Finite Group Databasebeta

← #27366988

Subgroups — #27366988

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285089C211yesmaximal, characteristic
2#20284754C211yesmaximal, characteristic, complemented
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274281C212yesmaximal, complemented
5#20274253C212yesmaximal, complemented
6#20274157C212yesmaximal, complemented
7#20274129C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273403C211yesmaximal, characteristic, complemented
10#20273383C211yesmaximal, characteristic
11C2×C53×#460C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810987—22no—
14#16810969C2212yes—
15#16810969C2212yes—
16#16810958—21no—
17#16810918C2211yescharacteristic, complemented
18#16810918C2211yescharacteristic, complemented
19#16810833C2211yescharacteristic, complemented
20#16810833C2211yescharacteristic, complemented
21#16810779—21no—
22#16810754—22no—
23#16810754C2212yescomplemented
24Q8×#88322C2212yescomplemented
25C4×#11719925—21no—
26C4×#11719925—21no—
27#16810676C2211yescharacteristic
28#16810647—21no—
29#16810647C2211yescharacteristic
30#16810628—21no—
31#16810628—21no—
32C4×#11719917—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810370C2212yescomplemented
38#16810367C2212yes—
39#16810367C2212yes—
40C22×SD16×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460—21no—
45C53×#460C2212yes—
46C53×#460C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719925—21no—
50C2×#11719925—21no—
51C2×#11719917—21no—
52C2×#11719917D411yescharacteristic
53C2×#11719917D411yescharacteristic
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133209—21no—
57#13133196—21no—
58#13133196—21no—
59C4×Dic106—21no—
60C4×Dic106—21no—
61#13133176—21no—
62#13133150—21no—
63#13133150—21no—
64C4⋊#88322—21no—
65C4⋊#88322—22no—
66C4⋊#88322—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70#13133141—21no—
71#13133141—21no—
72#13133140—21no—
73#13133140—21no—
74#13133138—22no—
75#13133138—22no—
76#13133138—22no—
77#13133138—22no—
78C22⋊#88322—21no—
79#13133128—22no—
80#13133128D412yes—
81C4×#88322—21no—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C22×Q8×C53D411yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C2×SD16×C53—22no—
89C2×SD16×C53—22no—
90C2×SD16×C53—44no—
91C22×C8×C53—21no—
92#127×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53C2312yes—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53D411yescharacteristic
99C4×Q8×C53D411yescharacteristic
100C4×Q8×C53C2311yescharacteristic
101C4×Q8×C53C2311yescharacteristic
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—82no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C2×Dic106—21no—
133C2×Dic106—21no—
134SD16×C53—44no—
135SD16×C53—44no—
136C2×C8×C53—22no—
137C2×C8×C53—42no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53C2×D411yescharacteristic
144C4⋊C4×C53C2×D411yescharacteristic
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153C22⋊Dic53—41no—
154#11719925—42no—
155#11719925—42no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163#11719917—22no—
164#11719917—22no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—21no—
168C2×#88322—21no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53—42no—
172C23×C53—82no—
173C23×C53—82no—
174C23×C53—82no—
175C23×C53#12111yescharacteristic
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—42no—
180Q8×C53AGL(1,ℤ/8)12yes—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184D4×C53—42no—
185D4×C53—82no—
186D4×C53—82no—
187D4×C53—82no—
188D4×C53—82no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—81no—
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#13512yes—
202C2×C4×C53#13712yes—
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×Dic53—41no—
207C2×Dic53—41no—
208Dic106—41no—
209Dic106—41no—
210Dic106—81no—
211C8×C53—42no—
212#88322—22no—
213#88322—22no—
214#88322—42no—
215#88322—41no—
216#88322—41no—
217#19174—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—82no—
223C22×C53—164no—
224C22×C53#46412yes—
225C22×C53C2×#12111yescharacteristic
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53#46412yes—
234C4×C53C2×#13511yescharacteristic
235C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C53C2×#13711yescharacteristic
237C4×C53#57511yescharacteristic
238Dic53—81no—
239Dic53—81no—
240#3138—531no—
241#3104—531no—
242C2×#460D5311yescharacteristic
243Q8×M16—531no—
244#1702—532no—
245#1698—532no—
246#1682—532no—
247#1678—532no—
248#1565—531no—
249#1550—531no—
250#1545—531no—
251C2×C53—22no—
252C2×C53—162no—
253C2×C53C2×#46411yescharacteristic
254C2×C53#305511yescharacteristic
255C2×C53#306811yescharacteristic
256C22×SD16—21no—
257#533—531no—
258C2×#127C2×D5311yescharacteristic
259C2×C4×Q8C2×D5311yescharacteristic
260#490—1062no—
261#486—532no—
262#486—532no—
263#484—1061no—
264#477—531no—
265#477—531no—
266#462—531no—
267#462—531no—
268#460—21no—
269#460—21no—
270#460C2×D5312yes—
271#460C2×D5312yes—
272#451—1061no—
273#446—532no—
274#446—1062no—
275C8×Q8—532no—
276C4×Q16—1061no—
277C4×Q16—1061no—
278#431—531no—
279#423—531no—
280#423—1061no—
281C2×C4⋊C8C2×D5311yescharacteristic
282#418—1061no—
283#418—1061no—
284C2×#102—21no—
285C4×M16—531no—
286C4×M16—1061no—
287#335—532no—
288#334—532no—
289#333—532no—
290#332—532no—
291#331—532no—
292#331—532no—
293C53#1917411yescharacteristic, complemented, Sylow
294C22×Q8C22⋊D5311yescharacteristic
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×Q16—1061no—
299C2×Q16—1061no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—44no—
303C2×M16—531no—
304C2×M16—531no—
305C2×M16—1061no—
306C22×C8—21no—
307C4⋊Q8—531no—
308C4⋊Q8—531no—
309#127—21no—
310#127—21no—
311#127—21no—
312#127C22×D5312yes—
313#125—1061no—
314C22⋊Q8—1061no—
315C22⋊Q8—1061no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—1061no—
319C4×Q8—1061no—
320C4×Q8C22⋊D5311yescharacteristic
321C4×Q8C22⋊D5311yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×Q8C22×D5311yescharacteristic
324#117—1061no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D5311yescharacteristic
327C2×C42—21no—
328C2×C42C22×D5311yescharacteristic
329#107—1061no—
330#107—1061no—
331C4⋊C8—21no—
332C4⋊C8—532no—
333C4⋊C8—531no—
334C4⋊C8—531no—
335C4⋊C8—1062no—
336C4⋊C8—1062no—
337C4⋊C8—1061no—
338C4⋊C8C22×D5312yes—
339#103—1061no—
340#103—1061no—
341#103—1061no—
342#103—1061no—
343#102—22no—
344#102—22no—
345#102—1062no—
346#102—1062no—
347#102—1062no—
348#102—1062no—
349C22⋊C8—1061no—
350#97—532no—
351#97—1062no—
352C4×C8—532no—
353C4×C8—1062no—
354C4×C8—1061no—
355C24—41no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—1061no—
360C2×Q8—1061no—
361C2×Q8C2×C22⋊D5311yescharacteristic
362C2×Q8C2×C22⋊D5311yescharacteristic
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D5311yescharacteristic
377C22×C4#1171994711yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—2122no—
380Q16—2122no—
381SD16—44no—
382SD16—44no—
383M16—1062no—
384M16—1062no—
385M16—2122no—
386C2×C8—22no—
387C2×C8—42no—
388C2×C8—532no—
389C2×C8—532no—
390C2×C8—1062no—
391C2×C8—1061no—
392C2×C8—1061no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C4⋊C4C2×C22⋊D5311yescharacteristic
405C4⋊C4C2×C22⋊D5311yescharacteristic
406C22⋊C4—2121no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—2121no—
412C42D4×D5311yescharacteristic
413C42#1171994711yescharacteristic
414C42C2×C22⋊D5311yescharacteristic
415C42C23×D5311yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1313318911yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—4241no—
426Q8—2121no—
427Q8—2121no—
428Q8#1313324312yes—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4#1313318911yescharacteristic
449C2×C4#1313318911yescharacteristic
450C2×C4#1313318911yescharacteristic
451C2×C4#1313323812yes—
452C2×C4#1313324612yes—
453C2×C4C2×D4×D5311yescharacteristic
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C2×#1171994711yescharacteristic
456C8—42no—
457C8—1062no—
458C8—1062no—
459C8—2122no—
460C8—2121no—
461C8—2121no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1681084312yes—
469C22C2×#1313318911yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—4241no—
478C4—4241no—
479C4#1681084312yes—
480C4C2×#1313323811yescharacteristic
481C4C2×#1313324311yescharacteristic
482C4C2×#1313324611yescharacteristic
483C4#1681142111yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1681084311yescharacteristic
487C2#2028433011yescharacteristic
488C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#460C2
socleC22×C53C2×#121
prime radicalC2×#127×C53C22
hypercenterC2×#460D53
hypocenterC53#19174
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20284754C2semidirect
3. Q8×#11719917C2semidirect
4. #20274281C2semidirect
5. #20274253C2semidirect
6. #20274157C2semidirect
7. #20274129C2semidirect
8. #20273465C2semidirect
9. #20273403C2semidirect
17. #16810918C22semidirect
18. #16810918C22semidirect
19. #16810833C22semidirect
20. #16810833C22semidirect
23. #16810754C22semidirect
24. Q8×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
37. #16810370C22semidirect
293. C53#19174semidirect

Maximal subgroups

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

subgroupindex
#202850892
#202847542
Q8×#117199172
#202742812
#202742532
#202741572
#202741292
#202734652
#202734032
#202733832
C2×C53×#4602
#1917453