Finite Group Databasebeta

← #27477075

Subgroups — #27477075

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284347C212yesmaximal
2C2×#16810864C211yesmaximal, characteristic, complemented
3#20276220C211yesmaximal, characteristic, complemented
4#20276155C211yesmaximal, characteristic, complemented
5#20275983C211yesmaximal, characteristic, complemented
6#20275836C211yesmaximal, characteristic, complemented
7#20275370C212yesmaximal, complemented
8#20275230C212yesmaximal, complemented
9#20275181C212yesmaximal, complemented
10C2×#16810388C211yesmaximal, characteristic, complemented
11C53×#2929C211yesmaximal, characteristic
12C2×#13133247C2212yes—
13C22×Dic212C2211yescharacteristic
14C22×#11719917C2212yes—
15C2×#13133196C2212yes—
16#16810879—22no—
17#16810867—22no—
18#16810864—21no—
19#16810864—21no—
20#16810864—21no—
21#16810864—21no—
22#16810862—22no—
23#16810850—22no—
24#16810666C2212yes—
25C2×#13133151C2211yescharacteristic
26#16810631C2212yes—
27#16810624C2212yes—
28C2×#13133142C2212yes—
29C2×#13133135C2212yes—
30C2×#13133135C2212yes—
31#16810391C2212yes—
32#16810391C2212yes—
33#16810388C2211yescharacteristic
34#16810388C2212yes—
35#16810388C2211yescharacteristic
36C2×#131×C53C2211yescharacteristic
37C23×C8×C53C2211yescharacteristic
38C53×#525C2211yescharacteristic
39C53×#435—21no—
40C53×#435—21no—
41C8×D4×C53—21no—
42C8×D4×C53—21no—
43C53×#424C2211yescharacteristic
44C53×#408C2211yescharacteristic
45C53×#407C2211yescharacteristic
46C53×#405C2211yescharacteristic
47C22×Dic106C2312yes—
48#13133247—24no—
49#13133247—42no—
50C2×Dic212—21no—
51C2×Dic212—21no—
52C2×Dic212—42no—
53C2×#11719925—22no—
54C2×#11719923—22no—
55C2×#11719917—24no—
56C2×#11719917—22no—
57C2×#11719917—42no—
58C2×#11719917C2312yes—
59C2×#11719917C2312yes—
60C22×#88322—22no—
61#13133196—22no—
62#13133196—22no—
63#13133196—22no—
64#13133196—22no—
65C2×C4⋊Dic53C2312yes—
66C2×#11719919—22no—
67C2×C22⋊Dic53—22no—
68#13133151—21no—
69#13133151—21no—
70#13133150—22no—
71#13133148—22no—
72#13133142—22no—
73#13133142—22no—
74#13133141—22no—
75#13133140—22no—
76#13133140—22no—
77#13133138—22no—
78#13133135—24no—
79#13133135—42no—
80C2×#41×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82#131×C53—42no—
83C2×M16×C53—21no—
84C2×M16×C53C2311yescharacteristic
85C2×M16×C53C2311yescharacteristic
86C22×C8×C53—21no—
87C22×C8×C53—42no—
88C22×C8×C53D411yescharacteristic
89C22×C8×C53D411yescharacteristic
90C22×C8×C53D411yescharacteristic
91C22×C8×C53D411yescharacteristic
92C22×C8×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94#123×C53—21no—
95C22⋊Q8×C53D411yescharacteristic
96C22⋊Q8×C53D411yescharacteristic
97#121×C53D411yescharacteristic
98#121×C53D411yescharacteristic
99#120×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102#117×C53C2311yescharacteristic
103C4⋊C8×C53—21no—
104C4⋊C8×C53—21no—
105C22⋊C8×C53—21no—
106C22⋊C8×C53—21no—
107C22⋊C8×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C4×C8×C53—21no—
111C24×C53#4111yescharacteristic
112#41×C53—22no—
113#41×C53—22no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53C2×D411yescharacteristic
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—22no—
129C22×Dic53—22no—
130C2×Dic106—22no—
131C2×Dic106—22no—
132C2×Dic106—22no—
133C2×Dic106—22no—
134C2×Dic106—22no—
135C2×Dic106—22no—
136M16×C53—22no—
137M16×C53—22no—
138M16×C53—42no—
139C2×C8×C53—22no—
140C2×C8×C53—22no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—41no—
144C2×C8×C53—41no—
145C2×C8×C53—41no—
146C2×C8×C53—41no—
147C2×C8×C53—42no—
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53C2×D411yescharacteristic
155C4⋊C4×C53C2×D411yescharacteristic
156C22⋊C4×C53—21no—
157C22⋊C4×C53—21no—
158C22⋊C4×C53—21no—
159C22⋊C4×C53—21no—
160C22⋊C4×C53—21no—
161C42×C53—21no—
162C22⋊Dic53—42no—
163C22⋊Dic53—42no—
164#11719925—44no—
165#11719923—44no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168#11719919—42no—
169#11719919—42no—
170#11719917—22no—
171#11719917—22no—
172#11719917—22no—
173#11719917—44no—
174#11719917#4114yes—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—44no—
178Dic212—42no—
179Dic212—42no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—21no—
183C23×C53—41no—
184C23×C53#12111yescharacteristic
185C23×C53#12111yescharacteristic
186C23×C53C2×#4111yescharacteristic
187Q8×C53—22no—
188D4×C53—22no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—42no—
192C2×C4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4112yes—
209C2×C4×C53H(5,𝔽2)12yes—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—82no—
215Dic106—42no—
216Dic106—42no—
217Dic106—42no—
218Dic106—42no—
219C8×C53—21no—
220C8×C53—21no—
221C8×C53—21no—
222C8×C53—21no—
223C8×C53—41no—
224C8×C53—41no—
225#88322—24no—
226#88322—42no—
227#28591—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—44no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53#47312yes—
240C22×C53C2×#12111yescharacteristic
241C22×C53#55312yes—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—41no—
245C4×C53—41no—
246C4×C53—41no—
247C4×C53#53211yescharacteristic
248C4×C53#53411yescharacteristic
249C4×C53#54612yes—
250Dic53—82no—
251Dic53—82no—
252#3056—532no—
253C2×#467—531no—
254#2929D5311yescharacteristic
255#2089—531no—
256#2078—531no—
257#2047—531no—
258#2020—531no—
259#1930—532no—
260#1908—532no—
261#1901—532no—
262C2×#341—531no—
263C2×C53—22no—
264C2×C53—22no—
265C2×C53—41no—
266C2×C53—81no—
267C2×C53#254311yescharacteristic
268C2×C53C2×#47311yescharacteristic
269C2×C53#306711yescharacteristic
270C2×#137—532no—
271C22×Q16—531no—
272C2×#131C2×D5311yescharacteristic
273C22×M16—532no—
274C23×C8C2×D5311yescharacteristic
275#525C2×D5311yescharacteristic
276C2×C22⋊Q8—532no—
277#470—1062no—
278#468—1062no—
279#467—1061no—
280#467—1061no—
281#467—1061no—
282#467—1061no—
283#467—1062no—
284#465—1062no—
285#435—21no—
286#435—21no—
287C8×D4—21no—
288C8×D4—21no—
289#428—532no—
290C2×#107—531no—
291#424C2×D5311yescharacteristic
292#419—532no—
293#417—532no—
294C2×#103—532no—
295C2×#101—532no—
296C2×#101—532no—
297#408C2×D5311yescharacteristic
298#407C2×D5311yescharacteristic
299#405C2×D5311yescharacteristic
300#343—532no—
301#343—532no—
302#341—531no—
303#341—531no—
304#341—532no—
305C53#2859111yescharacteristic, complemented, Sylow
306C2×#41C22×D5311yescharacteristic
307C22×Q8—532no—
308C23×C4C22×D5311yescharacteristic
309#137—1064no—
310#137—2122no—
311C2×Q16—1062no—
312C2×Q16—1061no—
313C2×Q16—1061no—
314C2×Q16—2122no—
315C2×SD16—1062no—
316#131—42no—
317C2×M16—21no—
318C2×M16—532no—
319C2×M16—532no—
320C2×M16—1062no—
321C2×M16—1064no—
322C2×M16—2122no—
323C2×M16C22×D5311yescharacteristic
324C2×M16C22×D5311yescharacteristic
325C22×C8—21no—
326C22×C8—42no—
327C22×C8—1062no—
328C22×C8C22⋊D5311yescharacteristic
329C22×C8C22⋊D5311yescharacteristic
330C22×C8C22⋊D5311yescharacteristic
331C22×C8C22⋊D5311yescharacteristic
332C22×C8C22×D5311yescharacteristic
333C22×C8C22×D5311yescharacteristic
334#123—21no—
335C22⋊Q8—1062no—
336C22⋊Q8—1062no—
337C22⋊Q8—1062no—
338C22⋊Q8—1062no—
339C22⋊Q8D21211yescharacteristic
340C22⋊Q8D21211yescharacteristic
341#121D21211yescharacteristic
342#121D21211yescharacteristic
343#120—21no—
344C4×D4—21no—
345C4×D4—21no—
346#117C22×D5311yescharacteristic
347C2×C4⋊C4—532no—
348C2×C4⋊C4—1062no—
349C2×C22⋊C4—1062no—
350#107—1061no—
351#107—1061no—
352#107—1062no—
353#106—1062no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356#103—1062no—
357#103—1062no—
358#103—1062no—
359#103—1062no—
360#103—1062no—
361#102—1062no—
362#101—1064no—
363#101—2122no—
364C22⋊C8—21no—
365C22⋊C8—21no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368#97—21no—
369C4×C8—21no—
370C24#1171994511yescharacteristic
371#41—22no—
372#41—22no—
373C2×Q8—1062no—
374C2×Q8—1062no—
375C2×Q8—1062no—
376C2×Q8—1062no—
377C2×Q8—1062no—
378C2×Q8—1062no—
379C2×Q8C2×D21211yescharacteristic
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4C2×D21211yescharacteristic
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4C2×D21211yescharacteristic
390C22×C4C2×D21211yescharacteristic
391C22×C4#1171994511yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394C22×C4C23×D5311yescharacteristic
395Q16—2122no—
396Q16—2122no—
397Q16—2124no—
398SD16—2124no—
399M16—22no—
400M16—22no—
401M16—42no—
402M16—534no—
403M16—1062no—
404M16—1062no—
405M16—1062no—
406M16—2124no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—1062no—
417C2×C8—1062no—
418C2×C8—2124no—
419C2×C8C2×C22⋊D5311yescharacteristic
420C2×C8C2×C22⋊D5311yescharacteristic
421C2×C8C2×C22⋊D5311yescharacteristic
422C2×C8C2×C22⋊D5311yescharacteristic
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—1062no—
426C4⋊C4—1062no—
427C4⋊C4—2122no—
428C4⋊C4—2122no—
429C4⋊C4C2×D21211yescharacteristic
430C4⋊C4C2×D21211yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—2122no—
437C22⋊C4—2122no—
438C42—21no—
439C23—22no—
440C23—22no—
441C23—21no—
442C23—41no—
443C23#1313319111yescharacteristic
444C23#1313319111yescharacteristic
445C23C2×#1171994511yescharacteristic
446Q8—22no—
447Q8—2122no—
448Q8—2122no—
449Q8—2122no—
450Q8—2122no—
451D4—22no—
452D4—42no—
453D4—42no—
454D4—42no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—4242no—
468C2×C4—2122no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4#1313319111yescharacteristic
473C2×C4#1313319111yescharacteristic
474C2×C4C22×C22⋊D5311yescharacteristic
475C2×C4C22×D21211yescharacteristic
476C2×C4#41×D5312yes—
477C2×C4#1313326212yes—
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—41no—
483C8—41no—
484C8—1064no—
485C8—2122no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—44no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22#1681089812yes—
498C22C2×#1313319111yescharacteristic
499C22#1681132212yes—
500C4—21no—
501C4—21no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—4242no—
506C4—4242no—
507C4#1681116911yescharacteristic
508C4#1681118211yescharacteristic
509C4#1681126812yes—
510C2—22no—
511C2—22no—
512C2—41no—
513C2—81no—
514C2#2027967111yescharacteristic
515C2C2×#1681089811yescharacteristic
516C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2929C2
socleC22×C53C2×#121
prime radicalC53×#525C22
hypercenter#2929D53
hypocenterC53#28591
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. C2×#16810864C2semidirect
3. #20276220C2semidirect
4. #20276155C2semidirect
5. #20275983C2semidirect
6. #20275836C2semidirect
7. #20275370C2semidirect
8. #20275230C2semidirect
9. #20275181C2semidirect
10. C2×#16810388C2semidirect
305. C53#28591semidirect

Maximal subgroups

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

subgroupindex
#202843472
C2×#168108642
#202762202
#202761552
#202759832
#202758362
#202753702
#202752302
#202751812
C2×#168103882
C53×#29292
#2859153