Finite Group Databasebeta

← #27358138

Subgroups — #27358138

Order 13568.

1168 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C212yesmaximal, complemented
2#20280536C211yesmaximal, characteristic, complemented
3#20273752C214yesmaximal, complemented
4#20273735C214yesmaximal, complemented
5#20273275C212yesmaximal, complemented
6#20273029C211yesmaximal, characteristic, complemented
7C53×#3449C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic, complemented
9#16811019—22no—
10#16810969—22no—
11#16810939C2218yescomplemented
12#16810913—24no—
13#16810676C2212yescomplemented
14#16810562C2212yescomplemented
15#16810559C2212yes—
16#16810543—21no—
17#16810535—21no—
18#16810521—21no—
19#16810489C2211yescharacteristic, complemented
20#16810483—21no—
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—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2312yes—
41C2×#11719919—22no—
42C2×C22⋊Dic53—22no—
43#13133148—28no—
44C4⋊#88322—22no—
45#13133138—28no—
46C22⋊#88322C2314yes—
47C4×#88322—22no—
48#13133124—22no—
49#13133124—21no—
50#13133124—22no—
51#13133124C2312yes—
52C2×#41×C53—22no—
53C2×#41×C53—22no—
54C2×#41×C53—22no—
55C2×#41×C53—22no—
56C2×#41×C53—44no—
57C2×#41×C53—44no—
58C22×Q8×C53—21no—
59C22×D4×C53—21no—
60C22×D4×C53—22no—
61C23×C4×C53—22no—
62C23×C4×C53D411yescharacteristic
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53D412yes—
65#127×C53—21no—
66#127×C53D412yes—
67#124×C53—22no—
68#124×C53—22no—
69#123×C53—24no—
70#123×C53—24no—
71C22⋊Q8×C53—24no—
72C22⋊Q8×C53—24no—
73#121×C53—24no—
74#121×C53—24no—
75#121×C53—22no—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53C2314yes—
79#120×C53—24no—
80#120×C53—24no—
81C4×D4×C53—22no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53D414yes—
85#117×C53—22no—
86#117×C53—22no—
87C2×C4⋊C4×C53D411yescharacteristic
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53D411yescharacteristic
90C2×C22⋊C4×C53C2311yescharacteristic
91C2×C42×C53D411yescharacteristic
92C2×C42×C53C2311yescharacteristic
93C24×C53—22no—
94C24×C53C2×D411yescharacteristic
95#41×C53—44no—
96#41×C53—48no—
97#41×C53—48no—
98#41×C53—44no—
99#41×C53—44no—
100#41×C53—44no—
101C2×Q8×C53—22no—
102C2×Q8×C53—22no—
103C2×Q8×C53—44no—
104C2×D4×C53—24no—
105C2×D4×C53—22no—
106C2×D4×C53—24no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—44no—
112C2×D4×C53—44no—
113C2×D4×C53—44no—
114C2×D4×C53C2×D414yes—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—24no—
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—48no—
122C22×C4×C53SD1614yes—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—22no—
131C2×Dic106—42no—
132C2×Dic106—82no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53C2×D414yes—
139C22⋊C4×C53—24no—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—24no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—24no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53C2×D412yes—
148C42×C53C2×D412yes—
149C22⋊Dic53—82no—
150C4⋊Dic53—24no—
151C4⋊Dic53—41no—
152C4⋊Dic53—41no—
153C4⋊Dic53—42no—
154#11719919—82no—
155C2×#88322—24no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—44no—
162C23×C53—82no—
163C23×C53—82no—
164C23×C53C2×SD1612yes—
165C23×C53C22×D411yescharacteristic
166Q8×C53—44no—
167Q8×C53—44no—
168D4×C53—44no—
169D4×C53—44no—
170D4×C53—48no—
171D4×C53—44no—
172D4×C53—44no—
173D4×C53—44no—
174D4×C53—82no—
175D4×C53—82no—
176D4×C53—82no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—24no—
180C2×C4×C53—24no—
181C2×C4×C53—24no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—48no—
187C2×C4×C53—44no—
188C2×C4×C53—48no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53#12012yes—
192C2×C4×C53#12011yescharacteristic
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53C2×SD1612yes—
195C2×C4×C53C2×SD1612yes—
196C2×C4×C53AGL(1,ℤ/8)12yes—
197C2×C4×C53C22×D412yes—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53—42no—
204Dic106—84no—
205#88322—44no—
206#18309—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—22no—
210C22×C53—42no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—44no—
214C22×C53—44no—
215C22×C53—44no—
216C22×C53—81no—
217C22×C53—88no—
218C22×C53C2×#12011yescharacteristic
219C22×C53C22×SD1611yescharacteristic
220C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—44no—
225C4×C53—41no—
226C4×C53—41no—
227C4×C53#45012yes—
228C4×C53#45012yes—
229Dic53—82no—
230Dic53—82no—
231#3449D5311yescharacteristic
232#3227—532no—
233#2628—531no—
234#1616—534no—
235#1613—534no—
236#1517—532no—
237#1458—531no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—42no—
241C2×C53—84no—
242C2×C53C2×#45011yescharacteristic
243C2×C53#302111yescharacteristic
244C2×C53#302811yescharacteristic
245C22×#41—21no—
246#537—21no—
247#537C2×D5312yes—
248#536—22no—
249#535—22no—
250#534—21no—
251#534C2×D5312yes—
252#532—21no—
253#532C2×D5312yes—
254C2×C4⋊Q8—531no—
255#525—24no—
256C2×#117C2×D5311yescharacteristic
257#498—1062no—
258#486—1062no—
259#481—538no—
260#476—1064no—
261#431—532no—
262#400—532no—
263#399—532no—
264#395—1061no—
265#393—1061no—
266#389—1061no—
267#382—531no—
268#380—1061no—
269#327—538no—
270#324—532no—
271#323—532no—
272C53#1830911yescharacteristic, complemented, Sylow
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—44no—
278C2×#41—44no—
279C22×Q8—21no—
280C22×Q8—1061no—
281C22×D4—21no—
282C22×D4—22no—
283C23×C4—22no—
284C23×C4C22⋊D5311yescharacteristic
285C4⋊Q8—21no—
286C4⋊Q8—1062no—
287C4⋊Q8—1062no—
288C4⋊Q8C22⋊D5312yes—
289#127—21no—
290#127C22⋊D5312yes—
291#124—22no—
292#124—22no—
293#123—24no—
294#123—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297#121—24no—
298#121—24no—
299#121—22no—
300#121—22no—
301#121—22no—
302#121C22×D5314yes—
303#120—24no—
304#120—24no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4C22⋊D5314yes—
309#117—22no—
310#117—22no—
311C2×C4⋊C4—532no—
312C2×C4⋊C4—1062no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4C22⋊D5311yescharacteristic
316C2×C4⋊C4C22×D5311yescharacteristic
317C2×C22⋊C4—1062no—
318C2×C22⋊C4C22⋊D5311yescharacteristic
319C2×C22⋊C4C22×D5311yescharacteristic
320C2×C42C22⋊D5311yescharacteristic
321C2×C42C22×D5311yescharacteristic
322#106—1068no—
323C4⋊C8—1062no—
324#102—1068no—
325C22⋊C8—534no—
326C4×C8—1062no—
327#95—532no—
328#95—1062no—
329#95—1061no—
330#95—1062no—
331C24—22no—
332C24C2×C22⋊D5311yescharacteristic
333#41—44no—
334#41—48no—
335#41—48no—
336#41—44no—
337#41—44no—
338#41—44no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—44no—
342C2×Q8—4242no—
343C2×Q8—2122no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4C2×C22⋊D5314yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4#1171992314yes—
365C22×C4C2×C22⋊D5311yescharacteristic
366C22×C4C2×C22⋊D5311yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368C22×C4C2×C22⋊D5311yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C23×D5311yescharacteristic
371C2×C8—1064no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—1064no—
378C4⋊C4—4242no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2122no—
382C4⋊C4C2×C22⋊D5314yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—4242no—
390C42—21no—
391C42—21no—
392C42C2×C22⋊D5312yes—
393C42C2×C22⋊D5312yes—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—44no—
400C23—82no—
401C23—82no—
402C23C2×#1171992312yes—
403C23C22×C22⋊D5311yescharacteristic
404Q8—44no—
405Q8—44no—
406Q8—4244no—
407D4—44no—
408D4—44no—
409D4—48no—
410D4—44no—
411D4—44no—
412D4—44no—
413D4—82no—
414D4—82no—
415D4—82no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—2122no—
431C2×C4—2122no—
432C2×C4—2122no—
433C2×C4—2122no—
434C2×C4—2122no—
435C2×C4—2122no—
436C2×C4#1313318612yes—
437C2×C4#1313318611yescharacteristic
438C2×C4#1313318611yescharacteristic
439C2×C4C2×#1171992312yes—
440C2×C4C2×#1171992312yes—
441C2×C4#1313324312yes—
442C2×C4C22×C22⋊D5312yes—
443C8—2124no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—81no—
454C22—88no—
455C22C2×#1313318611yescharacteristic
456C22C22×#1171992311yescharacteristic
457C22C2×#1313324311yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—41no—
462C4—41no—
463C4—44no—
464C4—4242no—
465C4—4242no—
466C4#1681077412yes—
467C4#1681077412yes—
468C2—21no—
469C2—21no—
470C2—42no—
471C2—84no—
472C2C2×#1681077411yescharacteristic
473C2#2028408711yescharacteristic
474C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#3449C2
socleC22×C53C2×#120
hypercenter#3449D53
hypocenterC53#18309
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. #20285840C2semidirect
2. #20280536C2semidirect
3. #20273752C2semidirect
4. #20273735C2semidirect
5. #20273275C2semidirect
6. #20273029C2semidirect
8. C2×#13133217C22semidirect
11. #16810939C22semidirect
13. #16810676C22semidirect
14. #16810562C22semidirect
19. #16810489C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
272. C53#18309semidirect

Maximal subgroups

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

subgroupindex
#202858402
#202805362
#202737522
#202737352
#202732752
#202730292
C53×#34492
#1830953