Finite Group Databasebeta

← #27542234

Subgroups — #27542234

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811052C211yesmaximal, characteristic, complemented
2#508×D53C211yesmaximal, characteristic, complemented
3#20277035C211yesmaximal, characteristic, complemented
4#20277032C211yesmaximal, characteristic, complemented
5#20277031C212yesmaximal, complemented
6#20277028C211yesmaximal, characteristic, complemented
7#20277026C212yesmaximal, complemented
8#20277023C212yesmaximal, complemented
9#20277021C212yesmaximal, complemented
10M64×D53C211yesmaximal, characteristic, complemented
11C53×#2263C211yesmaximal, characteristic, complemented
12C2×#13133236—21no—
13#135×D53—21no—
14#16811056C2211yescharacteristic, complemented
15#16811055C2211yescharacteristic
16#16811054—21no—
17#16811053—21no—
18#16811052—21no—
19#16811052—21no—
20#16811052C2212yes—
21#16811052C2211yescharacteristic, complemented
22#16811052C2212yescomplemented
23C2×Dic424C2211yescharacteristic, complemented
24Q32×D53C2211yescharacteristic, complemented
25C2×#13133161—21no—
26SD32×D53—21no—
27D16×D53C2211yescharacteristic, complemented
28C2×C16×D53C2211yescharacteristic
29Dic848C2212yescomplemented
30#16810463C2212yescomplemented
31#16810462C2212yescomplemented
32#16810461C2212yescomplemented
33#16810461C2212yescomplemented
34#16810458C2212yescomplemented
35#16810457C2211yescharacteristic, complemented
36#16810456C2212yescomplemented
37C32×D53C2212yescomplemented
38C53×#508C2211yescharacteristic
39C2×D16×C53C2211yescharacteristic, complemented
40C53×SD64C2212yescomplemented
41C53×D32C2212yescomplemented
42C53×M64C2211yescharacteristic, complemented
43C2×#11719947—41no—
44#41×D53—41no—
45#13133240—21no—
46#13133239—21no—
47#13133238—41no—
48#13133237—41no—
49#13133236—22no—
50#13133236—22no—
51#13133236—21no—
52#13133236—41no—
53#13133236—41no—
54C2×Dic212—21no—
55Q16×D53—21no—
56C2×#11719923—41no—
57SD16×D53—41no—
58D8×D53—21no—
59C2×C8×D53D411yescharacteristic
60Dic424—21no—
61Dic424C2311yescharacteristic
62Dic424C2312yes—
63#13133163—21no—
64#13133162—21no—
65#13133161—21no—
66#13133161—22no—
67#13133161—22no—
68#13133160—21no—
69D848C2311yescharacteristic
70#13133158—21no—
71C2×#11719908D411yescharacteristic
72C16×D53D411yescharacteristic
73C16×D53D411yescharacteristic
74C16×D53C2311yescharacteristic
75C16×D53C2311yescharacteristic
76#13133123C2312yes—
77#135×C53—21no—
78C2×D8×C53—21no—
79Q32×C53C2311yescharacteristic
80SD32×C53—21no—
81D16×C53—21no—
82D16×C53C2312yes—
83D16×C53C2311yescharacteristic
84C2×C16×C53C2311yescharacteristic
85C32×C53C2312yes—
86#41×C53—41no—
87C2×D4×C53—41no—
88C2×C22⋊D53—81no—
89C22×Dic53—81no—
90#11719949—41no—
91Q8×D53—41no—
92#11719947—42no—
93#11719947—42no—
94#11719947—41no—
95#11719947—81no—
96#11719947—81no—
97#11719947—81no—
98D4×D53—41no—
99D4×D53—81no—
100#11719945—41no—
101#11719945—81no—
102C2×C4×D53—81no—
103C2×C4×D53D811yescharacteristic
104C2×Dic106—41no—
105Q16×C53—21no—
106SD16×C53—41no—
107D8×C53—22no—
108D8×C53—21no—
109D8×C53—41no—
110C2×C8×C53C2×D411yescharacteristic
111#11719925—41no—
112#11719924—41no—
113#11719923—41no—
114#11719923—42no—
115#11719923—42no—
116#11719922—41no—
117C2×#88322D811yescharacteristic
118Dic212—22no—
119Dic212—21no—
120Dic212—41no—
121D424—21no—
122#11719913—41no—
123C8×D53D811yescharacteristic
124C8×D53D811yescharacteristic
125C8×D53C2×D411yescharacteristic
126C8×D53C2×D411yescharacteristic
127C16×C53C2×D411yescharacteristic
128C16×C53C2411yescharacteristic
129#11719908C2×D411yescharacteristic
130#11719908C2×D411yescharacteristic
131C23×C53—81no—
132C22×D53—81no—
133C22×D53D1611yescharacteristic
134Q8×C53—41no—
135D4×C53—41no—
136D4×C53—42no—
137D4×C53—81no—
138D4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53C2×D811yescharacteristic
141C22⋊D53—81no—
142C22⋊D53—81no—
143C22⋊D53—82no—
144C22⋊D53—82no—
145C22⋊D53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—82no—
149C2×Dic53—82no—
150C2×Dic53D1611yescharacteristic
151D212—41no—
152D212—81no—
153C4×D53—81no—
154C4×D53—81no—
155C4×D53—81no—
156C4×D53D1611yescharacteristic
157C4×D53D1611yescharacteristic
158C4×D53C2×D811yescharacteristic
159C4×D53C2×D811yescharacteristic
160Dic106—42no—
161Dic106—41no—
162Dic106—81no—
163Dic106—81no—
164C8×C53C2×D811yescharacteristic
165C8×C53C22×D411yescharacteristic
166#88322C2×D811yescharacteristic
167#88322C2×D811yescharacteristic
168#33981—531nomaximal, Sylow
169C22×C53—82no—
170C22×C53—81no—
171C22×C53—162no—
172C22×C53C2×D1611yescharacteristic
173C2×D53—41no—
174C2×D53—81no—
175C2×D53—162no—
176C2×D53C2×D1611yescharacteristic
177C2×D53C2×D1611yescharacteristic
178C4×C53—81no—
179C4×C53C2×D1611yescharacteristic
180C4×C53C22×D811yescharacteristic
181Dic53—81no—
182Dic53—82no—
183Dic53C2×D1611yescharacteristic
184Dic53C2×D1611yescharacteristic
185C2×#508—531no—
186C2×#508—531no—
187#2264—531no—
188#2264—532no—
189#2264—531no—
190#2263—531no—
191#2263—532no—
192#2263D5311yescharacteristic, complemented
193#2262—532no—
194#2262—532no—
195C2×M64—531no—
196C2×C53—21no—
197C2×C53—161no—
198C2×C53—162no—
199C2×C53C22×D1611yescharacteristic
200D53—21no—
201D53—21no—
202D53—161no—
203C2×#135—1061no—
204C2×#135—1061no—
205#508—531no—
206#508—532no—
207#508—532no—
208#508—531no—
209#508—531no—
210#508—1061no—
211#508—1061no—
212#508—1061no—
213#508—1061no—
214#508C2×D5311yescharacteristic
215C2×Q32—531no—
216C2×Q32—531no—
217C2×SD32—1061no—
218C2×SD32—1061no—
219C2×D16—531no—
220C2×D16C2×D5311yescharacteristic, complemented
221C22×C16—531no—
222Q64—532no—
223Q64—532no—
224SD64—532no—
225SD64—532no—
226SD64—532no—
227SD64C2×D5312yescomplemented
228D32—532no—
229D32C2×D5312yescomplemented
230M64—531no—
231M64—532no—
232M64C2×D5311yescharacteristic, complemented
233C2×C32—532no—
234C53#3398111yescharacteristic, complemented, Sylow
235C2×#41—2121no—
236C2×#41—2121no—
237#135—21no—
238#135—1062no—
239#135—1062no—
240#135—1061no—
241#135—1061no—
242#135—1061no—
243#135—2121no—
244#135—2121no—
245#135—2121no—
246#135—2121no—
247C2×Q16—1061no—
248C2×Q16—1061no—
249C2×SD16—2121no—
250C2×SD16—2121no—
251C2×D8—21no—
252C2×D8—1061no—
253C22×C8—531no—
254Q32—531no—
255Q32—532no—
256Q32—1061no—
257Q32—1061no—
258Q32C22×D5311yescharacteristic
259SD32—21no—
260SD32—1062no—
261SD32—1062no—
262SD32—1061no—
263SD32—1061no—
264SD32—1061no—
265D16—21no—
266D16—531no—
267D16—1061no—
268D16C22×D5312yes—
269D16C22×D5311yescharacteristic
270C2×C16—531no—
271C2×C16—531no—
272C2×C16—531no—
273C2×C16—531no—
274C2×C16—531no—
275C2×C16C22×D5311yescharacteristic
276C32—532no—
277C32C22×D5312yes—
278#41—41no—
279#41—4241no—
280#41—4241no—
281#41—4241no—
282#41—4241no—
283#41—2122no—
284#41—2122no—
285#41—2121no—
286#41—2121no—
287#41—2121no—
288C2×Q8—2121no—
289C2×Q8—2121no—
290C2×D4—41no—
291C2×D4—4241no—
292C2×D4—4241no—
293C2×D4—2121no—
294C22×C4—531no—
295C22×C4—4241no—
296C22×C4—4241no—
297Q16—21no—
298Q16—1062no—
299Q16—1061no—
300Q16—2121no—
301Q16—2121no—
302SD16—41no—
303SD16—2122no—
304SD16—2122no—
305SD16—2121no—
306SD16—2121no—
307SD16—2121no—
308D8—22no—
309D8—21no—
310D8—41no—
311D8—1061no—
312D8—2121no—
313C2×C8—531no—
314C2×C8—531no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8D4×D5311yescharacteristic
319C16—531no—
320C16—531no—
321C16D4×D5311yescharacteristic
322C16C23×D5311yescharacteristic
323C23—81no—
324C23—531no—
325C23—4241no—
326Q8—41no—
327Q8—4241no—
328Q8—4241no—
329Q8—2122no—
330Q8—2121no—
331D4—42no—
332D4—41no—
333D4—81no—
334D4—81no—
335D4—4241no—
336D4—4241no—
337D4—4241no—
338D4—4242no—
339D4—4242no—
340D4—4241no—
341D4—2121no—
342C2×C4—81no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—531no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—4242no—
352C2×C4—4242no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4D8×D5311yescharacteristic
356C8—531no—
357C8—531no—
358C8D8×D5311yescharacteristic
359C8C2×D4×D5311yescharacteristic
360C22—81no—
361C22—82no—
362C22—162no—
363C22—531no—
364C22—531no—
365C22—8482no—
366C22—4241no—
367C22—2121no—
368C22D16×D5311yescharacteristic
369C4—81no—
370C4—531no—
371C4—531no—
372C4—4241no—
373C4—4242no—
374C4D16×D5311yescharacteristic
375C4C2×D8×D5311yescharacteristic
376C2—21no—
377C2—162no—
378C2—161no—
379C2—8481no—
380C2—1061no—
381C2—1061no—
382C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2263C2
socleC2×C53C22×D16
hypercenter#2263D53
hypocenterC53#33981
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. C2×#16811052C2semidirect
2. #508×D53C2semidirect
3. #20277035C2semidirect
4. #20277032C2semidirect
5. #20277031C2semidirect
6. #20277028C2semidirect
7. #20277026C2semidirect
8. #20277023C2semidirect
9. #20277021C2semidirect
10. M64×D53C2semidirect
11. C53×#2263C2semidirect
14. #16811056C22semidirect
21. #16811052C22semidirect
22. #16811052C22semidirect
23. C2×Dic424C22semidirect
24. Q32×D53C22semidirect
27. D16×D53C22semidirect
29. Dic848C22semidirect
30. #16810463C22semidirect
31. #16810462C22semidirect
32. #16810461C22semidirect
33. #16810461C22semidirect
34. #16810458C22semidirect
35. #16810457C22semidirect
36. #16810456C22semidirect
37. C32×D53C22semidirect
39. C2×D16×C53C22semidirect
40. C53×SD64C22semidirect
41. C53×D32C22semidirect
42. C53×M64C22semidirect
192. #2263D53semidirect
220. C2×D16C2×D53semidirect
227. SD64C2×D53semidirect
229. D32C2×D53semidirect
232. M64C2×D53semidirect
234. C53#33981semidirect

Maximal subgroups

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

subgroupindex
C2×#168110522
#508×D532
#202770352
#202770322
#202770312
#202770282
#202770262
#202770232
#202770212
M64×D532
C53×#22632
#3398153