Finite Group Databasebeta

← #27542247

Subgroups — #27542247

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#20287853C211yesmaximal, characteristic, complemented
3#20277034C212yesmaximal, complemented
4#20277030C211yesmaximal, characteristic, complemented
5#20277029C212yesmaximal, complemented
6#20277025C212yesmaximal, complemented
7#20277022C211yesmaximal, characteristic, complemented
8C2×#16810460C211yesmaximal, characteristic, complemented
9SD64×D53C212yesmaximal, complemented
10#20277010C211yesmaximal, characteristic, complemented
11C53×#2264C211yesmaximal, characteristic, complemented
12#16811436—21no—
13#16811425—21no—
14#16811065—22no—
15#16811064—21no—
16#16811060—22no—
17#16811058—21no—
18#16811056C2212yes—
19#16811056C2212yescomplemented
20#16811055C2211yescharacteristic, complemented
21Q32×D53C2212yescomplemented
22C2×D848C2211yescharacteristic
23D16×D53C2212yes—
24#16811039C2211yescharacteristic
25#16810463C2211yescharacteristic, complemented
26#16810462C2212yescomplemented
27#16810461C2211yescharacteristic, complemented
28#16810460C2212yescomplemented
29#16810460C2212yescomplemented
30#16810460C2211yescharacteristic, complemented
31D1696C2212yescomplemented
32#16810458C2211yescharacteristic, complemented
33#16810456C2212yescomplemented
34C2×#13133123C2211yescharacteristic, complemented
35C32×D53C2212yescomplemented
36C53×#508C2211yescharacteristic
37C2×Q32×C53C2211yescharacteristic, complemented
38C53×Q64C2212yescomplemented
39C53×SD64C2212yescomplemented
40C53×M64C2211yescharacteristic, complemented
41#13133265—41no—
42#13133263—41no—
43#13133249—42no—
44#13133248—41no—
45#13133244—42no—
46#13133242—41no—
47#13133240—22no—
48#13133240—22no—
49#13133239—21no—
50Q16×D53—22no—
51C2×D424—21no—
52D8×D53—22no—
53#13133223D411yescharacteristic
54Dic424C2311yescharacteristic
55#13133163—22no—
56#13133162—21no—
57#13133160—22no—
58#13133160—22no—
59D848—21no—
60D848C2312yes—
61D848C2311yescharacteristic
62#13133158—22no—
63#13133157D411yescharacteristic
64#13133156D412yes—
65C16×D53C2312yes—
66#13133123C2311yescharacteristic
67#13133123C2311yescharacteristic
68#135×C53—21no—
69C2×Q16×C53—21no—
70Q32×C53—21no—
71Q32×C53C2312yes—
72Q32×C53C2311yescharacteristic
73SD32×C53—21no—
74D16×C53C2311yescharacteristic
75C2×C16×C53C2311yescharacteristic
76C32×C53C2312yes—
77#41×C53—41no—
78C2×Q8×C53—41no—
79#11719949—42no—
80#11719949—42no—
81#11719949—81no—
82Q8×D53—42no—
83Q8×D53—81no—
84D4×D53—42no—
85D4×D53—82no—
86#11719945—41no—
87#11719945—82no—
88#11719945—81no—
89#11719945D811yescharacteristic
90C2×D212—41no—
91C2×D212—81no—
92Q16×C53—21no—
93Q16×C53—22no—
94Q16×C53—41no—
95SD16×C53—41no—
96D8×C53—21no—
97C2×C8×C53C2×D411yescharacteristic
98#11719925—42no—
99#11719924—42no—
100#11719924—42no—
101#11719922—42no—
102#11719917D811yescharacteristic
103Dic212—21no—
104D424—22no—
105D424—21no—
106D424—41no—
107#11719913—41no—
108#11719912D812yes—
109C8×D53C2×D412yes—
110C16×C53C2×D411yescharacteristic
111C16×C53C2411yescharacteristic
112#11719908C2×D412yes—
113C22×D53—81no—
114C22×D53—82no—
115Q8×C53—42no—
116Q8×C53—41no—
117Q8×C53—81no—
118D4×C53—41no—
119D4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53C2×D811yescharacteristic
123C22⋊D53—81no—
124C22⋊D53—82no—
125C22⋊D53D1612yes—
126D212—42no—
127D212—41no—
128D212—82no—
129D212—82no—
130D212—81no—
131D212D1611yescharacteristic
132C4×D53—82no—
133C4×D53—81no—
134C4×D53—82no—
135C4×D53—82no—
136C4×D53C2×D812yes—
137Dic106—41no—
138Dic106—82no—
139Dic106D1611yescharacteristic
140C8×C53C2×D811yescharacteristic
141C8×C53C22×D411yescharacteristic
142#88322C2×D812yes—
143#33983—531nomaximal, Sylow
144C22×C53—81no—
145C22×C53C2×D1611yescharacteristic
146C2×D53—81no—
147C2×D53—82no—
148C2×D53—322no—
149C2×D53—321no—
150C2×D53C2×D1612yes—
151C4×C53—82no—
152C4×C53—81no—
153C4×C53C2×D1611yescharacteristic
154C4×C53C22×D811yescharacteristic
155Dic53—81no—
156Dic53C2×D1612yes—
157#3417—531no—
158#3415—531no—
159#2264—532no—
160#2264D5311yescharacteristic, complemented
161#2263—532no—
162#2263—531no—
163#2262—531no—
164#2262—532no—
165C2×SD64—531no—
166C2×SD64—532no—
167#2258—531no—
168C2×C53—21no—
169C2×C53—161no—
170C2×C53C22×D1611yescharacteristic
171D53—22no—
172D53—161no—
173D53—162no—
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#508C2×D5311yescharacteristic
184C2×Q32—532no—
185C2×Q32C2×D5311yescharacteristic, complemented
186C2×D16—532no—
187C2×D16—531no—
188#504—531no—
189Q64—531no—
190Q64C2×D5312yescomplemented
191SD64—532no—
192SD64—532no—
193SD64—531no—
194SD64—531no—
195SD64—532no—
196SD64C2×D5312yescomplemented
197D32—531no—
198D32—532no—
199M64—532no—
200M64C2×D5311yescharacteristic, complemented
201C2×C32—531no—
202C2×C32—532no—
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—21no—
211#135—1062no—
212#135—1062no—
213#135—1061no—
214C2×Q16—21no—
215C2×Q16—1062no—
216C2×D8—1062no—
217C2×D8—1061no—
218#131—531no—
219Q32—21no—
220Q32—531no—
221Q32—1062no—
222Q32C22×D5312yes—
223Q32C22×D5311yescharacteristic
224SD32—21no—
225SD32—1062no—
226SD32—1062no—
227SD32—1061no—
228D16—531no—
229D16—532no—
230D16—1062no—
231D16—1061no—
232D16C22×D5311yescharacteristic
233M32—531no—
234M32—532no—
235C2×C16—532no—
236C2×C16C22×D5311yescharacteristic
237C32—531no—
238C32—531no—
239C32C22×D5312yes—
240#41—41no—
241#41—531no—
242#41—4242no—
243#41—4241no—
244#41—4241no—
245#41—2122no—
246#41—2122no—
247#41—2121no—
248C2×Q8—41no—
249C2×Q8—4241no—
250C2×Q8—2122no—
251C2×D4—4241no—
252C2×D4—4242no—
253C2×D4—2122no—
254C2×D4—2121no—
255Q16—21no—
256Q16—22no—
257Q16—41no—
258Q16—1061no—
259Q16—2122no—
260SD16—41no—
261SD16—2122no—
262SD16—2122no—
263SD16—2121no—
264D8—21no—
265D8—1062no—
266D8—1061no—
267D8—2122no—
268D8—2121no—
269M16—531no—
270M16—532no—
271C2×C8—532no—
272C2×C8D4×D5311yescharacteristic
273C16—532no—
274C16D4×D5311yescharacteristic
275C16C23×D5311yescharacteristic
276C23—4242no—
277C23—4241no—
278Q8—41no—
279Q8—42no—
280Q8—81no—
281Q8—531no—
282Q8—4242no—
283Q8—2121no—
284D4—41no—
285D4—81no—
286D4—532no—
287D4—531no—
288D4—4242no—
289D4—4242no—
290D4—4242no—
291D4—4241no—
292D4—4241no—
293D4—2122no—
294D4—2121no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—532no—
298C2×C4—4242no—
299C2×C4—4242no—
300C2×C4—4242no—
301C2×C4—4241no—
302C2×C4D8×D5311yescharacteristic
303C8—532no—
304C8D8×D5311yescharacteristic
305C8C2×D4×D5311yescharacteristic
306C22—81no—
307C22—532no—
308C22—16961no—
309C22—16962no—
310C22—4241no—
311C22—4242no—
312C22D16×D5311yescharacteristic
313C4—81no—
314C4—82no—
315C4—532no—
316C4—4241no—
317C4D16×D5311yescharacteristic
318C4C2×D8×D5311yescharacteristic
319C2—21no—
320C2—161no—
321C2—8482no—
322C2—8481no—
323C2—1062no—
324C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2264C2
socleC2×C53C22×D16
hypercenter#2264D53
hypocenterC53#33983
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
1. #20287864C2semidirect
2. #20287853C2semidirect
3. #20277034C2semidirect
4. #20277030C2semidirect
5. #20277029C2semidirect
6. #20277025C2semidirect
7. #20277022C2semidirect
8. C2×#16810460C2semidirect
9. SD64×D53C2semidirect
10. #20277010C2semidirect
11. C53×#2264C2semidirect
19. #16811056C22semidirect
20. #16811055C22semidirect
21. Q32×D53C22semidirect
25. #16810463C22semidirect
26. #16810462C22semidirect
27. #16810461C22semidirect
28. #16810460C22semidirect
29. #16810460C22semidirect
30. #16810460C22semidirect
31. D1696C22semidirect
32. #16810458C22semidirect
33. #16810456C22semidirect
34. C2×#13133123C22semidirect
35. C32×D53C22semidirect
37. C2×Q32×C53C22semidirect
38. C53×Q64C22semidirect
39. C53×SD64C22semidirect
40. C53×M64C22semidirect
160. #2264D53semidirect
185. C2×Q32C2×D53semidirect
190. Q64C2×D53semidirect
196. SD64C2×D53semidirect
200. M64C2×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
#202878532
#202770342
#202770302
#202770292
#202770252
#202770222
C2×#168104602
SD64×D532
#202770102
C53×#22642
#3398353