Finite Group Databasebeta

← #27358131

Subgroups — #27358131

Order 13568.

1188 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285833C212yesmaximal, complemented
2#20280481C211yesmaximal, characteristic, complemented
3#20273718C214yesmaximal, complemented
4#20273701C214yesmaximal, complemented
5#20273275C212yesmaximal, complemented
6#20273029C211yesmaximal, characteristic, complemented
7C53×#3449C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic, complemented
9C2×#13133196—21no—
10#16811026—22no—
11#16810969—22no—
12#16810934C2218yescomplemented
13#16810903—24no—
14#16810676C2212yescomplemented
15#16810553—21no—
16#16810553C2212yescomplemented
17#16810548—21no—
18#16810548C2212yes—
19#16810529—21no—
20#16810489C2211yescharacteristic, complemented
21#16810357C2218yescomplemented
22#16810349C2212yescomplemented
23#16810347C2212yes—
24C22×#41×C53—21no—
25C53×#537—21no—
26C53×#537C2212yes—
27C53×#536—22no—
28C53×#535—22no—
29C53×#534—21no—
30C53×#534C2212yes—
31C53×#532—21no—
32C53×#532C2212yes—
33C53×#525—24no—
34C2×#117×C53C2211yescharacteristic
35C22×Dic106—21no—
36#13133217—22no—
37#13133217—22no—
38#13133196—44no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2312yes—
42C2×#11719919—22no—
43C2×C22⋊Dic53—22no—
44#13133150—28no—
45C4⋊#88322—22no—
46#13133138—28no—
47C22⋊#88322C2314yes—
48C4×#88322—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124—21no—
52#13133124C2312yes—
53C2×#41×C53—22no—
54C2×#41×C53—22no—
55C2×#41×C53—22no—
56C2×#41×C53—22no—
57C2×#41×C53—44no—
58C2×#41×C53—44no—
59C22×Q8×C53—21no—
60C22×D4×C53—21no—
61C22×D4×C53—22no—
62C23×C4×C53—22no—
63C23×C4×C53D411yescharacteristic
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53D412yes—
66#127×C53—21no—
67#127×C53D412yes—
68#124×C53—22no—
69#124×C53—22no—
70#123×C53—24no—
71#123×C53—24no—
72C22⋊Q8×C53—24no—
73C22⋊Q8×C53—24no—
74#121×C53—24no—
75#121×C53—24no—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53—22no—
79#121×C53C2314yes—
80#120×C53—24no—
81#120×C53—24no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53—22no—
85C4×D4×C53D414yes—
86#117×C53—22no—
87#117×C53—22no—
88C2×C4⋊C4×C53D411yescharacteristic
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53C2311yescharacteristic
92C2×C42×C53D411yescharacteristic
93C2×C42×C53C2311yescharacteristic
94C24×C53—22no—
95C24×C53C2×D411yescharacteristic
96#41×C53—44no—
97#41×C53—48no—
98#41×C53—48no—
99#41×C53—44no—
100#41×C53—44no—
101#41×C53—44no—
102C2×Q8×C53—22no—
103C2×Q8×C53—22no—
104C2×Q8×C53—44no—
105C2×D4×C53—24no—
106C2×D4×C53—22no—
107C2×D4×C53—24no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—44no—
113C2×D4×C53—44no—
114C2×D4×C53—44no—
115C2×D4×C53C2×D414yes—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—24no—
122C22×C4×C53—48no—
123C22×C4×C53D814yes—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—22no—
131C22×Dic53—22no—
132C2×Dic106—41no—
133C2×Dic106—41no—
134C2×Dic106—42no—
135C2×Dic106—42no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53C2×D414yes—
142C22⋊C4×C53—24no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—24no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53C2×D412yes—
151C42×C53C2×D412yes—
152C22⋊Dic53—44no—
153C4⋊Dic53—24no—
154C4⋊Dic53—42no—
155C4⋊Dic53—42no—
156#11719919—44no—
157C2×#88322—24no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—24no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—44no—
164C23×C53—82no—
165C23×C53—82no—
166C23×C53C2×D812yes—
167C23×C53C22×D411yescharacteristic
168Q8×C53—44no—
169Q8×C53—44no—
170D4×C53—44no—
171D4×C53—44no—
172D4×C53—48no—
173D4×C53—44no—
174D4×C53—44no—
175D4×C53—44no—
176D4×C53—82no—
177D4×C53—82no—
178D4×C53—82no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—24no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—48no—
189C2×C4×C53—44no—
190C2×C4×C53—48no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53#12012yes—
194C2×C4×C53#12011yescharacteristic
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53C2×D812yes—
197C2×C4×C53C2×D812yes—
198C2×C4×C53AGL(1,ℤ/8)12yes—
199C2×C4×C53C22×D412yes—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53—42no—
204C2×Dic53—42no—
205C2×Dic53—42no—
206Dic106—82no—
207Dic106—82no—
208#88322—44no—
209#18308—531nomaximal, Sylow
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—22no—
213C22×C53—42no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—44no—
217C22×C53—44no—
218C22×C53—44no—
219C22×C53—81no—
220C22×C53—88no—
221C22×C53C2×#12011yescharacteristic
222C22×C53C22×D811yescharacteristic
223C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
224C4×C53—22no—
225C4×C53—22no—
226C4×C53—22no—
227C4×C53—44no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53#44712yes—
231C4×C53#44712yes—
232Dic53—82no—
233Dic53—82no—
234#3449D5311yescharacteristic
235#3226—532no—
236#2621—531no—
237#1610—534no—
238#1607—534no—
239#1517—532no—
240#1458—531no—
241C2×C53—21no—
242C2×C53—21no—
243C2×C53—42no—
244C2×C53—84no—
245C2×C53C2×#44711yescharacteristic
246C2×C53#302211yescharacteristic
247C2×C53#302811yescharacteristic
248C22×#41—21no—
249#537—21no—
250#537C2×D5312yes—
251#536—22no—
252#535—22no—
253#534—21no—
254#534C2×D5312yes—
255#532—21no—
256#532C2×D5312yes—
257C2×C4⋊Q8—531no—
258#525—24no—
259C2×C22⋊Q8—1061no—
260C2×#117C2×D5311yescharacteristic
261#500—1062no—
262#486—1062no—
263#480—538no—
264#474—1064no—
265#431—532no—
266#398—532no—
267#398—1061no—
268#397—532no—
269#397—1061no—
270#391—1061no—
271#382—531no—
272#327—538no—
273#324—532no—
274#323—532no—
275C53#1830811yescharacteristic, complemented, Sylow
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41—44no—
281C2×#41—44no—
282C22×Q8—21no—
283C22×Q8—1061no—
284C22×D4—21no—
285C22×D4—22no—
286C23×C4—22no—
287C23×C4C22⋊D5311yescharacteristic
288C4⋊Q8—21no—
289C4⋊Q8—1062no—
290C4⋊Q8—1062no—
291C4⋊Q8C22⋊D5312yes—
292#127—21no—
293#127C22⋊D5312yes—
294#124—22no—
295#124—22no—
296#123—24no—
297#123—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—2124no—
301#121—24no—
302#121—24no—
303#121—22no—
304#121—22no—
305#121—22no—
306#121C22×D5314yes—
307#120—24no—
308#120—24no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4C22⋊D5314yes—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—532no—
316C2×C4⋊C4—1062no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4C22⋊D5311yescharacteristic
320C2×C4⋊C4C22×D5311yescharacteristic
321C2×C22⋊C4—1062no—
322C2×C22⋊C4C22⋊D5311yescharacteristic
323C2×C22⋊C4C22×D5311yescharacteristic
324C2×C42C22⋊D5311yescharacteristic
325C2×C42C22×D5311yescharacteristic
326#107—1068no—
327C4⋊C8—1062no—
328#102—1068no—
329C22⋊C8—534no—
330C4×C8—1062no—
331#95—532no—
332#95—1061no—
333#95—1062no—
334#95—1062no—
335C24—22no—
336C24C2×C22⋊D5311yescharacteristic
337#41—44no—
338#41—48no—
339#41—48no—
340#41—44no—
341#41—44no—
342#41—44no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—44no—
346C2×Q8—2121no—
347C2×Q8—2121no—
348C2×Q8—2122no—
349C2×Q8—2122no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4C2×C22⋊D5314yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4—1062no—
369C22×C4—1062no—
370C22×C4#1171992214yes—
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C2×C22⋊D5311yescharacteristic
373C22×C4C2×C22⋊D5311yescharacteristic
374C22×C4C2×C22⋊D5311yescharacteristic
375C22×C4C2×C22⋊D5311yescharacteristic
376C22×C4C23×D5311yescharacteristic
377C2×C8—1064no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—1064no—
384C4⋊C4—2122no—
385C4⋊C4—2124no—
386C4⋊C4—2122no—
387C4⋊C4C2×C22⋊D5314yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—24no—
394C22⋊C4—2124no—
395C42—21no—
396C42—21no—
397C42C2×C22⋊D5312yes—
398C42C2×C22⋊D5312yes—
399C23—21no—
400C23—21no—
401C23—24no—
402C23—41no—
403C23—41no—
404C23—44no—
405C23—82no—
406C23—82no—
407C23C2×#1171992212yes—
408C23C22×C22⋊D5311yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—4242no—
412Q8—4242no—
413D4—44no—
414D4—44no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—44no—
419D4—82no—
420D4—82no—
421D4—82no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4#1313318612yes—
443C2×C4#1313318611yescharacteristic
444C2×C4#1313318611yescharacteristic
445C2×C4C2×#1171992212yes—
446C2×C4C2×#1171992212yes—
447C2×C4#1313324312yes—
448C2×C4C22×C22⋊D5312yes—
449C8—2124no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—81no—
460C22—88no—
461C22C2×#1313318611yescharacteristic
462C22C22×#1171992211yescharacteristic
463C22C2×#1313324311yescharacteristic
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—44no—
470C4—4242no—
471C4—4242no—
472C4#1681075712yes—
473C4#1681075712yes—
474C2—21no—
475C2—21no—
476C2—42no—
477C2—84no—
478C2C2×#1681075711yescharacteristic
479C2#2028409411yescharacteristic
480C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#3449C2
socleC22×C53C2×#120
hypercenter#3449D53
hypocenterC53#18308
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #20285833C2semidirect
2. #20280481C2semidirect
3. #20273718C2semidirect
4. #20273701C2semidirect
5. #20273275C2semidirect
6. #20273029C2semidirect
8. C2×#13133217C22semidirect
12. #16810934C22semidirect
14. #16810676C22semidirect
16. #16810553C22semidirect
20. #16810489C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
275. C53#18308semidirect

Maximal subgroups

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

subgroupindex
#202858332
#202804812
#202737182
#202737012
#202732752
#202730292
C53×#34492
#1830853