Finite Group Databasebeta

← #27356501

Subgroups — #27356501

Order 13568.

1398 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1398 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284651C212yesmaximal, complemented
2#20278989C211yesmaximal, characteristic, complemented
3#20273735C214yesmaximal, complemented
4#20273701C214yesmaximal, complemented
5#20273244C212yesmaximal, complemented
6#20273042C211yesmaximal, characteristic, complemented
7C2×C53×#534C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic, complemented
9#16810944—24no—
10#16810939C2214yescomplemented
11#16810934C2214yescomplemented
12#16810663—22no—
13#16810628—22no—
14#16810587C2212yescomplemented
15#16810559C2212yescomplemented
16#16810548C2212yescomplemented
17#16810511—22no—
18#16810496—21no—
19#16810486C2211yescharacteristic, complemented
20#16810479—21no—
21#16810357C2218yescomplemented
22#16810347C2214yescomplemented
23C23×D4×C53—21no—
24C53×#534—24no—
25C53×#534—22no—
26C53×#534C2214yes—
27C2×#123×C53—22no—
28C2×#121×C53—21no—
29C2×#121×C53C2212yes—
30C2×#120×C53—22no—
31C22×C22⋊C4×C53C2211yescharacteristic
32C2×#11719917—22no—
33#13133176—22no—
34#13133176—22no—
35C2×C4⋊Dic53C2311yescharacteristic, complemented
36C2×C4⋊Dic53C2311yescharacteristic, complemented
37C2×#11719919—22no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42C2×C4×Dic53—21no—
43#13133150—24no—
44#13133148—24no—
45#13133138—24no—
46#13133138—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yescomplemented
49#13133124—22no—
50#13133124—21no—
51#13133124C2312yescomplemented
52C25×C53—21no—
53C25×C53D411yescharacteristic
54C22×D4×C53—22no—
55C22×D4×C53—24no—
56C22×D4×C53—22no—
57C22×D4×C53—24no—
58C22×D4×C53—21no—
59C22×D4×C53—48no—
60C22×D4×C53D412yes—
61C23×C4×C53—21no—
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64#123×C53—28no—
65#123×C53—28no—
66#121×C53—24no—
67#121×C53—24no—
68#121×C53—22no—
69#121×C53—22no—
70#121×C53D414yes—
71#121×C53C2314yes—
72#120×C53—28no—
73#120×C53—28no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53D412yes—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—22no—
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53D412yes—
81C2×C22⋊C4×C53C2312yes—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—24no—
85C24×C53—24no—
86C24×C53—22no—
87C24×C53—44no—
88C24×C53—42no—
89C24×C53—81no—
90C24×C53—82no—
91C24×C53—82no—
92C24×C53C2×D411yescharacteristic
93C24×C53C2×D412yes—
94C2×D4×C53—28no—
95C2×D4×C53—24no—
96C2×D4×C53—24no—
97C2×D4×C53—24no—
98C2×D4×C53—22no—
99C2×D4×C53—44no—
100C2×D4×C53—48no—
101C2×D4×C53—416no—
102C2×D4×C53—48no—
103C2×D4×C53—48no—
104C2×D4×C53—48no—
105C2×D4×C53—82no—
106C2×D4×C53C2×D414yes—
107C22×C4×C53—24no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—42no—
112C22×C4×C53—44no—
113C22×C4×C53—81no—
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2×D412yes—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2411yescharacteristic
119C22×Dic53—22no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C4⋊C4×C53—24no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53C2×D414yes—
125C22⋊C4×C53—28no—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C22⋊Dic53—42no—
131C22⋊Dic53—41no—
132C22⋊Dic53—41no—
133C22⋊Dic53—44no—
134C22⋊Dic53—82no—
135C4⋊Dic53—22no—
136C4⋊Dic53—22no—
137#11719919—82no—
138C4×Dic53—42no—
139#11719917—44no—
140C2×#88322—24no—
141C23×C53—24no—
142C23×C53—24no—
143C23×C53—22no—
144C23×C53—22no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53—24no—
148C23×C53—41no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—44no—
153C23×C53—48no—
154C23×C53—44no—
155C23×C53—42no—
156C23×C53—44no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53—88no—
160C23×C53—84no—
161C23×C53—84no—
162C23×C53—816no—
163C23×C53#12011yescharacteristic
164C23×C53#12011yescharacteristic
165C23×C53AGL(1,ℤ/8)14yes—
166C23×C53C22×D411yescharacteristic
167D4×C53—44no—
168D4×C53—416no—
169D4×C53—48no—
170D4×C53—48no—
171D4×C53—82no—
172D4×C53—84no—
173C2×C4×C53—24no—
174C2×C4×C53—24no—
175C2×C4×C53—22no—
176C2×C4×C53—42no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—44no—
180C2×C4×C53—48no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—81no—
184C2×C4×C53#12012yes—
185C2×C4×C53C22×D412yes—
186C2×Dic53—42no—
187C2×Dic53—42no—
188C2×Dic53—42no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193#88322—44no—
194#18137—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—24no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—44no—
204C22×C53—44no—
205C22×C53—48no—
206C22×C53—42no—
207C22×C53—42no—
208C22×C53—44no—
209C22×C53—44no—
210C22×C53—44no—
211C22×C53—82no—
212C22×C53—88no—
213C22×C53—84no—
214C22×C53—816no—
215C22×C53#45812yes—
216C22×C53C2×#12011yescharacteristic
217C22×C53C2×AGL(1,ℤ/8)12yes—
218C4×C53—22no—
219C4×C53—22no—
220C4×C53—44no—
221C4×C53—42no—
222Dic53—82no—
223Dic53—81no—
224Dic53—81no—
225C2×#534D5311yescharacteristic
226#3094—532no—
227#2482—531no—
228#1613—534no—
229#1607—534no—
230#1510—532no—
231#1462—531no—
232C2×C53—22no—
233C2×C53—22no—
234C2×C53—41no—
235C2×C53—44no—
236C2×C53—84no—
237C2×C53#300311yescharacteristic
238C2×C53#301911yescharacteristic
239C2×C53C2×#45811yescharacteristic
240C23×D4—21no—
241#534—24no—
242#534—22no—
243#534C2×D5314yes—
244C2×#123—22no—
245C2×#121—21no—
246C2×#121C2×D5312yes—
247C2×#120—22no—
248C2×#117—531no—
249C22×C22⋊C4C2×D5311yescharacteristic
250#482—1064no—
251#481—534no—
252#480—534no—
253#428—1062no—
254#418—1062no—
255#407—532no—
256#399—532no—
257#397—532no—
258#388—1062no—
259#385—1061no—
260#381—531no—
261#379—1061no—
262#327—538no—
263#323—534no—
264C53#1813711yescharacteristic, complemented, Sylow
265C25—21no—
266C25C22⋊D5311yescharacteristic
267C22×D4—22no—
268C22×D4—24no—
269C22×D4—22no—
270C22×D4—24no—
271C22×D4—21no—
272C22×D4—48no—
273C22×D4C22⋊D5312yes—
274C23×C4—21no—
275C23×C4C22⋊D5311yescharacteristic
276C23×C4C22×D5311yescharacteristic
277C2×M16—1062no—
278#123—28no—
279#123—28no—
280#121—24no—
281#121—24no—
282#121—22no—
283#121—22no—
284#121C22⋊D5314yes—
285#121C22×D5314yes—
286#120—28no—
287#120—28no—
288#117—1062no—
289#117—1062no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4—531no—
293C2×C4⋊C4—1062no—
294C2×C4⋊C4C22⋊D5312yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—1062no—
300C2×C22⋊C4—1061no—
301C2×C22⋊C4—1061no—
302C2×C22⋊C4—1062no—
303C2×C22⋊C4C22⋊D5312yes—
304C2×C22⋊C4C22×D5312yes—
305C2×C42—1061no—
306#107—1064no—
307#106—1064no—
308#102—1064no—
309#102—1064no—
310C22⋊C8—534no—
311C22⋊C8—1062no—
312#95—532no—
313#95—1061no—
314#95—1062no—
315C24—21no—
316C24—21no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—44no—
321C24—42no—
322C24—81no—
323C24—82no—
324C24—82no—
325C24C2×C22⋊D5311yescharacteristic
326C24C2×C22⋊D5312yes—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—82no—
339C2×D4C2×C22⋊D5314yes—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—81no—
347C22×C4—1062no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4C2×C22⋊D5312yes—
351C22×C4C2×C22⋊D5312yes—
352C22×C4C2×C22⋊D5311yescharacteristic
353C22×C4C2×C22⋊D5312yes—
354C22×C4C23×D5311yescharacteristic
355M16—2124no—
356C2×C8—1064no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—1062no—
360C4⋊C4—1062no—
361C4⋊C4—4242no—
362C4⋊C4C2×C22⋊D5314yes—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—4242no—
369C22⋊C4—2122no—
370C22⋊C4—2121no—
371C22⋊C4—2121no—
372C22⋊C4—2124no—
373C42—2122no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—48no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—81no—
391C23—81no—
392C23—88no—
393C23—84no—
394C23—84no—
395C23—816no—
396C23#1313318611yescharacteristic
397C23#1313318611yescharacteristic
398C23#1313324314yes—
399C23C22×C22⋊D5311yescharacteristic
400D4—44no—
401D4—416no—
402D4—48no—
403D4—48no—
404D4—82no—
405D4—84no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—2122no—
418C2×C4—2122no—
419C2×C4—2122no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2122no—
423C2×C4—2122no—
424C2×C4#1313318612yes—
425C2×C4C22×C22⋊D5312yes—
426C8—2124no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—82no—
444C22—88no—
445C22—84no—
446C22—816no—
447C22#1681081312yes—
448C22C2×#1313318611yescharacteristic
449C22C2×#1313324312yes—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—42no—
454C4—4242no—
455C4—4241no—
456C4—4241no—
457C2—22no—
458C2—22no—
459C2—41no—
460C2—44no—
461C2—84no—
462C2#2028394211yescharacteristic
463C2#2028407211yescharacteristic
464C2C2×#1681081311yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#534C2
socleC22×C53C2×#120
hypercenterC2×#534D53
hypocenterC53#18137
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #20284651C2semidirect
2. #20278989C2semidirect
3. #20273735C2semidirect
4. #20273701C2semidirect
5. #20273244C2semidirect
6. #20273042C2semidirect
8. C2×#13133176C22semidirect
10. #16810939C22semidirect
11. #16810934C22semidirect
14. #16810587C22semidirect
15. #16810559C22semidirect
16. #16810548C22semidirect
19. #16810486C22semidirect
21. #16810357C22semidirect
22. #16810347C22semidirect
35. C2×C4⋊Dic53C23semidirect
36. C2×C4⋊Dic53C23semidirect
48. C22⋊#88322C23semidirect
51. #13133124C23semidirect
264. C53#18137semidirect

Maximal subgroups

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

subgroupindex
#202846512
#202789892
#202737352
#202737012
#202732442
#202730422
C2×C53×#5342
#1813753