Finite Group Databasebeta

← #27367959

Subgroups — #27367959

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285290C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic, complemented
3C2×Q8×#88322C211yesmaximal, characteristic, complemented
4#20274133C212yesmaximal, complemented
5#20274129C212yesmaximal, complemented
6#20274117C212yesmaximal, complemented
7#20274073C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273411C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3112C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811026C2211yescharacteristic
14#16810999C2211yescharacteristic
15#16810971C2211yescharacteristic
16#16810969C2211yescharacteristic
17#16810936—21no—
18#16810918C2212yes—
19#16810857—21no—
20#16810857—21no—
21#16810823C2212yes—
22#16810762—21no—
23Q8×#88322—21no—
24Q8×#88322—21no—
25Q8×#88322C2212yes—
26Q8×#88322C2212yes—
27#16810721—22no—
28C4×#11719922—21no—
29C4×#11719922—21no—
30#16810680C2211yescharacteristic
31C2×C4⋊#88322—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×C4×#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yes—
36#16810373C2212yes—
37#16810372C2212yes—
38#16810371C2212yes—
39#16810367C2212yes—
40#16810366C2212yes—
41C53×#532C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#480—21no—
44C53×#477C2212yes—
45C53×#460C2212yes—
46C53×#448—21no—
47C53×#440—22no—
48C53×#432C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719924—22no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C22×#88322—21no—
54C22×#88322D411yescharacteristic
55C22×#88322D411yescharacteristic
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191—21no—
61C4×D212—21no—
62C4×D212—21no—
63#13133150—22no—
64#13133150—21no—
65#13133150—21no—
66C4⋊#88322—21no—
67C4⋊#88322—22no—
68C4⋊#88322—22no—
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133141—22no—
72#13133138—21no—
73#13133138—21no—
74#13133137—22no—
75#13133137—21no—
76#13133137—21no—
77C22⋊#88322—21no—
78#13133128—21no—
79C4×#88322—22no—
80C4×#88322—22no—
81C4×#88322D411yescharacteristic
82C4×#88322D411yescharacteristic
83C4×#88322C2311yescharacteristic
84C4×#88322C2311yescharacteristic
85C2×#41×C53—21no—
86C22×Q8×C53D411yescharacteristic
87C2×SD16×C53—22no—
88C4⋊Q8×C53C2311yescharacteristic
89#127×C53C2311yescharacteristic
90#124×C53—21no—
91#121×C53—21no—
92#121×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53D412yes—
96C4×Q8×C53C2312yes—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#107×C53—22no—
104C4⋊C8×C53C2312yes—
105#103×C53—22no—
106#102×C53—22no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109#41×C53—82no—
110C2×Q8×C53—24no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53C2×D412yes—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—41no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125#11719945—82no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131SD16×C53—44no—
132C2×C8×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147#11719924—44no—
148#11719922—42no—
149#11719922—42no—
150#11719921—41no—
151#11719921—41no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—41no—
158C2×#88322—42no—
159C2×#88322C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162C2×#88322C2×D411yescharacteristic
163C23×C53—41no—
164C23×C53#12111yescharacteristic
165C22×D53—41no—
166Q8×C53—24no—
167Q8×C53—22no—
168Q8×C53—81no—
169Q8×C53#13514yes—
170D4×C53—42no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—81no—
174D4×C53—82no—
175C2×C4×C53—24no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—82no—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C22⋊D53—82no—
192C2×Dic53—41no—
193D212—42no—
194D212—41no—
195D212—41no—
196D212—81no—
197C4×D53—82no—
198Dic106—81no—
199C8×C53—42no—
200#88322—21no—
201#88322—21no—
202#88322—21no—
203#88322—42no—
204#88322—42no—
205#88322#13712yes—
206#19264—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53#47212yes—
213C22×C53C2×#12111yescharacteristic
214C2×D53—81no—
215C2×D53—81no—
216C2×D53—81no—
217C4×C53—21no—
218C4×C53—24no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—42no—
223C4×C53—81no—
224C4×C53C2×#13512yes—
225C4×C53C2×#13711yescharacteristic
226C4×C53#57511yescharacteristic
227Dic53—81no—
228#3169—531no—
229#3112—531no—
230#3112D5311yescharacteristic, complemented
231C2×C8×Q8—531no—
232#1678—532no—
233#1678—532no—
234#1676—532no—
235#1668—532no—
236#1568—531no—
237#1551—531no—
238#1551—531no—
239C2×C53—22no—
240C2×C53—161no—
241C2×C53#303711yescharacteristic
242C2×C53#305011yescharacteristic
243C2×C53C2×#47211yescharacteristic
244D53—161no—
245#532—531no—
246#532C2×D5311yescharacteristic
247C2×C4×Q8C2×D5311yescharacteristic
248#500—531no—
249#493—531no—
250#486—531no—
251#486—531no—
252#480—21no—
253#480—1061no—
254#477—532no—
255#477C2×D5312yes—
256#466—1061no—
257#466—1061no—
258#460—532no—
259#460C2×D5312yes—
260#448—21no—
261#448—1061no—
262C8×Q8—532no—
263C8×Q8—532no—
264C8×Q8—1061no—
265C8×Q8—1061no—
266#440—22no—
267#440—1062no—
268C4×D8—1061no—
269C4×D8—1061no—
270#432—531no—
271#432C2×D5311yescharacteristic
272C2×C4⋊C8—531no—
273C2×C4⋊C8—1061no—
274C2×C4×C8—531no—
275C2×C4×C8—1061no—
276#335—532no—
277#334—532no—
278#333—532no—
279#333—532no—
280#331—532no—
281#331—532no—
282C53#1926411yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1061no—
285C22×Q8C22⋊D5311yescharacteristic
286C2×SD16—22no—
287C2×SD16—1062no—
288C2×D8—1061no—
289C2×D8—1061no—
290C22×C8—531no—
291C22×C8—531no—
292C22×C8—1061no—
293C4⋊Q8—531no—
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×Q8—21no—
304C4×Q8—21no—
305C4×Q8C22⋊D5312yes—
306C4×Q8C22×D5312yes—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—1061no—
310C4×D4—1061no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C22⋊D5311yescharacteristic
313C2×C42—21no—
314C2×C42C22×D5311yescharacteristic
315#107—22no—
316#107—1061no—
317#107—1061no—
318#107—1062no—
319C4⋊C8—532no—
320C4⋊C8—532no—
321C4⋊C8—1062no—
322C4⋊C8—1062no—
323C4⋊C8—1061no—
324C4⋊C8C22×D5312yes—
325#103—22no—
326#103—1062no—
327#102—22no—
328#102—1061no—
329#102—1061no—
330#102—1061no—
331#102—1061no—
332#102—1062no—
333C22⋊C8—21no—
334C22⋊C8—1061no—
335#97—21no—
336#97—1061no—
337C4×C8—531no—
338C4×C8—531no—
339C4×C8—531no—
340C4×C8—531no—
341C4×C8—1062no—
342C4×C8—1062no—
343#41—82no—
344#41—4242no—
345C2×Q8—24no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—1061no—
349C2×Q8C2×C22⋊D5312yes—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1061no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—2121no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—2121no—
362C22×C4D4×D5311yescharacteristic
363C22×C4#1171994711yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365SD16—44no—
366SD16—2124no—
367D8—2122no—
368D8—2122no—
369C2×C8—22no—
370C2×C8—531no—
371C2×C8—531no—
372C2×C8—531no—
373C2×C8—531no—
374C2×C8—1062no—
375C2×C8—1062no—
376C2×C8—1062no—
377C2×C8—2121no—
378C2×C8—2122no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4C2×C22⋊D5312yes—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C42—22no—
392C42—22no—
393C42D4×D5311yescharacteristic
394C42#1171994711yescharacteristic
395C42C2×C22⋊D5311yescharacteristic
396C42C23×D5311yescharacteristic
397C23—41no—
398C23—2121no—
399C23#1313318911yescharacteristic
400Q8—24no—
401Q8—22no—
402Q8—81no—
403Q8—4241no—
404Q8#1313323714yes—
405D4—42no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—82no—
410D4—4241no—
411D4—4242no—
412D4—2122no—
413D4—2121no—
414D4—2121no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—82no—
425C2×C4—4242no—
426C2×C4—2121no—
427C2×C4#1313318911yescharacteristic
428C2×C4#1313318911yescharacteristic
429C2×C4#1313318911yescharacteristic
430C2×C4C2×D4×D5311yescharacteristic
431C2×C4C22×C22⋊D5311yescharacteristic
432C2×C4C2×#1171994711yescharacteristic
433C8—42no—
434C8—532no—
435C8—1061no—
436C8—1061no—
437C8—1061no—
438C8—2122no—
439C8—2122no—
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—4241no—
446C22—4241no—
447C22—4241no—
448C22#1681089112yes—
449C22C2×#1313318911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—24no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—4241no—
458C4C2×#1313323712yes—
459C4#137×D5311yescharacteristic
460C4#1681142211yescharacteristic
461C2—22no—
462C2—161no—
463C2—8481no—
464C2#2028418411yescharacteristic
465C2#2028429311yescharacteristic
466C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3112C2
socleC22×C53C2×#121
prime radical#20285290C2
hypercenter#3112D53
hypocenterC53#19264
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. C2×Q8×#88322C2semidirect
4. #20274133C2semidirect
5. #20274129C2semidirect
6. #20274117C2semidirect
7. #20274073C2semidirect
8. #20273476C2semidirect
9. #20273411C2semidirect
10. #20273410C2semidirect
11. C53×#3112C2semidirect
230. #3112D53semidirect
282. C53#19264semidirect

Maximal subgroups

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

subgroupindex
#202852902
#202848372
C2×Q8×#883222
#202741332
#202741292
#202741172
#202740732
#202734762
#202734112
#202734102
C53×#31122
#1926453