Finite Group Databasebeta

← #27366501

Subgroups — #27366501

Order 13568.

628 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#20285038C211yesmaximal, characteristic
2#20284754C211yesmaximal, characteristic, complemented
3#20284746C211yesmaximal, characteristic
4#20274173C212yesmaximal, complemented
5#20274159C212yesmaximal, complemented
6#20274145C212yesmaximal, complemented
7#20274131C212yesmaximal, complemented
8#20273463C211yesmaximal, characteristic, complemented
9#20273373C211yesmaximal, characteristic, complemented
10#20273365C211yesmaximal, characteristic
11C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16810978—21no—
15#16810978—21no—
16#16810974C2211yescharacteristic
17#16810974C2211yescharacteristic
18#16810970C2211yescharacteristic
19#16810970C2211yescharacteristic
20#16810958—21no—
21#16810941—21no—
22#16810918C2212yes—
23#16810908C2212yes—
24#16810833C2212yes—
25#16810823C2212yes—
26#16810779—21no—
27#16810762—21no—
28C4×#11719925—22no—
29C4×#11719924—22no—
30#16810676C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810620—21no—
33#16810620—21no—
34#16810376C2212yes—
35#16810374C2212yes—
36#16810371C2212yes—
37#16810370C2212yes—
38#16810368C2212yes—
39#16810366C2212yes—
40C2×C4×Q8×C53C2211yescharacteristic
41C8×Q8×C53—21no—
42C8×Q8×C53—21no—
43C8×Q8×C53C2212yes—
44C8×Q8×C53C2212yes—
45C2×C4⋊C8×C53—21no—
46C2×C4⋊C8×C53C2211yescharacteristic
47C2×C4×C8×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719925—22no—
51C2×#11719924—22no—
52#13133217C2311yescharacteristic
53#13133217C2311yescharacteristic
54#13133217C2311yescharacteristic
55#13133214C2311yescharacteristic
56#13133209—21no—
57#13133205—21no—
58#13133196—21no—
59#13133196—21no—
60#13133191—21no—
61#13133191—21no—
62C4×Dic106—21no—
63C4×Dic106—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133176—21no—
67#13133150—22no—
68#13133148—22no—
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133142—21no—
72#13133142—21no—
73#13133142—21no—
74#13133142—21no—
75#13133141—22no—
76#13133141—22no—
77#13133140—22no—
78#13133139—21no—
79#13133139—21no—
80#13133139—21no—
81#13133139—21no—
82#13133137—22no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85C4×#88322—21no—
86C4×#88322—21no—
87C22×Q8×C53D411yescharacteristic
88C22×C8×C53—21no—
89C22×C8×C53D411yescharacteristic
90C22×C8×C53D411yescharacteristic
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53D412yes—
94C4×Q8×C53C2312yes—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—22no—
100C4⋊C8×C53—22no—
101C4⋊C8×C53—21no—
102C4⋊C8×C53C2312yes—
103C4×C8×C53—22no—
104C4×C8×C53—22no—
105C4×C8×C53D411yescharacteristic
106C4×C8×C53D411yescharacteristic
107C4×C8×C53C2311yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C2×Q8×C53—24no—
110C2×Q8×C53—21no—
111C2×Q8×C53C2×D412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—41no—
118#11719945—42no—
119#11719945—42no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212—21no—
123C2×C4×D53—41no—
124C2×Dic106—21no—
125C2×Dic106—21no—
126C2×Dic106—21no—
127C2×C8×C53—22no—
128C2×C8×C53—22no—
129C2×C8×C53—22no—
130C2×C8×C53—42no—
131C2×C8×C53C2×D411yescharacteristic
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53C2×D412yes—
139C42×C53—22no—
140C42×C53—22no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—41no—
146#11719925—44no—
147#11719924—44no—
148#11719921—41no—
149#11719921—41no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156#11719919—41no—
157#11719919—41no—
158C4×Dic53—41no—
159C2×#88322—22no—
160C2×#88322—22no—
161C23×C53#12111yescharacteristic
162C22×D53—41no—
163Q8×C53—24no—
164Q8×C53—22no—
165Q8×C53#13514yes—
166C2×C4×C53—24no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#13512yes—
176C2×C4×C53#13712yes—
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C22⋊D53—82no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184D212—41no—
185D212—42no—
186D212—41no—
187D212—81no—
188C4×D53—82no—
189Dic106—41no—
190Dic106—42no—
191Dic106—41no—
192Dic106—81no—
193C8×C53—21no—
194C8×C53—21no—
195C8×C53—21no—
196C8×C53—21no—
197C8×C53—42no—
198#88322—42no—
199#88322—42no—
200#19123—531nomaximal, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53#46412yes—
204C22×C53C2×#12111yescharacteristic
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—24no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—42no—
213C4×C53#46412yes—
214C4×C53C2×#13511yescharacteristic
215C4×C53C2×#13512yes—
216C4×C53C2×#13711yescharacteristic
217Dic53—81no—
218Dic53—81no—
219Dic53—81no—
220#3131—531no—
221#3104—531no—
222#3103—531no—
223C2×C8×Q8D5311yescharacteristic, complemented
224#1684—532no—
225#1682—532no—
226#1680—532no—
227#1678—532no—
228#1564—531no—
229#1542—531no—
230#1540—531no—
231C2×C53—22no—
232C2×C53C2×#46411yescharacteristic
233C2×C53#305011yescharacteristic
234C2×C53#306711yescharacteristic
235D53—161no—
236#533—531no—
237#532—531no—
238C2×C4×Q8C2×D5311yescharacteristic
239#488—1061no—
240#488—1061no—
241#487—531no—
242#487—531no—
243#486—531no—
244#486—531no—
245#484—1061no—
246#481—1061no—
247#477—532no—
248#475—532no—
249#462—532no—
250#460—532no—
251#451—1061no—
252#448—1061no—
253C8×Q8—21no—
254C8×Q8—21no—
255C8×Q8C2×D5312yes—
256C8×Q8C2×D5312yes—
257C4×Q16—1062no—
258C4×SD16—1062no—
259#431—531no—
260#431—531no—
261C2×C4⋊C8—21no—
262C2×C4⋊C8C2×D5311yescharacteristic
263#416—1061no—
264#416—1061no—
265C2×C4×C8—21no—
266C2×C4×C8C2×D5311yescharacteristic
267#335—532no—
268#334—532no—
269#333—532no—
270#332—532no—
271#332—532no—
272#331—532no—
273C53#1912311yescharacteristic, complemented, Sylow
274C2×#41—1061no—
275C22×Q8D21211yescharacteristic
276C2×Q16—1062no—
277C2×SD16—1062no—
278C22×C8—21no—
279C22×C8C22⋊D5311yescharacteristic
280C22×C8C22⋊D5311yescharacteristic
281C4⋊Q8—531no—
282C4⋊Q8—531no—
283C4⋊Q8—531no—
284#127—531no—
285#125—1061no—
286#124—1061no—
287C22⋊Q8—1061no—
288C22⋊Q8—1061no—
289#121—1061no—
290#121—1061no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—1061no—
294C4×Q8—1061no—
295C4×Q8D21212yes—
296C4×Q8C22×D5312yes—
297C4×D4—1061no—
298C4×D4—1061no—
299#117—1061no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4D21211yescharacteristic
302C2×C42—21no—
303C2×C42C22×D5311yescharacteristic
304#107—1062no—
305#106—1062no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8—532no—
310C4⋊C8—532no—
311C4⋊C8C22×D5312yes—
312#103—1062no—
313#103—1062no—
314#103—1061no—
315#103—1061no—
316#103—1061no—
317#103—1061no—
318#103—1062no—
319#102—1061no—
320#102—1061no—
321#102—1061no—
322#102—1061no—
323#102—1062no—
324C22⋊C8—1061no—
325C22⋊C8—1061no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—1061no—
329C4×C8—1061no—
330C4×C8C22⋊D5311yescharacteristic
331C4×C8C22⋊D5311yescharacteristic
332C4×C8C22×D5311yescharacteristic
333C4×C8C22×D5311yescharacteristic
334#41—2122no—
335#41—2122no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—1061no—
339C2×Q8—1061no—
340C2×Q8—1061no—
341C2×Q8C2×D21212yes—
342C2×D4—1061no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—2121no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2121no—
349C22×C4C2×D21211yescharacteristic
350C22×C4#1171994511yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—2124no—
353SD16—2124no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—42no—
358C2×C8—1062no—
359C2×C8—1062no—
360C2×C8C2×C22⋊D5311yescharacteristic
361C2×C8C2×C22⋊D5311yescharacteristic
362C2×C8C2×C22⋊D5311yescharacteristic
363C2×C8C2×C22⋊D5311yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—1061no—
368C4⋊C4—1061no—
369C4⋊C4—1061no—
370C4⋊C4—1061no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4C2×D21212yes—
376C22⋊C4—2121no—
377C22⋊C4—2121no—
378C22⋊C4—2121no—
379C42—22no—
380C42—22no—
381C42—2121no—
382C42C2×D21211yescharacteristic
383C42#1171994511yescharacteristic
384C42C2×C22⋊D5311yescharacteristic
385C42C23×D5311yescharacteristic
386C23—2121no—
387C23#1313319111yescharacteristic
388Q8—24no—
389Q8—22no—
390Q8—4241no—
391Q8—2121no—
392Q8—2122no—
393Q8—2121no—
394Q8#1313323914yes—
395D4—4241no—
396D4—4242no—
397D4—2121no—
398D4—2122no—
399D4—2121no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—4242no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4#1313319111yescharacteristic
411C2×C4#1313319111yescharacteristic
412C2×C4#1313319111yescharacteristic
413C2×C4#1313323712yes—
414C2×C4#1313324812yes—
415C2×C4C22×C22⋊D5311yescharacteristic
416C2×C4C22×D21211yescharacteristic
417C2×C4C2×#1171994511yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—42no—
423C8—2122no—
424C8—2122no—
425C22—21no—
426C22—21no—
427C22—4241no—
428C22—4241no—
429C22—4241no—
430C22#1681084512yes—
431C22C2×#1313319111yescharacteristic
432C4—21no—
433C4—21no—
434C4—24no—
435C4—21no—
436C4—42no—
437C4—4241no—
438C4—4241no—
439C4—4241no—
440C4#1681084512yes—
441C4C2×#1313323711yescharacteristic
442C4C2×#1313323912yes—
443C4C2×#1313324811yescharacteristic
444C2—22no—
445C2—8481no—
446C2C2×#1681084511yescharacteristic
447C2#2028429611yescharacteristic
448C2#2028443511yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C8×Q8×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×C8×Q8D53
hypocenterC53#19123
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20284754C2semidirect
4. #20274173C2semidirect
5. #20274159C2semidirect
6. #20274145C2semidirect
7. #20274131C2semidirect
8. #20273463C2semidirect
9. #20273373C2semidirect
11. C2×C8×Q8×C53C2semidirect
223. C2×C8×Q8D53semidirect
273. C53#19123semidirect

Maximal subgroups

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

subgroupindex
#202850382
#202847542
#202847462
#202741732
#202741592
#202741452
#202741312
#202734632
#202733732
#202733652
C2×C8×Q8×C532
#1912353