Finite Group Databasebeta

← #27393248

Subgroups — #27393248

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288359C211yesmaximal, characteristic
2#20283173C212yesmaximal, complemented
3#20275254C214yesmaximal, complemented
4#20275125C214yesmaximal, complemented
5#20274821C212yesmaximal, complemented
6#20274633C211yesmaximal, characteristic, complemented
7C53×#1782C211yesmaximal, characteristic, complemented
8C2×#13133262C2211yescharacteristic
9#16811272—24no—
10#16811261—24no—
11#16811197C412yescomplemented
12#16811182C412yescomplemented
13#16811182C2212yes—
14#16811134C412yescomplemented
15C2×#13133200C412yescomplemented
16C2×#13133200C2212yes—
17#16811098C2211yescharacteristic
18#16811096C2214yes—
19#16810595—24no—
20C2×#13133132C2212yes—
21C2×#13133132C2212yescomplemented
22C2×#13133131C2214yes—
23#99×D53—24no—
24#16810486C2212yescomplemented
25#16810390C2214yescomplemented
26#16810389C2214yescomplemented
27C22×#41×C53C2211yescharacteristic
28C2×#99×C53C2212yes—
29C2×#99×C53C2212yescomplemented
30C53×#381C2212yescomplemented
31#13133262—42no—
32#13133262—42no—
33C2×#11719945C2×C412yes—
34C2×#11719947D414yes—
35C22×C22⋊D53C2×C412yes—
36C22×C22⋊D53C2312yes—
37C2×D4×D53D414yes—
38#13133215—22no—
39#13133206—22no—
40#13133200—24no—
41#13133200—24no—
42#13133200C2×C414yescomplemented
43#13133200C2×C414yescomplemented
44#13133200C2×C414yescomplemented
45#13133198—22no—
46#13133198—22no—
47#13133196—22no—
48#13133193—24no—
49#13133191—22no—
50#13133189—24no—
51#13133189—24no—
52#13133186C2×C414yescomplemented
53#13133185—22no—
54#13133185—22no—
55D4×Dic53—24no—
56C4×C22⋊D53—24no—
57C4×C22⋊D53—44no—
58#13133177—44no—
59#13133176C2×C412yescomplemented
60#13133173—24no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×C22⋊Dic53C2×C412yescomplemented
64C2×C22⋊Dic53C2314yes—
65C2×C22⋊Dic53C2312yes—
66C2×#11719921—22no—
67C2×#11719921—22no—
68C22⋊C4×D53—24no—
69#13133132—24no—
70#13133132—42no—
71#13133131—28no—
72#13133124—42no—
73C2×#41×C53—24no—
74C2×#41×C53—42no—
75C2×#41×C53—42no—
76C22×Q8×C53C2×C411yescharacteristic, complemented
77C22×D4×C53C2×C411yescharacteristic, complemented
78C22×D4×C53C2311yescharacteristic
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53C2×C411yescharacteristic, complemented
81C23×C4×C53C2×C411yescharacteristic, complemented
82C23×C4×C53C2311yescharacteristic
83C2×C22⋊C4×C53C2314yes—
84#99×C53—24no—
85#99×C53—42no—
86#95×C53—42no—
87C24×C53C22×C411yescharacteristic
88C24×C53C22×C411yescharacteristic
89C24×C53C2411yescharacteristic
90C23×D53C22⋊C412yes—
91C23×D53C2×D412yes—
92#41×C53—416no—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×D4×C53—22no—
97C2×D4×C53—22no—
98C2×D4×C53—22no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53C2×D414yes—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—24no—
111C22×C4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C2×C22⋊D53—24no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53—24no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C2×C22⋊D53C22⋊C414yes—
125C2×C22⋊D53C22×C414yes—
126C2×C22⋊D53C2×D414yes—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—24no—
130C22×Dic53C22⋊C412yes—
131C22×Dic53C2×D412yes—
132#11719947—48no—
133D4×D53—48no—
134#11719945—42no—
135#11719945—42no—
136C2×D212C22⋊C412yes—
137C2×C4×D53—44no—
138C2×C4×D53C22⋊C414yes—
139C2×Dic106C22⋊C412yes—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—24no—
142C22⋊Dic53—22no—
143C22⋊Dic53—22no—
144C22⋊Dic53—24no—
145C22⋊Dic53—24no—
146C22⋊Dic53—24no—
147C22⋊Dic53C22×C414yes—
148#11719921—24no—
149#11719921—24no—
150#11719921—24no—
151#11719921—44no—
152C4⋊Dic53—22no—
153#11719919—24no—
154#11719919—24no—
155#11719919—24no—
156#11719919—44no—
157C4×Dic53—22no—
158C4×Dic53—44no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—81no—
165C23×C53—81no—
166C23×C53—81no—
167C23×C53C2×C22⋊C412yes—
168C23×C53C23×C411yescharacteristic
169C23×C53C22×D412yes—
170C23×C53H(5,𝔽2)12yes—
171C23×C53H(5,𝔽2)12yes—
172C22×D53—24no—
173C22×D53—42no—
174C22×D53—82no—
175C22×D53—82no—
176C22×D53C2×C22⋊C414yes—
177Q8×C53—44no—
178D4×C53—28no—
179D4×C53—44no—
180D4×C53—44no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
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—21no—
191C2×C4×C53—21no—
192C2×C4×C53—44no—
193C2×C4×C53—44no—
194C2×C4×C53—44no—
195C2×C4×C53—44no—
196C2×C4×C53—84no—
197C2×C4×C53C2×C22⋊C412yes—
198C22⋊D53—24no—
199C22⋊D53—24no—
200C22⋊D53—28no—
201C22⋊D53—44no—
202C22⋊D53—84no—
203C2×Dic53—24no—
204C2×Dic53—24no—
205C2×Dic53—42no—
206C2×Dic53—44no—
207C2×Dic53—82no—
208C2×Dic53—82no—
209C2×Dic53—84no—
210C2×Dic53C2×C22⋊C414yes—
211D212—42no—
212C4×D53—44no—
213C4×D53—164no—
214Dic106—42no—
215#21777—531nomaximal, Sylow
216C22×C53—22no—
217C22×C53—42no—
218C22×C53—42no—
219C22×C53—42no—
220C22×C53—41no—
221C22×C53—42no—
222C22×C53—42no—
223C22×C53—41no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53C22×C22⋊C411yescharacteristic
229C22×C53#51811yescharacteristic
230C22×C53#51811yescharacteristic
231C22×C53#53414yes—
232C2×D53—24no—
233C2×D53—44no—
234C2×D53—84no—
235C2×D53—162no—
236C4×C53—22no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53—41no—
240C4×C53—84no—
241Dic53—24no—
242Dic53—44no—
243Dic53—84no—
244#3485—531no—
245#2883—532no—
246#1913—534no—
247#1890—534no—
248#1828—532no—
249#1782—531no—
250#1782D5311yescharacteristic, complemented
251C2×C53—21no—
252C2×C53—24no—
253C2×C53—21no—
254C2×C53—21no—
255C2×C53—42no—
256C2×C53—81no—
257C2×C53#230811yescharacteristic
258D53—44no—
259C2×H(5,𝔽2)—531no—
260C22×#41C2×D5311yescharacteristic
261#547—1064no—
262#546—1064no—
263#536—532no—
264#534—532no—
265#534—532no—
266#525—532no—
267C2×#123—532no—
268C2×#123—532no—
269#518—531no—
270#518—534no—
271#410—1064no—
272C2×#99—532no—
273C2×#99—534no—
274C2×#99—532no—
275C2×#99—1064no—
276C2×#99C2×D5312yes—
277C2×#99C2×D5312yescomplemented
278#381—532no—
279#381C2×D5312yescomplemented
280#342—534no—
281#342—534no—
282C53#2177711yescharacteristic, complemented, Sylow
283H(5,𝔽2)—2122no—
284H(5,𝔽2)—2122no—
285C2×#41—24no—
286C2×#41—42no—
287C2×#41—42no—
288C2×#41—534no—
289C2×#41—532no—
290C22×Q8C4×D5311yescharacteristic, complemented
291C22×D4—532no—
292C22×D4—532no—
293C22×D4—534no—
294C22×D4C4×D5311yescharacteristic, complemented
295C22×D4C22×D5311yescharacteristic
296C22×D4C22×D5311yescharacteristic
297C23×C4C4×D5311yescharacteristic, complemented
298C23×C4C4×D5311yescharacteristic, complemented
299C23×C4C22×D5311yescharacteristic
300C4⋊Q8—1062no—
301#124—1062no—
302#123—534no—
303#123—534no—
304#123—534no—
305#123—1064no—
306#123—1064no—
307#123—1062no—
308#123—1062no—
309C22⋊Q8—1062no—
310C22⋊Q8—1064no—
311#121—1062no—
312#121—1064no—
313#121—1064no—
314#120—534no—
315#120—1062no—
316#120—1062no—
317C4×D4—1064no—
318C4×D4—1064no—
319C4×D4—2124no—
320C4×D4—2124no—
321#117—532no—
322#117—1064no—
323C2×C4⋊C4—1062no—
324C2×C4⋊C4—1062no—
325C2×C22⋊C4—532no—
326C2×C22⋊C4—532no—
327C2×C22⋊C4—534no—
328C2×C22⋊C4—1064no—
329C2×C22⋊C4—1062no—
330C2×C22⋊C4—1062no—
331C2×C22⋊C4C22×D5314yes—
332#99—24no—
333#99—42no—
334#99—1068no—
335#99—1064no—
336#99—2122no—
337#95—42no—
338#95—2122no—
339C24—532no—
340C24—532no—
341C24C2×C4×D5311yescharacteristic
342C24C2×C4×D5311yescharacteristic
343C24C23×D5311yescharacteristic
344#41—416no—
345#41—2122no—
346#41—2122no—
347#41—2128no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—532no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—534no—
361C2×D4—534no—
362C2×D4—534no—
363C2×D4—532no—
364C2×D4—1064no—
365C2×D4—1064no—
366C2×D4—1064no—
367C2×D4—2128no—
368C2×D4—2122no—
369C2×D4—2122no—
370C2×D4D4×D5314yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—532no—
382C22×C4—532no—
383C22×C4—534no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4—1064no—
387C22×C4—2124no—
388C22×C4C2×C4×D5311yescharacteristic
389C22×C4C2×C4×D5311yescharacteristic
390C22×C4C2×C4×D5311yescharacteristic
391C22×C4C2×C4×D5311yescharacteristic
392C4⋊C4—1064no—
393C4⋊C4—1064no—
394C4⋊C4—1064no—
395C4⋊C4—1062no—
396C4⋊C4—2124no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—534no—
400C22⋊C4—1064no—
401C22⋊C4—1064no—
402C22⋊C4—1064no—
403C22⋊C4—1064no—
404C22⋊C4—1062no—
405C22⋊C4—1062no—
406C22⋊C4—1064no—
407C22⋊C4—1064no—
408C22⋊C4—2124no—
409C42—1062no—
410C42—2124no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—534no—
420C23—1064no—
421C23—4242no—
422C23—4242no—
423C23—2122no—
424C23C22⋊C4×D5312yes—
425C23C22×C4×D5311yescharacteristic
426C23C2×D4×D5312yes—
427C23#1313326212yes—
428C23#1313326212yes—
429Q8—44no—
430Q8—2122no—
431D4—28no—
432D4—44no—
433D4—44no—
434D4—1064no—
435D4—1064no—
436D4—1068no—
437D4—4244no—
438D4—2122no—
439D4—2124no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—84no—
456C2×C4—534no—
457C2×C4—8484no—
458C2×C4—1064no—
459C2×C4—1064no—
460C2×C4—4244no—
461C2×C4—4242no—
462C2×C4—4242no—
463C2×C4—2124no—
464C2×C4—2122no—
465C2×C4—2124no—
466C2×C4C22⋊C4×D5312yes—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—8482no—
480C22—1064no—
481C22—4244no—
482C22—2124no—
483C22C2×C22⋊C4×D5311yescharacteristic
484C22#1681109611yescharacteristic
485C22#1681109611yescharacteristic
486C22#1681118314yes—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—84no—
492C4—1064no—
493C4—4244no—
494C4—2124no—
495C2—24no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—81no—
501C2—2124no—
502C2#2027729111yescharacteristic
subgroupG/H
centerC2#20277291
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1782C2
socleC2×C53#2308
prime radical#20288359C2
cosocleC24×C53C24
hypercenter#1782D53
hypocenterC53#21777
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #20283173C2semidirect
3. #20275254C2semidirect
4. #20275125C2semidirect
5. #20274821C2semidirect
6. #20274633C2semidirect
7. C53×#1782C2semidirect
11. #16811197C4semidirect
12. #16811182C4semidirect
14. #16811134C4semidirect
15. C2×#13133200C4semidirect
21. C2×#13133132C22semidirect
24. #16810486C22semidirect
25. #16810390C22semidirect
26. #16810389C22semidirect
29. C2×#99×C53C22semidirect
30. C53×#381C22semidirect
42. #13133200C2×C4semidirect
43. #13133200C2×C4semidirect
44. #13133200C2×C4semidirect
52. #13133186C2×C4semidirect
59. #13133176C2×C4semidirect
63. C2×C22⋊Dic53C2×C4semidirect
76. C22×Q8×C53C2×C4semidirect
77. C22×D4×C53C2×C4semidirect
80. C23×C4×C53C2×C4semidirect
81. C23×C4×C53C2×C4semidirect
250. #1782D53semidirect
277. C2×#99C2×D53semidirect
279. #381C2×D53semidirect
282. C53#21777semidirect
290. C22×Q8C4×D53semidirect
294. C22×D4C4×D53semidirect
297. C23×C4C4×D53semidirect
298. C23×C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202883592
#202831732
#202752542
#202751252
#202748212
#202746332
C53×#17822
#2177753