Finite Group Databasebeta

← #27542242

Subgroups — #27542242

Order 13568.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#20287853C211yesmaximal, characteristic, complemented
2#20287852C211yesmaximal, characteristic, complemented
3#20277030C211yesmaximal, characteristic, complemented
4#20277029C212yesmaximal, complemented
5#20277026C212yesmaximal, complemented
6#20277023C212yesmaximal, complemented
7#20277022C211yesmaximal, characteristic, complemented
8C2×#16810458C211yesmaximal, characteristic, complemented
9D32×D53C212yesmaximal, complemented
10#20277010C211yesmaximal, characteristic, complemented
11C53×#2263C211yesmaximal, characteristic, complemented
12#16811425—21no—
13#16811424—21no—
14#16811060—22no—
15#16811059—21no—
16#16811058—21no—
17#16811057—22no—
18#16811056C2212yescomplemented
19#16811055C2211yescharacteristic, complemented
20#16811052C2212yescomplemented
21C2×D848C2211yescharacteristic, complemented
22D16×D53C2212yescomplemented
23D16×D53C2212yescomplemented
24#16811039C2211yescharacteristic
25#16810463C2211yescharacteristic, complemented
26#16810462C2212yescomplemented
27#16810461C2211yescharacteristic, complemented
28#16810460C2211yescharacteristic, complemented
29D1696C2212yescomplemented
30#16810458C2211yescharacteristic, complemented
31#16810458C2212yescomplemented
32#16810458C2212yescomplemented
33#16810456C2212yescomplemented
34C2×#13133123C2211yescharacteristic, complemented
35C32×D53C2212yescomplemented
36C53×#508C2211yescharacteristic, complemented
37C2×D16×C53C2211yescharacteristic, complemented
38C53×SD64C2212yescomplemented
39C53×D32C2212yescomplemented
40C53×M64C2211yescharacteristic, complemented
41#13133263—41no—
42#13133262—41no—
43#13133244—42no—
44#13133243—41no—
45#13133242—41no—
46#13133241—42no—
47#13133240—22no—
48#13133239—21no—
49#13133236—22no—
50C2×D424—21no—
51D8×D53—22no—
52D8×D53—22no—
53#13133223D411yescharacteristic
54Dic424C2311yescharacteristic
55#13133162—21no—
56#13133161—22no—
57#13133160—22no—
58D848—21no—
59D848C2311yescharacteristic
60D848C2312yes—
61#13133158—22no—
62#13133158—22no—
63#13133157D411yescharacteristic
64#13133156D412yes—
65C16×D53C2312yes—
66#13133123C2311yescharacteristic
67#13133123C2311yescharacteristic
68#135×C53—21no—
69C2×D8×C53—21no—
70Q32×C53C2311yescharacteristic
71SD32×C53—21no—
72D16×C53—21no—
73D16×C53C2312yes—
74D16×C53C2311yescharacteristic
75C2×C16×C53C2311yescharacteristic
76C32×C53C2312yes—
77#41×C53—41no—
78C2×D4×C53—41no—
79C2×C22⋊D53—82no—
80#11719949—42no—
81#11719947—42no—
82#11719947—81no—
83D4×D53—42no—
84D4×D53—42no—
85D4×D53—82no—
86D4×D53—81no—
87#11719945—41no—
88#11719945—81no—
89#11719945D811yescharacteristic
90C2×D212—41no—
91C2×D212—81no—
92Q16×C53—21no—
93SD16×C53—41no—
94D8×C53—22no—
95D8×C53—21no—
96D8×C53—41no—
97C2×C8×C53C2×D411yescharacteristic
98#11719924—42no—
99#11719923—42no—
100#11719922—42no—
101#11719922—42no—
102#11719917D811yescharacteristic
103Dic212—21no—
104D424—21no—
105D424—22no—
106D424—41no—
107#11719913—41no—
108#11719912D812yes—
109C8×D53C2×D412yes—
110C16×C53C2×D411yescharacteristic
111C16×C53C2411yescharacteristic
112#11719908C2×D412yes—
113C23×C53—81no—
114C22×D53—81no—
115C22×D53—82no—
116C22×D53—82no—
117Q8×C53—41no—
118D4×C53—41no—
119D4×C53—42no—
120D4×C53—81no—
121D4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53C2×D811yescharacteristic
124C22⋊D53—81no—
125C22⋊D53—82no—
126C22⋊D53—82no—
127C22⋊D53—82no—
128C22⋊D53D1612yes—
129C2×Dic53—82no—
130D212—42no—
131D212—41no—
132D212—82no—
133D212—81no—
134D212D1611yescharacteristic
135C4×D53—81no—
136C4×D53—82no—
137C4×D53C2×D812yes—
138Dic106—41no—
139Dic106D1611yescharacteristic
140C8×C53C2×D811yescharacteristic
141C8×C53C22×D411yescharacteristic
142#88322C2×D812yes—
143#33982—531nomaximal, Sylow
144C22×C53—82no—
145C22×C53—81no—
146C22×C53—321no—
147C22×C53C2×D1611yescharacteristic
148C2×D53—82no—
149C2×D53—81no—
150C2×D53—322no—
151C2×D53—322no—
152C2×D53—321no—
153C2×D53C2×D1612yes—
154C4×C53—81no—
155C4×C53C2×D1611yescharacteristic
156C4×C53C22×D811yescharacteristic
157Dic53—81no—
158Dic53C2×D1612yes—
159#3415—531no—
160#3415—531no—
161#2263—532no—
162#2263—532no—
163#2263—531no—
164#2263D5311yescharacteristic, complemented
165#2262—531no—
166#2262—532no—
167C2×D32—531no—
168C2×D32—532no—
169#2258—531no—
170C2×C53—21no—
171C2×C53—161no—
172C2×C53—162no—
173C2×C53C22×D1611yescharacteristic
174D53—22no—
175D53—162no—
176D53—161no—
177#576—1061no—
178#576—1061no—
179#509—1062no—
180#509—1062no—
181#509—1061no—
182#509—1061no—
183#508—532no—
184#508—532no—
185#508—531no—
186#508C2×D5311yescharacteristic, complemented
187C2×D16—532no—
188C2×D16—532no—
189C2×D16—531no—
190C2×D16C2×D5311yescharacteristic, complemented
191#504—531no—
192Q64—531no—
193SD64—531no—
194SD64—531no—
195SD64—532no—
196SD64C2×D5312yescomplemented
197D32—532no—
198D32—531no—
199D32—532no—
200D32—532no—
201D32C2×D5312yescomplemented
202M64—532no—
203M64C2×D5311yescharacteristic, complemented
204C2×C32—531no—
205C2×C32—532no—
206C53#3398211yescharacteristic, complemented, Sylow
207H(5,𝔽2)—2121no—
208H(5,𝔽2)—2121no—
209AGL(1,ℤ/8)—2122no—
210AGL(1,ℤ/8)—2122no—
211AGL(1,ℤ/8)—2121no—
212AGL(1,ℤ/8)—2121no—
213#135—21no—
214#135—1062no—
215#135—1062no—
216#135—1061no—
217C2×D8—21no—
218C2×D8—1062no—
219C2×D8—1062no—
220C2×D8—1061no—
221#131—531no—
222Q32—531no—
223Q32C22×D5311yescharacteristic
224SD32—21no—
225SD32—1062no—
226SD32—1062no—
227SD32—1061no—
228D16—21no—
229D16—531no—
230D16—532no—
231D16—1062no—
232D16—1062no—
233D16—1061no—
234D16C22×D5312yes—
235D16C22×D5311yescharacteristic
236M32—531no—
237M32—532no—
238C2×C16—532no—
239C2×C16C22×D5311yescharacteristic
240C32—531no—
241C32—531no—
242C32C22×D5312yes—
243#41—41no—
244#41—531no—
245#41—4241no—
246#41—4241no—
247#41—2122no—
248#41—2122no—
249#41—2121no—
250C2×D4—41no—
251C2×D4—4241no—
252C2×D4—4242no—
253C2×D4—4241no—
254C2×D4—4242no—
255C2×D4—2122no—
256C2×D4—2122no—
257C2×D4—2121no—
258Q16—21no—
259Q16—1061no—
260SD16—41no—
261SD16—2122no—
262SD16—2122no—
263SD16—2121no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—1061no—
268D8—1062no—
269D8—2122no—
270D8—2122no—
271D8—2121no—
272M16—531no—
273M16—532no—
274C2×C8—532no—
275C2×C8D4×D5311yescharacteristic
276C16—532no—
277C16D4×D5311yescharacteristic
278C16C23×D5311yescharacteristic
279C23—81no—
280C23—4242no—
281C23—4242no—
282C23—4241no—
283Q8—41no—
284Q8—531no—
285Q8—2121no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—532no—
291D4—531no—
292D4—4242no—
293D4—4242no—
294D4—4242no—
295D4—4242no—
296D4—4241no—
297D4—4241no—
298D4—2122no—
299D4—2121no—
300C2×C4—81no—
301C2×C4—532no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—4241no—
305C2×C4D8×D5311yescharacteristic
306C8—532no—
307C8D8×D5311yescharacteristic
308C8C2×D4×D5311yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—532no—
313C22—16961no—
314C22—16962no—
315C22—16962no—
316C22—4242no—
317C22—4241no—
318C22D16×D5311yescharacteristic
319C4—81no—
320C4—532no—
321C4—4241no—
322C4D16×D5311yescharacteristic
323C4C2×D8×D5311yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—8482no—
328C2—8481no—
329C2—1062no—
330C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2263C2
socleC2×C53C22×D16
hypercenter#2263D53
hypocenterC53#33982
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #20287853C2semidirect
2. #20287852C2semidirect
3. #20277030C2semidirect
4. #20277029C2semidirect
5. #20277026C2semidirect
6. #20277023C2semidirect
7. #20277022C2semidirect
8. C2×#16810458C2semidirect
9. D32×D53C2semidirect
10. #20277010C2semidirect
11. C53×#2263C2semidirect
18. #16811056C22semidirect
19. #16811055C22semidirect
20. #16811052C22semidirect
21. C2×D848C22semidirect
22. D16×D53C22semidirect
23. D16×D53C22semidirect
25. #16810463C22semidirect
26. #16810462C22semidirect
27. #16810461C22semidirect
28. #16810460C22semidirect
29. D1696C22semidirect
30. #16810458C22semidirect
31. #16810458C22semidirect
32. #16810458C22semidirect
33. #16810456C22semidirect
34. C2×#13133123C22semidirect
35. C32×D53C22semidirect
36. C53×#508C22semidirect
37. C2×D16×C53C22semidirect
38. C53×SD64C22semidirect
39. C53×D32C22semidirect
40. C53×M64C22semidirect
164. #2263D53semidirect
186. #508C2×D53semidirect
190. C2×D16C2×D53semidirect
196. SD64C2×D53semidirect
201. D32C2×D53semidirect
203. M64C2×D53semidirect
206. C53#33982semidirect

Maximal subgroups

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

subgroupindex
#202878532
#202878522
#202770302
#202770292
#202770262
#202770232
#202770222
C2×#168104582
D32×D532
#202770102
C53×#22632
#3398253