Finite Group Databasebeta

← #27370041

Subgroups — #27370041

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic
3#20283749C211yesmaximal, characteristic
4#20274259C212yesmaximal, complemented
5#20274201C212yesmaximal, complemented
6#20274135C212yesmaximal, complemented
7#20274077C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273432C211yesmaximal, characteristic, complemented
10#20273415C211yesmaximal, characteristic, complemented
11C53×#3209C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810956—21no—
14#16810900—21no—
15#16810900—21no—
16#16810900C2211yescharacteristic
17#16810900C2212yes—
18#16810900C2212yes—
19#16810900C2211yescharacteristic
20#16810825C2211yescharacteristic
21#16810825C2211yescharacteristic
22#16810773—21no—
23#16810721—21no—
24#16810721—21no—
25#16810702C2212yes—
26#16810693—22no—
27C8×D212C2212yes—
28#16810680C2211yescharacteristic
29#16810680C2211yescharacteristic
30C2×#13133150—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×#13133137—21no—
33#16810585—21no—
34C22⋊C8×D53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810369C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#494C2212yescomplemented
44C53×#487C2212yescomplemented
45C53×#462—22no—
46C53×#451—21no—
47C53×#451—21no—
48C53×#431C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719924—21no—
52C2×#11719924—21no—
53C2×#11719912—22no—
54C22×#88322—21no—
55C2×C8×D53—22no—
56C4×D212—21no—
57C4×D212—21no—
58C4×D212D411yescharacteristic
59C4×D212D411yescharacteristic
60C4×D212C2311yescharacteristic
61C4×D212C2311yescharacteristic
62C2×C4⋊Dic53D411yescharacteristic
63C2×#11719921—21no—
64#13133150—22no—
65#13133150—22no—
66#13133150—21no—
67#13133150—21no—
68C4⋊#88322—21no—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71C4⋊#88322C2312yes—
72C4⋊#88322C2312yes—
73#13133141—21no—
74#13133141—21no—
75#13133137—21no—
76#13133137—21no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—21no—
80C22⋊#88322—21no—
81#13133129—22no—
82#13133129—22no—
83#13133128—21no—
84#13133128—21no—
85C22×Q8×C53—21no—
86C2×Q16×C53—22no—
87C2×Q16×C53—22no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2311yescharacteristic
92C4⋊Q8×C53C2312yes—
93C4⋊Q8×C53C2311yescharacteristic
94#125×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99#117×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53D412yes—
104#103×C53—22no—
105#103×C53—22no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108C4×C8×C53C2312yes—
109C23×D53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—41no—
114C2×Q8×C53—82no—
115C22×C4×C53—41no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×Dic53—21no—
120C2×D212—22no—
121C2×D212—21no—
122C2×D212—41no—
123C2×D212C2×D411yescharacteristic
124C2×D212C2×D411yescharacteristic
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—82no—
130Q16×C53—82no—
131Q16×C53—82no—
132C2×C8×C53C2×D412yes—
133C2×C8×C53C2×D412yes—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719924—42no—
151#11719924—42no—
152#11719921—21no—
153#11719921—21no—
154#11719921—21no—
155#11719921—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53C2×D411yescharacteristic
158C4⋊Dic53C2×D411yescharacteristic
159C2×#88322—22no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—22no—
163C2×#88322—42no—
164#11719912—82no—
165C8×D53—82no—
166C23×C53#12111yescharacteristic
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—42no—
170C22×D53—42no—
171C22×D53—81no—
172Q8×C53—41no—
173Q8×C53—41no—
174Q8×C53—42no—
175Q8×C53—42no—
176Q8×C53—42no—
177Q8×C53—82no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—82no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—81no—
195D212—21no—
196D212—21no—
197D212—21no—
198D212—42no—
199D212#13712yes—
200C4×D53—42no—
201C4×D53—42no—
202C4×D53—81no—
203C4×D53—81no—
204C8×C53—22no—
205C8×C53—22no—
206#88322—41no—
207#88322—42no—
208#88322—41no—
209#88322—42no—
210#19459—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53#47312yes—
214C22×C53C2×#12111yescharacteristic
215C2×D53—22no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—84no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—82no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53C2×#13711yescharacteristic
230C4×C53#57511yescharacteristic
231C4×C53#57612yes—
232Dic53—41no—
233Dic53—41no—
234#3209D5311yescharacteristic, complemented
235#3110—531no—
236C2×#474—531no—
237#2974—531no—
238#1699—532no—
239#1689—532no—
240#1679—532no—
241#1669—532no—
242#1573—531no—
243#1556—531no—
244#1552—531no—
245C2×C53—22no—
246C2×C53#304111yescharacteristic
247C2×C53#305611yescharacteristic
248C2×C53C2×#47311yescharacteristic
249D53—42no—
250D53—81no—
251#533C2×D5311yescharacteristic
252C2×C4⋊Q8C2×D5311yescharacteristic
253C2×C4×D4—531no—
254#494C2×D5312yescomplemented
255#487C2×D5312yescomplemented
256#484—1061no—
257#474—531no—
258#474—531no—
259#474—532no—
260#474—532no—
261#474—1061no—
262#474—1061no—
263#462—22no—
264#461—531no—
265#461—531no—
266#451—21no—
267#451—21no—
268#450—1061no—
269#440—1061no—
270#440—1061no—
271#436—532no—
272#435—1062no—
273C8×D4—532no—
274#432—531no—
275#432—531no—
276#431C2×D5311yescharacteristic
277C2×#107—1061no—
278C2×C4⋊C8—531no—
279C2×#102—1061no—
280#407—1061no—
281C2×C22⋊C8—1061no—
282#335—532no—
283#334—532no—
284#332—532no—
285#332—532no—
286#332—532no—
287#332—532no—
288C53#1945911yescharacteristic, complemented, Sylow
289C22×Q8—21no—
290C22×D4—531no—
291C23×C4—1061no—
292C2×Q16—22no—
293C2×Q16—22no—
294C2×SD16—1061no—
295C2×SD16—1061no—
296C2×M16—1062no—
297C22×C8—1061no—
298C22×C8—1062no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8C22×D5312yes—
303C4⋊Q8C22×D5311yescharacteristic
304C4⋊Q8C22×D5311yescharacteristic
305#125—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×D4—531no—
311C4×D4—531no—
312C4×D4—531no—
313C4×D4—531no—
314C4×D4—1061no—
315C4×D4—1061no—
316#117—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—531no—
320C2×C22⋊C4—1061no—
321C2×C42C22×D5311yescharacteristic
322#107—1062no—
323#107—1062no—
324#107—1061no—
325#107—1061no—
326C4⋊C8—532no—
327C4⋊C8—531no—
328C4⋊C8—532no—
329C4⋊C8—531no—
330C4⋊C8—1061no—
331C4⋊C8C22⋊D5312yes—
332#103—22no—
333#103—22no—
334#103—1061no—
335#103—1061no—
336#102—1062no—
337#102—1062no—
338#102—1061no—
339#102—1061no—
340C22⋊C8—1061no—
341C22⋊C8—1061no—
342C22⋊C8—1062no—
343C22⋊C8—1062no—
344C22⋊C8C22⋊D5311yescharacteristic
345C22⋊C8C22⋊D5311yescharacteristic
346#97—1061no—
347#97—1061no—
348C4×C8C22×D5312yes—
349C24—1061no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—41no—
354C2×Q8—82no—
355C2×D4—531no—
356C2×D4—531no—
357C2×D4—1062no—
358C2×D4—1061no—
359C2×D4—2121no—
360C22×C4—41no—
361C22×C4—1061no—
362C22×C4—1061no—
363C22×C4—1061no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—4242no—
367C22×C4D4×D5311yescharacteristic
368C22×C4#1171994711yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370Q16—82no—
371Q16—82no—
372SD16—2122no—
373SD16—2122no—
374M16—4242no—
375C2×C8—1061no—
376C2×C8—1061no—
377C2×C8—1062no—
378C2×C8—1062no—
379C2×C8—4242no—
380C2×C8—2122no—
381C2×C8C2×C22⋊D5312yes—
382C2×C8C2×C22⋊D5312yes—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—531no—
394C4⋊C4—531no—
395C4⋊C4—1061no—
396C22⋊C4—41no—
397C22⋊C4—1061no—
398C22⋊C4—1061no—
399C22⋊C4—1061no—
400C22⋊C4—1061no—
401C42—41no—
402C42D4×D5311yescharacteristic
403C42#1171994711yescharacteristic
404C42C2×C22⋊D5311yescharacteristic
405C42C23×D5311yescharacteristic
406C23—1061no—
407C23—1061no—
408C23—4241no—
409C23—2122no—
410C23—2122no—
411C23#1313318811yescharacteristic
412Q8—41no—
413Q8—41no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—82no—
418D4—532no—
419D4—1061no—
420D4—1061no—
421D4—1061no—
422D4—2122no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—4241no—
434C2×C4—4241no—
435C2×C4—4241no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4#1313318811yescharacteristic
439C2×C4#1313318811yescharacteristic
440C2×C4#1313318811yescharacteristic
441C2×C4C2×D4×D5311yescharacteristic
442C2×C4C22×C22⋊D5311yescharacteristic
443C2×C4C2×#1171994711yescharacteristic
444C8—22no—
445C8—22no—
446C8—2121no—
447C8—2122no—
448C8—2121no—
449C8—2122no—
450C22—21no—
451C22—21no—
452C22—1062no—
453C22—4244no—
454C22—2121no—
455C22—2121no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22#1681089512yes—
460C22C2×#1313318811yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4—2121no—
469C4—2121no—
470C4#137×D5311yescharacteristic
471C4#1681142211yescharacteristic
472C4#1681142612yes—
473C2—22no—
474C2—4241no—
475C2—2122no—
476C2#2028421411yescharacteristic
477C2#2028433911yescharacteristic
478C2C2×#1681089511yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3209C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3209D53
hypocenterC53#19459
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #20274259C2semidirect
5. #20274201C2semidirect
6. #20274135C2semidirect
7. #20274077C2semidirect
8. #20273500C2semidirect
9. #20273432C2semidirect
10. #20273415C2semidirect
11. C53×#3209C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810369C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810369C22semidirect
43. C53×#494C22semidirect
44. C53×#487C22semidirect
234. #3209D53semidirect
254. #494C2×D53semidirect
255. #487C2×D53semidirect
288. C53#19459semidirect

Maximal subgroups

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

subgroupindex
#202848102
C2×#168109002
#202837492
#202742592
#202742012
#202741352
#202740772
#202735002
#202734322
#202734152
C53×#32092
#1945953