Finite Group Databasebeta

← #27357947

Subgroups — #27357947

Order 13568.

1064 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285487C212yesmaximal, complemented
2#20280520C211yesmaximal, characteristic
3#20273740C214yesmaximal, complemented
4#20273672C214yesmaximal, complemented
5#20273287C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic
7C2×C53×#543C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810956C2214yescomplemented
11#16810862—24no—
12#16810779—22no—
13#16810778C2214yescomplemented
14C2×C22⋊#88322C2212yes—
15#16810559C2212yes—
16#16810546—22no—
17#16810535C2212yes—
18#16810496—21no—
19#16810483C2211yescharacteristic
20#16810479—21no—
21#16810360C2218yescomplemented
22#16810347C2214yes—
23C53×#543—24no—
24C53×#543—22no—
25C53×#543C2214yes—
26C2×C22⋊Q8×C53—22no—
27C2×C22⋊Q8×C53—21no—
28C2×C22⋊Q8×C53—22no—
29C2×C22⋊Q8×C53C2212yes—
30C22×C22⋊C4×C53—21no—
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719925—24no—
34C22×#88322—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45#13133150—24no—
46#13133141—24no—
47#13133140—24no—
48C22⋊#88322—22no—
49C22⋊#88322C2314yes—
50#13133124—22no—
51#13133124C2312yes—
52C25×C53D411yescharacteristic
53C22×Q8×C53—21no—
54C22×Q8×C53D412yes—
55C23×C4×C53—21no—
56C23×C4×C53—21no—
57C23×C4×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C22⋊Q8×C53—28no—
60C22⋊Q8×C53—28no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—28no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—24no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53D414yes—
68C22⋊Q8×C53C2314yes—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53D412yes—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—24no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—24no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—22no—
79C2×C22⋊C4×C53D412yes—
80C2×C22⋊C4×C53C2312yes—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—22no—
84C24×C53—81no—
85C24×C53—81no—
86C24×C53Q1614yescomplemented
87C24×C53C2×D412yes—
88C24×C53C2×D411yescharacteristic
89C2×Q8×C53—24no—
90C2×Q8×C53—22no—
91C2×Q8×C53—82no—
92C2×Q8×C53—82no—
93C2×Q8×C53C2×D414yes—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—24no—
99C22×C4×C53—21no—
100C22×C4×C53—81no—
101C22×C4×C53—81no—
102C22×C4×C53—82no—
103C22×C4×C53—82no—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2411yescharacteristic
109C22×Dic53—21no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C2×Dic106—22no—
113C2×Dic106—81no—
114C4⋊C4×C53—28no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—24no—
119C4⋊C4×C53C2×D414yes—
120C22⋊C4×C53—28no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—24no—
124C22⋊C4×C53—24no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127C22⋊Dic53—42no—
128C22⋊Dic53—82no—
129C22⋊Dic53—82no—
130#11719925—84no—
131C4⋊Dic53—22no—
132C4⋊Dic53—81no—
133#11719919—42no—
134#11719919—82no—
135C2×#88322—24no—
136C2×#88322—82no—
137C23×C53—22no—
138C23×C53—24no—
139C23×C53—22no—
140C23×C53—24no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53—84no—
150C23×C53—84no—
151C23×C53#12011yescharacteristic
152C23×C53#12011yescharacteristic
153C23×C53C2×Q1614yes—
154C23×C53C2×Q1612yes—
155C23×C53#13712yes—
156C23×C53C22×D411yescharacteristic
157Q8×C53—44no—
158Q8×C53—82no—
159Q8×C53—84no—
160C2×C4×C53—24no—
161C2×C4×C53—24no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—44no—
169C2×C4×C53—44no—
170C2×C4×C53—42no—
171C2×C4×C53—81no—
172C2×C4×C53—84no—
173C2×C4×C53—84no—
174C2×C4×C53#12012yes—
175C2×C4×C53C22×D412yes—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—42no—
183Dic106—44no—
184#88322—44no—
185#18287—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—24no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—44no—
196C22×C53—48no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—84no—
201C22×C53#45112yes—
202C22×C53#45112yes—
203C22×C53C2×#12011yescharacteristic
204C22×C53C22×Q1611yescharacteristic
205C22×C53C2×#13711yescharacteristic
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—42no—
209C4×C53—44no—
210C4×C53—44no—
211Dic53—82no—
212Dic53—81no—
213Dic53—81no—
214C2×#543D5311yescharacteristic
215#3189—532no—
216#2626—531no—
217#1614—534no—
218#1602—534no—
219#1520—532no—
220C2×#323—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53—22no—
225C2×C53—44no—
226C2×C53C2×#45111yescharacteristic
227C2×C53#301911yescharacteristic
228C2×C53#302611yescharacteristic
229#543—24no—
230#543—22no—
231#543C2×D5314yes—
232C2×C22⋊Q8—22no—
233C2×C22⋊Q8—21no—
234C2×C22⋊Q8—22no—
235C2×C22⋊Q8—531no—
236C2×C22⋊Q8C2×D5312yes—
237C22×C22⋊C4—21no—
238C22×C22⋊C4C2×D5311yescharacteristic
239#484—534no—
240#484—1062no—
241#467—1064no—
242#451—534no—
243#451—1062no—
244C2×C22⋊C8—532no—
245#399—532no—
246#396—1062no—
247#393—532no—
248#385—1061no—
249#380—531no—
250#379—1061no—
251#328—538no—
252#323—534no—
253C53#1828711yescharacteristic, complemented, Sylow
254C25C22⋊D5311yescharacteristic
255C22×Q8—21no—
256C22×Q8—531no—
257C22×Q8C22⋊D5312yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C22⋊D5311yescharacteristic
261C23×C4C22×D5311yescharacteristic
262C2×Q16—1064no—
263C22×C8—1062no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—24no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—28no—
272C22⋊Q8—1062no—
273C22⋊Q8—1062no—
274C22⋊Q8C22⋊D5314yes—
275C22⋊Q8C22×D5314yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4C22⋊D5312yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—1062no—
291C2×C22⋊C4—1062no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4C22⋊D5312yes—
295C2×C22⋊C4C22×D5312yes—
296#107—1064no—
297#103—1064no—
298#103—1064no—
299C22⋊C8—534no—
300C22⋊C8—1062no—
301#95—532no—
302#95—1062no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—81no—
307C24—81no—
308C24#1171992514yescomplemented
309C24C2×C22⋊D5312yes—
310C24C2×C22⋊D5311yescharacteristic
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—1062no—
316C2×Q8—4241no—
317C2×Q8C2×C22⋊D5314yes—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—1062no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4C2×C22⋊D5312yes—
332C22×C4C2×C22⋊D5311yescharacteristic
333C22×C4C2×C22⋊D5312yes—
334C22×C4C2×C22⋊D5312yes—
335C22×C4C23×D5311yescharacteristic
336Q16—4244no—
337C2×C8—1064no—
338C2×C8—4242no—
339C4⋊C4—24no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—1062no—
345C4⋊C4—4242no—
346C4⋊C4—4241no—
347C4⋊C4—2122no—
348C4⋊C4C2×C22⋊D5314yes—
349C22⋊C4—28no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—4242no—
355C22⋊C4—4242no—
356C22⋊C4—2121no—
357C22⋊C4—2121no—
358C22⋊C4—2122no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23#1313318611yescharacteristic
374C23#1313318611yescharacteristic
375C23C2×#1171992514yes—
376C23C2×#1171992512yes—
377C23#1313324812yes—
378C23C22×C22⋊D5311yescharacteristic
379Q8—44no—
380Q8—82no—
381Q8—84no—
382Q8—2124no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2122no—
404C2×C4#1313318612yes—
405C2×C4C22×C22⋊D5312yes—
406C8—2124no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#1681077912yes—
423C22#1681077912yes—
424C22C2×#1313318611yescharacteristic
425C22C22×#1171992511yescharacteristic
426C22C2×#1313324811yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—4242no—
433C4—4241no—
434C4—4241no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—22no—
439C2—44no—
440C2C2×#1681077911yescharacteristic
441C2#2028407211yescharacteristic
442C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radicalC25×C53D4
hypercenterC2×#543D53
hypocenterC53#18287
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #20285487C2semidirect
3. #20273740C2semidirect
4. #20273672C2semidirect
5. #20273287C2semidirect
10. #16810956C22semidirect
13. #16810778C22semidirect
21. #16810360C22semidirect
86. C24×C53Q16semidirect
253. C53#18287semidirect
308. C24#11719925semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202805202
#202737402
#202736722
#202732872
C2×#168103472
C2×C53×#5432
#1828753