Finite Group Databasebeta

← #27361279

Subgroups — #27361279

Order 13568.

992 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C211yesmaximal, characteristic, complemented
2C2×#16811002C211yesmaximal, characteristic, complemented
3#20284530C211yesmaximal, characteristic, complemented
4#20273860C214yesmaximal, complemented
5#20273849C214yesmaximal, complemented
6#20273396C211yesmaximal, characteristic, complemented
7#20273166C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3511C211yesmaximal, characteristic
10C22×#11719925—22no—
11C2×#13133217C2211yescharacteristic
12#16811019—21no—
13#16811002—22no—
14#16811002C2214yes—
15#16810969—21no—
16#16810939—22no—
17#16810918C2212yes—
18#16810913C2212yes—
19#16810913C2214yes—
20#16810908—21no—
21#16810903—21no—
22#16810676C2211yescharacteristic
23#16810666—21no—
24#16810647C2211yescharacteristic
25#16810624—21no—
26C2×C4×#88322C2211yescharacteristic
27#16810372C2214yes—
28Q8⋊#88322C2212yes—
29D4⋊#88322C2214yes—
30D4⋊#88322C2212yes—
31C53×#556—21no—
32C53×#555C2212yes—
33C53×#554C2212yes—
34C53×#553—22no—
35C53×#548—22no—
36C53×#547—21no—
37C53×#542—22no—
38C53×#537—21no—
39C53×#533—21no—
40C53×#532—21no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#520—21no—
44C4×#41×C53C2211yescharacteristic
45C22×Dic106—21no—
46C2×#11719925—28no—
47C2×#11719925—48no—
48C2×#11719917—21no—
49C22×#88322—21no—
50#13133217—21no—
51#13133217—21no—
52#13133217—21no—
53#13133217C2312yes—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56#13133150—22no—
57#13133148—24no—
58#13133148—22no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133138—22no—
64#13133138—24no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322—21no—
68C4×#88322C2312yes—
69C2×#41×C53—22no—
70C2×#41×C53—21no—
71C2×#41×C53D411yescharacteristic
72C22×Q8×C53D412yes—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53D411yescharacteristic
76C4⋊Q8×C53D412yes—
77C4⋊Q8×C53C2312yes—
78#127×C53D411yescharacteristic
79#126×C53—24no—
80#125×C53—22no—
81#124×C53—22no—
82#124×C53—21no—
83#124×C53—24no—
84#123×C53—24no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—24no—
89#121×C53—22no—
90#121×C53—24no—
91#121×C53—22no—
92#121×C53—22no—
93#121×C53—21no—
94#121×C53D412yes—
95C4×Q8×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53D411yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C4×D4×C53—22no—
100C4×D4×C53—24no—
101C4×D4×C53—21no—
102C4×D4×C53—22no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53D411yescharacteristic
106C4×D4×C53C2311yescharacteristic
107C4×D4×C53C2312yes—
108#117×C53—22no—
109#117×C53—22no—
110#117×C53—21no—
111#117×C53D411yescharacteristic
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#41×C53—22no—
118#41×C53—22no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—84no—
122C2×Q8×C53—21no—
123C2×Q8×C53—22no—
124C2×Q8×C53—44no—
125C2×Q8×C53C2×D411yescharacteristic
126C2×Q8×C53C2×D414yes—
127C2×D4×C53—21no—
128C2×D4×C53—22no—
129C2×D4×C53—24no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C2×D412yes—
133C2×D4×C53C2×D411yescharacteristic
134C22×C4×C53—24no—
135C22×C4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×Dic53—41no—
144C2×Dic106—22no—
145C2×Dic106—44no—
146C2×Dic106—41no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×D411yescharacteristic
157C4⋊C4×C53C2×D411yescharacteristic
158C4⋊C4×C53C2×D412yes—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—21no—
163C22⋊C4×C53—21no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—21no—
166C22⋊C4×C53—22no—
167C42×C53—24no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53C2×D411yescharacteristic
172C42×C53C2×D411yescharacteristic
173C42×C53C2×D411yescharacteristic
174C42×C53C2411yescharacteristic
175#11719925—416no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—41no—
179C4⋊Dic53—41no—
180#11719917—42no—
181C2×#88322—24no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53#12011yescharacteristic
188Q8×C53—21no—
189Q8×C53—21no—
190Q8×C53—44no—
191Q8×C53—42no—
192Q8×C53—44no—
193D4×C53—21no—
194D4×C53—21no—
195D4×C53—24no—
196D4×C53—42no—
197D4×C53—44no—
198D4×C53—42no—
199D4×C53—42no—
200D4×C53—84no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—24no—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—82no—
214C2×C4×C53—84no—
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#13512yes—
219C2×C4×C53#13712yes—
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×Dic53—42no—
224C2×Dic53—82no—
225Dic106—44no—
226Dic106—82no—
227#88322—24no—
228#88322—42no—
229#88322—42no—
230#18627—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—41no—
234C22×C53—42no—
235C22×C53—41no—
236C22×C53—42no—
237C22×C53—41no—
238C22×C53—42no—
239C22×C53#45212yes—
240C22×C53C2×#12011yescharacteristic
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—41no—
244C4×C53—42no—
245C4×C53—41no—
246C4×C53—41no—
247C4×C53—44no—
248C4×C53#45212yes—
249C4×C53#45312yes—
250C4×C53C2×#13511yescharacteristic
251C4×C53C2×#13711yescharacteristic
252C4×C53#57812yes—
253Dic53—82no—
254#3511D5311yescharacteristic
255#3227—531no—
256C2×#494—531no—
257#3079—531no—
258#1634—534no—
259#1632—534no—
260#1548—531no—
261#1492—531no—
262#1476—531no—
263C2×C53—22no—
264C2×C53—81no—
265C2×C53—82no—
266C2×C53C2×#45211yescharacteristic
267C2×C53#301111yescharacteristic
268C2×C53C2×#45311yescharacteristic
269C22×Q16—1062no—
270#556—21no—
271#555C2×D5312yes—
272#554C2×D5312yes—
273#553—22no—
274#548—22no—
275#547—21no—
276#542—22no—
277#537—21no—
278#533—21no—
279#532—21no—
280#532C2×D5311yescharacteristic
281C2×C4⋊Q8—531no—
282C2×C4⋊Q8C2×D5311yescharacteristic
283#520—21no—
284C4×#41C2×D5311yescharacteristic
285#498—1061no—
286#494—534no—
287#494—1062no—
288#486—1061no—
289#481—1062no—
290#477—532no—
291#476—532no—
292#476—534no—
293#475—1061no—
294#474—1061no—
295#431—531no—
296#428—1061no—
297#423—531no—
298#417—1061no—
299C2×C4×C8—531no—
300#333—534no—
301Q8⋊C8—532no—
302D4⋊C8—534no—
303D4⋊C8—532no—
304C53#1862711yescharacteristic, complemented, Sylow
305C2×#41—22no—
306C2×#41—21no—
307C2×#41C22⋊D5311yescharacteristic
308C22×Q8—1061no—
309C22×Q8C22⋊D5312yes—
310C2×Q16—1068no—
311C2×Q16—2128no—
312C2×M16—1061no—
313C22×C8—1061no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—532no—
317C4⋊Q8—1061no—
318C4⋊Q8—1061no—
319C4⋊Q8—1061no—
320C4⋊Q8C22⋊D5312yes—
321C4⋊Q8C22⋊D5311yescharacteristic
322C4⋊Q8C22×D5312yes—
323#127C22⋊D5311yescharacteristic
324#126—24no—
325#125—22no—
326#124—22no—
327#124—24no—
328#124—21no—
329#123—24no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334#121—22no—
335#121—24no—
336#121—22no—
337#121—22no—
338#121—21no—
339#121C22⋊D5312yes—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8C22⋊D5311yescharacteristic
343C4×Q8C22×D5311yescharacteristic
344C4×D4—22no—
345C4×D4—24no—
346C4×D4—22no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4C22⋊D5311yescharacteristic
351C4×D4C22×D5311yescharacteristic
352C4×D4C22×D5312yes—
353#117—22no—
354#117—22no—
355#117—21no—
356#117C22⋊D5311yescharacteristic
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C42—21no—
363C2×C42C22×D5311yescharacteristic
364#107—1062no—
365#106—1062no—
366#106—1064no—
367C4⋊C8—532no—
368C4⋊C8—532no—
369C4⋊C8—1061no—
370#103—1062no—
371#102—1062no—
372#102—1064no—
373C22⋊C8—1061no—
374C4×C8—532no—
375C4×C8—1061no—
376C4×C8—1061no—
377#41—22no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41—84no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—1062no—
386C2×Q8—2124no—
387C2×Q8—2121no—
388C2×Q8C2×C22⋊D5311yescharacteristic
389C2×Q8C2×C22⋊D5314yes—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×C22⋊D5312yes—
396C2×D4C2×C22⋊D5311yescharacteristic
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—2121no—
404C22×C4C2×C22⋊D5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407Q16—21216no—
408M16—2122no—
409C2×C8—1064no—
410C2×C8—1062no—
411C2×C8—1062no—
412C2×C8—1062no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—1062no—
423C4⋊C4—1062no—
424C4⋊C4—2121no—
425C4⋊C4—2121no—
426C4⋊C4C2×C22⋊D5311yescharacteristic
427C4⋊C4C2×C22⋊D5312yes—
428C4⋊C4C2×C22⋊D5311yescharacteristic
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—22no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42C2×C22⋊D5311yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C42C2×C22⋊D5311yescharacteristic
444C42C23×D5311yescharacteristic
445C23—21no—
446C23—22no—
447C23#1313318611yescharacteristic
448Q8—21no—
449Q8—21no—
450Q8—44no—
451Q8—42no—
452Q8—44no—
453Q8—4242no—
454Q8—2124no—
455D4—21no—
456D4—21no—
457D4—24no—
458D4—42no—
459D4—44no—
460D4—42no—
461D4—42no—
462D4—84no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—4242no—
478C2×C4—2122no—
479C2×C4#1313318611yescharacteristic
480C2×C4#1313318611yescharacteristic
481C2×C4#1313318611yescharacteristic
482C2×C4#1313323712yes—
483C2×C4#1313324712yes—
484C2×C4C22×C22⋊D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4C22×C22⋊D5311yescharacteristic
487C8—1064no—
488C8—2122no—
489C8—2122no—
490C22—21no—
491C22—21no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—41no—
497C22—42no—
498C22#1681078512yes—
499C22C2×#1313318611yescharacteristic
500C4—21no—
501C4—21no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—41no—
506C4—44no—
507C4—4242no—
508C4#1681078512yes—
509C4#1681079112yes—
510C4C2×#1313323711yescharacteristic
511C4C2×#1313324711yescharacteristic
512C4#1681143812yes—
513C2—22no—
514C2—81no—
515C2—82no—
516C2C2×#1681078511yescharacteristic
517C2#2028401011yescharacteristic
518C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3511C2
socleC22×C53C2×#120
hypercenter#3511D53
hypocenterC53#18627
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. C2×#16811002C2semidirect
3. #20284530C2semidirect
4. #20273860C2semidirect
5. #20273849C2semidirect
6. #20273396C2semidirect
7. #20273166C2semidirect
8. #20273097C2semidirect
304. C53#18627semidirect

Maximal subgroups

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

subgroupindex
#202858402
C2×#168110022
#202845302
#202738602
#202738492
#202733962
#202731662
#202730972
C53×#35112
#1862753