Finite Group Databasebeta

← #27538746

Subgroups — #27538746

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285227C211yesmaximal, characteristic, complemented
2C2×#16810705C211yesmaximal, characteristic
3#20276875C212yesmaximal, complemented
4#20276871C212yesmaximal, complemented
5#20276861C212yesmaximal, complemented
6#20276851C212yesmaximal, complemented
7#20276501C211yesmaximal, characteristic, complemented
8#20276470C211yesmaximal, characteristic, complemented
9#20276432C211yesmaximal, characteristic
10#20276430C211yesmaximal, characteristic
11C53×#2159C211yesmaximal, characteristic, complemented
12C22×D424C2211yescharacteristic
13#16811176—21no—
14C2×#13133179—21no—
15C2×#13133157C2211yescharacteristic
16#16811027C2211yescharacteristic, complemented
17#16811027C2211yescharacteristic, complemented
18#16811024C2211yescharacteristic, complemented
19#16811024C2211yescharacteristic, complemented
20#16811023—21no—
21#16811020—21no—
22#16810705—21no—
23#16810705—21no—
24#16810705C2212yes—
25#16810705C2212yes—
26#16810661C2211yescharacteristic
27#16810659—21no—
28C2×#13133137—21no—
29C2×C8×Dic53C2211yescharacteristic
30#16810450C2212yescomplemented
31#16810448C2212yescomplemented
32#16810441C2212yescomplemented
33#16810441C2212yescomplemented
34#16810425C2212yes—
35#16810425C2212yes—
36#16810423C2212yes—
37#16810423C2212yes—
38C2×M32×C53C2211yescharacteristic
39C53×#427C2211yescharacteristic
40C2×#107×C53C2211yescharacteristic
41C53×#367C2212yescomplemented
42C53×#366C2212yescomplemented
43C22×D212—21no—
44C22×C4×D53—41no—
45C2×D424—21no—
46C2×D424—22no—
47C2×D424Q814yescomplemented
48C2×D424C2312yes—
49C22×#88322D411yescharacteristic
50#13133216—21no—
51#13133216—21no—
52#13133208—21no—
53#13133208—21no—
54#13133194—41no—
55#13133194—41no—
56C4×Dic106—41no—
57C4×Dic106—41no—
58#13133179—22no—
59#13133179—22no—
60#13133179—41no—
61#13133179—41no—
62#13133176—21no—
63C2×#11719921—41no—
64C2×C4×Dic53D411yescharacteristic
65#13133157—22no—
66C2×#11719908C2312yes—
67#13133151C2311yescharacteristic
68#13133151C2311yescharacteristic
69#13133150—21no—
70#13133150—21no—
71#13133149—21no—
72#13133148—21no—
73#13133148—21no—
74#13133137—22no—
75#13133137—22no—
76C8×Dic53D411yescharacteristic
77C8×Dic53D411yescharacteristic
78C8×Dic53C2311yescharacteristic
79C8×Dic53C2311yescharacteristic
80C22×C8×C53C2311yescharacteristic
81#117×C53—21no—
82C2×C4⋊C4×C53—21no—
83M32×C53—22no—
84C2×C16×C53C2312yes—
85#107×C53—21no—
86#107×C53C2311yescharacteristic
87#107×C53C2311yescharacteristic
88#107×C53C2312yes—
89#106×C53—21no—
90C23×D53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53D811yescharacteristic
94C22×Dic53D811yescharacteristic
95C2×D212—24no—
96C2×D212—22no—
97C2×D212—41no—
98C2×D212—42no—
99C2×C4×D53—42no—
100C2×C4×D53—42no—
101C2×C4×D53—84no—
102C2×Dic106—41no—
103C2×Dic106—41no—
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×Q812yes—
106C2×C8×C53#4112yes—
107C2×C8×C53C2411yescharacteristic
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53—41no—
114C22⋊Dic53—41no—
115#11719921—42no—
116#11719921—42no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121#11719919—41no—
122#11719919—41no—
123#11719919—41no—
124#11719919—41no—
125C4×Dic53—41no—
126C4×Dic53D811yescharacteristic
127C4×Dic53D811yescharacteristic
128C4×Dic53C2×D411yescharacteristic
129C4×Dic53C2×D411yescharacteristic
130C2×#88322—22no—
131C2×#88322C2×D411yescharacteristic
132C2×#88322C2×D411yescharacteristic
133D424—24no—
134D424—22no—
135D424C2×Q814yes—
136C16×C53—22no—
137#11719908—22no—
138C23×C53C2×D811yescharacteristic
139C22×D53—42no—
140C22×D53—44no—
141C22×D53—82no—
142C22×D53—82no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—82no—
147C2×C4×C53C22⋊Q812yes—
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53#13712yes—
150C2×C4×C53C22×D411yescharacteristic
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53C2×D811yescharacteristic
156C2×Dic53C2×D811yescharacteristic
157C2×Dic53C2×D811yescharacteristic
158C2×Dic53C2×D811yescharacteristic
159D212—24no—
160D212—42no—
161D212—44no—
162C4×D53—84no—
163C4×D53—84no—
164Dic106—82no—
165Dic106—82no—
166C8×C53C22⋊Q812yes—
167C8×C53C22×Q811yescharacteristic
168C8×C53C2×#4111yescharacteristic
169#88322—21no—
170#88322—21no—
171#33627—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53#47412yes—
175C22×C53C22×D811yescharacteristic
176C2×D53—44no—
177C2×D53—88no—
178C2×D53—82no—
179C2×D53—82no—
180C4×C53—81no—
181C4×C53—82no—
182C4×C53—81no—
183C4×C53#47412yes—
184C4×C53C2×C22⋊Q811yescharacteristic
185C4×C53C2×#13711yescharacteristic
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189Dic53—81no—
190Dic53—81no—
191Dic53#50912yes—
192#3160—531no—
193C2×C4×D8—531no—
194#2228—532no—
195#2228—532no—
196#2226—532no—
197#2224—532no—
198#2159—531no—
199#2159D5311yescharacteristic, complemented
200#2150—531no—
201#2140—531no—
202#2140—531no—
203C2×C53—22no—
204C2×C53C2×#47411yescharacteristic
205C2×C53C2×#50911yescharacteristic
206C2×C53#341411yescharacteristic
207D53—84no—
208C22×D8—531no—
209#533—1061no—
210C2×C4×D4—1061no—
211C2×M32—531no—
212C2×M32C2×D5311yescharacteristic
213#500—531no—
214#500—531no—
215#499—531no—
216#499—531no—
217#499—1061no—
218#498—1061no—
219C4×D8—532no—
220C4×D8—532no—
221C4×D8—1061no—
222C4×D8—1061no—
223#427—531no—
224#427—1061no—
225#427C2×D5311yescharacteristic
226C2×#107C2×D5311yescharacteristic
227C2×#102—1061no—
228C2×C4×C8—531no—
229#367—532no—
230#367C2×D5312yescomplemented
231#366—532no—
232#366C2×D5312yescomplemented
233C4⋊C16—532no—
234C4⋊C16—532no—
235#357—532no—
236#357—532no—
237#357—532no—
238#357—532no—
239C53#3362711yescharacteristic, complemented, Sylow
240C22×D4—1061no—
241C23×C4—2121no—
242C2×D8—534no—
243C2×D8—532no—
244C2×D8—1062no—
245C2×D8—1061no—
246C22×C8—531no—
247C22×C8C22×D5311yescharacteristic
248C4⋊Q8—1061no—
249C4⋊Q8—1061no—
250#125—1061no—
251#125—1061no—
252C22⋊Q8—2121no—
253C22⋊Q8—2121no—
254C4×Q8—2121no—
255C4×Q8—2121no—
256C4×D4—1062no—
257C4×D4—1062no—
258C4×D4—2121no—
259C4×D4—2121no—
260#117—21no—
261#117—1061no—
262C2×C4⋊C4—21no—
263C2×C22⋊C4—2121no—
264C2×C42—531no—
265M32—22no—
266M32—1062no—
267C2×C16—532no—
268C2×C16C22×D5312yes—
269#107—21no—
270#107—531no—
271#107—531no—
272#107—1061no—
273#107—1061no—
274#107C22×D5311yescharacteristic
275#107C22×D5311yescharacteristic
276#107C22×D5312yes—
277#106—21no—
278#106—1061no—
279#106—1061no—
280#106—1061no—
281#102—1062no—
282#102—1062no—
283C4×C8—531no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C24—2121no—
288C2×Q8—2121no—
289C2×Q8—2121no—
290C2×D4—1064no—
291C2×D4—1062no—
292C2×D4—2121no—
293C2×D4—2122no—
294C22×C4—41no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—4244no—
298C22×C4—2122no—
299C22×C4—2122no—
300C22×C4D4×D5311yescharacteristic
301D8—534no—
302D8—1064no—
303D8—1062no—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—1062no—
307C2×C8#1171994512yes—
308C2×C8D4×D5311yescharacteristic
309C2×C8Q8×D5312yes—
310C2×C8C23×D5311yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—41no—
324C22⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4—2121no—
327C42—41no—
328C42—531no—
329C42—531no—
330C42—531no—
331C42—531no—
332C42—2121no—
333C16—22no—
334C16—1062no—
335C23—4242no—
336C23—4242no—
337C23—2122no—
338C23—2124no—
339C23D8×D5311yescharacteristic
340Q8—4242no—
341Q8—4242no—
342D4—1064no—
343D4—2124no—
344D4—2122no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—531no—
353C2×C4—1062no—
354C2×C4—4244no—
355C2×C4—4244no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4#1313319512yes—
360C2×C4D8×D5311yescharacteristic
361C2×C4#1313324912yes—
362C2×C4C2×D4×D5311yescharacteristic
363C8—1061no—
364C8—1061no—
365C8#1313319512yes—
366C8C2×Q8×D5311yescharacteristic
367C8C2×#1171994511yescharacteristic
368C22—21no—
369C22—21no—
370C22—4248no—
371C22—4242no—
372C22—4242no—
373C22—2124no—
374C22#1681090212yes—
375C22C2×D8×D5311yescharacteristic
376C4—82no—
377C4—81no—
378C4—81no—
379C4—532no—
380C4—1061no—
381C4—1061no—
382C4—1061no—
383C4—4241no—
384C4—4241no—
385C4#1681090212yes—
386C4C2×#1313319511yescharacteristic
387C4C2×#1313324911yescharacteristic
388C2—22no—
389C2—4244no—
390C2C2×#1681090211yescharacteristic
391C2#509×D5311yescharacteristic
392C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC22×D424C22
hypercenter#2159D53
hypocenterC53#33627
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20276875C2semidirect
4. #20276871C2semidirect
5. #20276861C2semidirect
6. #20276851C2semidirect
7. #20276501C2semidirect
8. #20276470C2semidirect
11. C53×#2159C2semidirect
16. #16811027C22semidirect
17. #16811027C22semidirect
18. #16811024C22semidirect
19. #16811024C22semidirect
30. #16810450C22semidirect
31. #16810448C22semidirect
32. #16810441C22semidirect
33. #16810441C22semidirect
41. C53×#367C22semidirect
42. C53×#366C22semidirect
47. C2×D424Q8semidirect
199. #2159D53semidirect
230. #367C2×D53semidirect
232. #366C2×D53semidirect
239. C53#33627semidirect

Maximal subgroups

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

subgroupindex
#202852272
C2×#168107052
#202768752
#202768712
#202768612
#202768512
#202765012
#202764702
#202764322
#202764302
C53×#21592
#3362753