Finite Group Databasebeta

← #27356508

Subgroups — #27356508

Order 13568.

1054 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1054 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#20284659C212yesmaximal, complemented
2#20278989C211yesmaximal, characteristic
3#20273740C214yesmaximal, complemented
4#20273706C214yesmaximal, complemented
5#20273248C212yesmaximal, complemented
6#20273042C211yesmaximal, characteristic
7C2×C53×#543C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810961—24no—
10#16810956C2214yescomplemented
11#16810951C2214yescomplemented
12#16810663—22no—
13#16810632—22no—
14#16810587C2212yes—
15#16810559C2212yes—
16#16810548C2212yes—
17#16810511—22no—
18#16810496—21no—
19#16810486C2211yescharacteristic
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
32C2×#11719917—22no—
33#13133176—22no—
34#13133176—22no—
35C2×C4⋊Dic53C2311yescharacteristic
36C2×C4⋊Dic53C2311yescharacteristic
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#13133141—24no—
46#13133141—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yes—
49#13133124—22no—
50#13133124—21no—
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—24no—
84C24×C53—42no—
85C24×C53—81no—
86C24×C53C2×D411yescharacteristic
87C24×C53C2×D412yes—
88C2×Q8×C53—24no—
89C2×Q8×C53—22no—
90C2×Q8×C53—44no—
91C2×Q8×C53—82no—
92C2×Q8×C53C2×D414yes—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—24no—
97C22×C4×C53—24no—
98C22×C4×C53—21no—
99C22×C4×C53—42no—
100C22×C4×C53—81no—
101C22×C4×C53—82no—
102C22×C4×C53—82no—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2411yescharacteristic
108C22×Dic53—22no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C4⋊C4×C53—28no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53C2×D414yes—
117C22⋊C4×C53—28no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊Dic53—42no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—44no—
126C22⋊Dic53—82no—
127C4⋊Dic53—22no—
128C4⋊Dic53—22no—
129#11719919—82no—
130C4×Dic53—42no—
131#11719917—44no—
132C2×#88322—24no—
133C23×C53—24no—
134C23×C53—24no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—22no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—42no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53—88no—
147C23×C53#12011yescharacteristic
148C23×C53#12011yescharacteristic
149C23×C53#13714yes—
150C23×C53C22×D411yescharacteristic
151Q8×C53—44no—
152Q8×C53—82no—
153Q8×C53—84no—
154C2×C4×C53—24no—
155C2×C4×C53—24no—
156C2×C4×C53—24no—
157C2×C4×C53—22no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—44no—
163C2×C4×C53—44no—
164C2×C4×C53—42no—
165C2×C4×C53—81no—
166C2×C4×C53—84no—
167C2×C4×C53—84no—
168C2×C4×C53#12012yes—
169C2×C4×C53C22×D412yes—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177#88322—44no—
178#18138—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—24no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—44no—
188C22×C53—44no—
189C22×C53—48no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—82no—
193C22×C53—84no—
194C22×C53#45812yes—
195C22×C53C2×#12011yescharacteristic
196C22×C53C2×#13712yes—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—42no—
200C4×C53—44no—
201C4×C53—44no—
202Dic53—82no—
203Dic53—81no—
204Dic53—81no—
205C2×#543D5311yescharacteristic
206#3095—532no—
207#2482—531no—
208#1614—534no—
209#1608—534no—
210#1511—532no—
211#1462—531no—
212C2×C53—22no—
213C2×C53—22no—
214C2×C53—41no—
215C2×C53—44no—
216C2×C53#300411yescharacteristic
217C2×C53#301911yescharacteristic
218C2×C53C2×#45811yescharacteristic
219#543—24no—
220#543—22no—
221#543C2×D5314yes—
222C2×C22⋊Q8—22no—
223C2×C22⋊Q8—21no—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8C2×D5312yes—
226C2×#117—531no—
227C22×C22⋊C4—21no—
228C22×C22⋊C4C2×D5311yescharacteristic
229#485—1064no—
230#484—534no—
231#483—534no—
232#428—1062no—
233#419—1062no—
234#407—532no—
235#399—532no—
236#397—532no—
237#388—1062no—
238#385—1061no—
239#381—531no—
240#379—1061no—
241#328—538no—
242#323—534no—
243C53#1813811yescharacteristic, complemented, Sylow
244C25C22⋊D5311yescharacteristic
245C22×Q8—21no—
246C22×Q8C22⋊D5312yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22⋊D5311yescharacteristic
250C23×C4C22×D5311yescharacteristic
251C2×M16—1062no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—28no—
260C22⋊Q8C22⋊D5314yes—
261C22⋊Q8C22×D5314yes—
262#117—1062no—
263#117—1062no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4C22⋊D5312yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4C22⋊D5312yes—
282C2×C22⋊C4C22×D5312yes—
283C2×C42—1061no—
284#107—1064no—
285#106—1064no—
286#103—1064no—
287#103—1064no—
288C22⋊C8—534no—
289C22⋊C8—1062no—
290#95—532no—
291#95—1061no—
292#95—1062no—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D5311yescharacteristic
299C24C2×C22⋊D5312yes—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊D5314yes—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—1062no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4C2×C22⋊D5312yes—
319C22×C4C2×C22⋊D5312yes—
320C22×C4C2×C22⋊D5311yescharacteristic
321C22×C4C2×C22⋊D5312yes—
322C22×C4C23×D5311yescharacteristic
323M16—2124no—
324C2×C8—1064no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—1062no—
331C4⋊C4—1062no—
332C4⋊C4—4242no—
333C4⋊C4C2×C22⋊D5314yes—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—4242no—
340C22⋊C4—2122no—
341C22⋊C4—2121no—
342C22⋊C4—2121no—
343C22⋊C4—2124no—
344C42—2122no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1313318611yescharacteristic
360C23#1313318611yescharacteristic
361C23#1313324814yes—
362C23C22×C22⋊D5311yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4#1313318612yes—
388C2×C4C22×C22⋊D5312yes—
389C8—2124no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—24no—
396C22—21no—
397C22—21no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1681081312yes—
406C22C2×#1313318611yescharacteristic
407C22C2×#1313324812yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—4242no—
414C4—4241no—
415C4—4241no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2028394911yescharacteristic
421C2#2028407211yescharacteristic
422C2C2×#1681081311yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radicalC25×C53D4
hypercenterC2×#543D53
hypocenterC53#18138
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. #20284659C2semidirect
3. #20273740C2semidirect
4. #20273706C2semidirect
5. #20273248C2semidirect
10. #16810956C22semidirect
11. #16810951C22semidirect
21. #16810360C22semidirect
243. C53#18138semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202789892
#202737402
#202737062
#202732482
#202730422
C2×C53×#5432
#1813853