Finite Group Databasebeta

← #27276058

Subgroups — #27276058

Order 13568.

1020 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274725C211yesmaximal, characteristic, complemented
2#20273364C211yesmaximal, characteristic, complemented
3#20273275C211yesmaximal, characteristic, complemented
4#20273275C211yesmaximal, characteristic, complemented
5#20273114C211yesmaximal, characteristic, complemented
6#20273106C211yesmaximal, characteristic, complemented
7C53×#3496C211yesmaximal, characteristic
8#16810676C2211yescharacteristic, complemented
9#16810676C2211yescharacteristic, complemented
10C2×C4⋊#88322C2211yescharacteristic, complemented
11C2×#13133138—21no—
12#16810489C2211yescharacteristic, complemented
13#16810385—21no—
14#16810369—21no—
15#16810367—21no—
16#16810357—22no—
17#16810357—22no—
18D4⋊#88322—22no—
19D4⋊#88322—22no—
20C2×H(5,𝔽2)×C53—21no—
21C53×#547—21no—
22C53×#547C412yescomplemented
23C53×#546—22no—
24C53×#546—22no—
25D42×C53—21no—
26D42×C53C412yescomplemented
27C53×#537—21no—
28C53×#534—22no—
29C53×#532C2211yescharacteristic
30C53×#532C2211yescharacteristic
31C2×#121×C53—22no—
32C2×C4×D4×C53C2211yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39#13133138—42no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C4×#88322—21no—
43C4×#88322—21no—
44#13133124—21no—
45#13133124—21no—
46H(5,𝔽2)×C53—42no—
47H(5,𝔽2)×C53—42no—
48H(5,𝔽2)×C53—42no—
49H(5,𝔽2)×C53—42no—
50C2×#41×C53—22no—
51C2×#41×C53—22no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C22×D4×C53—24no—
55C22×D4×C53—22no—
56C22×D4×C53—22no—
57C22×D4×C53—22no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C4⋊Q8×C53C2×C411yescharacteristic
61C4⋊Q8×C53D411yescharacteristic
62#127×C53C2×C411yescharacteristic
63#127×C53D411yescharacteristic
64#124×C53—21no—
65#124×C53—21no—
66#123×C53—24no—
67#123×C53—24no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70#121×C53—24no—
71#121×C53—24no—
72#121×C53—22no—
73#121×C53—22no—
74#121×C53—22no—
75#121×C53—22no—
76#121×C53—21no—
77#121×C53—21no—
78#121×C53D412yes—
79#121×C53D412yes—
80#120×C53—24no—
81#120×C53—24no—
82C4×D4×C53—21no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53C2×C412yes—
87C4×D4×C53C2×C412yes—
88C4×D4×C53D412yes—
89C4×D4×C53D412yes—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53D411yescharacteristic
92C2×C22⋊C4×C53—24no—
93C2×C22⋊C4×C53—21no—
94C2×C42×C53C2311yescharacteristic
95C24×C53—21no—
96C24×C53—24no—
97#41×C53—44no—
98#41×C53—44no—
99#41×C53—44no—
100#41×C53—44no—
101#41×C53—41no—
102#41×C53—41no—
103#41×C53—41no—
104#41×C53—41no—
105#41×C53—41no—
106#41×C53—41no—
107#41×C53—41no—
108#41×C53—41no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×D4×C53—24no—
112C2×D4×C53—24no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—44no—
124C2×D4×C53—44no—
125C2×D4×C53—44no—
126C2×D4×C53—48no—
127C2×D4×C53—48no—
128C2×D4×C53—44no—
129C2×D4×C53—84no—
130C2×D4×C53C22⋊C412yes—
131C2×D4×C53C22⋊C412yes—
132C2×D4×C53C22⋊C412yes—
133C2×D4×C53D812yes—
134C2×D4×C53SD1612yes—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—82no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×Dic53—41no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C22⋊C412yes—
151C4⋊C4×C53C22⋊C412yes—
152C4⋊C4×C53C22⋊C412yes—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C42×C53C22×C411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53C2×D411yescharacteristic
164C42×C53C2×D411yescharacteristic
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167C2×#88322—41no—
168C2×#88322—41no—
169C2×#88322—41no—
170C2×#88322—41no—
171C2×#88322—41no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—48no—
178C23×C53—81no—
179C23×C53—81no—
180C23×C53—84no—
181C23×C53—84no—
182C23×C53#12011yescharacteristic
183Q8×C53—41no—
184Q8×C53—41no—
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—22no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—48no—
194D4×C53—48no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—81no—
204D4×C53—84no—
205D4×C53#10214yes—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—44no—
212C2×C4×C53—48no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—82no—
218C2×C4×C53—81no—
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53#12011yescharacteristic
223C2×C4×C53#12011yescharacteristic
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53C2×D811yescharacteristic
226C2×C4×C53C2×SD1611yescharacteristic
227C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
228C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
229C2×Dic53—81no—
230C2×Dic53—81no—
231C2×Dic53—81no—
232#88322—81no—
233#88322—81no—
234#88322—81no—
235#8428—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—24no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—82no—
248C22×C53—82no—
249C22×C53—168no—
250C22×C53#37911yescharacteristic
251C22×C53#44711yescharacteristic
252C22×C53#45011yescharacteristic
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—42no—
260C4×C53C2×#10211yescharacteristic
261C4×C53#41811yescharacteristic
262C4×C53#44711yescharacteristic
263C4×C53#45011yescharacteristic
264Dic53—161no—
265#3496D5311yescharacteristic
266#1804—531no—
267#1540—531no—
268#1517—531no—
269#1517—531no—
270#1480—531no—
271#1478—531no—
272C2×C53—21no—
273C2×C53—21no—
274C2×C53—44no—
275C2×C53—82no—
276C2×C53—82no—
277C2×C53#178611yescharacteristic
278C2×C53#179411yescharacteristic
279C2×C53#179511yescharacteristic
280C2×H(5,𝔽2)—21no—
281#547—21no—
282#547Dic5312yescomplemented
283#546—22no—
284#546—22no—
285D42—21no—
286D42Dic5312yescomplemented
287#537—21no—
288#534—22no—
289#532C2×D5311yescharacteristic
290#532C2×D5311yescharacteristic
291C2×#121—22no—
292C2×C4×D4C2×D5311yescharacteristic
293#431—531no—
294#431—531no—
295C2×C4⋊C8—531no—
296C2×#102—1061no—
297#382—531no—
298#340—1061no—
299#332—1061no—
300#331—1061no—
301#327—1062no—
302#327—1062no—
303D4⋊C8—1062no—
304D4⋊C8—1062no—
305C53#842811yescharacteristic, complemented, Sylow
306H(5,𝔽2)—42no—
307H(5,𝔽2)—42no—
308H(5,𝔽2)—42no—
309H(5,𝔽2)—42no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—21no—
313C2×#41—21no—
314C22×D4—24no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—22no—
318C22×D4C22⋊D5311yescharacteristic
319C23×C4—21no—
320C22×C8—1061no—
321C4⋊Q8C2×Dic5311yescharacteristic
322C4⋊Q8C22⋊D5311yescharacteristic
323#127C2×Dic5311yescharacteristic
324#127C22⋊D5311yescharacteristic
325#124—21no—
326#124—21no—
327#123—24no—
328#123—24no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331#121—24no—
332#121—24no—
333#121—22no—
334#121—22no—
335#121—21no—
336#121—21no—
337#121—22no—
338#121—22no—
339#121C22⋊D5312yes—
340#121C22⋊D5312yes—
341#120—24no—
342#120—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4C2×Dic5312yes—
348C4×D4C2×Dic5312yes—
349C4×D4C22⋊D5312yes—
350C4×D4C22⋊D5312yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4C22⋊D5311yescharacteristic
354C2×C22⋊C4—24no—
355C2×C22⋊C4—21no—
356C2×C42C22×D5311yescharacteristic
357C4⋊C8—1061no—
358C4⋊C8—1061no—
359C4⋊C8—1061no—
360C4⋊C8—1061no—
361#102—2122no—
362C22⋊C8—1061no—
363C22⋊C8—1061no—
364C4×C8—1061no—
365C4×C8—1061no—
366#95—1061no—
367#95—1061no—
368C24—21no—
369C24—24no—
370#41—44no—
371#41—44no—
372#41—44no—
373#41—44no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—84no—
403C2×D4#1171992212yes—
404C2×D4#1171992312yes—
405C2×D4C22⋊Dic5312yes—
406C2×D4C22⋊Dic5312yes—
407C2×D4C22⋊Dic5312yes—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—82no—
415C22×C4—2121no—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C2×C8—2121no—
420C2×C8—2121no—
421C2×C8—2121no—
422C2×C8—2121no—
423C2×C8—2121no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—2121no—
429C4⋊C4—2121no—
430C4⋊C4C22⋊Dic5312yes—
431C4⋊C4C22⋊Dic5312yes—
432C4⋊C4C22⋊Dic5312yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C42C22×Dic5311yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C42C2×C22⋊D5311yescharacteristic
444C42C2×C22⋊D5311yescharacteristic
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—48no—
451C23—81no—
452C23—81no—
453C23—84no—
454C23—84no—
455C23#1313318611yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—48no—
467D4—48no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—81no—
477D4—84no—
478D4#1313313814yes—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—4241no—
493C2×C4—4241no—
494C2×C4—4241no—
495C2×C4C2×C22⋊Dic5311yescharacteristic
496C2×C4C2×C22⋊Dic5311yescharacteristic
497C2×C4C2×C22⋊Dic5311yescharacteristic
498C2×C4#1313318611yescharacteristic
499C2×C4#1313318611yescharacteristic
500C2×C4#1313318611yescharacteristic
501C2×C4C2×#1171992211yescharacteristic
502C2×C4C2×#1171992311yescharacteristic
503C2×C4#1313324311yescharacteristic
504C2×C4#1313324311yescharacteristic
505C8—4241no—
506C8—4241no—
507C8—4241no—
508C22—21no—
509C22—21no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—82no—
521C22—168no—
522C22#1681047911yescharacteristic
523C22#1681075711yescharacteristic
524C22#1681077411yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—8481no—
533C4C2×#1313313811yescharacteristic
534C4#1681062811yescharacteristic
535C4#1681075711yescharacteristic
536C4#1681077411yescharacteristic
537C2—21no—
538C2—21no—
539C2—44no—
540C2—82no—
541C2—82no—
542C2#2027464611yescharacteristic
543C2#2027467911yescharacteristic
544C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3496C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3496D53
hypocenterC53#8428
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #20274725C2semidirect
2. #20273364C2semidirect
3. #20273275C2semidirect
4. #20273275C2semidirect
5. #20273114C2semidirect
6. #20273106C2semidirect
8. #16810676C22semidirect
9. #16810676C22semidirect
10. C2×C4⋊#88322C22semidirect
12. #16810489C22semidirect
22. C53×#547C4semidirect
26. D42×C53C4semidirect
282. #547Dic53semidirect
286. D42Dic53semidirect
305. C53#8428semidirect

Maximal subgroups

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

subgroupindex
#202747252
#202733642
#202732752
#202731142
#202731062
C53×#34962
#842853