Finite Group Databasebeta

← #27541078

Subgroups — #27541078

Order 13568.

554 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#20287118C211yesmaximal, characteristic, complemented
2#20276960C211yesmaximal, characteristic, complemented
3#20276920C211yesmaximal, characteristic, complemented
4#20276910C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276830C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276757C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276638C211yesmaximal, characteristic, complemented
11#20276589C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14#20276422C211yesmaximal, characteristic, complemented
15C53×#3283C211yesmaximal, characteristic, complemented
16#16811319—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810948—21no—
24#16810942—21no—
25#16810924—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47#16810399C2211yescharacteristic
48C2×#135×C53C2211yescharacteristic
49C2×#135×C53C2211yescharacteristic
50C53×#547—21no—
51C53×#494C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#467C2211yescharacteristic
54C53×#461—21no—
55C53×#452—21no—
56C53×#452—21no—
57C4×D8×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×D424C2311yescharacteristic
60#13133212—41no—
61#13133209—21no—
62#13133199—41no—
63#13133191—21no—
64#13133191—21no—
65#13133189—41no—
66C4×D212—21no—
67#13133177—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70#13133160—41no—
71#13133160—41no—
72#13133158—22no—
73#13133157—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133149—21no—
84#13133149—21no—
85#13133139—21no—
86#13133139—21no—
87#13133139—21no—
88C2×#41×C53—21no—
89C2×#41×C53—21no—
90#135×C53—22no—
91#135×C53—22no—
92#135×C53—41no—
93#135×C53—41no—
94C2×Q16×C53—21no—
95C2×Q16×C53C2311yescharacteristic
96C2×Q16×C53C2311yescharacteristic
97C2×SD16×C53—21no—
98C2×SD16×C53—21no—
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102C4⋊Q8×C53—21no—
103#123×C53—41no—
104#123×C53—41no—
105C22⋊Q8×C53—21no—
106C22⋊Q8×C53—21no—
107#121×C53—41no—
108C4×D4×C53—21no—
109C4×D4×C53D411yescharacteristic
110C2×C4⋊C4×C53—41no—
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—81no—
122#41×C53—81no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53D811yescharacteristic
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C2×C22⋊D53—41no—
135C2×C22⋊D53—41no—
136C22×Dic53—41no—
137C2×D212—21no—
138C2×C4×D53—41no—
139Q16×C53—21no—
140Q16×C53—21no—
141Q16×C53—21no—
142Q16×C53—21no—
143Q16×C53—81no—
144SD16×C53—42no—
145SD16×C53—42no—
146D8×C53—21no—
147D8×C53#4112yes—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2×D411yescharacteristic
154C2×C8×C53C2×D411yescharacteristic
155C2×C8×C53C2411yescharacteristic
156C4⋊C4×C53—21no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53—41no—
161C4⋊C4×C53D811yescharacteristic
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53D811yescharacteristic
166C22⋊C4×C53D811yescharacteristic
167C42×C53C2×D411yescharacteristic
168C22⋊Dic53—41no—
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179D424—21no—
180D424—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53—41no—
192Q8×C53—41no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53—82no—
197D4×C53—82no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—82no—
206C2×C4×C53—82no—
207C2×C4×C53—82no—
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C22⋊D53—82no—
214C22⋊D53—161no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—161no—
219D212—41no—
220D212—41no—
221C4×D53—161no—
222C8×C53—21no—
223C8×C53—21no—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53H(5,𝔽2)11yescharacteristic
227#33874—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—42no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53C22×D811yescharacteristic
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—81no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53#54611yescharacteristic
249C4×C53#57711yescharacteristic
250Dic53—81no—
251Dic53—81no—
252Dic53—81no—
253#3339—531no—
254#3283D5311yescharacteristic, complemented
255#2245—531no—
256#2237—531no—
257#2235—531no—
258#2228—531no—
259#2220—531no—
260#2217—531no—
261#2208—531no—
262#2186—531no—
263#2186—531no—
264#2177—531no—
265#2168—531no—
266#2139—531no—
267#2138—531no—
268C2×C53—41no—
269C2×C53—41no—
270C2×C53—82no—
271C2×C53#329411yescharacteristic
272C2×C53#341411yescharacteristic
273C2×C53#341611yescharacteristic
274D53—161no—
275C2×#135C2×D5311yescharacteristic
276C2×#135C2×D5311yescharacteristic
277#553—1061no—
278#547—21no—
279C2×SD32—531no—
280C2×SD32—531no—
281C2×D16—531no—
282C2×M32—531no—
283C22×C16—531no—
284#499—531no—
285#494C2×D5311yescharacteristic
286#482—1061no—
287#481—1061no—
288#478—1061no—
289#467C2×D5311yescharacteristic
290#467C2×D5311yescharacteristic
291#466—531no—
292#466—531no—
293#461—21no—
294#452—21no—
295#452—21no—
296C4×D8—531no—
297C4×D8C2×D5311yescharacteristic
298C8×D4C2×D5311yescharacteristic
299#427—531no—
300C2×#107—531no—
301#367—531no—
302#367—531no—
303#366—531no—
304C4⋊C16—531no—
305#358—531no—
306#358—531no—
307#358—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315C22⋊C16—531no—
316C22⋊C16—531no—
317#346—531no—
318C53#3387411yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C2×#41—21no—
321#135—22no—
322#135—22no—
323#135—41no—
324#135—41no—
325C2×Q16—21no—
326C2×Q16C22×D5311yescharacteristic
327C2×Q16C22×D5311yescharacteristic
328C2×SD16—21no—
329C2×SD16—21no—
330C2×D8—531no—
331C2×D8C22×D5311yescharacteristic
332C22×C8C22×D5311yescharacteristic
333C22×C8C22×D5311yescharacteristic
334C4⋊Q8—21no—
335#126—2121no—
336#125—1061no—
337#123—41no—
338#123—41no—
339#123—2121no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342#121—41no—
343#121—1061no—
344#121—1061no—
345#121—2121no—
346C4×D4—21no—
347C4×D4—1061no—
348C4×D4—2121no—
349C4×D4C22⋊D5311yescharacteristic
350#117—1061no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—1061no—
353SD32—2121no—
354SD32—2121no—
355D16—1062no—
356M32—2121no—
357C2×C16—531no—
358C2×C16—531no—
359C2×C16—531no—
360C2×C16—531no—
361C2×C16—2121no—
362#107—531no—
363#107—531no—
364#107—531no—
365#107—1061no—
366#107C22×D5311yescharacteristic
367#106—1061no—
368#106—1061no—
369C4⋊C8C22⋊D5311yescharacteristic
370#103—21no—
371#103—21no—
372#102—21no—
373#102—1061no—
374#102—1061no—
375#102—1061no—
376C22⋊C8C22⋊D5311yescharacteristic
377C22⋊C8C22⋊D5311yescharacteristic
378C4×C8C22×D5311yescharacteristic
379#41—42no—
380#41—42no—
381#41—81no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—1061no—
389C2×D4—2121no—
390C2×D4—2121no—
391C2×D4#1171992211yescharacteristic
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—2121no—
396C22×C4—2121no—
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399Q16—21no—
400Q16—21no—
401Q16—21no—
402Q16—21no—
403Q16—81no—
404SD16—42no—
405SD16—42no—
406D8—21no—
407D8—1061no—
408D8—1061no—
409D8#1171994712yes—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×C22⋊D5311yescharacteristic
415C2×C8C2×C22⋊D5311yescharacteristic
416C2×C8C2×C22⋊D5311yescharacteristic
417C2×C8C23×D5311yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4#1171992211yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4#1171992211yescharacteristic
438C22⋊C4#1171992211yescharacteristic
439C42—2121no—
440C42C2×C22⋊D5311yescharacteristic
441C16—1061no—
442C16—1061no—
443C16—1061no—
444C16—1061no—
445C23—41no—
446C23—2121no—
447C23C2×#1171992211yescharacteristic
448C23C2×#1171992211yescharacteristic
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453D4—22no—
454D4—41no—
455D4—41no—
456D4—82no—
457D4—82no—
458D4—8481no—
459D4—4242no—
460D4—2121no—
461D4—2121no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—8481no—
473C2×C4—8481no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4C2×#1171992211yescharacteristic
478C2×C4C2×#1171992211yescharacteristic
479C2×C4C2×#1171992211yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4C22×C22⋊D5311yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8C2×#1171994711yescharacteristic
486C8#1313326211yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—4241no—
497C22—4241no—
498C22—4241no—
499C22C22×#1171992211yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—4241no—
508C4—4241no—
509C4—4241no—
510C4#1681126711yescharacteristic
511C4#1681143411yescharacteristic
512C2—41no—
513C2—41no—
514C2—82no—
515C2—8481no—
516C2#2028652811yescharacteristic
517C2#2028784911yescharacteristic
518C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3283C2
socleC22×C53C22×D8
hypercenter#3283D53
hypocenterC53#33874
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #20287118C2semidirect
2. #20276960C2semidirect
3. #20276920C2semidirect
4. #20276910C2semidirect
5. #20276872C2semidirect
6. #20276830C2semidirect
7. #20276813C2semidirect
8. #20276757C2semidirect
9. #20276638C2semidirect
10. #20276638C2semidirect
11. #20276589C2semidirect
12. #20276541C2semidirect
13. #20276427C2semidirect
14. #20276422C2semidirect
15. C53×#3283C2semidirect
254. #3283D53semidirect
318. C53#33874semidirect

Maximal subgroups

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

subgroupindex
#202871182
#202769602
#202769202
#202769102
#202768722
#202768302
#202768132
#202767572
#202766382
#202765892
#202765412
#202764272
#202764222
C53×#32832
#3387453