Finite Group Databasebeta

← #27361381

Subgroups — #27361381

Order 13568.

940 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#20285637C211yesmaximal, characteristic, complemented
2C2×#16810973C211yesmaximal, characteristic, complemented
3#20284530C211yesmaximal, characteristic, complemented
4#20273943C214yesmaximal, complemented
5#20273922C214yesmaximal, complemented
6#20273396C211yesmaximal, characteristic, complemented
7#20273166C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3511C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16810995—21no—
12#16810975—21no—
13#16810973—22no—
14#16810973C2214yes—
15#16810918—21no—
16#16810913—21no—
17#16810908C2212yes—
18#16810903C2212yes—
19#16810828C2214yes—
20#16810761—22no—
21#16810676C2211yescharacteristic
22#16810666—21no—
23#16810647C2211yescharacteristic
24#16810624—21no—
25C2×#13133141—22no—
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×#11719923—24no—
47C2×#11719917—21no—
48C22×#88322—21no—
49#13133217—21no—
50#13133217—21no—
51#13133217—21no—
52#13133217C2312yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55#13133150—22no—
56#13133148—22no—
57C4⋊#88322—21no—
58C4⋊#88322C2312yes—
59C4⋊#88322C2312yes—
60#13133141—28no—
61#13133141—22no—
62#13133140—24no—
63#13133138—22no—
64C22⋊#88322—21no—
65C4×#88322—21no—
66C4×#88322—21no—
67C4×#88322C2312yes—
68C2×#41×C53—22no—
69C2×#41×C53—21no—
70C2×#41×C53D411yescharacteristic
71C22×Q8×C53D412yes—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53D411yescharacteristic
75C4⋊Q8×C53D412yes—
76C4⋊Q8×C53C2312yes—
77#127×C53D411yescharacteristic
78#126×C53—24no—
79#125×C53—22no—
80#124×C53—22no—
81#124×C53—21no—
82#124×C53—24no—
83#123×C53—24no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—24no—
88#121×C53—22no—
89#121×C53—24no—
90#121×C53—22no—
91#121×C53—22no—
92#121×C53—21no—
93#121×C53D412yes—
94C4×Q8×C53—22no—
95C4×Q8×C53—22no—
96C4×Q8×C53D411yescharacteristic
97C4×Q8×C53C2311yescharacteristic
98C4×D4×C53—22no—
99C4×D4×C53—24no—
100C4×D4×C53—21no—
101C4×D4×C53—22no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53D411yescharacteristic
105C4×D4×C53C2311yescharacteristic
106C4×D4×C53C2312yes—
107#117×C53—22no—
108#117×C53—22no—
109#117×C53—21no—
110#117×C53D411yescharacteristic
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116#41×C53—22no—
117#41×C53—22no—
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—84no—
121C2×Q8×C53—21no—
122C2×Q8×C53—22no—
123C2×Q8×C53—44no—
124C2×Q8×C53C2×D411yescharacteristic
125C2×Q8×C53C2×D414yes—
126C2×D4×C53—21no—
127C2×D4×C53—22no—
128C2×D4×C53—24no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C2×D412yes—
132C2×D4×C53C2×D411yescharacteristic
133C22×C4×C53—24no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—22no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×Dic53—41no—
143C2×Dic106—22no—
144C2×Dic106—41no—
145C2×Dic106—82no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53C2×D411yescharacteristic
156C4⋊C4×C53C2×D411yescharacteristic
157C4⋊C4×C53C2×D412yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—21no—
162C22⋊C4×C53—21no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—21no—
165C22⋊C4×C53—22no—
166C42×C53—24no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53C2×D411yescharacteristic
171C42×C53C2×D411yescharacteristic
172C42×C53C2×D411yescharacteristic
173C42×C53C2411yescharacteristic
174#11719923—84no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—41no—
178C4⋊Dic53—41no—
179#11719917—42no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—42no—
184C23×C53—21no—
185C23×C53—22no—
186C23×C53#12011yescharacteristic
187Q8×C53—21no—
188Q8×C53—21no—
189Q8×C53—44no—
190Q8×C53—42no—
191Q8×C53—44no—
192D4×C53—21no—
193D4×C53—21no—
194D4×C53—24no—
195D4×C53—42no—
196D4×C53—44no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—84no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—82no—
213C2×C4×C53—84no—
214C2×C4×C53#12011yescharacteristic
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53#13512yes—
218C2×C4×C53#13712yes—
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×Dic53—42no—
223C2×Dic53—82no—
224Dic106—44no—
225Dic106—82no—
226#88322—42no—
227#88322—42no—
228#88322—42no—
229#18637—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—41no—
233C22×C53—42no—
234C22×C53—41no—
235C22×C53—42no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53#45212yes—
239C22×C53C2×#12011yescharacteristic
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—41no—
243C4×C53—42no—
244C4×C53—41no—
245C4×C53—41no—
246C4×C53—44no—
247C4×C53#45212yes—
248C4×C53#45512yes—
249C4×C53C2×#13511yescharacteristic
250C4×C53C2×#13711yescharacteristic
251C4×C53#57712yes—
252Dic53—82no—
253#3511D5311yescharacteristic
254#3206—531no—
255C2×#487—531no—
256#3079—531no—
257#1646—534no—
258#1643—534no—
259#1548—531no—
260#1492—531no—
261#1476—531no—
262C2×C53—22no—
263C2×C53—81no—
264C2×C53—82no—
265C2×C53C2×#45211yescharacteristic
266C2×C53#301011yescharacteristic
267C2×C53C2×#45511yescharacteristic
268#556—21no—
269#555C2×D5312yes—
270#554C2×D5312yes—
271#553—22no—
272#548—22no—
273#547—21no—
274#542—22no—
275#537—21no—
276#533—21no—
277#532—21no—
278#532C2×D5311yescharacteristic
279C2×C4⋊Q8—531no—
280C2×C4⋊Q8C2×D5311yescharacteristic
281#520—21no—
282C4×#41C2×D5311yescharacteristic
283#492—1061no—
284#487—534no—
285#487—1062no—
286#487—1061no—
287#477—1061no—
288#476—1061no—
289#475—532no—
290#474—532no—
291#461—534no—
292#448—1062no—
293#431—531no—
294#428—1061no—
295#423—531no—
296#417—1061no—
297C2×#103—1062no—
298C2×C4×C8—531no—
299#333—534no—
300Q8⋊C8—532no—
301D4⋊C8—534no—
302D4⋊C8—532no—
303C53#1863711yescharacteristic, complemented, Sylow
304C2×#41—22no—
305C2×#41—21no—
306C2×#41C22⋊D5311yescharacteristic
307C22×Q8—1061no—
308C22×Q8C22⋊D5312yes—
309C2×SD16—1064no—
310C2×M16—1061no—
311C22×C8—1061no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—532no—
315C4⋊Q8—1061no—
316C4⋊Q8—1061no—
317C4⋊Q8—1061no—
318C4⋊Q8C22⋊D5312yes—
319C4⋊Q8C22⋊D5311yescharacteristic
320C4⋊Q8C22×D5312yes—
321#127C22⋊D5311yescharacteristic
322#126—24no—
323#125—22no—
324#124—22no—
325#124—24no—
326#124—21no—
327#123—24no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332#121—22no—
333#121—24no—
334#121—22no—
335#121—22no—
336#121—21no—
337#121C22⋊D5312yes—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8C22⋊D5311yescharacteristic
341C4×Q8C22×D5311yescharacteristic
342C4×D4—22no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4C22⋊D5311yescharacteristic
349C4×D4C22×D5311yescharacteristic
350C4×D4C22×D5312yes—
351#117—22no—
352#117—22no—
353#117—21no—
354#117C22⋊D5311yescharacteristic
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C42—21no—
361C2×C42C22×D5311yescharacteristic
362#107—1062no—
363#106—1062no—
364C4⋊C8—532no—
365C4⋊C8—532no—
366C4⋊C8—1061no—
367#103—1068no—
368#103—1062no—
369#103—1064no—
370#102—1062no—
371C22⋊C8—1061no—
372C4×C8—532no—
373C4×C8—1061no—
374C4×C8—1061no—
375#41—22no—
376#41—22no—
377#41—42no—
378#41—42no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—44no—
383C2×Q8—1062no—
384C2×Q8—4242no—
385C2×Q8—2121no—
386C2×Q8C2×C22⋊D5311yescharacteristic
387C2×Q8C2×C22⋊D5314yes—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×C22⋊D5311yescharacteristic
394C2×D4C2×C22⋊D5312yes—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—2121no—
402C22×C4C2×C22⋊D5311yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C2×C22⋊D5311yescharacteristic
405SD16—4244no—
406M16—2122no—
407C2×C8—1062no—
408C2×C8—1062no—
409C2×C8—1062no—
410C2×C8—2122no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—1062no—
421C4⋊C4—1062no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C4⋊C4C2×C22⋊D5312yes—
425C4⋊C4C2×C22⋊D5311yescharacteristic
426C4⋊C4C2×C22⋊D5311yescharacteristic
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C42—24no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42C2×C22⋊D5311yescharacteristic
440C42C2×C22⋊D5311yescharacteristic
441C42C2×C22⋊D5311yescharacteristic
442C42C23×D5311yescharacteristic
443C23—21no—
444C23—22no—
445C23#1313318611yescharacteristic
446Q8—21no—
447Q8—21no—
448Q8—44no—
449Q8—42no—
450Q8—44no—
451Q8—4242no—
452Q8—2124no—
453D4—21no—
454D4—21no—
455D4—24no—
456D4—42no—
457D4—44no—
458D4—42no—
459D4—42no—
460D4—84no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—84no—
475C2×C4—4242no—
476C2×C4—2122no—
477C2×C4#1313318611yescharacteristic
478C2×C4#1313318611yescharacteristic
479C2×C4#1313318611yescharacteristic
480C2×C4#1313323712yes—
481C2×C4#1313324712yes—
482C2×C4C22×C22⋊D5311yescharacteristic
483C2×C4C22×C22⋊D5311yescharacteristic
484C2×C4C22×C22⋊D5311yescharacteristic
485C8—2122no—
486C8—2122no—
487C8—2122no—
488C22—21no—
489C22—21no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22#1681078512yes—
497C22C2×#1313318611yescharacteristic
498C4—21no—
499C4—21no—
500C4—41no—
501C4—42no—
502C4—41no—
503C4—41no—
504C4—44no—
505C4—4242no—
506C4#1681078512yes—
507C4#1681080112yes—
508C4C2×#1313323711yescharacteristic
509C4C2×#1313324711yescharacteristic
510C4#1681143112yes—
511C2—22no—
512C2—81no—
513C2—82no—
514C2C2×#1681078511yescharacteristic
515C2#2028400111yescharacteristic
516C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3511C2
socleC22×C53C2×#120
hypercenter#3511D53
hypocenterC53#18637
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. #20285637C2semidirect
2. C2×#16810973C2semidirect
3. #20284530C2semidirect
4. #20273943C2semidirect
5. #20273922C2semidirect
6. #20273396C2semidirect
7. #20273166C2semidirect
8. #20273097C2semidirect
303. C53#18637semidirect

Maximal subgroups

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

subgroupindex
#202856372
C2×#168109732
#202845302
#202739432
#202739222
#202733962
#202731662
#202730972
C53×#35112
#1863753