Finite Group Databasebeta

← #27542248

Subgroups — #27542248

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#20287865C211yesmaximal, characteristic, complemented
2#20287852C211yesmaximal, characteristic, complemented
3#20277035C211yesmaximal, characteristic, complemented
4#20277031C212yesmaximal, complemented
5#20277026C212yesmaximal, complemented
6#20277022C211yesmaximal, characteristic, complemented
7#20277021C212yesmaximal, complemented
8C2×#16810461C211yesmaximal, characteristic, complemented
9SD64×D53C212yesmaximal, complemented
10#20277010C211yesmaximal, characteristic, complemented
11C53×#2263C211yesmaximal, characteristic, complemented
12#16811437—21no—
13#16811424—21no—
14#16811063—21no—
15#16811062—22no—
16#16811059—21no—
17#16811057—22no—
18#16811055C2211yescharacteristic
19#16811052C2212yes—
20#16811052C2212yescomplemented
21C2×Dic424C2211yescharacteristic, complemented
22Q32×D53C2212yescomplemented
23D16×D53C2212yes—
24#16811039C2211yescharacteristic
25Dic848C2212yescomplemented
26#16810463C2211yescharacteristic, complemented
27#16810462C2212yescomplemented
28#16810461C2212yescomplemented
29#16810461C2212yescomplemented
30#16810461C2211yescharacteristic, complemented
31#16810460C2211yescharacteristic, complemented
32#16810458C2211yescharacteristic, complemented
33#16810456C2212yescomplemented
34C2×#13133123C2211yescharacteristic, complemented
35C32×D53C2212yescomplemented
36C53×#508C2211yescharacteristic, complemented
37C2×D16×C53C2211yescharacteristic
38C53×SD64C2212yescomplemented
39C53×D32C2212yescomplemented
40C53×M64C2211yescharacteristic, complemented
41#13133264—41no—
42#13133262—41no—
43#13133247—41no—
44#13133246—42no—
45#13133243—41no—
46#13133241—42no—
47#13133239—21no—
48#13133236—22no—
49#13133236—22no—
50C2×Dic212—21no—
51Q16×D53—22no—
52D8×D53—22no—
53#13133223D411yescharacteristic
54Dic424—21no—
55Dic424C2312yes—
56Dic424C2311yescharacteristic
57#13133163—22no—
58#13133162—21no—
59#13133161—22no—
60#13133161—22no—
61D848C2311yescharacteristic
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—
80Q8×D53—42no—
81#11719947—42no—
82#11719947—42no—
83#11719947—82no—
84#11719947—81no—
85D4×D53—42no—
86D4×D53—81no—
87#11719945—41no—
88#11719945—81no—
89#11719945D811yescharacteristic
90C2×Dic106—41no—
91C2×Dic106—81no—
92Q16×C53—21no—
93SD16×C53—41no—
94D8×C53—22no—
95D8×C53—21no—
96D8×C53—41no—
97C2×C8×C53C2×D411yescharacteristic
98#11719925—42no—
99#11719923—42no—
100#11719923—42no—
101#11719922—42no—
102#11719917D811yescharacteristic
103Dic212—21no—
104Dic212—22no—
105Dic212—41no—
106D424—21no—
107#11719913—41no—
108#11719912D812yes—
109C8×D53C2×D412yes—
110C16×C53C2×D411yescharacteristic
111C16×C53C2411yescharacteristic
112#11719908C2×D412yes—
113C23×C53—81no—
114C22×D53—82no—
115Q8×C53—41no—
116D4×C53—42no—
117D4×C53—41no—
118D4×C53—81no—
119D4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53C2×D811yescharacteristic
122C22⋊D53—81no—
123C22⋊D53—82no—
124C22⋊D53—82no—
125C22⋊D53—82no—
126C22⋊D53D1612yes—
127C2×Dic53—81no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130D212—41no—
131D212D1611yescharacteristic
132C4×D53—82no—
133C4×D53—81no—
134C4×D53C2×D812yes—
135Dic106—42no—
136Dic106—41no—
137Dic106—81no—
138Dic106—82no—
139Dic106D1611yescharacteristic
140C8×C53C2×D811yescharacteristic
141C8×C53C22×D411yescharacteristic
142#88322C2×D812yes—
143#33983—531nomaximal, Sylow
144C22×C53—81no—
145C22×C53—82no—
146C22×C53—321no—
147C22×C53C2×D1611yescharacteristic
148C2×D53—81no—
149C2×D53—322no—
150C2×D53C2×D1612yes—
151C4×C53—81no—
152C4×C53C2×D1611yescharacteristic
153C4×C53C22×D811yescharacteristic
154Dic53—82no—
155Dic53—81no—
156Dic53C2×D1612yes—
157#3417—531no—
158#3415—531no—
159#2264—532no—
160#2264—531no—
161#2263—532no—
162#2263D5311yescharacteristic, complemented
163#2262—531no—
164#2262—532no—
165C2×SD64—531no—
166C2×SD64—532no—
167#2258—531no—
168C2×C53—21no—
169C2×C53—162no—
170C2×C53—161no—
171C2×C53C22×D1611yescharacteristic
172D53—22no—
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#508C2×D5311yescharacteristic, complemented
184C2×Q32—532no—
185C2×Q32—531no—
186C2×D16—532no—
187C2×D16C2×D5311yescharacteristic
188#504—531no—
189Q64—531no—
190Q64—532no—
191SD64—531no—
192SD64—532no—
193SD64—532no—
194SD64—531no—
195SD64—532no—
196SD64C2×D5312yescomplemented
197D32—531no—
198D32C2×D5312yescomplemented
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—1062no—
215C2×Q16—1061no—
216C2×D8—21no—
217C2×D8—1062no—
218#131—531no—
219Q32—532no—
220Q32—531no—
221Q32—1062no—
222Q32—1061no—
223Q32C22×D5311yescharacteristic
224SD32—21no—
225SD32—1062no—
226SD32—1062no—
227SD32—1061no—
228D16—21no—
229D16—531no—
230D16—1062no—
231D16C22×D5311yescharacteristic
232D16C22×D5312yes—
233M32—531no—
234M32—532no—
235C2×C16—532no—
236C2×C16C22×D5311yescharacteristic
237C32—531no—
238C32—531no—
239C32C22×D5312yes—
240#41—41no—
241#41—531no—
242#41—4241no—
243#41—4242no—
244#41—4241no—
245#41—2122no—
246#41—2122no—
247#41—2121no—
248C2×Q8—4241no—
249C2×Q8—2122no—
250C2×Q8—2121no—
251C2×D4—41no—
252C2×D4—4241no—
253C2×D4—4242no—
254C2×D4—2122no—
255Q16—21no—
256Q16—1061no—
257Q16—1062no—
258Q16—2122no—
259Q16—2121no—
260SD16—41no—
261SD16—2122no—
262SD16—2122no—
263SD16—2121no—
264D8—22no—
265D8—21no—
266D8—41no—
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—41no—
279Q8—531no—
280Q8—4242no—
281Q8—4241no—
282Q8—2121no—
283Q8—2122no—
284D4—42no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—532no—
289D4—531no—
290D4—4242no—
291D4—4242no—
292D4—4242no—
293D4—4241no—
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—81no—
307C22—82no—
308C22—321no—
309C22—532no—
310C22—16962no—
311C22—4241no—
312C22D16×D5311yescharacteristic
313C4—81no—
314C4—532no—
315C4—4241no—
316C4—4242no—
317C4D16×D5311yescharacteristic
318C4C2×D8×D5311yescharacteristic
319C2—21no—
320C2—162no—
321C2—161no—
322C2—8481no—
323C2—1062no—
324C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2263C2
socleC2×C53C22×D16
hypercenter#2263D53
hypocenterC53#33983
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. #20287865C2semidirect
2. #20287852C2semidirect
3. #20277035C2semidirect
4. #20277031C2semidirect
5. #20277026C2semidirect
6. #20277022C2semidirect
7. #20277021C2semidirect
8. C2×#16810461C2semidirect
9. SD64×D53C2semidirect
10. #20277010C2semidirect
11. C53×#2263C2semidirect
20. #16811052C22semidirect
21. C2×Dic424C22semidirect
22. Q32×D53C22semidirect
25. Dic848C22semidirect
26. #16810463C22semidirect
27. #16810462C22semidirect
28. #16810461C22semidirect
29. #16810461C22semidirect
30. #16810461C22semidirect
31. #16810460C22semidirect
32. #16810458C22semidirect
33. #16810456C22semidirect
34. C2×#13133123C22semidirect
35. C32×D53C22semidirect
36. C53×#508C22semidirect
38. C53×SD64C22semidirect
39. C53×D32C22semidirect
40. C53×M64C22semidirect
162. #2263D53semidirect
183. #508C2×D53semidirect
196. SD64C2×D53semidirect
198. D32C2×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
#202878652
#202878522
#202770352
#202770312
#202770262
#202770222
#202770212
C2×#168104612
SD64×D532
#202770102
C53×#22632
#3398353