Finite Group Databasebeta

← #27434489

Subgroups — #27434489

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285111C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic
3#20283458C211yesmaximal, characteristic
4#20276156C212yesmaximal, complemented
5#20276015C212yesmaximal, complemented
6#20275705C211yesmaximal, characteristic, complemented
7#20275491C211yesmaximal, characteristic, complemented
8#20275233C212yesmaximal, complemented
9#20274780C212yesmaximal, complemented
10C4×#13133136C211yesmaximal, characteristic, complemented
11C2×C53×#477C211yesmaximal, characteristic, complemented
12C2×#13133248C2212yes—
13C2×#13133223C2212yes—
14#16811173C2211yescharacteristic
15#16810985—22no—
16#16810975—21no—
17#16810971—21no—
18#16810845—22no—
19#16810833—21no—
20#16810823—21no—
21#16810700—21no—
22#16810693—22no—
23#16810688—21no—
24C2×#13133153C2212yes—
25#16810647C2211yescharacteristic
26#16810622C2212yes—
27#16810622C2212yes—
28#16810622C2212yes—
29C2×#13133136C2211yescharacteristic
30C2×#13133136C2211yescharacteristic
31#16810589C2212yes—
32C4×#11719917C2211yescharacteristic
33C4×#11719917C2211yescharacteristic
34#16810392C2212yes—
35#16810392C2212yes—
36#16810392C2211yescharacteristic
37#16810392C2211yescharacteristic
38#16810388C2212yes—
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×C4×Q8×C53C2211yescharacteristic
41C53×#477—21no—
42C53×#477—22no—
43C53×#477—21no—
44C2×#107×C53C2212yes—
45C2×C4⋊C8×C53C2211yescharacteristic
46C2×#103×C53C2212yes—
47C2×#11719945C2312yes—
48#13133248—24no—
49#13133248—42no—
50C2×#11719925—22no—
51C2×#11719924—22no—
52#13133223—24no—
53#13133223—42no—
54C2×#11719917C2311yescharacteristic
55C2×#11719917C2311yescharacteristic
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2312yes—
58C2×#11719917C2311yescharacteristic
59C2×#11719912—22no—
60C2×C8×D53—22no—
61#13133217D411yescharacteristic
62#13133214D411yescharacteristic
63#13133205D412yes—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133153—42no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71#13133140—22no—
72#13133140—22no—
73#13133140—22no—
74#13133137—22no—
75#13133137—22no—
76#13133137—22no—
77#13133136—22no—
78#13133136—22no—
79#13133129—22no—
80#13133129—22no—
81#13133128—21no—
82#13133128—21no—
83C4×#88322—21no—
84C4×#88322—21no—
85C22×Q8×C53—21no—
86C22×Q8×C53C2311yescharacteristic
87C22×C8×C53C2312yes—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#107×C53—22no—
103#107×C53—22no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D412yes—
106C4⋊C8×C53D411yescharacteristic
107#103×C53—22no—
108#103×C53—22no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—41no—
115C2×Q8×C53—41no—
116C2×Q8×C53—82no—
117C2×Q8×C53#4112yes—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53C2411yescharacteristic
124C2×C22⋊D53—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212C2×D412yes—
132C2×C4×D53—22no—
133C2×Dic106C2×D412yes—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C42×C53—41no—
147C42×C53—41no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152#11719925—44no—
153#11719924—44no—
154#11719921—22no—
155#11719921—22no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158#11719917—21no—
159#11719917—21no—
160#11719917—21no—
161#11719917—21no—
162#11719917—22no—
163#11719917—21no—
164#11719917—22no—
165#11719917—21no—
166#11719917—21no—
167#11719917#4112yes—
168C2×#88322—21no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—21no—
172C2×#88322—22no—
173#11719912—44no—
174C8×D53—44no—
175C23×C53C2×#4111yescharacteristic
176C22×D53—22no—
177Q8×C53—22no—
178Q8×C53—22no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—82no—
182Q8×C53—82no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—81no—
191C2×C4×C53—82no—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200C2×C4×C53H(5,𝔽2)11yescharacteristic
201C2×C4×C532−1+411yescharacteristic
202C22⋊D53—42no—
203C22⋊D53—42no—
204C2×Dic53—22no—
205D212—22no—
206D212—22no—
207D212—42no—
208D212—81no—
209C4×D53—42no—
210C4×D53—42no—
211Dic106—22no—
212Dic106—22no—
213C8×C53—22no—
214C8×C53—22no—
215#88322—22no—
216#88322—41no—
217#88322—42no—
218#88322—41no—
219#88322—41no—
220#25360—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53C2×#12111yescharacteristic
224C22×C53#52911yescharacteristic
225C22×C53#54211yescharacteristic
226C2×D53—42no—
227C2×D53—42no—
228C2×D53—42no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—42no—
233C4×C53—81no—
234C4×C53—82no—
235C4×C53#47312yes—
236C4×C53#54611yescharacteristic
237C4×C53#54611yescharacteristic
238C4×C53#54711yescharacteristic
239C4×C53#54811yescharacteristic
240Dic53—42no—
241#3141—531no—
242C2×#477D5311yescharacteristic, complemented
243#3037—531no—
244#2935—531no—
245#2078—532no—
246#2052—532no—
247#1997—531no—
248#1950—531no—
249#1908—532no—
250#1816—532no—
251C4×#101—531no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53#242711yescharacteristic
255C2×C53C2×#47311yescharacteristic
256C2×C53#306811yescharacteristic
257D53—82no—
258C2×#137—532no—
259C2×#131—532no—
260#532—531no—
261C2×C4⋊Q8C2×D5311yescharacteristic
262C2×C4×Q8C2×D5311yescharacteristic
263#489—1062no—
264#487—1061no—
265#486—1061no—
266#477—21no—
267#477—22no—
268#477—21no—
269#464—1062no—
270#462—1061no—
271#460—1061no—
272#436—1061no—
273#435—1062no—
274C8×D4—1061no—
275C2×#108—532no—
276C2×#107C2×D5312yes—
277#423—531no—
278C2×C4⋊C8C2×D5311yescharacteristic
279#417—532no—
280#417—532no—
281#417—532no—
282C2×#103C2×D5312yes—
283C2×#101—531no—
284C2×#101—531no—
285#408—532no—
286C4×M16—531no—
287C4×M16—531no—
288#343—532no—
289#343—531no—
290#343—532no—
291#343—531no—
292#341—532no—
293C53#2536011yescharacteristic, complemented, Sylow
294C2×#41—532no—
295C22×Q8—21no—
296C22×Q8C22×D5311yescharacteristic
297#137—1064no—
298#137—2122no—
299C2×Q16—1062no—
300C2×SD16—1062no—
301#131—1064no—
302#131—2122no—
303C2×M16—531no—
304C2×M16—532no—
305C2×M16—531no—
306C2×M16—531no—
307C2×M16—531no—
308C2×M16—1062no—
309C22×C8—1062no—
310C22×C8C22×D5312yes—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—531no—
316#127—531no—
317#124—532no—
318#121—1061no—
319#121—1061no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×D4—1061no—
325C4×D4—1061no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4C22×D5311yescharacteristic
329C2×C4⋊C4C22×D5312yes—
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#108—2122no—
333#107—22no—
334#107—22no—
335C4⋊C8—1061no—
336C4⋊C8—1061no—
337C4⋊C8C22⋊D5311yescharacteristic
338C4⋊C8C22⋊D5312yes—
339C4⋊C8C22⋊D5311yescharacteristic
340#103—22no—
341#103—22no—
342#103—1062no—
343#103—1062no—
344#103—1062no—
345#102—1062no—
346#102—1062no—
347#102—1062no—
348#101—1062no—
349#101—1062no—
350C22⋊C8—1062no—
351C22⋊C8—1062no—
352#97—1061no—
353#97—1061no—
354C4×C8—1061no—
355C4×C8—1061no—
356#41—1062no—
357#41—1062no—
358#41—1062no—
359#41—1062no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—82no—
368C2×Q8—532no—
369C2×Q8#1171994912yes—
370C2×D4—532no—
371C2×D4—1062no—
372C2×D4—1061no—
373C2×D4—1061no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—1062no—
378C22×C4#1171994711yescharacteristic
379C22×C4#1171994711yescharacteristic
380C22×C4C23×D5311yescharacteristic
381Q16—2124no—
382SD16—2124no—
383M16—532no—
384M16—1061no—
385M16—1061no—
386M16—1061no—
387M16—1061no—
388M16—1062no—
389M16—1061no—
390M16—1062no—
391M16—1061no—
392M16—1061no—
393M16—2124no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1061no—
397C2×C8—1061no—
398C2×C8—1061no—
399C2×C8—1062no—
400C2×C8—1061no—
401C2×C8—2124no—
402C2×C8C2×C22⋊D5312yes—
403C2×C8C2×C22⋊D5312yes—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C22⋊C4—1062no—
415C22⋊C4—1062no—
416C42—41no—
417C42—41no—
418C42D4×D5311yescharacteristic
419C42D4×D5311yescharacteristic
420C42C2×C22⋊D5311yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C23—1062no—
423C23C2×#1171994711yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—82no—
429Q8—82no—
430Q8—1062no—
431Q8—1062no—
432D4—1062no—
433D4—1062no—
434D4—4241no—
435D4—2122no—
436D4—2122no—
437D4—2122no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—1062no—
448C2×C4—2122no—
449C2×C4—2122no—
450C2×C4#1313318811yescharacteristic
451C2×C4#1313318811yescharacteristic
452C2×C4#1313318811yescharacteristic
453C2×C4#1313318811yescharacteristic
454C2×C4C2×D4×D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C2×C4#41×D5311yescharacteristic
457C2×C4C2×#1171994911yescharacteristic
458C2×C4#1313326311yescharacteristic
459C2×C4#1313326511yescharacteristic
460C8—22no—
461C8—22no—
462C8—1062no—
463C8—2121no—
464C8—2122no—
465C8—2121no—
466C8—2121no—
467C22—21no—
468C22—21no—
469C22—2122no—
470C22—2122no—
471C22—2122no—
472C22C2×#1313318811yescharacteristic
473C22#1681115911yescharacteristic
474C22#1681124611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—81no—
480C4—82no—
481C4—2122no—
482C4#1681089512yes—
483C4#1681126111yescharacteristic
484C4#1681126411yescharacteristic
485C4#1681127511yescharacteristic
486C4#1681128411yescharacteristic
487C2—21no—
488C2—21no—
489C2—4242no—
490C2#2027828811yescharacteristic
491C2C2×#1681089511yescharacteristic
492C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#477C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#477D53
hypocenterC53#25360
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
4. #20276156C2semidirect
5. #20276015C2semidirect
6. #20275705C2semidirect
7. #20275491C2semidirect
8. #20275233C2semidirect
9. #20274780C2semidirect
10. C4×#13133136C2semidirect
11. C2×C53×#477C2semidirect
242. C2×#477D53semidirect
293. C53#25360semidirect

Maximal subgroups

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

subgroupindex
#202851112
#202841872
#202834582
#202761562
#202760152
#202757052
#202754912
#202752332
#202747802
C4×#131331362
C2×C53×#4772
#2536053