Finite Group Databasebeta

← #27541944

Subgroups — #27541944

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287704C211yesmaximal, characteristic
2#20276962C211yesmaximal, characteristic, complemented
3#20276954C211yesmaximal, characteristic
4#20276949C211yesmaximal, characteristic
5#20276876C211yesmaximal, characteristic, complemented
6#20276869C211yesmaximal, characteristic
7#20276864C211yesmaximal, characteristic, complemented
8#20276777C211yesmaximal, characteristic
9#20276777C211yesmaximal, characteristic
10#20276765C211yesmaximal, characteristic, complemented
11#20276624C211yesmaximal, characteristic, complemented
12#20276584C211yesmaximal, characteristic, complemented
13#20276577C211yesmaximal, characteristic
14C4×#13133163C211yesmaximal, characteristic, complemented
15C53×#3383C211yesmaximal, characteristic, complemented
16#16811346—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20#16811007C2211yescharacteristic
21#16811007C2211yescharacteristic
22#16811003C2211yescharacteristic
23#16810846—21no—
24#16810846—21no—
25#16810829—21no—
26C4×Dic212C2211yescharacteristic
27C4×Dic212C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810448C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#557—21no—
49C53×#500C2211yescharacteristic
50C53×#499C2211yescharacteristic
51C53×#499C2211yescharacteristic
52C53×#479—21no—
53C53×#479—21no—
54C53×#475—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×Q16×C53C2211yescharacteristic
58C4×Q16×C53C2211yescharacteristic
59C2×Dic212—21no—
60C2×Dic212C2311yescharacteristic
61C2×Dic212C2311yescharacteristic
62C2×#11719913—21no—
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65#13133217—21no—
66#13133212—41no—
67#13133212—41no—
68#13133205—21no—
69#13133205—21no—
70#13133193—41no—
71C4×Dic106—21no—
72C4×Dic106—21no—
73C4×D212—21no—
74#13133175—41no—
75#13133163—22no—
76#13133163—22no—
77#13133160—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133142—21no—
85#13133139—21no—
86C2×Q16×C53C2311yescharacteristic
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—41no—
89#125×C53—21no—
90#125×C53—21no—
91#125×C53—41no—
92#125×C53—41no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—41no—
97C4×Q8×C53D411yescharacteristic
98#107×C53—21no—
99#107×C53C2311yescharacteristic
100#107×C53C2311yescharacteristic
101#107×C53C2311yescharacteristic
102#106×C53—21no—
103#106×C53—21no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#103×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—41no—
115C2×Q8×C53D811yescharacteristic
116C2×D212—21no—
117C2×C4×D53—41no—
118C2×Dic106—21no—
119C2×Dic106—21no—
120Q16×C53—21no—
121Q16×C53#4112yes—
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2411yescharacteristic
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53D811yescharacteristic
138C4⋊C4×C53D811yescharacteristic
139C4⋊C4×C53D811yescharacteristic
140C42×C53—41no—
141C42×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146#11719921—41no—
147#11719921—41no—
148#11719921—41no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—41no—
153C4⋊Dic53—41no—
154#11719919—41no—
155#11719919—41no—
156C4×Dic53—41no—
157C4×Dic53—41no—
158Dic212—21no—
159Dic212—21no—
160Dic212—21no—
161Dic212—21no—
162Dic212—81no—
163D424—21no—
164D424—21no—
165#11719913—81no—
166#11719913—81no—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C22×D53—41no—
172Q8×C53—22no—
173Q8×C53—41no—
174Q8×C53—41no—
175Q8×C53—82no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×D811yescharacteristic
181C2×C4×C53C2×D811yescharacteristic
182C2×C4×C53C2×D811yescharacteristic
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190D212—41no—
191D212—41no—
192C4×D53—82no—
193Dic106—41no—
194Dic106—41no—
195Dic106—41no—
196Dic106—41no—
197C8×C53—21no—
198C8×C53—21no—
199C8×C53—21no—
200C8×C53C2×#4111yescharacteristic
201C8×C532−1+411yescharacteristic
202#33944—531nomaximal, Sylow
203C22×C53C22×D811yescharacteristic
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53#54811yescharacteristic
218C4×C53#57711yescharacteristic
219Dic53—81no—
220Dic53—81no—
221Dic53—81no—
222#3393—531no—
223#3383D5311yescharacteristic, complemented
224#2245—531no—
225#2243—531no—
226#2242—531no—
227#2229—531no—
228#2228—531no—
229#2227—531no—
230#2211—531no—
231#2211—531no—
232#2209—531no—
233#2183—531no—
234#2176—531no—
235#2175—531no—
236C4×Q32—531no—
237C2×C53#335811yescharacteristic
238C2×C53#341411yescharacteristic
239C2×C53#341711yescharacteristic
240D53—161no—
241#557—21no—
242#556—1061no—
243C2×Q32—531no—
244C2×Q32—531no—
245C2×SD32—531no—
246#500C2×D5311yescharacteristic
247#499C2×D5311yescharacteristic
248#499C2×D5311yescharacteristic
249#495—531no—
250#495—531no—
251#494—531no—
252#479—21no—
253#479—21no—
254#475—21no—
255#464—1061no—
256#464—1061no—
257#461—1061no—
258C8×Q8C2×D5311yescharacteristic
259C4×Q16—531no—
260C4×Q16—531no—
261C4×Q16C2×D5311yescharacteristic
262C4×Q16C2×D5311yescharacteristic
263C4×Q16C2×D5311yescharacteristic
264C4×D8—531no—
265#367—531no—
266#366—531no—
267#366—531no—
268C4⋊C16—531no—
269C4⋊C16—531no—
270C4⋊C16—531no—
271#358—531no—
272#358—531no—
273#358—531no—
274#358—531no—
275#358—531no—
276#358—531no—
277#358—531no—
278#357—531no—
279#357—531no—
280#357—531no—
281#346—531no—
282#346—531no—
283C4×C16—531no—
284C53#3394411yescharacteristic, complemented, Sylow
285C2×Q16—531no—
286C2×Q16—531no—
287C2×Q16—1061no—
288C2×Q16C22×D5311yescharacteristic
289C2×SD16—1061no—
290C2×SD16—1061no—
291C2×D8—531no—
292C4⋊Q8—21no—
293C4⋊Q8—41no—
294C4⋊Q8—1061no—
295#126—2121no—
296#126—2121no—
297#125—21no—
298#125—21no—
299#125—41no—
300#125—41no—
301#124—1061no—
302#124—1061no—
303C22⋊Q8—2121no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—41no—
308C4×Q8—1061no—
309C4×Q8—1061no—
310C4×Q8C22⋊D5311yescharacteristic
311C4×D4—1061no—
312#117—2121no—
313Q32—1062no—
314Q32—1062no—
315SD32—1062no—
316C2×C16—531no—
317C2×C16—531no—
318C2×C16—531no—
319C2×C16—531no—
320#107—21no—
321#107—531no—
322#107C22×D5311yescharacteristic
323#107C22×D5311yescharacteristic
324#107C22×D5311yescharacteristic
325#106—21no—
326#106—21no—
327C4⋊C8C22⋊D5311yescharacteristic
328C4⋊C8C22⋊D5311yescharacteristic
329C4⋊C8C22⋊D5311yescharacteristic
330#103—21no—
331#103—21no—
332#103—21no—
333#103—1061no—
334#103—1061no—
335#102—1061no—
336C4×C8C22×D5311yescharacteristic
337C4×C8C22×D5311yescharacteristic
338C4×C8C22×D5311yescharacteristic
339C2×Q8—21no—
340C2×Q8—41no—
341C2×Q8—1061no—
342C2×Q8—1061no—
343C2×Q8#1171992211yescharacteristic
344C2×D4—1061no—
345C22×C4—2121no—
346Q16—21no—
347Q16—1061no—
348Q16—1061no—
349Q16—1061no—
350Q16—1061no—
351Q16—4241no—
352Q16#1171994912yes—
353SD16—4241no—
354SD16—4241no—
355D8—1061no—
356D8—1061no—
357C2×C8C2×C22⋊D5311yescharacteristic
358C2×C8C2×C22⋊D5311yescharacteristic
359C2×C8C2×C22⋊D5311yescharacteristic
360C2×C8C23×D5311yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—1061no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C4⋊C4#1171992211yescharacteristic
378C4⋊C4#1171992211yescharacteristic
379C4⋊C4#1171992211yescharacteristic
380C22⋊C4—2121no—
381C22⋊C4—2121no—
382C22⋊C4—2121no—
383C22⋊C4—2121no—
384C22⋊C4—2121no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—2121no—
389C42—2121no—
390C42C2×C22⋊D5311yescharacteristic
391C42C2×C22⋊D5311yescharacteristic
392C42C2×C22⋊D5311yescharacteristic
393C16—1061no—
394C16—1061no—
395C16—1061no—
396C16—1061no—
397C23—2121no—
398Q8—22no—
399Q8—41no—
400Q8—41no—
401Q8—82no—
402Q8—2121no—
403Q8—2121no—
404Q8—2121no—
405Q8—2121no—
406D4—2121no—
407D4—2121no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—4242no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C2×#1171992211yescharacteristic
417C2×C4C2×#1171992211yescharacteristic
418C2×C4C2×#1171992211yescharacteristic
419C2×C4C2×#1171992211yescharacteristic
420C2×C4C2×#1171992211yescharacteristic
421C2×C4C2×#1171992211yescharacteristic
422C2×C4C22×C22⋊D5311yescharacteristic
423C8—21no—
424C8—21no—
425C8—21no—
426C8C2×#1171994911yescharacteristic
427C8#1313326511yescharacteristic
428C22—4241no—
429C22—4241no—
430C22—4241no—
431C22C22×#1171992211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—4241no—
443C4—4241no—
444C4—4241no—
445C4#1681128411yescharacteristic
446C4#1681143111yescharacteristic
447C2—8481no—
448C2#2028735011yescharacteristic
449C2#2028784911yescharacteristic
450C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3383C2
socleC22×C53C22×D8
prime radicalC2×D424C23
hypercenter#3383D53
hypocenterC53#33944
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
2. #20276962C2semidirect
5. #20276876C2semidirect
7. #20276864C2semidirect
10. #20276765C2semidirect
11. #20276624C2semidirect
12. #20276584C2semidirect
14. C4×#13133163C2semidirect
15. C53×#3383C2semidirect
223. #3383D53semidirect
284. C53#33944semidirect

Maximal subgroups

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

subgroupindex
#202877042
#202769622
#202769542
#202769492
#202768762
#202768692
#202768642
#202767772
#202767652
#202766242
#202765842
#202765772
C4×#131331632
C53×#33832
#3394453