Finite Group Databasebeta

← #27469558

Subgroups — #27469558

Order 13568.

898 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283066C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20276193C211yesmaximal, characteristic, complemented
4#20276192C211yesmaximal, characteristic, complemented
5#20276191C211yesmaximal, characteristic, complemented
6#20275674C212yesmaximal, complemented
7#20274909C212yesmaximal
8#20274906C212yesmaximal
9#20274870C212yesmaximal, complemented
10#20274472C211yesmaximal, characteristic
11C53×#2084C211yesmaximal, characteristic, complemented
12C2×#13133216C2211yescharacteristic
13C2×#13133195—21no—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53C2212yes—
17C2×C42×D53C2211yescharacteristic
18C2×#13133149C2212yes—
19C2×#13133148C2212yes—
20C2×#13133148C2212yes—
21C2×#13133129C2212yes—
22C2×C8×Dic53C2212yes—
23#16810527—21no—
24#16810521C2211yescharacteristic
25#16810518—21no—
26#16810493C2211yescharacteristic
27#16810492—21no—
28#16810492—21no—
29#16810492C2212yes—
30#16810492C2211yescharacteristic
31#16810482—21no—
32#16810480C2211yescharacteristic
33#16810480C2211yescharacteristic
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38C2×#106×C53C2212yes—
39C53×#389C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#340C2212yes—
42C53×#336C2211yescharacteristic
43C22×Dic106—21no—
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49#13133216—24no—
50#13133195—44no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C4⋊C4×D53—42no—
64C4⋊C4×D53—42no—
65C4⋊C4×D53—44no—
66C4⋊C4×D53—44no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C42×D53—24no—
73#13133149—42no—
74#13133148—42no—
75#13133148—42no—
76#13133129Q818yescomplemented
77C8×Dic53—24no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C42×C53C2311yescharacteristic
89#106×C53—42no—
90#95×C53—21no—
91C23×D53C2×D411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—22no—
101C22×Dic53C2×D411yescharacteristic
102C22×Dic53C2×D411yescharacteristic
103C22×Dic53C2×D411yescharacteristic
104C2×C4×D53—22no—
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—22no—
111C2×C4×D53—44no—
112C2×C4×D53—44no—
113C2×C4×D53—44no—
114C2×C4×D53SD1614yes—
115C2×C4×D53SD1614yes—
116C2×C4×D53C2×Q814yes—
117C2×Dic106—42no—
118C2×Dic106—42no—
119C2×Dic106—42no—
120C2×C8×C53—22no—
121C2×C8×C53—22no—
122C2×C8×C53C2×Q814yes—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—42no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C42×C53—22no—
132#11719921—42no—
133#11719921—42no—
134C4⋊Dic53—22no—
135C4⋊Dic53—21no—
136C4⋊Dic53—22no—
137C4⋊Dic53—21no—
138C4⋊Dic53—22no—
139C4⋊Dic53—42no—
140C4⋊Dic53—42no—
141#11719919—44no—
142#11719919—42no—
143#11719919—42no—
144#11719919—44no—
145#11719919—42no—
146#11719919—44no—
147#11719919—42no—
148C4×Dic53—22no—
149C4×Dic53—22no—
150C4×Dic53#4114yes—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322C2×Q814yes—
154C23×C53C22×D411yescharacteristic
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53C4⋊Q812yes—
160C22×D53C2×SD1612yes—
161C22×D53C2×SD1612yes—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53C2×SD1612yes—
176C2×C4×C53C2×SD1612yes—
177C2×C4×C53C22×Q812yes—
178C2×C4×C53C2×#4112yes—
179C2×C4×C53H(5,𝔽2)11yescharacteristic
180C2×C4×C53H(5,𝔽2)11yescharacteristic
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—41no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53#12312yes—
193C2×Dic53#12312yes—
194C2×Dic53C4⋊Q812yes—
195C2×Dic53C2×SD1612yes—
196C2×Dic53C2×SD1612yes—
197C2×Dic53AGL(1,ℤ/8)12yes—
198C2×Dic53#13712yes—
199C4×D53—24no—
200C4×D53—24no—
201C4×D53—28no—
202C4×D53—28no—
203C4×D53—88no—
204Dic106—84no—
205C8×C53—24no—
206#88322—24no—
207#27948—531nomaximal, Sylow
208C22×C53C2×#12311yescharacteristic
209C22×C53C2×C4⋊Q811yescharacteristic
210C22×C53#53511yescharacteristic
211C22×C53C22×SD1611yescharacteristic
212C22×C53C22×SD1611yescharacteristic
213C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C53C2×#13711yescharacteristic
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53#49814yes—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—81no—
225C4×C53—81no—
226C4×C53—82no—
227C4×C53#55412yes—
228C4×C53#55412yes—
229Dic53—22no—
230Dic53—81no—
231Dic53—81no—
232Dic53—82no—
233Dic53#48114yes—
234Dic53#48314yes—
235#2857—531no—
236C2×#384—531no—
237#2084—531no—
238#2084—531no—
239#2084—531no—
240#2084D5311yescharacteristic, complemented
241#1990—532no—
242#1853—532no—
243#1853—532no—
244#1844—532no—
245C4×C22⋊C8—531no—
246C2×C53#250211yescharacteristic
247C2×C53C2×#48311yescharacteristic
248C2×C53C2×#48111yescharacteristic
249C2×C53C2×#49811yescharacteristic
250C2×C53#316511yescharacteristic
251C2×C53#320611yescharacteristic
252C2×C53#320811yescharacteristic
253D53—24no—
254C2×C4⋊Q8—531no—
255C2×C22⋊Q8—1061no—
256C22×C4⋊C4—532no—
257C22×C4⋊C4—1061no—
258C22×C4⋊C4—1061no—
259C22×C42—531no—
260C2×#106—532no—
261C2×#106—532no—
262C2×#106—532no—
263C2×#106C2×D5312yes—
264C2×C22⋊C8—532no—
265C2×C4×C8—532no—
266#391—1061no—
267#389—531no—
268#389—1061no—
269#389C2×D5311yescharacteristic
270#384—531no—
271#384—532no—
272#384—531no—
273#384—1061no—
274#384—1061no—
275#384C2×D5311yescharacteristic
276#380—531no—
277#380—531no—
278#380—1061no—
279#340—532no—
280#340—532no—
281#340—532no—
282#340C2×D5312yes—
283#336—531no—
284#336C2×D5311yescharacteristic
285C53#2794811yescharacteristic, complemented, Sylow
286C22×Q8—1061no—
287C23×C4—531no—
288C23×C4—531no—
289C23×C4—531no—
290C23×C4—1062no—
291C22×C8—532no—
292C22×C8C22×D5312yes—
293C4⋊Q8—1064no—
294C22⋊Q8—2124no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—532no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—1062no—
302C2×C4⋊C4—1062no—
303C2×C4⋊C4—1062no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—2122no—
311C2×C4⋊C4—2122no—
312C2×C4⋊C4—2124no—
313C2×C4⋊C4—2124no—
314C2×C4⋊C4C22×D5312yes—
315C2×C4⋊C4C22×D5311yescharacteristic
316C2×C4⋊C4C22×D5311yescharacteristic
317C2×C22⋊C4—1061no—
318C2×C22⋊C4—1061no—
319C2×C42—531no—
320C2×C42—531no—
321C2×C42—531no—
322C2×C42—1064no—
323C2×C42C22×D5311yescharacteristic
324#106—42no—
325#106—2122no—
326#106—2122no—
327#106—2122no—
328C22⋊C8—538no—
329C4×C8—1064no—
330#95—21no—
331#95—1061no—
332#95—1061no—
333#95—1061no—
334C24—531no—
335C2×Q8—2122no—
336C2×Q8—2122no—
337C2×Q8—2122no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—531no—
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—534no—
345C22×C4—534no—
346C22×C4—534no—
347C22×C4—1062no—
348C22×C4—1062no—
349C22×C4—1062no—
350C22×C4—1062no—
351C22×C4—1062no—
352C22×C4—1061no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—1062no—
357C22×C4—2124no—
358C22×C4—2124no—
359C22×C4—2124no—
360C22×C4D4×D5311yescharacteristic
361C22×C4D4×D5311yescharacteristic
362C22×C4C23×D5311yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—534no—
366C2×C8—1062no—
367C2×C8—1062no—
368C2×C8Q8×D5314yes—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—1062no—
378C4⋊C4—1061no—
379C4⋊C4—1062no—
380C4⋊C4—1061no—
381C4⋊C4—1062no—
382C4⋊C4—2122no—
383C4⋊C4—2124no—
384C4⋊C4—2124no—
385C4⋊C4—2122no—
386C4⋊C4—2122no—
387C4⋊C4—2122no—
388C4⋊C4—2124no—
389C4⋊C4—2122no—
390C4⋊C4—2122no—
391C22⋊C4—2122no—
392C22⋊C4—2122no—
393C42—22no—
394C42—534no—
395C42—1062no—
396C42—1062no—
397C23—532no—
398C23—532no—
399C23—532no—
400C23—1061no—
401C23—1061no—
402C23—1061no—
403C23—1061no—
404C23C2×D4×D5311yescharacteristic
405Q8—4244no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—532no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—1068no—
427C2×C4—1068no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1064no—
431C2×C4—1064no—
432C2×C4—4248no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2122no—
441C2×C4—2121no—
442C2×C4SD16×D5312yes—
443C2×C4SD16×D5312yes—
444C2×C4C2×Q8×D5312yes—
445C2×C4#41×D5312yes—
446C2×C4#1313326211yescharacteristic
447C2×C4#1313326311yescharacteristic
448C8—24no—
449C8—1064no—
450C22—534no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—1062no—
457C22#123×D5311yescharacteristic
458C22C4⋊Q8×D5311yescharacteristic
459C22#1681118911yescharacteristic
460C22C2×SD16×D5311yescharacteristic
461C22C2×SD16×D5311yescharacteristic
462C22AGL(1,ℤ/8)×D5311yescharacteristic
463C22#137×D5311yescharacteristic
464C4—22no—
465C4—22no—
466C4—81no—
467C4—81no—
468C4—82no—
469C4—534no—
470C4—534no—
471C4—1062no—
472C4—4241no—
473C4—4241no—
474C4—4242no—
475C4#1681132712yes—
476C4#1681132912yes—
477C2—1064no—
478C2#2027918811yescharacteristic
479C2#483×D5311yescharacteristic
480C2#481×D5311yescharacteristic
481C2#498×D5311yescharacteristic
482C2#2028526011yescharacteristic
483C2#2028563911yescharacteristic
484C2#2028565511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2084C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2084D53
hypocenterC53#27948
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
3. #20276193C2semidirect
4. #20276192C2semidirect
5. #20276191C2semidirect
6. #20275674C2semidirect
9. #20274870C2semidirect
11. C53×#2084C2semidirect
76. #13133129Q8semidirect
240. #2084D53semidirect
285. C53#27948semidirect

Maximal subgroups

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

subgroupindex
#202830662
#384×D532
#202761932
#202761922
#202761912
#202756742
#202749092
#202749062
#202748702
#202744722
C53×#20842
#2794853