Finite Group Databasebeta

← #27541087

Subgroups — #27541087

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286458C211yesmaximal, characteristic, complemented
2#20276954C211yesmaximal, characteristic, complemented
3#20276917C211yesmaximal, characteristic, complemented
4#20276895C211yesmaximal, characteristic, complemented
5#20276864C211yesmaximal, characteristic, complemented
6#20276826C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276770C211yesmaximal, characteristic, complemented
9#20276636C211yesmaximal, characteristic, complemented
10#20276631C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276425C211yesmaximal, characteristic, complemented
14C2×#16810426C211yesmaximal, characteristic, complemented
15C53×#3334C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×Dic212C2211yescharacteristic
18#16811282—21no—
19C2×#13133161C2211yescharacteristic
20C2×#13133161C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22C2×#13133157C2211yescharacteristic
23C22×#11719908C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810846—21no—
28#16810787—21no—
29#16810764—21no—
30C4×Dic212C2211yescharacteristic
31#16810448C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810426C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47#16810399C2211yescharacteristic
48C53×#547—21no—
49C53×#500C2211yescharacteristic
50C53×#480—21no—
51C53×#480—21no—
52C53×#474—21no—
53C53×#466C2211yescharacteristic
54C53×#466C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C8×D4×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×#11719945—21no—
60C22×Dic106—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212—21no—
64C2×Dic212—22no—
65C2×Dic212—41no—
66C2×Dic212C2311yescharacteristic
67C2×Dic212C2311yescharacteristic
68C2×#11719913—21no—
69C2×#11719913—21no—
70C2×D424C2311yescharacteristic
71#13133205—21no—
72#13133203—41no—
73#13133196—21no—
74#13133195—41no—
75#13133191—21no—
76#13133189—41no—
77C4×Dic106—21no—
78#13133177—41no—
79#13133161—22no—
80#13133161—41no—
81#13133158—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133142—21no—
91#13133139—21no—
92C2×#41×C53—41no—
93C2×D8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C22×C8×C53C2311yescharacteristic
96C4⋊Q8×C53—21no—
97#123×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—41no—
100#121×C53—21no—
101#121×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53D411yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53—21no—
108#107×C53—21no—
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#102×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119#41×C53—84no—
120C2×Q8×C53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53D811yescharacteristic
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C22×Dic53—41no—
133#11719945—42no—
134#11719945—81no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×Dic106—21no—
138C2×Dic106—21no—
139C2×Dic106—42no—
140C2×Dic106—41no—
141C2×Dic106—81no—
142D8×C53—21no—
143D8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2411yescharacteristic
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53D811yescharacteristic
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53D811yescharacteristic
163C22⋊C4×C53D811yescharacteristic
164C42×C53C2×D411yescharacteristic
165C22⋊Dic53—41no—
166#11719921—41no—
167#11719921—41no—
168#11719921—41no—
169C4⋊Dic53—21no—
170#11719919—41no—
171#11719919—41no—
172C4×Dic53—41no—
173Dic212—21no—
174Dic212—21no—
175Dic212—21no—
176Dic212—42no—
177Dic212#4112yescomplemented
178D424—21no—
179D424—21no—
180#11719913—42no—
181#11719913—81no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190Q8×C53—82no—
191D4×C53—22no—
192D4×C53—41no—
193D4×C53—41no—
194D4×C53—82no—
195D4×C53—82no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—82no—
204C2×C4×C53—82no—
205C2×C4×C53—82no—
206C2×C4×C53C2×D811yescharacteristic
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C22⋊D53—82no—
212C22⋊D53—82no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—82no—
217D212—41no—
218D212—41no—
219C4×D53—82no—
220Dic106—22no—
221Dic106—41no—
222Dic106—41no—
223Dic106—41no—
224Dic106—82no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53—21no—
228C8×C53C2×#4111yescharacteristic
229C8×C53H(5,𝔽2)11yescharacteristic
230#33875—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53#50912yes—
240C22×C53C22×D811yescharacteristic
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53—81no—
251C4×C53#54611yescharacteristic
252C4×C53#57711yescharacteristic
253Dic53—42no—
254Dic53—81no—
255Dic53—81no—
256#3334D5311yescharacteristic, complemented
257#3285—531no—
258#2243—531no—
259#2236—531no—
260#2232—531no—
261#2227—531no—
262#2220—531no—
263#2215—531no—
264#2210—531no—
265#2186—531no—
266#2185—531no—
267#2175—531no—
268#2168—531no—
269#2139—531no—
270C2×#358—531no—
271C2×C53—22no—
272C2×C53—41no—
273C2×C53—161no—
274C2×C53#329411yescharacteristic
275C2×C53C2×#50911yescharacteristic
276C2×C53#341711yescharacteristic
277D53—161no—
278C2×#135—531no—
279C22×Q16—531no—
280#548—1061no—
281#547—21no—
282C2×SD32—531no—
283C2×SD32—531no—
284C2×D16—531no—
285C2×M32—531no—
286C22×C16—531no—
287#500C2×D5311yescharacteristic
288#495—531no—
289#480—21no—
290#480—21no—
291#474—21no—
292#467—531no—
293#466—531no—
294#466C2×D5311yescharacteristic
295#466C2×D5311yescharacteristic
296#464—1061no—
297#452—1061no—
298#448—1061no—
299C4×Q16—531no—
300C4×D8C2×D5311yescharacteristic
301C8×D4C2×D5311yescharacteristic
302C2×#107C2×D5311yescharacteristic
303C2×#107C2×D5311yescharacteristic
304#366—531no—
305#366—531no—
306#366—531no—
307C4⋊C16—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#358—531no—
312#358—531no—
313#358—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318C22⋊C16—531no—
319C22⋊C16—531no—
320#346—531no—
321C53#3387511yescharacteristic, complemented, Sylow
322C2×#41—41no—
323C2×#41—1061no—
324C22×Q8—1061no—
325#135—1062no—
326#135—2121no—
327C2×Q16—531no—
328C2×Q16—531no—
329C2×Q16—1062no—
330C2×Q16—1061no—
331C2×Q16—2121no—
332C2×SD16—1061no—
333C2×SD16—1061no—
334C2×D8—531no—
335C2×D8C22×D5311yescharacteristic
336C22×C8C22×D5311yescharacteristic
337C22×C8C22×D5311yescharacteristic
338C4⋊Q8—21no—
339#124—1061no—
340#124—2121no—
341#123—41no—
342#123—41no—
343C22⋊Q8—41no—
344C22⋊Q8—1061no—
345C22⋊Q8—2121no—
346#121—21no—
347#121—21no—
348#121—1061no—
349#121—2121no—
350C4×Q8—1061no—
351C4×D4—21no—
352C4×D4—2121no—
353C4×D4C22⋊D5311yescharacteristic
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356SD32—1062no—
357SD32—2121no—
358D16—2121no—
359M32—2121no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—531no—
363C2×C16—531no—
364C2×C16—2121no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—531no—
369#107C22×D5311yescharacteristic
370#107C22×D5311yescharacteristic
371#107C22×D5311yescharacteristic
372C4⋊C8C22⋊D5311yescharacteristic
373#103—1061no—
374#103—1061no—
375#102—21no—
376#102—21no—
377#102—21no—
378#102—1061no—
379C22⋊C8C22⋊D5311yescharacteristic
380C22⋊C8C22⋊D5311yescharacteristic
381C4×C8C22×D5311yescharacteristic
382#41—84no—
383#41—4241no—
384#41—2122no—
385C2×Q8—41no—
386C2×Q8—1061no—
387C2×Q8—1061no—
388C2×Q8—4241no—
389C2×Q8—2122no—
390C2×Q8—2121no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—1061no—
395C2×D4—2121no—
396C2×D4—2121no—
397C2×D4#1171992211yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C2×C22⋊D5311yescharacteristic
405Q16—532no—
406Q16—1061no—
407Q16—1061no—
408Q16—1061no—
409Q16—2122no—
410SD16—4241no—
411SD16—2122no—
412D8—21no—
413D8—21no—
414D8—1061no—
415D8—1061no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D5311yescharacteristic
421C2×C8C2×C22⋊D5311yescharacteristic
422C2×C8C2×C22⋊D5311yescharacteristic
423C2×C8C23×D5311yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1061no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C4⋊C4#1171992211yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C22⋊C4#1171992211yescharacteristic
442C22⋊C4#1171992211yescharacteristic
443C42—2121no—
444C42C2×C22⋊D5311yescharacteristic
445C16—1061no—
446C16—1061no—
447C16—1061no—
448C16—1061no—
449C23—41no—
450C23—2121no—
451C23C2×#1171992211yescharacteristic
452C23C2×#1171992211yescharacteristic
453Q8—82no—
454Q8—1062no—
455Q8—4242no—
456Q8—2121no—
457Q8—2121no—
458Q8—2121no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—4242no—
465D4—4242no—
466D4—2121no—
467D4—2121no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4C2×#1171992211yescharacteristic
484C2×C4C2×#1171992211yescharacteristic
485C2×C4C2×#1171992211yescharacteristic
486C2×C4C2×#1171992211yescharacteristic
487C2×C4C22×C22⋊D5311yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8#41×D5311yescharacteristic
492C8#1313326311yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4241no—
502C22—4241no—
503C22—4241no—
504C22#1681105912yes—
505C22C22×#1171992211yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—4241no—
514C4—4241no—
515C4—2122no—
516C4#1681126111yescharacteristic
517C4#1681143111yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—8481no—
522C2#2028652211yescharacteristic
523C2C2×#1681105911yescharacteristic
524C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3334C2
socleC22×C53C22×D8
hypercenter#3334D53
hypocenterC53#33875
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #20286458C2semidirect
2. #20276954C2semidirect
3. #20276917C2semidirect
4. #20276895C2semidirect
5. #20276864C2semidirect
6. #20276826C2semidirect
7. #20276799C2semidirect
8. #20276770C2semidirect
9. #20276636C2semidirect
10. #20276631C2semidirect
11. #20276579C2semidirect
12. #20276541C2semidirect
13. #20276425C2semidirect
14. C2×#16810426C2semidirect
15. C53×#3334C2semidirect
177. Dic212#41semidirect
256. #3334D53semidirect
321. C53#33875semidirect

Maximal subgroups

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

subgroupindex
#202864582
#202769542
#202769172
#202768952
#202768642
#202768262
#202767992
#202767702
#202766362
#202766312
#202765792
#202765412
#202764252
C2×#168104262
C53×#33342
#3387553