Finite Group Databasebeta

← #27538916

Subgroups — #27538916

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810856C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic, complemented
3#20276905C212yesmaximal, complemented
4#20276904C212yesmaximal, complemented
5#20276899C212yesmaximal, complemented
6#20276894C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8#20276503C211yesmaximal, characteristic, complemented
9#20276431C211yesmaximal, characteristic, complemented
10#20276325C211yesmaximal, characteristic, complemented
11C53×#2140C211yesmaximal, characteristic, complemented
12C22×C8×D53C2211yescharacteristic
13C2×#13133188—21no—
14#117×D53—21no—
15C2×#13133157C2211yescharacteristic
16#16810856—21no—
17#16810856—21no—
18#16810856C2212yes—
19#16810856C2212yes—
20#16810661C2211yescharacteristic
21#16810660C2211yescharacteristic, complemented
22#16810660C2211yescharacteristic, complemented
23#16810659—21no—
24#16810658—21no—
25C2×#13133151C2211yescharacteristic
26#107×D53C2211yescharacteristic, complemented
27#107×D53C2211yescharacteristic, complemented
28#106×D53—21no—
29C2×#13133138—21no—
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810449C2212yescomplemented
33#16810448C2212yescomplemented
34#16810424C2212yescomplemented
35#16810424C2212yescomplemented
36#16810402C2212yescomplemented
37#16810402C2212yescomplemented
38C22×D8×C53C2211yescharacteristic
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C53×#357C2212yescomplemented
42C53×#357C2212yescomplemented
43C22×C22⋊D53—41no—
44C22×C4×D53D411yescharacteristic
45C22×#88322D411yescharacteristic
46C2×C8×D53—22no—
47C2×C8×D53—22no—
48C2×C8×D53D411yescharacteristic
49C2×C8×D53D411yescharacteristic
50C2×C8×D53C2311yescharacteristic
51C2×C8×D53C2311yescharacteristic
52#13133188—22no—
53#13133188—22no—
54#13133188—41no—
55#13133188—41no—
56#13133176—21no—
57#13133175—21no—
58#13133175—21no—
59#13133174—41no—
60#13133173—41no—
61C2×C4⋊Dic53—21no—
62C4⋊C4×D53—21no—
63C4⋊C4×D53—21no—
64C2×C22⋊Dic53—41no—
65C22⋊C4×D53—41no—
66C42×D53—41no—
67#13133157—22no—
68C2×#11719908C2312yes—
69#13133151—21no—
70#13133151C2312yes—
71#13133151C2311yescharacteristic
72#13133151C2311yescharacteristic
73#13133150—21no—
74#13133150—21no—
75#13133149—21no—
76#13133148—21no—
77#13133148—21no—
78#13133138—22no—
79#13133138—22no—
80C22×D4×C53—21no—
81C2×D8×C53—21no—
82C2×D8×C53—22no—
83C2×D8×C53—22no—
84C2×D8×C53C2312yes—
85C22×C8×C53C2311yescharacteristic
86#117×C53—21no—
87M32×C53—22no—
88C2×C16×C53C2312yes—
89#107×C53C2311yescharacteristic
90#107×C53C2311yescharacteristic
91#106×C53—21no—
92C24×C53—41no—
93C23×D53D811yescharacteristic
94C2×D4×C53—22no—
95C2×D4×C53—42no—
96C2×D4×C53—41no—
97C2×D4×C53—42no—
98C22×C4×C53C2×D411yescharacteristic
99C2×C22⋊D53—42no—
100C2×C22⋊D53—42no—
101C2×C22⋊D53—84no—
102C22×Dic53—41no—
103C22×Dic53D811yescharacteristic
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53D811yescharacteristic
109C2×C4×D53D811yescharacteristic
110C2×C4×D53C2×D411yescharacteristic
111C2×C4×D53C2×D411yescharacteristic
112D8×C53—22no—
113D8×C53—22no—
114D8×C53—42no—
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53#4112yes—
117C2×C8×C53#4112yes—
118C2×C8×C53C2411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—42no—
125C22⋊Dic53—42no—
126#11719921—41no—
127#11719921—41no—
128C4⋊Dic53—22no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131C4⋊Dic53—41no—
132#11719919—41no—
133#11719919—41no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C4×Dic53—41no—
137C2×#88322—22no—
138C2×#88322C2×D411yescharacteristic
139C2×#88322C2×D411yescharacteristic
140C8×D53—22no—
141C8×D53—22no—
142C8×D53—21no—
143C8×D53—21no—
144C8×D53—21no—
145C8×D53—21no—
146C16×C53—22no—
147#11719908—22no—
148C23×C53—42no—
149C23×C53—82no—
150C23×C53—82no—
151C23×C53—82no—
152C23×C53C2×D811yescharacteristic
153C22×D53—22no—
154C22×D53—22no—
155C22×D53—41no—
156C22×D53C2×D811yescharacteristic
157C22×D53C2×D811yescharacteristic
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—82no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53#12312yes—
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53#13712yes—
166C2×C4×C53C22×D411yescharacteristic
167C22⋊D53—84no—
168C22⋊D53—84no—
169C2×Dic53—22no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—42no—
173C2×Dic53—82no—
174C2×Dic53C2×D811yescharacteristic
175C2×Dic53C2×D811yescharacteristic
176C4×D53—21no—
177C4×D53—21no—
178C4×D53—21no—
179C4×D53—21no—
180C4×D53—22no—
181C4×D53—22no—
182C4×D53—82no—
183C4×D53—82no—
184C8×C53#12312yes—
185C8×C53C2×#4111yescharacteristic
186C8×C53C2×#4111yescharacteristic
187#88322—21no—
188#88322—21no—
189#33648—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—164no—
196C22×C53#48012yes—
197C22×C53C22×D811yescharacteristic
198C2×D53—21no—
199C2×D53—21no—
200C2×D53—21no—
201C2×D53—21no—
202C2×D53—21no—
203C2×D53—44no—
204C2×D53#50912yes—
205C4×C53—81no—
206C4×C53—81no—
207C4×C53#48012yes—
208C4×C53C2×#12311yescharacteristic
209C4×C53C2×#13711yescharacteristic
210Dic53—21no—
211Dic53—21no—
212Dic53—81no—
213Dic53—82no—
214Dic53—81no—
215C2×#466—531no—
216C2×#427—531no—
217#2234—532no—
218#2234—532no—
219#2233—532no—
220#2232—532no—
221#2159—531no—
222#2159—531no—
223#2140—531no—
224#2140D5311yescharacteristic, complemented
225#2112—531no—
226C2×C53—22no—
227C2×C53—162no—
228C2×C53C2×#48011yescharacteristic
229C2×C53C2×#50911yescharacteristic
230C2×C53#341411yescharacteristic
231D53—22no—
232D53—41no—
233C22×D8C2×D5311yescharacteristic
234C23×C8—531no—
235C2×#121—1061no—
236C2×#117—1061no—
237C2×M32—531no—
238C2×M32C2×D5311yescharacteristic
239#466—532no—
240#466—532no—
241#466—1061no—
242#466—1061no—
243#427—531no—
244#427—531no—
245#427—531no—
246#427—1061no—
247#427—1061no—
248#427C2×D5311yescharacteristic
249C2×#107—531no—
250C2×#107—531no—
251C2×#107—531no—
252C2×#106—1061no—
253C2×#102—1061no—
254#367—532no—
255#367—532no—
256#366—532no—
257#366—532no—
258#357—532no—
259#357—532no—
260#357C2×D5312yescomplemented
261#357C2×D5312yescomplemented
262C22⋊C16—532no—
263C22⋊C16—532no—
264C53#3364811yescharacteristic, complemented, Sylow
265C22×D4—21no—
266C22×D4—2121no—
267C23×C4—531no—
268C2×D8—22no—
269C2×D8—21no—
270C2×D8—22no—
271C2×D8C22×D5312yes—
272C22×C8—531no—
273C22×C8—531no—
274C22×C8—531no—
275C22×C8—531no—
276C22×C8—531no—
277C22×C8—1062no—
278C22×C8—1062no—
279C22×C8C22×D5311yescharacteristic
280#121—1062no—
281#121—1062no—
282#121—2121no—
283#121—2121no—
284#117—21no—
285#117—1061no—
286#117—1061no—
287#117—1061no—
288#117—2121no—
289#117—2121no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4—1061no—
293C2×C22⋊C4—2121no—
294C2×C22⋊C4—2121no—
295C2×C42—2121no—
296M32—22no—
297M32—1062no—
298C2×C16—532no—
299C2×C16C22×D5312yes—
300#107—531no—
301#107—531no—
302#107—532no—
303#107—1061no—
304#107—1061no—
305#107—1061no—
306#107C22×D5311yescharacteristic
307#107C22×D5311yescharacteristic
308#106—21no—
309#106—1061no—
310#106—1061no—
311#106—1061no—
312#102—1062no—
313#102—1062no—
314C24—41no—
315C24—531no—
316C2×D4—22no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—4244no—
321C2×D4—2122no—
322C2×D4—2122no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4D4×D5311yescharacteristic
334D8—22no—
335D8—22no—
336D8—42no—
337C2×C8—531no—
338C2×C8—531no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—1061no—
343C2×C8—1061no—
344C2×C8—1061no—
345C2×C8—1061no—
346C2×C8#1171994512yes—
347C2×C8D4×D5311yescharacteristic
348C2×C8#1171994712yes—
349C2×C8C23×D5311yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—1062no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4—2121no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C22⋊C4—41no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2122no—
362C22⋊C4—2122no—
363C22⋊C4—2121no—
364C42—41no—
365C42—2121no—
366C42—2121no—
367C42—2121no—
368C16—22no—
369C16—1062no—
370C23—42no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—531no—
375C23—531no—
376C23—1062no—
377C23—1062no—
378C23—2121no—
379C23D8×D5311yescharacteristic
380D4—42no—
381D4—42no—
382D4—82no—
383D4—4244no—
384D4—4244no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—531no—
388C2×C4—531no—
389C2×C4—1062no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—4242no—
397C2×C4—4242no—
398C2×C4—4242no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2122no—
402C2×C4#1313319912yes—
403C2×C4D8×D5311yescharacteristic
404C2×C4#1313324612yes—
405C2×C4C2×D4×D5311yescharacteristic
406C8—1061no—
407C8—1061no—
408C8#1313319912yes—
409C8C2×#1171994711yescharacteristic
410C8C2×#1171994511yescharacteristic
411C22—21no—
412C22—21no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—164no—
417C22—532no—
418C22—1061no—
419C22—1061no—
420C22—1061no—
421C22—1061no—
422C22—1061no—
423C22—2124no—
424C22#1681093512yes—
425C22C2×D8×D5311yescharacteristic
426C4—81no—
427C4—81no—
428C4—1061no—
429C4—1061no—
430C4—4241no—
431C4—4242no—
432C4—4241no—
433C4#1681093512yes—
434C4C2×#1313319911yescharacteristic
435C4C2×#1313324611yescharacteristic
436C2—22no—
437C2—162no—
438C2—1062no—
439C2—2121no—
440C2C2×#1681093511yescharacteristic
441C2#509×D5311yescharacteristic
442C2#2028784711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2140C2
socleC22×C53C22×D8
prime radicalC2×#16810856C2
hypercenter#2140D53
hypocenterC53#33648
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
2. #427×D53C2semidirect
3. #20276905C2semidirect
4. #20276904C2semidirect
5. #20276899C2semidirect
6. #20276894C2semidirect
7. #20276504C2semidirect
8. #20276503C2semidirect
9. #20276431C2semidirect
10. #20276325C2semidirect
11. C53×#2140C2semidirect
21. #16810660C22semidirect
22. #16810660C22semidirect
26. #107×D53C22semidirect
27. #107×D53C22semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810449C22semidirect
33. #16810448C22semidirect
34. #16810424C22semidirect
35. #16810424C22semidirect
36. #16810402C22semidirect
37. #16810402C22semidirect
41. C53×#357C22semidirect
42. C53×#357C22semidirect
224. #2140D53semidirect
260. #357C2×D53semidirect
261. #357C2×D53semidirect
264. C53#33648semidirect

Maximal subgroups

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

subgroupindex
C2×#168108562
#427×D532
#202769052
#202769042
#202768992
#202768942
#202765042
#202765032
#202764312
#202763252
C53×#21402
#3364853