Finite Group Databasebeta

← #27440242

Subgroups — #27440242

Order 13568.

1094 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#20279194C212yesmaximal, complemented
2#20279032C211yesmaximal, characteristic
3#20276067C214yesmaximal, complemented
4#20275701C212yesmaximal, complemented
5#20274996C214yesmaximal
6C4×#13133132C211yesmaximal, characteristic
7C53×#3501C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810562—21no—
11#16810559—21no—
12#16810553—24no—
13#16810530—22no—
14#16810521—21no—
15#16810503C2212yes—
16#16810495C2218yes—
17#16810495C2212yes—
18#16810492—22no—
19#16810483—21no—
20#16810483C2211yescharacteristic
21C4×C22⋊Dic53C2212yes—
22#16810390C2218yes—
23#16810390C2212yes—
24C53×#554—24no—
25C53×#554—22no—
26D4×Q8×C53—22no—
27C53×#544—21no—
28C53×#543—21no—
29C53×#543C2212yes—
30C53×#533—21no—
31C2×C22⋊Q8×C53—21no—
32C2×C22⋊Q8×C53C2212yes—
33C53×#518C2212yes—
34C2×C4×D4×C53C2211yescharacteristic
35C23×Dic53C2312yes—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—21no—
41C2×#11719919—22no—
42C2×#11719919—28no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53C2312yes—
45C2×C22⋊Dic53C2314yes—
46C2×C4×Dic53—22no—
47#13133132Q818yescomplemented
48#13133124—22no—
49#13133124—28no—
50#13133124—22no—
51#13133124—21no—
52#13133124—22no—
53C22×Q8×C53—21no—
54C22×Q8×C53D412yes—
55C22×D4×C53C2311yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53C2311yescharacteristic
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—24no—
61#125×C53—22no—
62#125×C53—24no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53—22no—
72C22⋊Q8×C53—22no—
73C22⋊Q8×C53—24no—
74C22⋊Q8×C53—24no—
75C4×Q8×C53—22no—
76C4×Q8×C53—22no—
77C4×D4×C53—22no—
78C4×D4×C53—22no—
79C4×D4×C53—24no—
80C4×D4×C53—24no—
81#117×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53D411yescharacteristic
85C2×C4⋊C4×C53D414yes—
86C2×C4⋊C4×C53D412yes—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53D411yescharacteristic
89C2×C22⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53C2314yes—
91C2×C42×C53D411yescharacteristic
92C24×C53C2×D411yescharacteristic
93C24×C53C2411yescharacteristic
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53—22no—
98C2×Q8×C53—44no—
99C2×Q8×C53—82no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53C2×Q814yes—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D414yes—
114C22×C4×C53C2×D414yes—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×Dic53—24no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—24no—
122C22×Dic53—82no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—24no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—24no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—24no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—24no—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—24no—
138C22⋊C4×C53—22no—
139C42×C53—24no—
140C42×C53—21no—
141C42×C53—21no—
142C22⋊Dic53—24no—
143C22⋊Dic53—24no—
144C22⋊Dic53—82no—
145C22⋊Dic53C2×Q818yescomplemented
146C4⋊Dic53—42no—
147C4⋊Dic53—41no—
148C4⋊Dic53—42no—
149C4⋊Dic53—41no—
150#11719919—88no—
151#11719919—82no—
152C4×Dic53—44no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53C4⋊Q812yes—
160C23×C53C22×D411yescharacteristic
161C23×C53C22×D412yes—
162C23×C53C22×Q812yes—
163C23×C532−1+412yes—
164Q8×C53—44no—
165Q8×C53—82no—
166Q8×C53—82no—
167Q8×C53—82no—
168D4×C53—28no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—22no—
172C2×C4×C53—24no—
173C2×C4×C53—21no—
174C2×C4×C53—24no—
175C2×C4×C53—24no—
176C2×C4×C53—21no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—44no—
182C2×C4×C53—44no—
183C2×C4×C53—41no—
184C2×C4×C53—81no—
185C2×C4×C53—82no—
186C2×C4×C53—82no—
187C2×C4×C53#12011yescharacteristic
188C2×C4×C53#12012yes—
189C2×C4×C53#12011yescharacteristic
190C2×C4×C53C4⋊Q812yes—
191C2×Dic53—28no—
192C2×Dic53—44no—
193C2×Dic53—44no—
194C2×Dic53—48no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201#25853—531nomaximal, Sylow
202C22×C53—22no—
203C22×C53—21no—
204C22×C53—22no—
205C22×C53—21no—
206C22×C53—22no—
207C22×C53—42no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—81no—
213C22×C53C2×#12011yescharacteristic
214C22×C53C2×C4⋊Q811yescharacteristic
215C22×C53#53611yescharacteristic
216C22×C53D4×Q814yes—
217C4×C53—22no—
218C4×C53—21no—
219C4×C53—22no—
220C4×C53—41no—
221C4×C53—44no—
222C4×C53—44no—
223C4×C53—41no—
224C4×C53#45812yes—
225Dic53—48no—
226Dic53—82no—
227Dic53—82no—
228#3501D5311yescharacteristic
229#2502—532no—
230#2487—531no—
231#2062—534no—
232#1996—532no—
233#1867—534no—
234C4×#99—531no—
235C2×C53—24no—
236C2×C53—21no—
237C2×C53—21no—
238C2×C53—42no—
239C2×C53#243411yescharacteristic
240C2×C53C2×#45811yescharacteristic
241C2×C53#302711yescharacteristic
242#554—22no—
243#554—24no—
244D4×Q8—22no—
245#544—21no—
246#543—21no—
247#543C2×D5312yes—
248#533—21no—
249C2×C22⋊Q8—21no—
250C2×C22⋊Q8C2×D5312yes—
251#518C2×D5312yes—
252C2×C4×D4C2×D5311yescharacteristic
253C22×C4⋊C4—531no—
254C2×#99—532no—
255#400—1061no—
256#399—1061no—
257#398—1064no—
258#391—1062no—
259#389—1061no—
260#386—532no—
261#385—532no—
262#385—538no—
263#384—1062no—
264#380—531no—
265#380—1061no—
266C4×C22⋊C4—532no—
267#342—532no—
268#342—538no—
269C53#2585311yescharacteristic, complemented, Sylow
270C22×Q8—21no—
271C22×Q8C22⋊D5312yes—
272C22×D4C22×D5311yescharacteristic
273C23×C4—532no—
274C23×C4C22⋊D5311yescharacteristic
275C23×C4C22×D5311yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—24no—
279#125—22no—
280#125—24no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—24no—
299#117—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—1068no—
303C2×C4⋊C4—1062no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1062no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4C22⋊D5311yescharacteristic
310C2×C4⋊C4C22⋊D5312yes—
311C2×C4⋊C4C22⋊D5314yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—534no—
314C2×C22⋊C4—532no—
315C2×C22⋊C4—1062no—
316C2×C22⋊C4C22⋊D5311yescharacteristic
317C2×C22⋊C4C22×D5311yescharacteristic
318C2×C22⋊C4C22×D5314yes—
319C2×C42—1062no—
320C2×C42C22⋊D5311yescharacteristic
321#99—538no—
322#95—1062no—
323#95—1061no—
324#95—1062no—
325#95—1068no—
326#95—1062no—
327C24C2×C22⋊D5311yescharacteristic
328C24C23×D5311yescharacteristic
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—82no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4Q8×D5314yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—1064no—
345C22×C4—1062no—
346C22×C4—1062no—
347C22×C4—1064no—
348C22×C4—4242no—
349C22×C4C2×C22⋊D5312yes—
350C22×C4C2×C22⋊D5311yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352C22×C4C2×C22⋊D5314yes—
353C22×C4C2×C22⋊D5311yescharacteristic
354C22×C4C2×C22⋊D5311yescharacteristic
355C22×C4C2×C22⋊D5314yes—
356C22×C4C2×C22⋊D5311yescharacteristic
357C22×C4C2×C22⋊D5312yes—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—4242no—
370C4⋊C4—4248no—
371C4⋊C4—2122no—
372C4⋊C4—2121no—
373C4⋊C4—2122no—
374C4⋊C4—2121no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—538no—
381C22⋊C4—1064no—
382C22⋊C4—1064no—
383C22⋊C4—4242no—
384C42—24no—
385C42—21no—
386C42—21no—
387C42—2124no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#1313321512yes—
395C23C22×C22⋊D5312yes—
396C23C22×C22⋊D5311yescharacteristic
397C23C2×Q8×D5312yes—
398C23#1313326512yes—
399Q8—44no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403D4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—1068no—
423C2×C4—2122no—
424C2×C4—2122no—
425C2×C4—2128no—
426C2×C4—2122no—
427C2×C4—2122no—
428C2×C4—2122no—
429C2×C4—2122no—
430C2×C4—2124no—
431C2×C4—2124no—
432C2×C4#1313318611yescharacteristic
433C2×C4#1313318612yes—
434C2×C4#1313318611yescharacteristic
435C2×C4#1313321512yes—
436C22—22no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#1313318611yescharacteristic
448C22C2×#1313321511yescharacteristic
449C22#1681119711yescharacteristic
450C22Q8×C22⋊D5314yes—
451C4—22no—
452C4—21no—
453C4—22no—
454C4—41no—
455C4—44no—
456C4—44no—
457C4—41no—
458C4—4242no—
459C4—4242no—
460C4—2128no—
461C4#1681081312yes—
462C2—21no—
463C2—24no—
464C2—21no—
465C2—42no—
466C2#2027835511yescharacteristic
467C2C2×#1681081311yescharacteristic
468C2#2028411711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#3501C2
socleC22×C53C2×#120
prime radicalC22×D4×C53C23
hypercenter#3501D53
hypocenterC53#25853
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #20279194C2semidirect
3. #20276067C2semidirect
4. #20275701C2semidirect
47. #13133132Q8semidirect
145. C22⋊Dic53C2×Q8semidirect
269. C53#25853semidirect

Maximal subgroups

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

subgroupindex
#202791942
#202790322
#202760672
#202757012
#202749962
C4×#131331322
C53×#35012
#2585353