Finite Group Databasebeta

← #27542244

Subgroups — #27542244

Order 13568.

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

normal subgroups only

This group has 462 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#20287864C211yesmaximal, characteristic, complemented
2#20287852C211yesmaximal, characteristic, complemented
3#20277033C211yesmaximal, characteristic, complemented
4#20277031C212yesmaximal, complemented
5#20277029C212yesmaximal, complemented
6#20277028C211yesmaximal, characteristic, complemented
7#20277024C211yesmaximal, characteristic, complemented
8#20277023C212yesmaximal, complemented
9SD64×D53C212yesmaximal, complemented
10#20277008C211yesmaximal, characteristic, complemented
11C2×C53×SD64C211yesmaximal, characteristic, complemented
12#16811436—21no—
13#16811424—21no—
14#16811065—22no—
15#16811064—21no—
16#16811059—21no—
17#16811057—22no—
18#16811056C2212yescomplemented
19#16811055C2211yescharacteristic
20#16811055C2211yescharacteristic, complemented
21#16811052C2212yes—
22Q32×D53C2212yescomplemented
23D16×D53C2212yes—
24#16811039C2211yescharacteristic
25Dic848C2211yescharacteristic, complemented
26#16810463C2212yescomplemented
27#16810462C2211yescharacteristic, complemented
28#16810462C2211yescharacteristic, complemented
29#16810461C2212yescomplemented
30#16810460C2212yescomplemented
31D1696C2211yescharacteristic, complemented
32#16810458C2212yescomplemented
33#16810457C2211yescharacteristic, complemented
34#16810456C2212yescomplemented
35C32×D53C2212yescomplemented
36C2×Q32×C53C2211yescharacteristic, complemented
37C2×D16×C53C2211yescharacteristic
38C53×SD64C2212yescomplemented
39C53×SD64C2212yescomplemented
40C2×C32×C53C2211yescharacteristic, complemented
41#13133265—41no—
42#13133262—41no—
43#13133249—42no—
44#13133248—41no—
45#13133243—41no—
46#13133241—42no—
47#13133240—22no—
48#13133239—21no—
49#13133239—21no—
50#13133236—22no—
51Q16×D53—22no—
52D8×D53—22no—
53#13133223D411yescharacteristic
54Dic424C2311yescharacteristic
55Dic424C2311yescharacteristic
56#13133163—22no—
57#13133162—21no—
58#13133162—21no—
59#13133161—22no—
60#13133160—22no—
61D848C2311yescharacteristic
62D848C2311yescharacteristic
63#13133158—22no—
64#13133157D411yescharacteristic
65#13133156D412yes—
66C16×D53C2312yes—
67#13133123C2312yes—
68C2×Q16×C53—21no—
69C2×D8×C53—21no—
70Q32×C53—21no—
71Q32×C53C2312yes—
72D16×C53—21no—
73D16×C53C2312yes—
74C2×C16×C53C2311yescharacteristic
75C32×C53C2311yescharacteristic
76C32×C53C2311yescharacteristic
77C2×Q8×C53—41no—
78C2×D4×C53—41no—
79C2×C22⋊D53—82no—
80#11719949—42no—
81#11719949—81no—
82Q8×D53—42no—
83Q8×D53—81no—
84#11719947—42no—
85#11719947—81no—
86D4×D53—42no—
87D4×D53—81no—
88#11719945—41no—
89#11719945—41no—
90#11719945—82no—
91#11719945D811yescharacteristic
92Q16×C53—22no—
93Q16×C53—41no—
94D8×C53—22no—
95D8×C53—41no—
96C2×C8×C53C2×D411yescharacteristic
97#11719925—42no—
98#11719924—42no—
99#11719923—42no—
100#11719922—42no—
101#11719917D811yescharacteristic
102Dic212—21no—
103Dic212—21no—
104D424—21no—
105D424—21no—
106#11719913—41no—
107#11719913—41no—
108#11719912D812yes—
109C8×D53C2×D412yes—
110C16×C53C2×D411yescharacteristic
111C16×C53C2411yescharacteristic
112#11719908C2×D412yes—
113C23×C53—81no—
114C22×D53—82no—
115Q8×C53—42no—
116Q8×C53—81no—
117D4×C53—42no—
118D4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D811yescharacteristic
121C22⋊D53—81no—
122C22⋊D53—81no—
123C22⋊D53—82no—
124C22⋊D53—82no—
125C22⋊D53D1612yes—
126C2×Dic53—82no—
127D212—41no—
128D212—41no—
129D212—82no—
130D212D1611yescharacteristic
131C4×D53—82no—
132C4×D53—81no—
133C4×D53—81no—
134C4×D53—82no—
135C4×D53C2×D812yes—
136Dic106—41no—
137Dic106—41no—
138Dic106—82no—
139Dic106D1611yescharacteristic
140C8×C53C2×D811yescharacteristic
141C8×C53C22×D411yescharacteristic
142#88322C2×D812yes—
143#33983—531nomaximal, Sylow
144C22×C53—82no—
145C22×C53—321no—
146C22×C53C2×D1611yescharacteristic
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—322no—
150C2×D53C2×D1612yes—
151C4×C53—82no—
152C4×C53C2×D1611yescharacteristic
153C4×C53C22×D811yescharacteristic
154Dic53—81no—
155Dic53—81no—
156Dic53C2×D1612yes—
157#3417—531no—
158#3415—531no—
159#2264—531no—
160#2264—532no—
161#2263—531no—
162#2263—532no—
163#2262—532no—
164#2262—531no—
165C2×SD64—532no—
166C2×SD64D5311yescharacteristic, complemented
167#2258—531no—
168C2×C53—21no—
169C2×C53—162no—
170C2×C53C22×D1611yescharacteristic
171D53—22no—
172D53—161no—
173D53—161no—
174#578—1061no—
175#576—1061no—
176#510—1062no—
177#510—1061no—
178#509—1062no—
179#509—1061no—
180#508—532no—
181#508—532no—
182#508—531no—
183#508—531no—
184C2×Q32—532no—
185C2×Q32C2×D5311yescharacteristic, complemented
186C2×D16—532no—
187C2×D16C2×D5311yescharacteristic
188#504—531no—
189Q64—532no—
190Q64—531no—
191SD64—532no—
192SD64—532no—
193SD64—531no—
194SD64—531no—
195SD64C2×D5312yescomplemented
196SD64C2×D5312yescomplemented
197D32—532no—
198D32—531no—
199M64—531no—
200M64—532no—
201C2×C32—532no—
202C2×C32C2×D5311yescharacteristic, complemented
203C53#3398311yescharacteristic, complemented, Sylow
2042−1+4—2121no—
205H(5,𝔽2)—2121no—
206#137—2122no—
207#137—2121no—
208AGL(1,ℤ/8)—2122no—
209AGL(1,ℤ/8)—2121no—
210#135—1062no—
211#135—1062no—
212#135—1061no—
213#135—1061no—
214C2×Q16—21no—
215C2×Q16—1062no—
216C2×D8—21no—
217C2×D8—1062no—
218#131—531no—
219Q32—21no—
220Q32—531no—
221Q32—531no—
222Q32—1062no—
223Q32C22×D5312yes—
224SD32—1062no—
225SD32—1062no—
226SD32—1061no—
227SD32—1061no—
228D16—21no—
229D16—531no—
230D16—531no—
231D16—1062no—
232D16C22×D5312yes—
233M32—531no—
234M32—532no—
235C2×C16—532no—
236C2×C16C22×D5311yescharacteristic
237C32—532no—
238C32C22×D5311yescharacteristic
239C32C22×D5311yescharacteristic
240#41—531no—
241#41—4242no—
242#41—4241no—
243#41—4241no—
244#41—2122no—
245#41—2122no—
246#41—2121no—
247#41—2121no—
248C2×Q8—41no—
249C2×Q8—4241no—
250C2×Q8—2122no—
251C2×D4—41no—
252C2×D4—4241no—
253C2×D4—4242no—
254C2×D4—2122no—
255Q16—22no—
256Q16—41no—
257Q16—1061no—
258Q16—1061no—
259Q16—2122no—
260SD16—2122no—
261SD16—2122no—
262SD16—2121no—
263SD16—2121no—
264D8—22no—
265D8—41no—
266D8—1061no—
267D8—1061no—
268D8—2122no—
269M16—531no—
270M16—532no—
271C2×C8—532no—
272C2×C8D4×D5311yescharacteristic
273C16—532no—
274C16D4×D5311yescharacteristic
275C16C23×D5311yescharacteristic
276C23—81no—
277C23—4242no—
278Q8—42no—
279Q8—81no—
280Q8—531no—
281Q8—4242no—
282Q8—2121no—
283Q8—2121no—
284D4—42no—
285D4—81no—
286D4—532no—
287D4—531no—
288D4—4242no—
289D4—4242no—
290D4—4242no—
291D4—4241no—
292D4—4241no—
293D4—2121no—
294D4—2121no—
295C2×C4—81no—
296C2×C4—532no—
297C2×C4—4242no—
298C2×C4—4242no—
299C2×C4—4242no—
300C2×C4—4241no—
301C2×C4—4241no—
302C2×C4D8×D5311yescharacteristic
303C8—532no—
304C8D8×D5311yescharacteristic
305C8C2×D4×D5311yescharacteristic
306C22—82no—
307C22—321no—
308C22—532no—
309C22—16962no—
310C22—4241no—
311C22—4241no—
312C22D16×D5311yescharacteristic
313C4—82no—
314C4—532no—
315C4—4241no—
316C4—4241no—
317C4D16×D5311yescharacteristic
318C4C2×D8×D5311yescharacteristic
319C2—21no—
320C2—162no—
321C2—8481no—
322C2—8481no—
323C2—1062no—
324C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC2×C53×SD64C2
socleC2×C53C22×D16
hypercenterC2×SD64D53
hypocenterC53#33983
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
1. #20287864C2semidirect
2. #20287852C2semidirect
3. #20277033C2semidirect
4. #20277031C2semidirect
5. #20277029C2semidirect
6. #20277028C2semidirect
7. #20277024C2semidirect
8. #20277023C2semidirect
9. SD64×D53C2semidirect
10. #20277008C2semidirect
11. C2×C53×SD64C2semidirect
18. #16811056C22semidirect
20. #16811055C22semidirect
22. Q32×D53C22semidirect
25. Dic848C22semidirect
26. #16810463C22semidirect
27. #16810462C22semidirect
28. #16810462C22semidirect
29. #16810461C22semidirect
30. #16810460C22semidirect
31. D1696C22semidirect
32. #16810458C22semidirect
33. #16810457C22semidirect
34. #16810456C22semidirect
35. C32×D53C22semidirect
36. C2×Q32×C53C22semidirect
38. C53×SD64C22semidirect
39. C53×SD64C22semidirect
40. C2×C32×C53C22semidirect
166. C2×SD64D53semidirect
185. C2×Q32C2×D53semidirect
195. SD64C2×D53semidirect
196. SD64C2×D53semidirect
202. C2×C32C2×D53semidirect
203. C53#33983semidirect

Maximal subgroups

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

subgroupindex
#202878642
#202878522
#202770332
#202770312
#202770292
#202770282
#202770242
#202770232
SD64×D532
#202770082
C2×C53×SD642
#3398353