Finite Group Databasebeta

← #27294324

Subgroups — #27294324

Order 13568.

504 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 89 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#20284298C211yesmaximal, characteristic
2#20276324C211yesmaximal, characteristic, complemented
3#340×D53C211yesmaximal, characteristic, complemented
4#20272727C211yesmaximal, characteristic, complemented
5#20272725C211yesmaximal, characteristic, complemented
6#20272723C211yesmaximal, characteristic, complemented
7C53×#1348C211yesmaximal, characteristic, complemented
8C2×#13133240C411yescharacteristic, complemented
9C22×C8×D53C2211yescharacteristic
10#16811136—21no—
11C2×C4⋊C4×D53—21no—
12C2×#13133156—21no—
13#16810861C411yescharacteristic, complemented
14#16810860C411yescharacteristic, complemented
15#16810859C2211yescharacteristic
16#16810855—21no—
17#16810854—21no—
18#16810660C411yescharacteristic, complemented
19#16810618—21no—
20C22⋊#11719908C2211yescharacteristic
21#16810402—21no—
22#16810386C2211yescharacteristic
23#16810385—21no—
24C53×#467C2211yescharacteristic
25C53×C22⋊C16C2211yescharacteristic
26C53×#340C2211yescharacteristic
27C2×#11719949—21no—
28C22×C4×D53—41no—
29C22×C4×D53D411yescharacteristic
30#13133240—42no—
31C2×#11719924—21no—
32C2×D424C2×C411yescharacteristic, complemented
33C22×#88322D411yescharacteristic
34C2×C8×D53—42no—
35C2×C8×D53C2×C411yescharacteristic
36C2×C8×D53C2×C411yescharacteristic
37C2×C8×D53D411yescharacteristic
38C2×C8×D53D411yescharacteristic
39#13133202—41no—
40#13133193—21no—
41#13133192—21no—
42#13133191—21no—
43#13133187—41no—
44#13133179—41no—
45#13133177—41no—
46#13133175—21no—
47C2×C4⋊Dic53—21no—
48C2×#11719919—41no—
49C4⋊C4×D53—24no—
50C4⋊C4×D53—42no—
51#13133156—42no—
52C2×#11719908—21no—
53#13133151C2×C411yescharacteristic, complemented
54#13133148—21no—
55#13133141—21no—
56#13133139—21no—
57#13133137—21no—
58C2×Q16×C53C2×C411yescharacteristic, complemented
59C22×C8×C53C2311yescharacteristic
60C22⋊Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C16×C53—21no—
63#107×C53C2×C411yescharacteristic, complemented
64#103×C53—21no—
65C23×D53D811yescharacteristic
66C2×Q8×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C2×C22⋊D53—41no—
70C22×Dic53—41no—
71C22×Dic53D811yescharacteristic
72#11719949—42no—
73#11719949—42no—
74C2×D212—21no—
75C2×D212—41no—
76C2×D212—41no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—44no—
80C2×C4×D53—41no—
81C2×C4×D53—82no—
82C2×C4×D53—82no—
83C2×C4×D53C22⋊C411yescharacteristic
84C2×C4×D53C22⋊C411yescharacteristic
85C2×C4×D53SD1611yescharacteristic
86C2×C4×D53SD1611yescharacteristic
87Q16×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53C22×C411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C4⋊C4×C53—22no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96#11719924—42no—
97#11719921—41no—
98#11719921—41no—
99#11719921—41no—
100#11719921—41no—
101C4⋊Dic53—22no—
102C4⋊Dic53—21no—
103C4⋊Dic53—41no—
104#11719919—44no—
105#11719919—41no—
106C4×Dic53—41no—
107C2×#88322—41no—
108C2×#88322C22⋊C411yescharacteristic
109C2×#88322C22⋊C411yescharacteristic
110D424—41no—
111C8×D53—22no—
112C8×D53—22no—
113C8×D53—41no—
114C8×D53—41no—
115C16×C53—41no—
116#11719908—41no—
117C23×C53C2×D811yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—22no—
121C22×D53—22no—
122C22×D53—41no—
123C22×D53#10211yescharacteristic
124C22×D53#10211yescharacteristic
125Q8×C53—41no—
126Q8×C53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C22⋊D53—81no—
137C22⋊D53—81no—
138C2×Dic53—22no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53#10211yescharacteristic
144C2×Dic53#10211yescharacteristic
145D212—41no—
146D212—41no—
147D212—82no—
148D212—82no—
149C4×D53—22no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—22no—
153C4×D53—88no—
154C4×D53—82no—
155C4×D53#9912yes—
156C4×D53#10412yes—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161#11704—531nomaximal, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53C2×#10211yescharacteristic
165C22×C53#50911yescharacteristic
166C22×C53#51011yescharacteristic
167C2×D53—24no—
168C2×D53—24no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53#32712yes—
177C2×D53#36012yes—
178C4×C53—21no—
179C4×C53—82no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C53C2×#9911yescharacteristic
183C4×C53C2×#10411yescharacteristic
184Dic53—21no—
185Dic53—82no—
186Dic53—81no—
187Dic53#32712yes—
188#3050—531no—
189#2112—531no—
190C2×#340—531no—
191#1348—531no—
192#1348D5311yescharacteristic, complemented
193#1347—531no—
194#1347—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53C2×#32711yescharacteristic
198C2×C53#213911yescharacteristic
199C2×C53C2×#36011yescharacteristic
200D53—22no—
201D53—22no—
202D53—161no—
203C2×#135—531no—
204C23×C8—531no—
205#525—1061no—
206C22×C4⋊C4—1061no—
207C2×M32—1061no—
208#467—531no—
209#467C2×D5311yescharacteristic
210#466—531no—
211#466—531no—
212#465—1061no—
213#465—1061no—
214#427—531no—
215#416—1061no—
216C22⋊C16—531no—
217C22⋊C16—1061no—
218C22⋊C16C2×D5311yescharacteristic
219#340—531no—
220#340—1061no—
221#340C2×D5311yescharacteristic
222C53#1170411yescharacteristic, complemented, Sylow
223C2×#41—1061no—
224C23×C4—531no—
225C23×C4—2121no—
226#135—2122no—
227C2×Q16C4×D5311yescharacteristic, complemented
228C2×SD16—1061no—
229C2×D8—531no—
230C22×C8—531no—
231C22×C8—531no—
232C22×C8—531no—
233C22×C8—531no—
234C22×C8—531no—
235C22×C8—2122no—
236C22×C8C22×D5311yescharacteristic
237#123—2121no—
238C22⋊Q8—21no—
239C22⋊Q8—1061no—
240#121—1061no—
241#121—1061no—
242#120—2121no—
243C4×D4—2121no—
244C4×D4—2121no—
245#117—1061no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1064no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—2122no—
250C2×C4⋊C4—2121no—
251M32—2122no—
252C2×C16—21no—
253C2×C16—1061no—
254#107—531no—
255#107C4×D5311yescharacteristic, complemented
256#106—1061no—
257#103—21no—
258#103—1061no—
259#102—1061no—
260#102—1061no—
261C24—531no—
262#41—2122no—
263#41—2122no—
264C2×Q8—21no—
265C2×D4—1061no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C22×C4—41no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—4242no—
278C22×C4—4242no—
279C22×C4—2121no—
280C22×C4—2124no—
281C22×C4—2121no—
282C22×C4D4×D5311yescharacteristic
283Q16—41no—
284SD16—2122no—
285D8—2121no—
286C2×C8—41no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—1062no—
290C2×C8—1062no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8C2×C4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—1062no—
301C4⋊C4—1061no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2124no—
305C22⋊C4—41no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C22⋊C4—2121no—
310C42—2121no—
311C16—41no—
312C16—2121no—
313C23—531no—
314C23—531no—
315C23—1062no—
316C23—1062no—
317C23—1061no—
318C23—1061no—
319C23—2121no—
320C23D8×D5311yescharacteristic
321Q8—41no—
322Q8—41no—
323D4—4242no—
324D4—4242no—
325D4—4241no—
326D4—4241no—
327D4—2121no—
328D4—2121no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—531no—
337C2×C4—532no—
338C2×C4—532no—
339C2×C4—531no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4242no—
346C2×C4—4248no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—2122no—
350C2×C4—2121no—
351C2×C4C22⋊C4×D5311yescharacteristic
352C2×C4SD16×D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1061no—
356C8—1061no—
357C22—21no—
358C22—21no—
359C22—532no—
360C22—532no—
361C22—1064no—
362C22—1064no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—4241no—
368C22—4241no—
369C22—4241no—
370C22#102×D5311yescharacteristic
371C22#1681106011yescharacteristic
372C22#1681106511yescharacteristic
373C4—21no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—532no—
378C4—1061no—
379C4—4242no—
380C4—4241no—
381C4#99×D5311yescharacteristic
382C4#104×D5311yescharacteristic
383C2—21no—
384C2—21no—
385C2—8481no—
386C2—1062no—
387C2—1062no—
388C2#327×D5311yescharacteristic
389C2#2027642611yescharacteristic
390C2#360×D5311yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1348C2
socleC22×C53C2×#102
prime radical#20284298C2
cosocleC22×C8×C53C23
hypercenter#1348D53
hypocenterC53#11704
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #20276324C2semidirect
3. #340×D53C2semidirect
4. #20272727C2semidirect
5. #20272725C2semidirect
6. #20272723C2semidirect
7. C53×#1348C2semidirect
8. C2×#13133240C4semidirect
13. #16810861C4semidirect
14. #16810860C4semidirect
18. #16810660C4semidirect
32. C2×D424C2×C4semidirect
53. #13133151C2×C4semidirect
58. C2×Q16×C53C2×C4semidirect
63. #107×C53C2×C4semidirect
192. #1348D53semidirect
222. C53#11704semidirect
227. C2×Q16C4×D53semidirect
255. #107C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202842982
#202763242
#340×D532
#202727272
#202727252
#202727232
C53×#13482
#1170453