Finite Group Databasebeta

← #27372025

Subgroups — #27372025

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic, complemented
2#20285833C211yesmaximal, characteristic, complemented
3#20285290C211yesmaximal, characteristic
4C2×#16811026C211yesmaximal, characteristic, complemented
5#20285174C211yesmaximal, characteristic, complemented
6#20274387C212yesmaximal, complemented
7#20274384C212yesmaximal, complemented
8#20274331C212yesmaximal, complemented
9#20274305C212yesmaximal, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#3152C211yesmaximal, characteristic, complemented
12C2×#13133250—22no—
13#16811173C2211yescharacteristic
14C2×#13133217C2211yescharacteristic
15#16811026—22no—
16#16811026C2212yes—
17#16811026C2212yes—
18#16811026C2211yescharacteristic
19#16811026C2212yes—
20#16811026C2212yes—
21#16811017—22no—
22#16811003C2212yes—
23#16810999C2211yescharacteristic
24#16810996C2212yes—
25#16810971C2211yescharacteristic
26#16810971C2212yes—
27#16810969C2211yescharacteristic
28#16810969C2212yes—
29#16810936—22no—
30#16810934—22no—
31#16810903—22no—
32#16810900—22no—
33#16810857—22no—
34C4×#11719922—22no—
35#16810676C2211yescharacteristic
36C2×#13133150—22no—
37C2×C4×#88322C2211yescharacteristic
38#16810344C2212yes—
39C8×#88322C2212yes—
40C2×#137×C53—22no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#497—22no—
44C53×#494C2212yes—
45C53×#492C2212yes—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×Dic106—21no—
49#13133250—48no—
50C2×Dic212—24no—
51C2×#11719913—24no—
52C2×#11719922—22no—
53C22×#88322D412yes—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217C2312yes—
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67#13133150—24no—
68#13133150—24no—
69#13133150—22no—
70#13133150—24no—
71#13133150—24no—
72C4⋊#88322D412yes—
73#13133138—22no—
74#13133138—24no—
75#13133137—22no—
76#13133137—24no—
77C22⋊#88322D412yes—
78C4×#88322D412yes—
79C4×#88322C2311yescharacteristic
80C4×#88322C2311yescharacteristic
81C4×#88322C2312yes—
82C2×#41×C53—21no—
83C22×Q8×C53—21no—
84#137×C53—48no—
85C2×Q16×C53—24no—
86C2×SD16×C53—24no—
87C2×M16×C53D412yes—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2312yes—
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53C2311yescharacteristic
94#124×C53—21no—
95#121×C53—21no—
96#121×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#97×C53D412yes—
103C4×C8×C53C2312yes—
104#41×C53—44no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—41no—
108C2×Q8×C53—44no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C2×C22⋊D53—41no—
119C22×Dic53—41no—
120#11719945—44no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126C2×Dic106—22no—
127C2×Dic106—44no—
128C2×Dic106—41no—
129Q16×C53—48no—
130SD16×C53—48no—
131M16×C53D818yescomplemented
132C2×C8×C53C2×D414yes—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53C2411yescharacteristic
145#11719922—44no—
146#11719921—41no—
147#11719921—41no—
148C4⋊Dic53—22no—
149C4⋊Dic53—21no—
150C4⋊Dic53—22no—
151C4⋊Dic53—21no—
152C4⋊Dic53—41no—
153C4⋊Dic53—41no—
154C2×#88322—24no—
155C2×#88322C2×D412yes—
156C2×#88322C2×D414yes—
157C2×#88322C2×D412yes—
158Dic212—48no—
159#11719913—48no—
160C23×C53—41no—
161C23×C53#12711yescharacteristic
162C22×D53—41no—
163Q8×C53—44no—
164Q8×C53—42no—
165Q8×C53—82no—
166D4×C53—42no—
167D4×C53—42no—
168D4×C53—42no—
169D4×C53—82no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53—82no—
176C2×C4×C53#12711yescharacteristic
177C2×C4×C53#12711yescharacteristic
178C2×C4×C53#12711yescharacteristic
179C2×C4×C53C2×D814yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C22×D411yescharacteristic
183C22⋊D53—82no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—82no—
187D212—42no—
188D212—42no—
189D212—42no—
190C4×D53—82no—
191Dic106—44no—
192Dic106—42no—
193Dic106—82no—
194C8×C53C2×D818yes—
195#88322—22no—
196#88322—24no—
197#88322#13714yescomplemented
198#19630—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53#49312yes—
205C22×C53C2×#12711yescharacteristic
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—81no—
212C4×C53—82no—
213C4×C53#49312yes—
214C4×C53C22×D812yes—
215C4×C53C2×#13712yes—
216C4×C53#57812yes—
217Dic53—82no—
218Dic53—81no—
219#3226—531no—
220#3226—531no—
221#3169—531no—
222C2×#500—531no—
223#3152—531no—
224#3152D5311yescharacteristic, complemented
225#1722—532no—
226#1722—532no—
227#1713—532no—
228#1706—532no—
229C8×M16—531no—
230C2×C53—22no—
231C2×C53—161no—
232C2×C53C2×#49311yescharacteristic
233C2×C53#315211yescharacteristic
234C2×C53#315611yescharacteristic
235D53—161no—
236C2×#137—22no—
237C2×#137—1062no—
238#532—531no—
239#532C2×D5311yescharacteristic
240C2×C4⋊Q8—531no—
241C2×C4⋊Q8C2×D5311yescharacteristic
242#500—532no—
243#500—532no—
244#500—531no—
245#500—532no—
246#500—532no—
247#500—1062no—
248#497—22no—
249#497—1062no—
250#494—532no—
251#494C2×D5312yes—
252#493—531no—
253#492—532no—
254#492C2×D5312yes—
255#486—531no—
256#486—532no—
257#486—531no—
258#486—532no—
259#480—1062no—
260#480—1062no—
261#474—1062no—
262#474—1062no—
263#466—1062no—
264C4×D8—1062no—
265#431—531no—
266C2×#107—1062no—
267C4×M16C2×D5311yescharacteristic
268C2×C4×C8—531no—
269#322—532no—
270C82—532no—
271C53#1963011yescharacteristic, complemented, Sylow
272C2×#41—21no—
273C2×#41—1061no—
274C22×Q8—21no—
275C22×Q8—1061no—
276#137—48no—
277#137—2128no—
278C2×Q16—24no—
279C2×Q16—1064no—
280C2×SD16—24no—
281C2×SD16—1064no—
282C2×D8—1062no—
283C2×M16C22⋊D5312yes—
284C22×C8—532no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—532no—
289C4⋊Q8—531no—
290C4⋊Q8—1061no—
291C4⋊Q8—1061no—
292C4⋊Q8—1061no—
293C4⋊Q8C22×D5312yes—
294C4⋊Q8C22×D5311yescharacteristic
295#127—531no—
296#127C22×D5311yescharacteristic
297#124—21no—
298#124—1061no—
299#121—21no—
300#121—21no—
301#121—1061no—
302#121—1061no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—1061no—
306C4×D4—1061no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—1061no—
311C2×C42C22×D5311yescharacteristic
312#107—1064no—
313#107—1064no—
314#107—1062no—
315#107—1064no—
316#107—1064no—
317C4⋊C8—532no—
318#102—1062no—
319#102—1064no—
320#102—1062no—
321#102—1064no—
322C22⋊C8—532no—
323#97C22⋊D5312yes—
324C4×C8—532no—
325C4×C8—531no—
326C4×C8—532no—
327C4×C8—531no—
328C4×C8C22×D5312yes—
329#41—44no—
330#41—2124no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—44no—
335C2×Q8—1061no—
336C2×Q8—1062no—
337C2×Q8—2124no—
338C2×Q8—2121no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—2121no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4D4×D5311yescharacteristic
352C22×C4D4×D5311yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354Q16—48no—
355Q16—2128no—
356SD16—48no—
357SD16—2128no—
358D8—2124no—
359M16#1171992218yescomplemented
360C2×C8—532no—
361C2×C8—532no—
362C2×C8—534no—
363C2×C8—1064no—
364C2×C8C2×C22⋊D5314yes—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1062no—
372C4⋊C4—1061no—
373C4⋊C4—1062no—
374C4⋊C4—1061no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42D4×D5311yescharacteristic
382C42D4×D5311yescharacteristic
383C42C2×C22⋊D5311yescharacteristic
384C42C23×D5311yescharacteristic
385C23—41no—
386C23—2121no—
387C23#1313321311yescharacteristic
388Q8—44no—
389Q8—42no—
390Q8—82no—
391Q8—4242no—
392Q8—2124no—
393Q8—2122no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—4242no—
399D4—2122no—
400D4—2122no—
401D4—2122no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—4242no—
409C2×C4—4242no—
410C2×C4—2121no—
411C2×C4—2122no—
412C2×C4#1313321311yescharacteristic
413C2×C4#1313321311yescharacteristic
414C2×C4#1313321311yescharacteristic
415C2×C4C2×#1171992214yes—
416C2×C4C2×D4×D5311yescharacteristic
417C2×C4C2×D4×D5311yescharacteristic
418C2×C4C22×C22⋊D5311yescharacteristic
419C8—534no—
420C8—1062no—
421C8—1064no—
422C8C2×#1171992218yes—
423C22—21no—
424C22—21no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—4241no—
429C22—4241no—
430C22—4241no—
431C22#1681099912yes—
432C22C2×#1313321311yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—4241no—
438C4—4242no—
439C4#1681099912yes—
440C4C22×#1171992212yes—
441C4#137×D5312yes—
442C4#1681143912yes—
443C2—22no—
444C2—161no—
445C2—8481no—
446C2C2×#1681099911yescharacteristic
447C2#2028517011yescharacteristic
448C2#2028520811yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#127
prime radical#20285290C2
hypercenter#3152D53
hypocenterC53#19630
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #20285836C2semidirect
2. #20285833C2semidirect
4. C2×#16811026C2semidirect
5. #20285174C2semidirect
6. #20274387C2semidirect
7. #20274384C2semidirect
8. #20274331C2semidirect
9. #20274305C2semidirect
10. M16×#88322C2semidirect
11. C53×#3152C2semidirect
131. M16×C53D8semidirect
197. #88322#137semidirect
224. #3152D53semidirect
271. C53#19630semidirect
359. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202858332
#202852902
C2×#168110262
#202851742
#202743872
#202743842
#202743312
#202743052
M16×#883222
C53×#31522
#1963053