Finite Group Databasebeta

← #27491267

Subgroups — #27491267

Order 13568.

726 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#20284655C211yesmaximal, characteristic
2C2×#16810854C211yesmaximal, characteristic
3C2×#16810650C211yesmaximal, characteristic, complemented
4#20276228C212yesmaximal, complemented
5#20276082C212yesmaximal, complemented
6#20276072C212yesmaximal, complemented
7#20275975C212yesmaximal, complemented
8#20275184C211yesmaximal, characteristic, complemented
9#20275176C211yesmaximal, characteristic
10#20274751C211yesmaximal, characteristic, complemented
11C53×#3095C211yesmaximal, characteristic, complemented
12C22×#11719924C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15C2×#13133191C2211yescharacteristic
16#16810947—22no—
17#16810941—21no—
18#16810936—21no—
19#16810854—22no—
20#16810854—22no—
21C2×#13133152C2212yes—
22#16810666C2212yes—
23C2×#13133148C2211yescharacteristic
24#16810650—22no—
25#16810650C2212yes—
26#16810650C2212yes—
27C2×C4⋊#88322—22no—
28#16810627C2212yes—
29C2×#13133141C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133136C2212yes—
32C2×#13133133C2212yes—
33#16810587C2211yescharacteristic
34#16810392C2212yes—
35#16810391C2212yes—
36#16810387C2212yes—
37C2×C22⋊Q8×C53C2211yescharacteristic
38C2×#117×C53C2211yescharacteristic
39C53×#485—22no—
40C53×#484—21no—
41C53×#483—21no—
42C53×#428C2212yes—
43C53×#419C2212yes—
44C53×#407C2211yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C2×#11719924—22no—
48C2×#11719924—42no—
49C2×#11719917—21no—
50C2×#11719917—42no—
51C2×#11719917C2312yes—
52C2×#11719917C2312yes—
53C22×#88322—21no—
54C22×#88322—42no—
55C22×#88322D412yes—
56C22×#88322D412yes—
57C22×#88322C2311yescharacteristic
58C22×#88322C2311yescharacteristic
59#13133191—21no—
60#13133191—21no—
61#13133191—21no—
62#13133191—21no—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719921—21no—
65#13133152—42no—
66#13133150—22no—
67#13133148—22no—
68#13133148—22no—
69C4⋊#88322—24no—
70C4⋊#88322—24no—
71#13133141—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75#13133136—42no—
76#13133133—42no—
77C22⋊#88322—21no—
78C22⋊#88322—21no—
79C22×Q8×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C2×M16×C53C2312yes—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53—21no—
85C22⋊Q8×C53—21no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53D412yes—
89#117×C53C2312yes—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53D411yescharacteristic
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—21no—
96C2×C42×C53—21no—
97#107×C53—22no—
98#106×C53—22no—
99#103×C53—22no—
100#103×C53—22no—
101C22⋊C8×C53—21no—
102C22⋊C8×C53—21no—
103C24×C53#4111yescharacteristic
104C23×D53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53C2411yescharacteristic
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—82no—
124C22×Dic53—21no—
125C2×D212—21no—
126C2×D212—21no—
127C2×D212—22no—
128C2×D212—22no—
129M16×C53—22no—
130M16×C53—22no—
131C2×C8×C53—22no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53C2×D412yes—
139C4⋊C4×C53C2×D412yes—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C42×C53—22no—
145C42×C53—22no—
146#11719924—44no—
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151#11719917—22no—
152#11719917—22no—
153#11719917—22no—
154#11719917—22no—
155#11719917—42no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—42no—
161C2×#88322—42no—
162C2×#88322—42no—
163C2×#88322C2×D412yes—
164C2×#88322C2×D412yes—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—41no—
168C23×C53#12111yescharacteristic
169C23×C53#12111yescharacteristic
170C23×C53C2×#4111yescharacteristic
171C22×D53—41no—
172C22×D53—42no—
173C22×D53—81no—
174C22×D53—81no—
175Q8×C53—41no—
176Q8×C53—41no—
177Q8×C53—42no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—81no—
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53H(5,𝔽2)12yes—
198C2×C4×C532−1+412yes—
199C22⋊D53—82no—
200C22⋊D53—82no—
201C2×Dic53—41no—
202C2×Dic53—81no—
203D212—41no—
204D212—41no—
205D212—42no—
206C8×C53—42no—
207#88322—22no—
208#88322—22no—
209#88322—42no—
210#88322—42no—
211#29790—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—44no—
220C22×C53#47112yes—
221C22×C53C2×#12111yescharacteristic
222C22×C53#53912yes—
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—162no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—42no—
230C4×C53—42no—
231C4×C53—81no—
232C4×C53—81no—
233C4×C53#53411yescharacteristic
234C4×C53#53611yescharacteristic
235C4×C53#53712yes—
236Dic53—81no—
237#3095D5311yescharacteristic, complemented
238#3094—531no—
239C2×#465—531no—
240C2×#424—531no—
241#2090—532no—
242#2064—532no—
243#2063—532no—
244#2046—532no—
245#1901—531no—
246#1900—531no—
247#1810—531no—
248C2×C53—22no—
249C2×C53—22no—
250C2×C53—41no—
251C2×C53#261811yescharacteristic
252C2×C53C2×#47111yescharacteristic
253C2×C53#306711yescharacteristic
254D53—161no—
255C22×SD16—531no—
256C22×M16—531no—
257C23×C8—531no—
258C2×C22⋊Q8C2×D5311yescharacteristic
259C2×#121—531no—
260C2×#117C2×D5311yescharacteristic
261#485—22no—
262#484—21no—
263#483—21no—
264#482—1062no—
265#481—1061no—
266#480—1061no—
267#465—1062no—
268#465—1062no—
269C2×#108—532no—
270#428—532no—
271#428C2×D5312yes—
272C2×#106—531no—
273#424—532no—
274#424—532no—
275#424—1062no—
276C2×C4⋊C8—1062no—
277#419C2×D5312yes—
278#418—532no—
279C2×#103—531no—
280C2×#102—531no—
281C2×#101—532no—
282C2×#100—532no—
283#407—531no—
284#407C2×D5311yescharacteristic
285#343—532no—
286#343—532no—
287#341—532no—
288C53#2979011yescharacteristic, complemented, Sylow
289C22×Q8C22×D5311yescharacteristic
290C22×D4—531no—
291C22×D4—1061no—
292C23×C4C22×D5311yescharacteristic
293C2×SD16—1062no—
294C2×SD16—2122no—
295C2×M16—532no—
296C2×M16—532no—
297C2×M16—1061no—
298C2×M16—2122no—
299C2×M16C22×D5312yes—
300C22×C8—531no—
301C22×C8—531no—
302C22×C8—532no—
303C22×C8—532no—
304C22×C8—1061no—
305C22×C8—2122no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—1061no—
311#121—1061no—
312#121—1061no—
313#121—1061no—
314#117—21no—
315#117—21no—
316#117C22⋊D5312yes—
317#117C22×D5312yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—531no—
320C2×C4⋊C4C22⋊D5311yescharacteristic
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C4⋊C4C22×D5311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—1061no—
326C2×C42—21no—
327#108—2122no—
328#107—22no—
329#107—1062no—
330#106—22no—
331#106—1062no—
332#106—1062no—
333C4⋊C8—1064no—
334C4⋊C8—1064no—
335#103—22no—
336#103—22no—
337#103—1062no—
338#102—1062no—
339#102—1062no—
340#102—1062no—
341#101—2122no—
342#100—2122no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—1061no—
346C22⋊C8—1061no—
347C24—1061no—
348C24#1171994711yescharacteristic
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×D4—1061no—
354C2×D4—1061no—
355C2×D4—1062no—
356C2×D4—1062no—
357C2×D4—4242no—
358C2×D4—2121no—
359C2×D4—2121no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—1061no—
367C22×C4D4×D5311yescharacteristic
368C22×C4D4×D5311yescharacteristic
369C22×C4#1171994711yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C23×D5311yescharacteristic
373SD16—2124no—
374M16—22no—
375M16—22no—
376M16—1062no—
377M16—1062no—
378M16—1062no—
379M16—1062no—
380M16—2122no—
381C2×C8—22no—
382C2×C8—532no—
383C2×C8—532no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—1062no—
387C2×C8—1062no—
388C2×C8—2122no—
389C2×C8—2122no—
390C2×C8—2122no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4C2×C22⋊D5312yes—
400C4⋊C4C2×C22⋊D5312yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C42—22no—
408C42—22no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—4241no—
413C23—4241no—
414C23—2122no—
415C23—2121no—
416C23#1313318911yescharacteristic
417C23#1313318911yescharacteristic
418C23C2×#1171994711yescharacteristic
419Q8—41no—
420Q8—41no—
421Q8—42no—
422D4—4242no—
423D4—4242no—
424D4—2121no—
425D4—2121no—
426D4—2122no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—81no—
442C2×C4—4241no—
443C2×C4—2121no—
444C2×C4#1313318911yescharacteristic
445C2×C4#1313318911yescharacteristic
446C2×C4C2×D4×D5311yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C2×C4#1313326312yes—
449C2×C4#1313326412yes—
450C8—42no—
451C8—1062no—
452C8—1062no—
453C8—2122no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—8482no—
464C22—4241no—
465C22—4241no—
466C22—4241no—
467C22#1681088812yes—
468C22C2×#1313318911yescharacteristic
469C22#1681121712yes—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—4241no—
477C4#1681118411yescharacteristic
478C4#1681119511yescharacteristic
479C4#1681120112yes—
480C2—22no—
481C2—22no—
482C2—41no—
483C2—8481no—
484C2#2028046111yescharacteristic
485C2C2×#1681088811yescharacteristic
486C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3095C2
socleC22×C53C2×#121
prime radicalC2×#13133191C22
hypercenter#3095D53
hypocenterC53#29790
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
3. C2×#16810650C2semidirect
4. #20276228C2semidirect
5. #20276082C2semidirect
6. #20276072C2semidirect
7. #20275975C2semidirect
8. #20275184C2semidirect
10. #20274751C2semidirect
11. C53×#3095C2semidirect
237. #3095D53semidirect
288. C53#29790semidirect

Maximal subgroups

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

subgroupindex
#202846552
C2×#168108542
C2×#168106502
#202762282
#202760822
#202760722
#202759752
#202751842
#202751762
#202747512
C53×#30952
#2979053