Finite Group Databasebeta

← #27284547

Subgroups — #27284547

Order 13568.

610 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 109 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#20285468C211yesmaximal, characteristic
2C2×#16810941C211yesmaximal, characteristic
3#20275551C211yesmaximal, characteristic
4#20274805C211yesmaximal, characteristic, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2502C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810951—21no—
10#16810941—22no—
11#16810941—21no—
12#16810941—22no—
13#16810854—22no—
14#16810773—21no—
15#16810762—21no—
16C2×#13133148—21no—
17C2×#13133137—21no—
18C2×#13133131—22no—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810483C2211yescharacteristic
22#16810385—22no—
23#16810360—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#398—22no—
27C53×#391—21no—
28C53×#385—22no—
29C53×#384—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C2×#11719924—22no—
34C22×#88322—21no—
35C22×#88322D412yes—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133148—22no—
45#13133148—24no—
46#13133141—22no—
47#13133137—22no—
48#13133137—24no—
49#13133131—44no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—21no—
55C22×Q8×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C2×C4⋊C4×C53—24no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53D412yes—
73C2×C42×C53—21no—
74#95×C53—24no—
75#95×C53—21no—
76C24×C53C2×D411yescharacteristic
77C23×D53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—42no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—82no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C2×C22⋊D53—41no—
96C2×C22⋊D53—41no—
97C2×C22⋊D53—42no—
98C2×C22⋊D53—42no—
99C22×Dic53—21no—
100C22×Dic53—41no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—42no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—84no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53SD1614yescomplemented
117C42×C53—42no—
118C22⋊Dic53—42no—
119#11719924—82no—
120#11719921—41no—
121#11719921—41no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—42no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—42no—
128C2×#88322—42no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53#12011yescharacteristic
132C23×C53#12111yescharacteristic
133C23×C53#12711yescharacteristic
134C23×C53C2×SD1612yes—
135C23×C53#13712yes—
136C22×D53—41no—
137C22×D53—42no—
138C22×D53—81no—
139C22×D53—81no—
140Q8×C53—42no—
141Q8×C53—81no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—44no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C22⋊D53—82no—
164C22⋊D53—81no—
165C22⋊D53—81no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171D212—42no—
172D212—42no—
173#88322—42no—
174#88322—42no—
175#9951—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—42no—
183C22×C53—42no—
184C22×C53#39211yescharacteristic
185C22×C53#44812yes—
186C22×C53#46112yes—
187C22×C53#49211yescharacteristic
188C22×C53#49711yescharacteristic
189C2×D53—81no—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—162no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—44no—
197C4×C53—81no—
198C4×C53—82no—
199C4×C53—81no—
200C4×C53#45812yes—
201Dic53—81no—
202Dic53—161no—
203#3187—531no—
204C2×#481—531no—
205#2502D5311yescharacteristic, complemented
206#1964—531no—
207#1824—531no—
208#1520—531no—
209#1282—531no—
210C2×C53—21no—
211C2×C53—22no—
212C2×C53—22no—
213C2×C53—21no—
214C2×C53#200811yescharacteristic
215C2×C53#201311yescharacteristic
216C2×C53#202211yescharacteristic
217D53—161no—
218C2×C22⋊Q8C2×D5311yescharacteristic
219C2×#121—531no—
220C22×C4⋊C4C2×D5311yescharacteristic
221#483—1061no—
222#481—1062no—
223#481—1062no—
224#481—1061no—
225#465—1062no—
226#450—1061no—
227#448—1061no—
228C2×#106—1061no—
229C2×#102—1061no—
230C2×#99—1062no—
231C2×C22⋊C8—531no—
232C2×C22⋊C8—531no—
233#398—22no—
234#391—21no—
235#385—22no—
236#384—21no—
237#380—531no—
238C4×C22⋊C4C2×D5311yescharacteristic
239#340—1062no—
240#328—1062no—
241C53#995111yescharacteristic, complemented, Sylow
242C22×Q8C22⋊D5311yescharacteristic
243C22×D4—531no—
244C22×D4—1061no—
245C23×C4—21no—
246C23×C4C22×D5311yescharacteristic
247C2×SD16—1062no—
248C22×C8—532no—
249C22×C8—1061no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254#121—1061no—
255#121—1061no—
256#121—1061no—
257#121—1061no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4C22⋊D5312yes—
268C2×C4⋊C4C22⋊D5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4C22⋊D5312yes—
273C2×C42—21no—
274#106—1064no—
275#106—1062no—
276#103—1062no—
277#102—1064no—
278#102—1062no—
279#99—2124no—
280C22⋊C8—532no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284#95—24no—
285#95—21no—
286#95—1061no—
287C24—1061no—
288C24C2×C22⋊D5311yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—2122no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2122no—
298C2×D4—2122no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—1061no—
308C22×C4—2121no—
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311C22×C4D4×D5311yescharacteristic
312C22×C4#1171994711yescharacteristic
313C22×C4C2×C22⋊D5311yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315SD16—4242no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—2122no—
319C2×C8—2122no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—84no—
329C4⋊C4—1061no—
330C4⋊C4—1061no—
331C4⋊C4—2122no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—2122no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C22⋊C4#1171992414yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—4241no—
343C23—4241no—
344C23—2122no—
345C23—2121no—
346C23#1313318611yescharacteristic
347C23#1313318811yescharacteristic
348C23#1313321311yescharacteristic
349C23C2×#1171992412yes—
350C23#1313324812yes—
351Q8—42no—
352Q8—81no—
353D4—4241no—
354D4—4242no—
355D4—4241no—
356D4—2122no—
357D4—2122no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—4241no—
376C2×C4—4241no—
377C2×C4—4241no—
378C2×C4—2121no—
379C2×C4—2122no—
380C2×C4#1313318511yescharacteristic
381C2×C4#1313318511yescharacteristic
382C2×C4#1313318911yescharacteristic
383C2×C4#1313318911yescharacteristic
384C8—2122no—
385C8—2122no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—8482no—
395C22—4241no—
396C22—4241no—
397C22—4241no—
398C22#1681053111yescharacteristic
399C22#1681076212yes—
400C22#1681082512yes—
401C22#1681099711yescharacteristic
402C22#1681101511yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—8481no—
411C4—4241no—
412C4#1681081212yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—8481no—
418C2#2027576811yescharacteristic
419C2#2027579811yescharacteristic
420C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2502C2
socleC22×C53#392
prime radicalC2×#13133191C22
hypercenter#2502D53
hypocenterC53#9951
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
4. #20274805C2semidirect
5. #20273287C2semidirect
6. #20272563C2semidirect
7. C53×#2502C2semidirect
116. C22⋊C4×C53SD16semidirect
205. #2502D53semidirect
241. C53#9951semidirect
338. C22⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202854682
C2×#168109412
#202755512
#202748052
#202732872
#202725632
C53×#25022
#995153