Finite Group Databasebeta

← #27301653

Subgroups — #27301653

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283198C211yesmaximal, characteristic
2#20276410C211yesmaximal, characteristic, complemented
3#20276369C211yesmaximal, characteristic, complemented
4#20274690C211yesmaximal, characteristic
5#20272844C211yesmaximal, characteristic, complemented
6#20272838C211yesmaximal, characteristic, complemented
7C53×#3011C211yesmaximal, characteristic
8#16811390—21no—
9#16810640—21no—
10#16810632—21no—
11#16810607—22no—
12#16810607C2211yescharacteristic
13#16810607C2211yescharacteristic
14#16810607C2211yescharacteristic
15C2×#13133136—22no—
16C2×#13133136—21no—
17#16810597C2211yescharacteristic
18#16810422—22no—
19#16810413—22no—
20C2×2−1+4×C53C2211yescharacteristic
21C2×#137×C53C411yescharacteristic, complemented
22C22×Q16×C53C411yescharacteristic, complemented
23C53×#536C2211yescharacteristic
24C53×#452—21no—
25C53×#452—21no—
26C53×#451—21no—
27C53×#451—21no—
28C53×#419C411yescharacteristic, complemented
29C53×#416C411yescharacteristic, complemented
30C53×#408C2211yescharacteristic
31#13133224—42no—
32#13133224—42no—
33C2×#11719917—24no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36C2×#11719917—21no—
37#13133176—21no—
38#13133144—42no—
39#13133141—42no—
40#13133136—24no—
41#13133136—21no—
42#13133136—21no—
43#13133136D412yes—
44#13133134—21no—
45#13133134—21no—
46#13133134D412yes—
47#13133132—21no—
48#13133132—21no—
492−1+4×C53—42no—
502−1+4×C53—42no—
51C2×#41×C53—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C2×#41×C53C2×C411yescharacteristic
55C2×#41×C53D411yescharacteristic
56C2×#41×C53D411yescharacteristic
57C2×#41×C53C2311yescharacteristic
58C22×Q8×C53—21no—
59C22×Q8×C53C2×C411yescharacteristic
60C22×Q8×C53C2×C411yescharacteristic
61C22×Q8×C53D411yescharacteristic
62C22×Q8×C53D411yescharacteristic
63#137×C53—42no—
64C2×Q16×C53—21no—
65C2×Q16×C53—21no—
66C2×Q16×C53—21no—
67C2×Q16×C53—42no—
68C2×SD16×C53—21no—
69C2×M16×C53C2×C411yescharacteristic
70C22×C8×C53C2×C411yescharacteristic
71C4⋊Q8×C53—21no—
72#124×C53—21no—
73#123×C53—21no—
74#123×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53—21no—
77#117×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84#41×C53—22no—
85#41×C53—22no—
86#41×C53—22no—
87#41×C53—22no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—44no—
93#41×C53—44no—
94#41×C53—44no—
95C2×Q8×C53—21no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53—22no—
99C2×Q8×C53—22no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—41no—
103C2×Q8×C53—41no—
104C2×Q8×C53—41no—
105C2×Q8×C53—41no—
106C2×Q8×C53—44no—
107C2×Q8×C53—41no—
108C2×Q8×C53—41no—
109C2×Q8×C53C22⋊C411yescharacteristic
110C2×Q8×C53C22⋊C411yescharacteristic
111C2×Q8×C53C22⋊C411yescharacteristic
112C2×Q8×C53C2×D411yescharacteristic
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53C22⋊C411yescharacteristic
117C2×D4×C53C2×D411yescharacteristic
118C2×D4×C53#4111yescharacteristic
119C2×D4×C53#4111yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130Q16×C53—42no—
131Q16×C53—42no—
132Q16×C53—42no—
133SD16×C53—42no—
134M16×C53—41no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—41no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C42×C53—21no—
145C22⋊Dic53—41no—
146C4⋊Dic53—41no—
147C4×Dic53—41no—
148#11719917—22no—
149#11719917—22no—
150#11719917—44no—
151#11719917—41no—
152#11719917—41no—
153#11719917—41no—
154#11719917—41no—
155C2×#88322—44no—
156C2×#88322—41no—
157C2×#88322—41no—
158C2×#88322—41no—
159C23×C53—21no—
160C23×C53C4×D411yescharacteristic
161C23×C53#12111yescharacteristic
162C23×C53#12311yescharacteristic
163Q8×C53—21no—
164Q8×C53—21no—
165Q8×C53—21no—
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—41no—
169Q8×C53—42no—
170Q8×C53—42no—
171Q8×C53—42no—
172Q8×C53—41no—
173Q8×C53—41no—
174Q8×C53—81no—
175Q8×C53#9912yes—
176D4×C53—24no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53#9912yes—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—81no—
202C2×C4×C53C2×C22⋊C411yescharacteristic
203C2×C4×C53C4×D411yescharacteristic
204C2×C4×C53#12011yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×Dic53—81no—
207C8×C53—41no—
208C8×C53—41no—
209#88322—42no—
210#88322—42no—
211#88322—81no—
212#13096—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—22no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—81no—
221C22×C53#38611yescharacteristic
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—22no—
225C4×C53—21no—
226C4×C53—41no—
227C4×C53—41no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53C2×#9911yescharacteristic
231C4×C53C2×#9911yescharacteristic
232Dic53—161no—
233#3011D5311yescharacteristic
234#2889—531no—
235#2135—531no—
236#2123—531no—
237#1796—531no—
238#1395—531no—
239#1392—531no—
240C2×C53—21no—
241C2×C53—41no—
242C2×C53—41no—
243C2×C53—41no—
244C2×C53—42no—
245C2×C53#189811yescharacteristic
246C2×2−1+4C2×D5311yescharacteristic
247C2×#137Dic5311yescharacteristic, complemented
248C22×Q16Dic5311yescharacteristic, complemented
249#568—1061no—
250#536C2×D5311yescharacteristic
251#452—21no—
252#452—21no—
253#451—21no—
254#451—21no—
255#421—1061no—
256#419—1061no—
257#419Dic5311yescharacteristic, complemented
258#416Dic5311yescharacteristic, complemented
259#413—531no—
260#413—531no—
261#413—531no—
262#413—1062no—
263C2×#101—1062no—
264C2×#101—1061no—
265#410—531no—
266#408C2×D5311yescharacteristic
267#356—1062no—
268#352—1062no—
269C53#1309611yescharacteristic, complemented, Sylow
2702−1+4—42no—
2712−1+4—42no—
272C2×#41—21no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41C2×Dic5311yescharacteristic
276C2×#41C22⋊D5311yescharacteristic
277C2×#41C22⋊D5311yescharacteristic
278C2×#41C22×D5311yescharacteristic
279C22×Q8—21no—
280C22×Q8C2×Dic5311yescharacteristic
281C22×Q8C2×Dic5311yescharacteristic
282C22×Q8C22⋊D5311yescharacteristic
283C22×Q8C22⋊D5311yescharacteristic
284#137—42no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16—21no—
288C2×Q16—42no—
289C2×SD16—21no—
290#131—2122no—
291#131—2122no—
292C2×M16—1064no—
293C2×M16—1061no—
294C2×M16—1061no—
295C2×M16—1061no—
296C2×M16C2×Dic5311yescharacteristic
297C22×C8C2×Dic5311yescharacteristic
298C4⋊Q8—21no—
299#124—21no—
300#123—21no—
301#123—21no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#117—1061no—
305#117C2×Dic5311yescharacteristic
306#104—2122no—
307#103—21no—
308#103—21no—
309#103—21no—
310#103—2122no—
311#102—21no—
312#101—532no—
313#101—1064no—
314#101—1061no—
315#101—1061no—
316#100—532no—
317#100—1061no—
318#100—1061no—
319#99—1061no—
320#99—1061no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8C22⋊Dic5311yescharacteristic
349C2×Q8C22⋊Dic5311yescharacteristic
350C2×Q8C22⋊Dic5311yescharacteristic
351C2×Q8D4×D5311yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C22⋊Dic5311yescharacteristic
356C2×D4D4×D5311yescharacteristic
357C2×D4#1171994711yescharacteristic
358C2×D4#1171994711yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4C22×Dic5311yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368C22×C4C2×C22⋊D5311yescharacteristic
369Q16—42no—
370Q16—42no—
371Q16—42no—
372SD16—42no—
373M16—41no—
374M16—1062no—
375M16—1062no—
376M16—2121no—
377M16—2121no—
378M16—2124no—
379M16—2121no—
380M16—2121no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—41no—
384C2×C8—2121no—
385C2×C8—2124no—
386C2×C8—2121no—
387C2×C8—2121no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—2121no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—2121no—
396C42—21no—
397C42—2121no—
398C23—21no—
399C23D4×Dic5311yescharacteristic
400C23#1313318911yescharacteristic
401C23#1313319811yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—21no—
405Q8—21no—
406Q8—21no—
407Q8—41no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8#1313313212yes—
415D4—24no—
416D4—21no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—42no—
422D4#1313313212yes—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—4241no—
442C2×C4C2×C22⋊Dic5311yescharacteristic
443C2×C4D4×Dic5311yescharacteristic
444C2×C4#1313318511yescharacteristic
445C2×C4#1313318911yescharacteristic
446C8—41no—
447C8—41no—
448C8—4241no—
449C8—2122no—
450C8—2122no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22#1681050311yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—81no—
467C4—81no—
468C4—8481no—
469C4C2×#1313313211yescharacteristic
470C4C2×#1313313211yescharacteristic
471C2—21no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2—42no—
476C2#2027516911yescharacteristic
subgroupG/H
centerC2#20275169
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3011C2
socleC2×C53#1898
prime radicalC2×2−1+4×C53C22
cosocleC2×#41×C53C23
hypercenter#3011D53
hypocenterC53#13096
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3011

Complements

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

NH ≅ G/Ntype
2. #20276410C2semidirect
3. #20276369C2semidirect
5. #20272844C2semidirect
6. #20272838C2semidirect
21. C2×#137×C53C4semidirect
22. C22×Q16×C53C4semidirect
28. C53×#419C4semidirect
29. C53×#416C4semidirect
247. C2×#137Dic53semidirect
248. C22×Q16Dic53semidirect
257. #419Dic53semidirect
258. #416Dic53semidirect
269. C53#13096semidirect

Maximal subgroups

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

subgroupindex
#202831982
#202764102
#202763692
#202746902
#202728442
#202728382
C53×#30112
#1309653