Finite Group Databasebeta

← #27367924

Subgroups — #27367924

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic
2#20284520C211yesmaximal, characteristic, complemented
3#20284187C211yesmaximal, characteristic
4#20274145C212yesmaximal, complemented
5#20274131C212yesmaximal, complemented
6#20274097C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273471C211yesmaximal, characteristic, complemented
9#20273403C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic
11C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13#16811180C2211yescharacteristic
14#16811173C2211yescharacteristic
15#16811021C2211yescharacteristic
16#16810996C2211yescharacteristic
17#16810974C2211yescharacteristic
18#16810970C2211yescharacteristic
19#16810930—22no—
20#16810918C2212yes—
21#16810908C2212yes—
22#16810852—21no—
23#16810852—21no—
24#16810845—22no—
25#16810833C2212yes—
26#16810823C2212yes—
27C4×#11719913—21no—
28C4×#11719913—21no—
29#16810663—21no—
30#16810647C2211yescharacteristic
31#16810647C2211yescharacteristic
32#16810632—21no—
33#16810622—21no—
34#16810374C2212yes—
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—
50#13133248—44no—
51C2×#11719925—22no—
52C2×#11719913—21no—
53C2×#11719913—21no—
54C2×#11719924—22no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57#13133217C2311yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133209—21no—
62#13133205—21no—
63#13133196—21no—
64#13133196—21no—
65#13133191—21no—
66#13133191—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69C4×D212—21no—
70C4×D212—21no—
71#13133176—21no—
72#13133150—22no—
73#13133149—21no—
74#13133149—21no—
75#13133148—22no—
76C4⋊#88322—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2312yes—
79C4⋊#88322C2312yes—
80#13133142—21no—
81#13133142—21no—
82#13133141—22no—
83#13133141—22no—
84#13133140—22no—
85#13133139—21no—
86#13133139—21no—
87#13133137—22no—
88C22×Q8×C53D411yescharacteristic
89C22×C8×C53—21no—
90C22×C8×C53D411yescharacteristic
91C22×C8×C53D411yescharacteristic
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53D412yes—
95C4×Q8×C53C2312yes—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100C4⋊C8×C53—22no—
101C4⋊C8×C53—22no—
102C4⋊C8×C53—21no—
103C4⋊C8×C53C2312yes—
104C4×C8×C53—22no—
105C4×C8×C53—22no—
106C4×C8×C53D411yescharacteristic
107C4×C8×C53D411yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C4×C8×C53C2311yescharacteristic
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53SD1614yescomplemented
113C2×Q8×C53C2×D412yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—41no—
120#11719945—42no—
121#11719945—42no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—21no—
127C2×Dic106—21no—
128C2×Dic106—21no—
129C2×C8×C53—22no—
130C2×C8×C53—22no—
131C2×C8×C53—41no—
132C2×C8×C53—42no—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53C2×D412yes—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C22⋊Dic53—41no—
148#11719925—44no—
149#11719924—44no—
150#11719921—41no—
151#11719921—41no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158#11719919—41no—
159#11719919—41no—
160C4×Dic53—41no—
161#11719917—42no—
162#11719917—42no—
163C2×#88322—22no—
164C2×#88322—22no—
165#11719913—42no—
166#11719913—42no—
167C23×C53#12111yescharacteristic
168C22×D53—41no—
169Q8×C53—22no—
170Q8×C53—24no—
171Q8×C53C2×SD1614yes—
172C2×C4×C53—22no—
173C2×C4×C53—24no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—22no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53C2×SD1612yes—
183C2×C4×C53#13512yes—
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C22×D411yescharacteristic
186C2×C4×C53C2×#4111yescharacteristic
187C22⋊D53—82no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191D212—41no—
192D212—41no—
193D212—41no—
194D212—41no—
195D212—81no—
196C4×D53—82no—
197Dic106—41no—
198Dic106—41no—
199Dic106—41no—
200Dic106—41no—
201Dic106—81no—
202C8×C53—21no—
203C8×C53—21no—
204C8×C53—21no—
205C8×C53—42no—
206C8×C53#13712yescomplemented
207#88322—42no—
208#88322—42no—
209#19260—531nomaximal, Sylow
210C22×C53—21no—
211C22×C53—21no—
212C22×C53#46512yes—
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—42no—
222C4×C53#46512yes—
223C4×C53C22×SD1611yescharacteristic
224C4×C53C2×#13511yescharacteristic
225C4×C53C2×#13711yescharacteristic
226C4×C53#57511yescharacteristic
227Dic53—81no—
228Dic53—81no—
229Dic53—81no—
230#3167—531no—
231#3078—531no—
232#3037—531no—
233C2×C8×Q8D5311yescharacteristic, complemented
234#1680—532no—
235#1678—532no—
236#1672—532no—
237#1666—532no—
238#1566—531no—
239#1550—531no—
240#1547—531no—
241C2×C53—22no—
242C2×C53#303711yescharacteristic
243C2×C53C2×#46511yescharacteristic
244C2×C53#306711yescharacteristic
245D53—161no—
246C2×#137—1061no—
247#533—531no—
248#532—531no—
249C2×C4×Q8C2×D5311yescharacteristic
250#498—531no—
251#492—531no—
252#487—531no—
253#486—531no—
254#479—1062no—
255#477—532no—
256#475—532no—
257#465—1061no—
258#465—1061no—
259#464—1062no—
260#462—532no—
261#460—532no—
262C8×Q8—21no—
263C8×Q8—21no—
264C8×Q8C2×D5312yes—
265C8×Q8C2×D5312yes—
266C4×SD16—1061no—
267C4×SD16—1061no—
268#428—1061no—
269#423—531no—
270#423—531no—
271C2×C4⋊C8—21no—
272C2×C4⋊C8C2×D5311yescharacteristic
273#419—1061no—
274#417—1061no—
275C2×C4×C8—21no—
276C2×C4×C8C2×D5311yescharacteristic
277#334—532no—
278#334—532no—
279#333—532no—
280#332—532no—
281#332—532no—
282#331—532no—
283C53#1926011yescharacteristic, complemented, Sylow
284C2×#41—1061no—
285C22×Q8D21211yescharacteristic
286#137—2124no—
287C2×Q16—1062no—
288C2×SD16—1062no—
289C2×SD16—1061no—
290C2×SD16—1061no—
291C2×M16—1061no—
292C2×M16—1061no—
293C22×C8—21no—
294C22×C8C22⋊D5311yescharacteristic
295C22×C8C22⋊D5311yescharacteristic
296C4⋊Q8—531no—
297C4⋊Q8—531no—
298C4⋊Q8—531no—
299#127—531no—
300#125—1061no—
301#124—1061no—
302C22⋊Q8—1061no—
303C22⋊Q8—1061no—
304#121—1061no—
305#121—1061no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—1061no—
309C4×Q8—1061no—
310C4×Q8D21212yes—
311C4×Q8C22×D5312yes—
312C4×D4—1061no—
313C4×D4—1061no—
314#117—1061no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4D21211yescharacteristic
317C2×C42—21no—
318C2×C42C22×D5311yescharacteristic
319#107—1062no—
320#106—1061no—
321#106—1061no—
322#106—1062no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—21no—
326C4⋊C8—532no—
327C4⋊C8—532no—
328C4⋊C8—1061no—
329C4⋊C8—1061no—
330C4⋊C8C22×D5312yes—
331#103—1062no—
332#103—1062no—
333#103—1061no—
334#103—1061no—
335#103—1062no—
336#102—1061no—
337#102—1061no—
338#102—1062no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8C22⋊D5311yescharacteristic
342C4×C8C22⋊D5311yescharacteristic
343C4×C8C22×D5311yescharacteristic
344C4×C8C22×D5311yescharacteristic
345#41—2122no—
346#41—2122no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—1061no—
350C2×Q8—1061no—
351C2×Q8—1061no—
352C2×Q8#1171991314yescomplemented
353C2×Q8C2×D21212yes—
354C2×D4—1061no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—2121no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—2121no—
361C22×C4C2×D21211yescharacteristic
362C22×C4#1171994511yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364Q16—2124no—
365SD16—2122no—
366SD16—2124no—
367SD16—2122no—
368M16—2122no—
369M16—2122no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—1062no—
375C2×C8—1062no—
376C2×C8C2×C22⋊D5311yescharacteristic
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C2×C22⋊D5311yescharacteristic
379C2×C8C2×C22⋊D5311yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—1061no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4—1061no—
387C4⋊C4—1061no—
388C4⋊C4—1061no—
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C4⋊C4C2×D21212yes—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—22no—
396C42—22no—
397C42—2121no—
398C42C2×D21211yescharacteristic
399C42#1171994511yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—2121no—
403C23#1313319111yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—4241no—
407Q8—2121no—
408Q8—2121no—
409Q8—2121no—
410Q8—2121no—
411Q8C2×#1171991314yes—
412D4—4241no—
413D4—4242no—
414D4—2121no—
415D4—2121no—
416D4—2121no—
417D4—2121no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—4242no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4#1313319111yescharacteristic
430C2×C4#1313319111yescharacteristic
431C2×C4#1313319111yescharacteristic
432C2×C4C2×#1171991312yes—
433C2×C4#1313323912yes—
434C2×C4C22×C22⋊D5311yescharacteristic
435C2×C4C22×D21211yescharacteristic
436C2×C4C2×#1171994511yescharacteristic
437C8—21no—
438C8—21no—
439C8—21no—
440C8—42no—
441C8—2122no—
442C8—2122no—
443C8#1313324812yescomplemented
444C22—21no—
445C22—21no—
446C22—4241no—
447C22—4241no—
448C22—4241no—
449C22#1681085212yes—
450C22C2×#1313319111yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—24no—
455C4—42no—
456C4—4241no—
457C4—4241no—
458C4—4241no—
459C4#1681085212yes—
460C4C22×#1171991311yescharacteristic
461C4C2×#1313323911yescharacteristic
462C4C2×#1313324811yescharacteristic
463C4#1681142111yescharacteristic
464C2—22no—
465C2—8481no—
466C2#2028418711yescharacteristic
467C2C2×#1681085211yescharacteristic
468C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C8×Q8×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×C8×Q8D53
hypocenterC53#19260
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20284520C2semidirect
4. #20274145C2semidirect
5. #20274131C2semidirect
6. #20274097C2semidirect
7. #20274063C2semidirect
8. #20273471C2semidirect
9. #20273403C2semidirect
11. C2×C8×Q8×C53C2semidirect
112. C2×Q8×C53SD16semidirect
206. C8×C53#137semidirect
233. C2×C8×Q8D53semidirect
283. C53#19260semidirect
352. C2×Q8#11719913semidirect
443. C8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202845202
#202841872
#202741452
#202741312
#202740972
#202740632
#202734712
#202734032
#202733902
C2×C8×Q8×C532
#1926053