Finite Group Databasebeta

← #27541855

Subgroups — #27541855

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287728C211yesmaximal, characteristic, complemented
2#20276958C211yesmaximal, characteristic, complemented
3#20276954C211yesmaximal, characteristic, complemented
4#20276949C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276869C211yesmaximal, characteristic, complemented
7#20276849C211yesmaximal, characteristic, complemented
8#20276770C211yesmaximal, characteristic, complemented
9#20276770C211yesmaximal, characteristic, complemented
10#20276750C211yesmaximal, characteristic
11#20276624C211yesmaximal, characteristic, complemented
12#20276589C211yesmaximal, characteristic, complemented
13#20276579C211yesmaximal, characteristic, complemented
14C4×#13133158C211yesmaximal, characteristic, complemented
15C53×#3380C211yesmaximal, characteristic, complemented
16#16811358—21no—
17C2×#13133161C2211yescharacteristic
18C2×#13133158C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20#16811007C2211yescharacteristic
21#16811007C2211yescharacteristic
22#16811000C2211yescharacteristic
23#16810839—21no—
24#16810839—21no—
25#16810824—21no—
26C4×Dic212C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810448C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810426C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810424C2211yescharacteristic
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#554—21no—
49C53×#500C2211yescharacteristic
50C53×#499C2211yescharacteristic
51C53×#499C2211yescharacteristic
52C53×#478—21no—
53C53×#478—21no—
54C53×#476—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×D8×C53C2211yescharacteristic
57C4×D8×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×#11719913—21no—
62C2×D424—21no—
63C2×D424C2311yescharacteristic
64C2×D424C2311yescharacteristic
65#13133214—21no—
66#13133206—41no—
67#13133205—21no—
68#13133205—21no—
69#13133202—41no—
70#13133202—41no—
71C4×Dic106—21no—
72C4×D212—21no—
73C4×D212—21no—
74#13133179—41no—
75#13133161—22no—
76#13133158—22no—
77#13133158—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133139—21no—
85#13133139—21no—
86C2×D8×C53C2311yescharacteristic
87C4⋊Q8×C53—21no—
88#125×C53—21no—
89#125×C53—21no—
90C22⋊Q8×C53—41no—
91C22⋊Q8×C53—41no—
92C22⋊Q8×C53—41no—
93C4×Q8×C53D411yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C2×C4⋊C4×C53—41no—
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#102×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—41no—
114C2×Q8×C53D811yescharacteristic
115C2×D4×C53—21no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—41no—
122C2×D212—41no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126D8×C53—21no—
127D8×C53#4112yes—
128C2×C8×C53C2×D411yescharacteristic
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2411yescharacteristic
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53D811yescharacteristic
141C4⋊C4×C53D811yescharacteristic
142C4⋊C4×C53D811yescharacteristic
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149#11719921—41no—
150#11719921—41no—
151#11719921—41no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156#11719919—41no—
157C4×Dic53—41no—
158Dic212—21no—
159Dic212—21no—
160D424—21no—
161D424—21no—
162D424—21no—
163D424—21no—
164D424—81no—
165#11719913—81no—
166#11719913—81no—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C22×D53—41no—
173C22×D53—41no—
174Q8×C53—41no—
175Q8×C53—161no—
176D4×C53—22no—
177D4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—82no—
182C2×C4×C53—161no—
183C2×C4×C53—161no—
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×Dic53—41no—
192C2×Dic53—41no—
193D212—41no—
194D212—41no—
195D212—41no—
196D212—41no—
197D212—82no—
198D212—161no—
199C4×D53—161no—
200C4×D53—161no—
201Dic106—41no—
202Dic106—41no—
203C8×C53—21no—
204C8×C53—21no—
205C8×C53—21no—
206C8×C53C2×#4111yescharacteristic
207C8×C532−1+411yescharacteristic
208#33937—531nomaximal, Sylow
209C22×C53—42no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53C22×D811yescharacteristic
213C2×D53—81no—
214C2×D53—81no—
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—81no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53#54811yescharacteristic
229C4×C53#57711yescharacteristic
230Dic53—81no—
231Dic53—81no—
232#3396—531no—
233#3380D5311yescharacteristic, complemented
234#2244—531no—
235#2243—531no—
236#2242—531no—
237#2229—531no—
238#2228—531no—
239#2224—531no—
240#2210—531no—
241#2210—531no—
242#2206—531no—
243#2183—531no—
244#2177—531no—
245#2175—531no—
246C4×D16—531no—
247C2×C53—82no—
248C2×C53#335811yescharacteristic
249C2×C53#341411yescharacteristic
250C2×C53#341511yescharacteristic
251D53—161no—
252D53—161no—
253#559—1061no—
254#554—21no—
255C2×SD32—531no—
256C2×D16—531no—
257C2×D16—531no—
258#500C2×D5311yescharacteristic
259#499C2×D5311yescharacteristic
260#499C2×D5311yescharacteristic
261#495—531no—
262#495—531no—
263#493—531no—
264#478—21no—
265#478—21no—
266#476—21no—
267#463—1061no—
268#463—1061no—
269#460—1061no—
270C8×Q8C2×D5311yescharacteristic
271C4×Q16—531no—
272C4×D8—531no—
273C4×D8—531no—
274C4×D8C2×D5311yescharacteristic
275C4×D8C2×D5311yescharacteristic
276C4×D8C2×D5311yescharacteristic
277#367—531no—
278#366—531no—
279#366—531no—
280C4⋊C16—531no—
281C4⋊C16—531no—
282C4⋊C16—531no—
283#358—531no—
284#358—531no—
285#358—531no—
286#357—531no—
287#357—531no—
288#357—531no—
289#357—531no—
290#357—531no—
291#357—531no—
292#357—531no—
293#346—531no—
294#346—531no—
295C4×C16—531no—
296C53#3393711yescharacteristic, complemented, Sylow
297C2×Q16—531no—
298C2×SD16—1061no—
299C2×SD16—1061no—
300C2×D8—531no—
301C2×D8—531no—
302C2×D8—1061no—
303C2×D8C22×D5311yescharacteristic
304C4⋊Q8—21no—
305#127—1061no—
306#125—21no—
307#125—21no—
308#124—1061no—
309#124—1061no—
310#124—2121no—
311#123—2121no—
312#123—2121no—
313C22⋊Q8—41no—
314C22⋊Q8—41no—
315C22⋊Q8—41no—
316C4×Q8—1061no—
317C4×Q8C22⋊D5311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—1061no—
322C4×D4—1061no—
323C4×D4—2121no—
324C2×C4⋊C4—41no—
325SD32—1062no—
326D16—1062no—
327D16—1062no—
328C2×C16—531no—
329C2×C16—531no—
330C2×C16—531no—
331C2×C16—531no—
332#107—21no—
333#107—531no—
334#107C22×D5311yescharacteristic
335#107C22×D5311yescharacteristic
336#107C22×D5311yescharacteristic
337#106—21no—
338#106—21no—
339C4⋊C8C22⋊D5311yescharacteristic
340C4⋊C8C22⋊D5311yescharacteristic
341C4⋊C8C22⋊D5311yescharacteristic
342#103—1061no—
343#102—21no—
344#102—21no—
345#102—21no—
346#102—1061no—
347#102—1061no—
348C4×C8C22×D5311yescharacteristic
349C4×C8C22×D5311yescharacteristic
350C4×C8C22×D5311yescharacteristic
351C2×Q8—41no—
352C2×Q8—1061no—
353C2×Q8#1171992211yescharacteristic
354C2×D4—21no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—2121no—
358C2×D4—2121no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—2121no—
363C22×C4—2121no—
364Q16—1061no—
365Q16—1061no—
366SD16—4241no—
367SD16—4241no—
368D8—21no—
369D8—1061no—
370D8—1061no—
371D8—1061no—
372D8—1061no—
373D8—4241no—
374D8#1171994912yes—
375C2×C8C2×C22⋊D5311yescharacteristic
376C2×C8C2×C22⋊D5311yescharacteristic
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C23×D5311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—1061no—
388C4⋊C4—2121no—
389C4⋊C4#1171992211yescharacteristic
390C4⋊C4#1171992211yescharacteristic
391C4⋊C4#1171992211yescharacteristic
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—2121no—
396C22⋊C4—2121no—
397C22⋊C4—2121no—
398C22⋊C4—2121no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C42—2121no—
402C42C2×C22⋊D5311yescharacteristic
403C42C2×C22⋊D5311yescharacteristic
404C42C2×C22⋊D5311yescharacteristic
405C16—1061no—
406C16—1061no—
407C16—1061no—
408C16—1061no—
409C23—41no—
410C23—2121no—
411C23—2121no—
412Q8—41no—
413Q8—161no—
414Q8—2121no—
415Q8—2121no—
416D4—22no—
417D4—41no—
418D4—8481no—
419D4—4242no—
420D4—2121no—
421D4—2121no—
422D4—2121no—
423D4—2121no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—161no—
429C2×C4—161no—
430C2×C4—8481no—
431C2×C4—8481no—
432C2×C4—2121no—
433C2×C4—2121no—
434C2×C4C2×#1171992211yescharacteristic
435C2×C4C2×#1171992211yescharacteristic
436C2×C4C2×#1171992211yescharacteristic
437C2×C4C2×#1171992211yescharacteristic
438C2×C4C2×#1171992211yescharacteristic
439C2×C4C2×#1171992211yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8C2×#1171994911yescharacteristic
445C8#1313326511yescharacteristic
446C22—42no—
447C22—81no—
448C22—81no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22—4241no—
454C22—4241no—
455C22C22×#1171992211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—4241no—
466C4—4241no—
467C4#1681128411yescharacteristic
468C4#1681143111yescharacteristic
469C2—82no—
470C2—8481no—
471C2—8481no—
472C2#2028735011yescharacteristic
473C2#2028784911yescharacteristic
474C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3380C2
socleC22×C53C22×D8
prime radical#20276750C2
hypercenter#3380D53
hypocenterC53#33937
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #20287728C2semidirect
2. #20276958C2semidirect
3. #20276954C2semidirect
4. #20276949C2semidirect
5. #20276876C2semidirect
6. #20276869C2semidirect
7. #20276849C2semidirect
8. #20276770C2semidirect
9. #20276770C2semidirect
11. #20276624C2semidirect
12. #20276589C2semidirect
13. #20276579C2semidirect
14. C4×#13133158C2semidirect
15. C53×#3380C2semidirect
233. #3380D53semidirect
296. C53#33937semidirect

Maximal subgroups

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

subgroupindex
#202877282
#202769582
#202769542
#202769492
#202768762
#202768692
#202768492
#202767702
#202767502
#202766242
#202765892
#202765792
C4×#131331582
C53×#33802
#3393753