Finite Group Databasebeta

← #27535103

Subgroups — #27535103

Order 13568.

768 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284663C211yesmaximal, characteristic, complemented
2#20284655C211yesmaximal, characteristic
3C2×#16810869C211yesmaximal, characteristic
4#20275618C212yesmaximal, complemented
5#20275500C212yesmaximal, complemented
6#20275377C212yesmaximal, complemented
7#20275370C212yesmaximal, complemented
8#20275183C211yesmaximal, characteristic, complemented
9#20275178C211yesmaximal, characteristic
10#20274757C211yesmaximal, characteristic, complemented
11C53×#2873C211yesmaximal, characteristic, complemented
12C22×#11719913C2211yescharacteristic
13C2×#13133196C2211yescharacteristic
14C2×#13133191C2211yescharacteristic
15#16810964—22no—
16#16810958—21no—
17#16810953—21no—
18#16810947—22no—
19#16810941—21no—
20#16810936—21no—
21#16810869—22no—
22#16810869—22no—
23C2×#13133153C2212yes—
24#16810666C2212yes—
25#16810666C2212yes—
26C2×#13133151C2211yescharacteristic
27#16810631C2212yes—
28#16810627C2212yes—
29C2×#13133142C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133135C2212yes—
32C2×#13133133C2212yes—
33#16810587C2211yescharacteristic
34#16810587C2211yescharacteristic
35#16810391C2212yes—
36#16810391C2212yes—
37#16810388C2212yes—
38C22×M16×C53C2211yescharacteristic
39C22×M16×C53C2211yescharacteristic
40C2×#117×C53C2211yescharacteristic
41C53×#405—22no—
42C53×#405C2212yes—
43C53×#405C2212yes—
44C4×M16×C53—22no—
45C22×Dic106C2311yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C2×#11719913—22no—
49C2×#11719913—24no—
50C2×#11719917C2312yes—
51C2×#11719917C2312yes—
52#13133196—21no—
53#13133196—21no—
54#13133196—21no—
55#13133196—21no—
56#13133191—21no—
57#13133191—21no—
58#13133191—21no—
59#13133191—21no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×#11719919—21no—
63C2×C22⋊Dic53—21no—
64C2×#11719921—21no—
65#13133153—24no—
66#13133151—22no—
67#13133150—22no—
68#13133150—22no—
69#13133148—22no—
70#13133148—22no—
71#13133142—22no—
72#13133140—22no—
73#13133140—22no—
74#13133139—22no—
75#13133137—22no—
76#13133137—22no—
77#13133135—24no—
78#13133133—24no—
79C22⋊#88322—21no—
80C22⋊#88322—21no—
81C22⋊#88322—21no—
82C22⋊#88322—21no—
83C23×C4×C53C2311yescharacteristic
84C2×M16×C53—24no—
85C2×M16×C53—24no—
86C2×M16×C53—21no—
87C2×M16×C53D412yes—
88C2×M16×C53D412yes—
89C2×M16×C53C2312yes—
90C22×C8×C53—21no—
91C22×C8×C53C2311yescharacteristic
92C22×C8×C53C2311yescharacteristic
93#117×C53—21no—
94#117×C53—21no—
95#117×C53D412yes—
96#117×C53C2312yes—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53—21no—
101#97×C53—22no—
102#97×C53—22no—
103C4×C8×C53—22no—
104C4×C8×C53—22no—
105C24×C53#4111yescharacteristic
106C23×D53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53C2411yescharacteristic
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—82no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C22×Dic53—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212—22no—
126C2×D212—22no—
127C2×Dic106—21no—
128C2×Dic106—21no—
129C2×Dic106—22no—
130C2×Dic106—22no—
131M16×C53—24no—
132M16×C53—22no—
133M16×C53—24no—
134M16×C53—24no—
135M16×C53#4114yes—
136C2×C8×C53—24no—
137C2×C8×C53—22no—
138C2×C8×C53—22no—
139C2×C8×C53C2×D412yes—
140C2×C8×C53C2×D412yes—
141C2×C8×C53#4114yes—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C2×D412yes—
145C4⋊C4×C53C2×D412yes—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C42×C53—22no—
149C42×C53—22no—
150C22⋊Dic53—41no—
151C22⋊Dic53—41no—
152#11719921—41no—
153#11719921—41no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158#11719919—41no—
159#11719919—41no—
160#11719917—22no—
161#11719917—22no—
162#11719917—22no—
163#11719917—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166#11719913—44no—
167C23×C53—22no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53#12111yescharacteristic
171C23×C53#12111yescharacteristic
172C23×C53C2×#4111yescharacteristic
173C22×D53—41no—
174C22×D53—42no—
175C22×D53—81no—
176C22×D53—81no—
177C2×C4×C53—22no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C2×#4112yes—
193C2×C4×C53C2×#4112yes—
194C22⋊D53—82no—
195C22⋊D53—82no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—42no—
200C2×Dic53—81no—
201C2×Dic53—81no—
202D212—41no—
203D212—41no—
204D212—42no—
205Dic106—41no—
206Dic106—41no—
207Dic106—42no—
208C8×C53—22no—
209C8×C53—24no—
210C8×C53#12314yes—
211#88322—42no—
212#88322—42no—
213#33127—531nomaximal, Sylow
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—82no—
223C22×C53C2×#12111yescharacteristic
224C22×C53#52912yes—
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—162no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—42no—
232C4×C53—42no—
233C4×C53C2×#12311yescharacteristic
234C4×C53#52511yescharacteristic
235C4×C53#53212yes—
236Dic53—81no—
237Dic53—81no—
238Dic53—81no—
239#3095—531no—
240#3094—531no—
241C2×#468—531no—
242#2873D5311yescharacteristic, complemented
243#1977—532no—
244#1952—532no—
245#1931—532no—
246#1930—532no—
247#1901—531no—
248#1900—531no—
249#1811—531no—
250C2×C53—22no—
251C2×C53—41no—
252C2×C53—41no—
253C2×C53#285411yescharacteristic
254C2×C53#306811yescharacteristic
255C2×C53#306911yescharacteristic
256D53—161no—
257C22×SD16—531no—
258C22×M16C2×D5311yescharacteristic
259C22×M16C2×D5311yescharacteristic
260C2×C22⋊Q8—531no—
261C2×#121—531no—
262C2×#117C2×D5311yescharacteristic
263#485—1062no—
264#484—1061no—
265#483—1061no—
266#482—1062no—
267#481—1061no—
268#480—1061no—
269#468—1062no—
270#468—1062no—
271C2×#108—532no—
272#428—532no—
273#428—532no—
274C2×#107—531no—
275#419—532no—
276#418—532no—
277C2×#103—531no—
278C2×#102—531no—
279C2×#101—532no—
280C2×#100—532no—
281#407—531no—
282#407—531no—
283#405—22no—
284#405C2×D5312yes—
285#405C2×D5312yes—
286C4×M16—22no—
287#343—532no—
288#343—532no—
289#341—532no—
290C53#3312711yescharacteristic, complemented, Sylow
291C22×Q8—531no—
292C22×D4—531no—
293C22×D4—1061no—
294C23×C4C22×D5311yescharacteristic
295C2×SD16—1064no—
296C2×SD16—1062no—
297C2×M16—21no—
298C2×M16—24no—
299C2×M16—24no—
300C2×M16—532no—
301C2×M16—532no—
302C2×M16C22⋊D5312yes—
303C2×M16C22⋊D5312yes—
304C2×M16C22×D5312yes—
305C22×C8—21no—
306C22×C8C22×D5311yescharacteristic
307C22×C8C22×D5311yescharacteristic
308C22⋊Q8—1061no—
309C22⋊Q8—1061no—
310C22⋊Q8—1061no—
311C22⋊Q8—1061no—
312#121—1061no—
313#121—1061no—
314#121—1061no—
315#121—1061no—
316#117—21no—
317#117—21no—
318#117D21212yes—
319#117C22×D5312yes—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—531no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4D21211yescharacteristic
324C2×C4⋊C4D21211yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—1061no—
327C2×C22⋊C4—1061no—
328C2×C42—21no—
329#108—1064no—
330#107—1062no—
331#107—1062no—
332#107—1062no—
333#106—1062no—
334#106—1062no—
335#103—1062no—
336#103—1062no—
337#103—1062no—
338#102—1062no—
339#102—1062no—
340#102—1062no—
341#101—1064no—
342#100—1064no—
343C22⋊C8—1061no—
344C22⋊C8—1061no—
345C22⋊C8—1061no—
346C22⋊C8—1061no—
347#97—22no—
348#97—22no—
349C4×C8—22no—
350C4×C8—22no—
351C24—1061no—
352C24#1171994511yescharacteristic
353C2×Q8—1061no—
354C2×Q8—1061no—
355C2×Q8—1062no—
356C2×Q8—1062no—
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1062no—
360C2×D4—1062no—
361C2×D4—4242no—
362C2×D4—2121no—
363C2×D4—2121no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4C2×D21211yescharacteristic
372C22×C4C2×D21211yescharacteristic
373C22×C4#1171994511yescharacteristic
374C22×C4C2×C22⋊D5311yescharacteristic
375C22×C4C2×C22⋊D5311yescharacteristic
376C22×C4C23×D5311yescharacteristic
377SD16—2124no—
378M16—24no—
379M16—22no—
380M16—24no—
381M16—24no—
382M16—1062no—
383M16—1062no—
384M16—1062no—
385M16—1062no—
386M16#1171994514yes—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—1062no—
391C2×C8—1062no—
392C2×C8#1171994514yes—
393C2×C8C2×C22⋊D5312yes—
394C2×C8C2×C22⋊D5312yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4—1061no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4C2×D21212yes—
404C4⋊C4C2×D21212yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—2121no—
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C42—22no—
412C42—22no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—4241no—
417C23—4241no—
418C23—2122no—
419C23—2121no—
420C23#1313319111yescharacteristic
421C23#1313319111yescharacteristic
422C23C2×#1171994511yescharacteristic
423Q8—2121no—
424Q8—2121no—
425Q8—2122no—
426D4—4242no—
427D4—4242no—
428D4—2121no—
429D4—2121no—
430D4—2122no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—4241no—
443C2×C4—4241no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2122no—
448C2×C4#1313319111yescharacteristic
449C2×C4#1313319111yescharacteristic
450C2×C4C22×C22⋊D5311yescharacteristic
451C2×C4C22×D21211yescharacteristic
452C2×C4C2×#1171994512yes—
453C2×C4C2×#1171994512yes—
454C8—22no—
455C8—24no—
456C8—2122no—
457C8—2122no—
458C8#1313320014yes—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—82no—
468C22—8482no—
469C22—4241no—
470C22—4241no—
471C22—4241no—
472C22C2×#1313319111yescharacteristic
473C22#1681115712yes—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—4241no—
479C4—4241no—
480C4—4241no—
481C4C2×#1313320011yescharacteristic
482C4#1681113411yescharacteristic
483C4#1681117312yes—
484C2—22no—
485C2—41no—
486C2—41no—
487C2—8481no—
488C2#2028304611yescharacteristic
489C2#2028444311yescharacteristic
490C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2873C2
socleC22×C53C2×#121
prime radicalC2×#13133191C22
hypercenter#2873D53
hypocenterC53#33127
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2873

Complements

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

NH ≅ G/Ntype
1. #20284663C2semidirect
4. #20275618C2semidirect
5. #20275500C2semidirect
6. #20275377C2semidirect
7. #20275370C2semidirect
8. #20275183C2semidirect
10. #20274757C2semidirect
11. C53×#2873C2semidirect
242. #2873D53semidirect
290. C53#33127semidirect

Maximal subgroups

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

subgroupindex
#202846632
#202846552
C2×#168108692
#202756182
#202755002
#202753772
#202753702
#202751832
#202751782
#202747572
C53×#28732
#3312753