Finite Group Databasebeta

← #28308858

Subgroups — #28308858

Order 13568.

504 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 86 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#20290943C211yesmaximal, characteristic, complemented
2C2×#16811497C211yesmaximal, characteristic, complemented
3#20289703C212yesmaximal, complemented
4#20289702C212yesmaximal, complemented
5#20284298C211yesmaximal, characteristic
6C22×C4⋊#6157—21no—
7C2×#13133292—21no—
8#16811518C2212yes—
9C22⋊#11719910—21no—
10#16811497—21no—
11#16811497C2212yes—
12C2×#13133240C411yescharacteristic, complemented
13C22×C8×D53C2211yescharacteristic
14#16811136—21no—
15#16810861C2211yescharacteristic
16#16810860C2211yescharacteristic
17#16810859C411yescharacteristic, complemented
18#16810855—21no—
19#16810854—21no—
20#16810660C411yescharacteristic, complemented
21#16810618—21no—
22C53×#467C411yescharacteristic, complemented
23C23×#6157—41no—
24C2×C4⋊#6157—24no—
25C2×C4⋊#6157—22no—
26C2×C4⋊#6157—42no—
27C2×C4⋊#6157—41no—
28#13133292—42no—
29C2×#11719910—22no—
30C2×#11719949—21no—
31C22×C4×D53D411yescharacteristic
32#13133240—42no—
33C2×#11719924—21no—
34C2×D424C2×C411yescharacteristic, complemented
35C22×#88322D411yescharacteristic
36C2×C8×D53—42no—
37C2×C8×D53C2×C411yescharacteristic
38C2×C8×D53D411yescharacteristic
39C2×C8×D53D411yescharacteristic
40C2×C8×D53C2311yescharacteristic
41#13133202—41no—
42#13133193—21no—
43#13133192—21no—
44#13133191—21no—
45#13133187—41no—
46#13133179—41no—
47#13133177—41no—
48#13133175—21no—
49#13133151C2×C411yescharacteristic, complemented
50#13133148—21no—
51#13133141—21no—
52#13133139—21no—
53#13133137—21no—
54C2×Q16×C53C2×C411yescharacteristic, complemented
55C22×C8×C53C2×C411yescharacteristic
56C22⋊Q8×C53—21no—
57#107×C53C2×C411yescharacteristic, complemented
58#103×C53—21no—
59C23×D53D811yescharacteristic
60C22×#6157—44no—
61C22×#6157—42no—
62C22×#6157—82no—
63C22×#6157—82no—
64C2×Q8×C53—21no—
65C22×C4×C53C22⋊C411yescharacteristic
66C2×C22⋊D53—41no—
67C22×Dic53SD1611yescharacteristic
68#11719949—42no—
69#11719949—42no—
70C2×D212—21no—
71C2×D212—41no—
72C2×D212—41no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53C22⋊C411yescharacteristic
77C2×C4×D53D811yescharacteristic
78C2×C4×D53SD1611yescharacteristic
79C2×C4×D53C2×D411yescharacteristic
80C4⋊#6157—24no—
81C4⋊#6157—42no—
82C4⋊#6157—44no—
83Q16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C22⋊C411yescharacteristic
86C2×C8×C53C22×C411yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90#11719924—42no—
91#11719921—41no—
92#11719921—41no—
93#11719921—41no—
94#11719921—41no—
95C4⋊Dic53—21no—
96#11719919—41no—
97C4×Dic53—41no—
98C2×#88322—41no—
99C2×#88322C22⋊C411yescharacteristic
100C2×#88322C2×D411yescharacteristic
101D424—41no—
102C8×D53—22no—
103C8×D53—22no—
104C8×D53—41no—
105C8×D53—41no—
106#11719910—42no—
107C23×C53#10211yescharacteristic
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—41no—
113C22×D53#10211yescharacteristic
114C22×D53C2×D811yescharacteristic
115C2×#6157—44no—
116C2×#6157—88no—
117C2×#6157—82no—
118C2×#6157—82no—
119Q8×C53—41no—
120Q8×C53—41no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10211yescharacteristic
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C22⋊D53—82no—
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53#10211yescharacteristic
131C2×Dic53C2×SD1611yescharacteristic
132D212—41no—
133D212—41no—
134D212—82no—
135D212—81no—
136D212—81no—
137C4×D53—22no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—22no—
141C4×D53—82no—
142C4×D53#9912yes—
143C4×D53#10412yes—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148#11704—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53#36012yes—
152C22×C53C2×#10211yescharacteristic
153C2×D53—24no—
154C2×D53—24no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—81no—
160C2×D53—81no—
161C2×D53—81no—
162C2×D53#32712yes—
163C2×D53#50911yescharacteristic
164C2×D53#51011yescharacteristic
165#6157—84no—
166C4×C53—21no—
167C4×C53—81no—
168C4×C53—81no—
169C4×C53C2×#9911yescharacteristic
170C4×C53C2×#10411yescharacteristic
171Dic53—21no—
172Dic53—81no—
173Dic53#32712yes—
174#3050—531no—
175#2112—531no—
176C2×#340—531no—
177#1348—532no—
178#1347—532no—
179C2×C53—22no—
180C2×C53C2×#32711yescharacteristic
181C2×C53#213911yescharacteristic
182C2×C53C2×#36011yescharacteristic
183D53—21no—
184D53—22no—
185D53—21no—
186D53—161no—
187C2×#135—531no—
188C23×C8—531no—
189#525—1061no—
190C22×C4⋊C4—1061no—
191C2×M32—1061no—
192#467—531no—
193#467#615711yescharacteristic, complemented
194#466—531no—
195#466—531no—
196#465—1061no—
197#465—1061no—
198#427—531no—
199#416—1061no—
200C22⋊C16—532no—
201C22⋊C16—1061no—
202#340—532no—
203#340—1061no—
204C53#1170411yescharacteristic, complemented, Sylow
205C2×#41—1061no—
206C23×C4—531no—
207C23×C4—2121no—
208#135—2122no—
209C2×Q16C2×#615711yescharacteristic, complemented
210C2×SD16—1061no—
211C2×D8—531no—
212C22×C8—531no—
213C22×C8—531no—
214C22×C8—531no—
215C22×C8—531no—
216C22×C8—531no—
217C22×C8—2122no—
218C22×C8C2×#615711yescharacteristic
219#123—2121no—
220C22⋊Q8—21no—
221C22⋊Q8—1061no—
222#121—1061no—
223#121—1061no—
224#120—2121no—
225C4×D4—2121no—
226C4×D4—2121no—
227#117—1061no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1064no—
230C2×C4⋊C4—2121no—
231C2×C4⋊C4—2122no—
232M32—2122no—
233C2×C16—1062no—
234#107—531no—
235#107C2×#615711yescharacteristic, complemented
236#106—1061no—
237#103—21no—
238#103—1061no—
239#102—1061no—
240#102—1061no—
241C24—531no—
242#41—2122no—
243#41—2122no—
244C2×Q8—21no—
245C2×D4—1061no—
246C2×D4—2121no—
247C2×D4—2121no—
248C2×D4—2121no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—1062no—
255C22×C4—1062no—
256C22×C4—4242no—
257C22×C4—4242no—
258C22×C4—2121no—
259C22×C4—2122no—
260C22×C4—2124no—
261C22×C4C22⋊#615711yescharacteristic
262Q16—41no—
263SD16—2122no—
264D8—2121no—
265C2×C8—41no—
266C2×C8—531no—
267C2×C8—531no—
268C2×C8—1062no—
269C2×C8—1062no—
270C2×C8—2121no—
271C2×C8—2121no—
272C2×C8—2121no—
273C2×C8C22⋊#615711yescharacteristic
274C2×C8C22×#615711yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—1064no—
278C4⋊C4—1061no—
279C4⋊C4—2121no—
280C4⋊C4—2124no—
281C4⋊C4—2122no—
282C22⋊C4—41no—
283C22⋊C4—2121no—
284C22⋊C4—2121no—
285C22⋊C4—2121no—
286C22⋊C4—2121no—
287C42—2121no—
288C16—2122no—
289C23—531no—
290C23—531no—
291C23—1062no—
292C23—1061no—
293C23—1061no—
294C23—1062no—
295C23—2121no—
296C23#1313327911yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—4241no—
300D4—4241no—
301D4—4242no—
302D4—4242no—
303D4—2121no—
304D4—2121no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—531no—
310C2×C4—532no—
311C2×C4—532no—
312C2×C4—531no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—4242no—
319C2×C4—4242no—
320C2×C4—4248no—
321C2×C4—4242no—
322C2×C4—2124no—
323C2×C4—2121no—
324C2×C4#1313327911yescharacteristic
325C2×C4C2×C22⋊#615711yescharacteristic
326C8—21no—
327C8—21no—
328C8—1061no—
329C8—1061no—
330C22—21no—
331C22—21no—
332C22—532no—
333C22—531no—
334C22—531no—
335C22—1064no—
336C22—1064no—
337C22—1061no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22—4241no—
342C22—4241no—
343C22—4241no—
344C22#1681154012yes—
345C22C2×#1313327911yescharacteristic
346C4—21no—
347C4—81no—
348C4—81no—
349C4—532no—
350C4—1061no—
351C4—4244no—
352C4—4241no—
353C4D53⋊#9911yescharacteristic
354C4D53⋊#10411yescharacteristic
355C2—22no—
356C2—8481no—
357C2—1061no—
358C2—1062no—
359C2—1061no—
360C2D53⋊#32711yescharacteristic
361C2#2029102311yescharacteristic
362C2C2×#1681154011yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#467C4
socleC22×C53C2×#102
prime radical#20284298C2
cosocleC2×C8×D53C23
hypercenter#467#6157
hypocenterC53#11704
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20290943C2semidirect
2. C2×#16811497C2semidirect
3. #20289703C2semidirect
4. #20289702C2semidirect
12. C2×#13133240C4semidirect
17. #16810859C4semidirect
20. #16810660C4semidirect
22. C53×#467C4semidirect
34. C2×D424C2×C4semidirect
49. #13133151C2×C4semidirect
54. C2×Q16×C53C2×C4semidirect
57. #107×C53C2×C4semidirect
193. #467#6157semidirect
204. C53#11704semidirect
209. C2×Q16C2×#6157semidirect
235. #107C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202909432
C2×#168114972
#202897032
#202897022
#202842982
#1170453