Finite Group Databasebeta

← #27366994

Subgroups — #27366994

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#20285091C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274285C212yesmaximal, complemented
5#20274257C212yesmaximal, complemented
6#20274161C212yesmaximal, complemented
7#20274133C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273382C211yesmaximal, characteristic
11C53×#3104C211yesmaximal, characteristic, complemented
12C22×#11719924—21no—
13C2×#13133214C2211yescharacteristic
14#16810989—22no—
15#16810971C2212yes—
16#16810971C2212yes—
17#16810823—21no—
18#16810823—21no—
19#16810823C2212yes—
20#16810823C2212yes—
21#16810754—22no—
22#16810754C2212yescomplemented
23Q8×#88322C2212yescomplemented
24#16810647—21no—
25#16810647C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810627—21no—
28#16810627—21no—
29C2×#13133137—21no—
30C4×#11719917—21no—
31C4×#11719917C2211yescharacteristic
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#533C2211yescharacteristic
39C2×C4×Q8×C53C2211yescharacteristic
40C53×#484—21no—
41C53×#477C2211yescharacteristic, complemented
42C53×#477C2211yescharacteristic, complemented
43C53×#462C2211yescharacteristic, complemented
44C53×#462C2211yescharacteristic, complemented
45C53×#451—21no—
46C4×Q16×C53—21no—
47C4×Q16×C53—21no—
48C53×#431C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719924—44no—
55C2×#11719917—21no—
56C2×#11719917D411yescharacteristic
57C2×#11719917D411yescharacteristic
58C22×#88322—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214C2312yes—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322—22no—
66C4⋊#88322—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75#13133128—22no—
76#13133128D412yes—
77C4×#88322—22no—
78C4×#88322C2312yes—
79C22×Q8×C53D411yescharacteristic
80C2×Q16×C53—21no—
81C2×Q16×C53—21no—
82C4⋊Q8×C53C2311yescharacteristic
83C4⋊Q8×C53C2311yescharacteristic
84#125×C53—21no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×Q8×C53C2311yescharacteristic
95#117×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#107×C53—21no—
101#107×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#103×C53—21no—
105#103×C53—21no—
106#103×C53—21no—
107#103×C53—21no—
108C22⋊C8×C53—21no—
109C4×C8×C53—21no—
110C23×D53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—41no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×Q8×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—42no—
127C2×D212—42no—
128C2×D212—42no—
129C2×D212—42no—
130C2×D212—41no—
131C2×D212—41no—
132C2×D212—41no—
133C2×D212—82no—
134Q16×C53—42no—
135Q16×C53—42no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—41no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—41no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719924—44no—
162#11719924—44no—
163#11719917—22no—
164#11719917—22no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—42no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53#12111yescharacteristic
172C22×D53—42no—
173C22×D53—82no—
174C22×D53—82no—
175C22×D53—82no—
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—42no—
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53—81no—
183Q8×C53AGL(1,ℤ/8)12yes—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—81no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#13512yes—
199C2×C4×C53#13712yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203D212—42no—
204D212—42no—
205D212—42no—
206D212—42no—
207D212—82no—
208D212—82no—
209D212—82no—
210D212—82no—
211C8×C53—41no—
212C8×C53—41no—
213#88322—22no—
214#88322—22no—
215#88322—42no—
216#88322—42no—
217#19174—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53#46412yes—
221C22×C53C2×#12111yescharacteristic
222C2×D53—82no—
223C2×D53—82no—
224C2×D53—82no—
225C2×D53—164no—
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53#46412yes—
236C4×C53C2×#13511yescharacteristic
237C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C53C2×#13711yescharacteristic
239C4×C53#57511yescharacteristic
240#3138—531no—
241#3104D5311yescharacteristic, complemented
242C2×#460—531no—
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×C53C2×#46411yescharacteristic
253C2×C53#305511yescharacteristic
254C2×C53#306811yescharacteristic
255D53—162no—
256C22×SD16—1061no—
257#533C2×D5311yescharacteristic
258C2×#127—531no—
259C2×C4×Q8C2×D5311yescharacteristic
260#490—1062no—
261#486—532no—
262#486—532no—
263#484—21no—
264#477C2×D5311yescharacteristic, complemented
265#477C2×D5311yescharacteristic, complemented
266#462C2×D5311yescharacteristic, complemented
267#462C2×D5311yescharacteristic, complemented
268#460—532no—
269#460—532no—
270#460—1061no—
271#460—1061no—
272#451—21no—
273#446—532no—
274#446—1062no—
275C8×Q8—532no—
276C4×Q16—21no—
277C4×Q16—21no—
278#431C2×D5311yescharacteristic
279#423—531no—
280#423—1061no—
281C2×C4⋊C8—531no—
282#418—1061no—
283#418—1061no—
284C2×#102—1061no—
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—1061no—
296C22×D4—1061no—
297C22×D4—1061no—
298C2×Q16—21no—
299C2×Q16—21no—
300C2×SD16—1062no—
301C2×SD16—1062no—
302C2×SD16—2124no—
303C2×M16—531no—
304C2×M16—531no—
305C2×M16—1061no—
306C22×C8—1061no—
307C4⋊Q8C22×D5311yescharacteristic
308C4⋊Q8C22×D5311yescharacteristic
309#127—532no—
310#127—1061no—
311#127—1061no—
312#127—1061no—
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D5311yescharacteristic
321C4×Q8C22⋊D5311yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×Q8C22×D5311yescharacteristic
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D5311yescharacteristic
327C2×C42—21no—
328C2×C42C22×D5311yescharacteristic
329#107—21no—
330#107—21no—
331C4⋊C8—532no—
332C4⋊C8—532no—
333C4⋊C8—1062no—
334C4⋊C8—1062no—
335C4⋊C8—1061no—
336C4⋊C8—1061no—
337C4⋊C8C22×D5311yescharacteristic
338C4⋊C8C22×D5311yescharacteristic
339#103—21no—
340#103—21no—
341#103—21no—
342#103—21no—
343#102—1062no—
344#102—1062no—
345#102—1062no—
346#102—1062no—
347#102—1062no—
348#102—1062no—
349C22⋊C8—21no—
350#97—532no—
351#97—1062no—
352C4×C8—21no—
353C4×C8—532no—
354C4×C8—1062no—
355C24—2121no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8C2×C22⋊D5311yescharacteristic
362C2×Q8C2×C22⋊D5311yescharacteristic
363C2×D4—1062no—
364C2×D4—1062no—
365C2×D4—1062no—
366C2×D4—4242no—
367C2×D4—2122no—
368C2×D4—2122no—
369C2×D4—2121no—
370C2×D4—2121no—
371C2×D4—2121no—
372C2×D4—2122no—
373C2×D4—2122no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D5311yescharacteristic
377C22×C4#1171994711yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—42no—
380Q16—42no—
381SD16—2124no—
382SD16—2124no—
383M16—1062no—
384M16—1062no—
385M16—2122no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—532no—
389C2×C8—532no—
390C2×C8—1062no—
391C2×C8—1062no—
392C2×C8—2122no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C2×C22⋊D5311yescharacteristic
405C4⋊C4C2×C22⋊D5311yescharacteristic
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42D4×D5311yescharacteristic
413C42#1171994711yescharacteristic
414C42C2×C22⋊D5311yescharacteristic
415C42C23×D5311yescharacteristic
416C23—4242no—
417C23—4242no—
418C23—4242no—
419C23—2122no—
420C23#1313318911yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8#1313324212yes—
429D4—4242no—
430D4—4242no—
431D4—4242no—
432D4—4242no—
433D4—2122no—
434D4—2122no—
435D4—2122no—
436D4—2122no—
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—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4#1313318911yescharacteristic
449C2×C4#1313318911yescharacteristic
450C2×C4#1313318911yescharacteristic
451C2×C4#1313324012yes—
452C2×C4#1313324912yes—
453C2×C4C2×D4×D5311yescharacteristic
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C2×#1171994711yescharacteristic
456C8—41no—
457C8—41no—
458C8—1062no—
459C8—1062no—
460C8—2122no—
461C8—2122no—
462C22—21no—
463C22—21no—
464C22—8484no—
465C22—4242no—
466C22—4242no—
467C22—4242no—
468C22#1681084712yes—
469C22C2×#1313318911yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4#1681084712yes—
480C4C2×#1313324011yescharacteristic
481C4C2×#1313324211yescharacteristic
482C4C2×#1313324911yescharacteristic
483C4#1681142111yescharacteristic
484C2—22no—
485C2—8482no—
486C2C2×#1681084711yescharacteristic
487C2#2028433611yescharacteristic
488C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3104C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#3104D53
hypocenterC53#19174
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. Q8×#11719917C2semidirect
4. #20274285C2semidirect
5. #20274257C2semidirect
6. #20274161C2semidirect
7. #20274133C2semidirect
8. #20273468C2semidirect
9. #20273405C2semidirect
11. C53×#3104C2semidirect
22. #16810754C22semidirect
23. Q8×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
41. C53×#477C22semidirect
42. C53×#477C22semidirect
43. C53×#462C22semidirect
44. C53×#462C22semidirect
241. #3104D53semidirect
264. #477C2×D53semidirect
265. #477C2×D53semidirect
266. #462C2×D53semidirect
267. #462C2×D53semidirect
293. C53#19174semidirect

Maximal subgroups

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

subgroupindex
#202850912
C2×#168108232
Q8×#117199172
#202742852
#202742572
#202741612
#202741332
#202734682
#202734052
#202733822
C53×#31042
#1917453