Finite Group Databasebeta

← #27538582

Subgroups — #27538582

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811063C211yesmaximal, characteristic, complemented
2C22×#13133157C211yesmaximal, characteristic, complemented
3C2×#16810864C211yesmaximal, characteristic, complemented
4#20276826C212yesmaximal, complemented
5#20276821C212yesmaximal, complemented
6#20276809C212yesmaximal, complemented
7#20276804C212yesmaximal, complemented
8#20276504C211yesmaximal, characteristic, complemented
9#20276434C211yesmaximal, characteristic, complemented
10#20276427C211yesmaximal, characteristic, complemented
11C53×#3050C211yesmaximal, characteristic
12C22×Dic212C2211yescharacteristic
13C2×#13133196—21no—
14#16811063—24no—
15C2×#13133163C2212yescomplemented
16C2×#13133161C2212yescomplemented
17C2×#13133157—24no—
18C2×#13133157C2212yes—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21C22×#11719908C2212yes—
22#16810864—21no—
23#16810864—21no—
24#16810864C2212yes—
25#16810864C2212yes—
26C2×#13133151C2211yescharacteristic
27C2×#13133142—21no—
28#16810450C2212yescomplemented
29#16810448C2212yescomplemented
30#16810427C2212yescomplemented
31#16810426C2212yescomplemented
32#16810426C2212yescomplemented
33#16810424C2212yescomplemented
34C2×#135×C53C2211yescharacteristic
35C23×C8×C53C2211yescharacteristic
36C53×#525—21no—
37C53×#467C2211yescharacteristic
38C53×#467C2211yescharacteristic
39C53×#466C2211yescharacteristic
40C53×#466C2211yescharacteristic
41C53×#465—21no—
42C53×#465—21no—
43C53×#427C2211yescharacteristic
44C53×#416—21no—
45C22×Dic106—21no—
46C2×Dic212—21no—
47C2×Dic212—22no—
48C2×Dic212—22no—
49C2×Dic212C2312yes—
50#13133196—22no—
51#13133196—22no—
52#13133196—41no—
53#13133196—41no—
54C2×C4⋊Dic53—21no—
55C2×#11719919—41no—
56C2×C22⋊Dic53—41no—
57#13133163—24no—
58#13133161—24no—
59#13133157—22no—
60#13133157—24no—
61#13133157D414yescomplemented
62C2×#11719908—24no—
63C2×#11719908C2312yes—
64C2×#11719908C2312yes—
65#13133151—21no—
66#13133151C2312yes—
67#13133142—22no—
68#13133142—22no—
69C2×#41×C53—21no—
70C23×C4×C53D411yescharacteristic
71#135×C53—22no—
72#135×C53—22no—
73C2×Q16×C53C2311yescharacteristic
74C2×SD16×C53—21no—
75C2×D8×C53C2311yescharacteristic
76C22×C8×C53—22no—
77C22×C8×C53—22no—
78C22×C8×C53D411yescharacteristic
79C22×C8×C53D411yescharacteristic
80C22×C8×C53D411yescharacteristic
81C22×C8×C53C2311yescharacteristic
82C22×C8×C53C2311yescharacteristic
83C22×C8×C53C2311yescharacteristic
84#123×C53—41no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89#120×C53—41no—
90C4×D4×C53—41no—
91C4×D4×C53—41no—
92#117×C53—21no—
93#107×C53C2311yescharacteristic
94#107×C53C2311yescharacteristic
95#106×C53—21no—
96#103×C53—21no—
97#103×C53—21no—
98#102×C53—21no—
99#102×C53—21no—
100C24×C53D811yescharacteristic
101#41×C53—42no—
102#41×C53—42no—
103C2×Q8×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—41no—
111C22×C4×C53D811yescharacteristic
112C22×C4×C53D811yescharacteristic
113C22×C4×C53D811yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×Dic53—41no—
118C22×Dic53—41no—
119C2×Dic106—22no—
120C2×Dic106—42no—
121C2×Dic106—42no—
122C2×Dic106—41no—
123Q16×C53—21no—
124Q16×C53—21no—
125SD16×C53—42no—
126D8×C53—21no—
127D8×C53—21no—
128C2×C8×C53—22no—
129C2×C8×C53—22no—
130C2×C8×C53—22no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D412yes—
139C2×C8×C53#4112yes—
140C2×C8×C53C2411yescharacteristic
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C22⋊Dic53—42no—
152C22⋊Dic53—42no—
153C4⋊Dic53—22no—
154C4⋊Dic53—41no—
155#11719919—42no—
156#11719919—42no—
157Dic212—22no—
158Dic212—22no—
159Dic212—42no—
160#11719908—22no—
161#11719908C2×D414yes—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53C2×D811yescharacteristic
167C23×C53C2×D811yescharacteristic
168C23×C53C2×D811yescharacteristic
169Q8×C53—41no—
170Q8×C53—41no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—82no—
174D4×C53—82no—
175D4×C53—82no—
176C2×C4×C53—22no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—82no—
187C2×C4×C53#12112yes—
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53#13512yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—82no—
196C2×Dic53—82no—
197Dic106—42no—
198Dic106—42no—
199Dic106—82no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53#12112yes—
203C8×C53C22×D411yescharacteristic
204C8×C53C2×#4111yescharacteristic
205#33606—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—44no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53#46612yes—
218C22×C53#51012yescomplemented
219C22×C53C22×D811yescharacteristic
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—81no—
223C4×C53—81no—
224C4×C53—81no—
225C4×C53#46612yes—
226C4×C53C2×#12111yescharacteristic
227C4×C53C2×#13511yescharacteristic
228Dic53—82no—
229Dic53—82no—
230C2×#510—531no—
231C22×M32—531no—
232#3050D5311yescharacteristic
233C2×#467—531no—
234#2220—532no—
235#2219—532no—
236#2217—532no—
237#2216—532no—
238#2159—531no—
239#2141—531no—
240#2139—531no—
241C2×C53—22no—
242C2×C53—22no—
243C2×C53—41no—
244C2×C53—161no—
245C2×C53C2×#46611yescharacteristic
246C2×C53C2×#51011yescharacteristic
247C2×C53#341411yescharacteristic
248C2×#135C2×D5311yescharacteristic
249C22×Q16—531no—
250C23×C8C2×D5311yescharacteristic
251#525—21no—
252C2×C22⋊Q8—1061no—
253#510—1064no—
254C2×Q32—532no—
255C2×SD32—532no—
256C2×M32—531no—
257C2×M32—531no—
258C2×M32—532no—
259C2×M32—1064no—
260C22×C16—532no—
261#467—532no—
262#467—532no—
263#467—1061no—
264#467—1061no—
265#467C2×D5311yescharacteristic
266#467C2×D5311yescharacteristic
267#466C2×D5311yescharacteristic
268#466C2×D5311yescharacteristic
269#465—21no—
270#465—21no—
271#427C2×D5311yescharacteristic
272C2×#107—531no—
273#416—21no—
274C2×#103—1061no—
275#367—532no—
276#366—532no—
277#358—532no—
278#358—532no—
279#358—532no—
280#357—532no—
281C53#3360611yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×Q8—1061no—
284C23×C4C22⋊D5311yescharacteristic
285#135—22no—
286#135—22no—
287C2×Q16—532no—
288C2×Q16—1062no—
289C2×Q16—1062no—
290C2×Q16—1061no—
291C2×Q16C22×D5311yescharacteristic
292C2×SD16—21no—
293C2×D8C22×D5311yescharacteristic
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D5311yescharacteristic
297C22×C8C22⋊D5311yescharacteristic
298C22×C8C22⋊D5311yescharacteristic
299C22×C8C22×D5311yescharacteristic
300C22×C8C22×D5311yescharacteristic
301C22×C8C22×D5311yescharacteristic
302#123—41no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—1062no—
306C22⋊Q8—1062no—
307C22⋊Q8—2121no—
308C22⋊Q8—2121no—
309#121—21no—
310#121—21no—
311#120—41no—
312C4×D4—41no—
313C4×D4—41no—
314#117—21no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—2121no—
317C2×C22⋊C4—2121no—
318Q32—1064no—
319SD32—1064no—
320M32—534no—
321M32—1064no—
322M32—1062no—
323C2×C16—532no—
324C2×C16—532no—
325C2×C16—1064no—
326#107—532no—
327#107—1061no—
328#107C22×D5311yescharacteristic
329#107C22×D5311yescharacteristic
330#106—21no—
331#103—21no—
332#103—21no—
333#103—1062no—
334#103—1062no—
335#102—21no—
336#102—21no—
337C24#1171992211yescharacteristic
338#41—42no—
339#41—42no—
340C2×Q8—21no—
341C2×Q8—1062no—
342C2×Q8—2122no—
343C2×Q8—2122no—
344C2×Q8—2121no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—2121no—
353C22×C4—2121no—
354C22×C4#1171992211yescharacteristic
355C22×C4#1171992211yescharacteristic
356C22×C4#1171992211yescharacteristic
357C22×C4C2×C22⋊D5311yescharacteristic
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C2×C22⋊D5311yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—1062no—
363Q16—1062no—
364Q16—2122no—
365SD16—42no—
366D8—21no—
367D8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8D4×D5312yes—
376C2×C8#1171994712yes—
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C2×C22⋊D5311yescharacteristic
379C2×C8C2×C22⋊D5311yescharacteristic
380C2×C8C23×D5311yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—1062no—
386C4⋊C4—2121no—
387C4⋊C4—2122no—
388C4⋊C4—2122no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—2122no—
395C22⋊C4—2122no—
396C42—41no—
397C16—534no—
398C16—1062no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23C2×#1171992211yescharacteristic
404C23C2×#1171992211yescharacteristic
405C23C2×#1171992211yescharacteristic
406Q8—41no—
407Q8—41no—
408Q8—4242no—
409Q8—2122no—
410Q8—2122no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415D4—82no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—4242no—
428C2×C4—4242no—
429C2×C4—2122no—
430C2×C4—2122no—
431C2×C4#1313318912yes—
432C2×C4C2×#1171992211yescharacteristic
433C2×C4C2×#1171992211yescharacteristic
434C2×C4C2×#1171992211yescharacteristic
435C2×C4#1313323712yes—
436C2×C4C22×C22⋊D5311yescharacteristic
437C8—21no—
438C8—21no—
439C8#1313318912yes—
440C8C2×D4×D5311yescharacteristic
441C8C2×#1171994711yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#1681085712yes—
454C22#1681106312yescomplemented
455C22C22×#1171992211yescharacteristic
456C4—21no—
457C4—21no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—4242no—
462C4—4242no—
463C4#1681085712yes—
464C4C2×#1313318911yescharacteristic
465C4C2×#1313323711yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—161no—
470C2C2×#1681085711yescharacteristic
471C2C2×#1681106311yescharacteristic
472C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3050C2
socleC22×C53C22×D8
hypercenter#3050D53
hypocenterC53#33606
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
2. C22×#13133157C2semidirect
3. C2×#16810864C2semidirect
4. #20276826C2semidirect
5. #20276821C2semidirect
6. #20276809C2semidirect
7. #20276804C2semidirect
8. #20276504C2semidirect
9. #20276434C2semidirect
10. #20276427C2semidirect
15. C2×#13133163C22semidirect
16. C2×#13133161C22semidirect
28. #16810450C22semidirect
29. #16810448C22semidirect
30. #16810427C22semidirect
31. #16810426C22semidirect
32. #16810426C22semidirect
33. #16810424C22semidirect
61. #13133157D4semidirect
218. C22×C53#510semidirect
281. C53#33606semidirect
454. C22#16811063semidirect

Maximal subgroups

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

subgroupindex
C2×#168110632
C22×#131331572
C2×#168108642
#202768262
#202768212
#202768092
#202768042
#202765042
#202764342
#202764272
C53×#30502
#3360653