Finite Group Databasebeta

← #27367271

Subgroups — #27367271

Order 13568.

660 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#20286022C211yesmaximal, characteristic
2#20285040C211yesmaximal, characteristic, complemented
3#20283427C211yesmaximal, characteristic, complemented
4#20274292C212yesmaximal, complemented
5#20274278C212yesmaximal, complemented
6#20274140C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8C2×#16810373C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic, complemented
11C53×#3246C211yesmaximal, characteristic, complemented
12C2×#13133224—21no—
13#16811173C2211yescharacteristic
14#16810980—21no—
15#16810980—21no—
16#16810975C2211yescharacteristic
17#16810975C2211yescharacteristic
18#16810971C2211yescharacteristic
19#16810971C2211yescharacteristic
20#16810903C2212yes—
21#16810900—21no—
22#16810857—22no—
23#16810818C2212yes—
24#16810815—21no—
25#16810701—21no—
26#16810701C2212yes—
27D4×#88322—21no—
28D4×#88322C2212yes—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810617—21no—
32#16810617—21no—
33#16810590—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810373C2212yes—
36#16810373C2212yes—
37#16810372C2212yes—
38#16810369C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C53×#533C2211yescharacteristic
42C53×#532C2211yescharacteristic
43C53×#476C2212yes—
44C53×#475—21no—
45C53×#465—22no—
46C53×#461C2212yes—
47C53×#460—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719922—22no—
51#13133224—44no—
52C2×#11719917—22no—
53C2×#11719917—21no—
54C22×#88322—22no—
55C22×#88322—21no—
56C22×#88322D411yescharacteristic
57C22×#88322D411yescharacteristic
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65#13133150—22no—
66C4⋊#88322—21no—
67C4⋊#88322—21no—
68C4⋊#88322C2312yes—
69C4⋊#88322C2312yes—
70#13133140—21no—
71#13133140—21no—
72#13133140—21no—
73#13133140—21no—
74#13133138—22no—
75#13133137—22no—
76#13133137—21no—
77#13133137—21no—
78#13133137—21no—
79#13133137—21no—
80C22⋊#88322—22no—
81C22⋊#88322—22no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C2×#41×C53—21no—
87C2×SD16×C53—22no—
88C22×C8×C53—21no—
89C4⋊Q8×C53D411yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91C4⋊Q8×C53C2311yescharacteristic
92#127×C53D411yescharacteristic
93#125×C53—21no—
94#124×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97#121×C53—21no—
98#121×C53D412yes—
99C4×Q8×C53—21no—
100C4×Q8×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53C2312yes—
103#117×C53—21no—
104C2×C42×C53C2311yescharacteristic
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#103×C53—22no—
109#102×C53—22no—
110#41×C53—42no—
111#41×C53—42no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53C2×D412yes—
119C22×C4×C53—22no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—42no—
125#11719945—42no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131SD16×C53—82no—
132C2×C8×C53—22no—
133C2×C8×C53—81no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719922—82no—
151#11719921—41no—
152#11719921—41no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155#11719917—44no—
156#11719917—42no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—44no—
162C2×#88322—81no—
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C23×C53—22no—
168C23×C53#12111yescharacteristic
169C22×D53—41no—
170Q8×C53—42no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—81no—
174D4×C53—22no—
175D4×C53—22no—
176D4×C53—42no—
177D4×C53—44no—
178D4×C53—42no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#13512yes—
191C2×C4×C53AGL(1,ℤ/8)12yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C22⋊D53—82no—
196C2×Dic53—41no—
197D212—41no—
198D212—41no—
199D212—41no—
200D212—41no—
201D212—81no—
202C4×D53—82no—
203Dic106—41no—
204Dic106—41no—
205C8×C53—42no—
206#88322—21no—
207#88322—21no—
208#88322—21no—
209#88322—21no—
210#88322—42no—
211#88322—42no—
212#19200—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53#46312yes—
219C22×C53C2×#12111yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—42no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53#46312yes—
230C4×C53C2×#13511yescharacteristic
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57711yescharacteristic
233C4×C53#57711yescharacteristic
234Dic53—81no—
235#3246D5311yescharacteristic, complemented
236#3245—531no—
237#3131—531no—
238#2931—531no—
239#1703—532no—
240#1701—532no—
241#1679—532no—
242#1677—532no—
243C2×#334—531no—
244#1550—531no—
245#1547—531no—
246C2×C53—22no—
247C2×C53—82no—
248C2×C53C2×#46311yescharacteristic
249C2×C53#306011yescharacteristic
250C2×C53#306711yescharacteristic
251D53—161no—
252C2×#131—1061no—
253#533C2×D5311yescharacteristic
254#532—531no—
255#532C2×D5311yescharacteristic
256#488—1061no—
257#488—1061no—
258#487—531no—
259#487—531no—
260#486—531no—
261#486—531no—
262#476C2×D5312yes—
263#475—21no—
264#474—532no—
265#474—1061no—
266#466—1062no—
267#465—22no—
268#461C2×D5312yes—
269#460—21no—
270#459—532no—
271#459—1061no—
272#436—532no—
273#436—1061no—
274C8×D4—532no—
275C8×D4—1061no—
276#423—531no—
277C2×C4⋊C8—531no—
278C2×C4⋊C8C2×D5311yescharacteristic
279#416—1061no—
280#416—1061no—
281#408—1061no—
282C2×C4×C8—531no—
283#334—532no—
284#334—532no—
285#333—532no—
286#332—532no—
287#332—532no—
288#331—532no—
289C53#1920011yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C2×#41—1061no—
292C2×SD16—22no—
293C2×D8—1062no—
294#131—2124no—
295C2×M16—1062no—
296C2×M16—1061no—
297C22×C8—21no—
298C22×C8—531no—
299C22×C8—531no—
300C22×C8—1062no—
301C22×C8—1061no—
302C4⋊Q8—531no—
303C4⋊Q8C22⋊D5311yescharacteristic
304C4⋊Q8C22×D5311yescharacteristic
305C4⋊Q8C22×D5311yescharacteristic
306#127—531no—
307#127C22⋊D5311yescharacteristic
308#125—21no—
309#124—21no—
310#124—1061no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313#121—21no—
314#121—1061no—
315#121—1061no—
316#121C22⋊D5312yes—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—1061no—
321C4×D4—1061no—
322C4×D4C22×D5312yes—
323#117—21no—
324C2×C42C22×D5311yescharacteristic
325#107—1062no—
326#106—22no—
327C4⋊C8—21no—
328C4⋊C8—532no—
329C4⋊C8—532no—
330C4⋊C8—1061no—
331C4⋊C8—1061no—
332C4⋊C8C22×D5312yes—
333#103—22no—
334#103—1061no—
335#103—1061no—
336#103—1061no—
337#103—1061no—
338#102—22no—
339#102—1061no—
340#102—1061no—
341#102—1061no—
342#102—1061no—
343#102—1062no—
344#102—1062no—
345C22⋊C8—1062no—
346C22⋊C8—1062no—
347C4×C8—531no—
348C4×C8—531no—
349C4×C8—531no—
350C4×C8—531no—
351#41—42no—
352#41—42no—
353#41—2122no—
354#41—2122no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—1061no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—1061no—
363C2×D4—1061no—
364C2×D4—1061no—
365C2×D4—2121no—
366C2×D4C2×C22⋊D5312yes—
367C22×C4—22no—
368C22×C4—2121no—
369C22×C4D4×D5311yescharacteristic
370C22×C4#1171994711yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372SD16—82no—
373D8—4242no—
374M16—2124no—
375M16—2122no—
376C2×C8—22no—
377C2×C8—81no—
378C2×C8—531no—
379C2×C8—531no—
380C2×C8—531no—
381C2×C8—531no—
382C2×C8—1062no—
383C2×C8—1062no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—4241no—
387C2×C8—2124no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4C2×C22⋊D5312yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—41no—
401C22⋊C4—2121no—
402C22⋊C4—2121no—
403C42—41no—
404C42D4×D5311yescharacteristic
405C42#1171994711yescharacteristic
406C42C2×C22⋊D5311yescharacteristic
407C42C23×D5311yescharacteristic
408C23—22no—
409C23—2121no—
410C23#1313318911yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8—2121no—
416Q8—2121no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—4241no—
423D4—4242no—
424D4—2121no—
425D4—2121no—
426D4—2121no—
427D4—2121no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—4242no—
437C2×C4—2121no—
438C2×C4#1313318911yescharacteristic
439C2×C4#1313318911yescharacteristic
440C2×C4#1313318911yescharacteristic
441C2×C4#1313323812yes—
442C2×C4#1313324412yes—
443C2×C4C2×D4×D5311yescharacteristic
444C2×C4C22×C22⋊D5311yescharacteristic
445C2×C4C2×#1171994711yescharacteristic
446C8—42no—
447C8—1061no—
448C8—1061no—
449C8—1061no—
450C8—1061no—
451C8—2122no—
452C8—2122no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—4241no—
459C22—4241no—
460C22—4241no—
461C22#1681084012yes—
462C22C2×#1313318911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—4241no—
470C4#1681084012yes—
471C4C2×#1313323811yescharacteristic
472C4C2×#1313324411yescharacteristic
473C4#1681143111yescharacteristic
474C4#1681143411yescharacteristic
475C2—22no—
476C2—82no—
477C2—8481no—
478C2C2×#1681084011yescharacteristic
479C2#2028437411yescharacteristic
480C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3246C2
socleC22×C53C2×#121
prime radical#20286022C2
hypercenter#3246D53
hypocenterC53#19200
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
2. #20285040C2semidirect
3. #20283427C2semidirect
4. #20274292C2semidirect
5. #20274278C2semidirect
6. #20274140C2semidirect
7. #20274126C2semidirect
8. C2×#16810373C2semidirect
9. #20273405C2semidirect
10. #20273390C2semidirect
11. C53×#3246C2semidirect
235. #3246D53semidirect
289. C53#19200semidirect

Maximal subgroups

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

subgroupindex
#202860222
#202850402
#202834272
#202742922
#202742782
#202741402
#202741262
C2×#168103732
#202734052
#202733902
C53×#32462
#1920053