278semidirect
379. C22⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202854682
#202755512
#202732872
C2×#168103562
#202725942
#202725632
C53×#23682
#927853