Finite Group Databasebeta

← #27368490

Subgroups — #27368490

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285090C211yesmaximal, characteristic, complemented
2#20284877C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic
4#20274283C212yesmaximal, complemented
5#20274255C212yesmaximal, complemented
6#20274152C212yesmaximal, complemented
7#20274124C212yesmaximal, complemented
8#20273482C211yesmaximal, characteristic, complemented
9#20273446C211yesmaximal, characteristic, complemented
10#20273428C211yesmaximal, characteristic, complemented
11C53×#2970C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16810988—21no—
16#16810988—21no—
17#16810970C2212yes—
18#16810970C2212yes—
19#16810964—21no—
20#16810918C2211yescharacteristic, complemented
21#16810913C2211yescharacteristic, complemented
22#16810833C2211yescharacteristic, complemented
23#16810828C2211yescharacteristic, complemented
24#16810823—21no—
25#16810823C2212yes—
26#16810818—21no—
27#16810818C2212yes—
28#16810785—21no—
29#16810728—21no—
30#16810723—21no—
31#16810680C2211yescharacteristic
32#16810647C2211yescharacteristic
33#16810629—22no—
34#16810627—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yes—
40#16810366C2212yes—
41C4×#41×C53C2211yescharacteristic
42C53×#446C2212yescomplemented
43C53×#435—22no—
44C8×D4×C53C2212yescomplemented
45C53×#432—21no—
46C53×#432C2211yescharacteristic
47C4×M16×C53C2211yescharacteristic
48C2×#97×C53—21no—
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53C2×#11719925—21no—
54C2×#11719923—21no—
55C2×#11719924—22no—
56C2×#11719922—22no—
57C2×#11719917—21no—
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133214—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214C2312yes—
64#13133209—21no—
65#13133196—21no—
66#13133196—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69#13133176—21no—
70#13133150—21no—
71#13133148—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76#13133141—21no—
77#13133140—21no—
78#13133140—21no—
79#13133139—22no—
80#13133139—22no—
81#13133139—22no—
82#13133139—22no—
83#13133138—21no—
84#13133137—22no—
85#13133137—22no—
86C22⋊#88322—21no—
87#13133128—21no—
88C2×#41×C53D411yescharacteristic
89C2×M16×C53D412yes—
90C22×C8×C53—22no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97#117×C53—21no—
98#117×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53—22no—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—21no—
105#97×C53—22no—
106#97×C53—21no—
107#97×C53D411yescharacteristic
108#97×C53D411yescharacteristic
109C4×C8×C53C2312yes—
110C23×D53—41no—
111#41×C53—22no—
112#41×C53—41no—
113C2×Q8×C53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—42no—
126C2×D212—42no—
127C2×D212—44no—
128C2×D212—41no—
129C2×D212—41no—
130C2×D212—41no—
131C2×D212—82no—
132C2×Dic106—21no—
133C2×Dic106—21no—
134M16×C53—42no—
135C2×C8×C53—22no—
136C2×C8×C53—44no—
137C2×C8×C53C2×D412yes—
138C2×C8×C53C2×D412yes—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153C22⋊Dic53—41no—
154#11719925—42no—
155#11719924—44no—
156#11719923—42no—
157#11719922—44no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165#11719917—42no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—22no—
169C23×C53—21no—
170C23×C53#12111yescharacteristic
171C22×D53—42no—
172C22×D53—82no—
173C22×D53—82no—
174C22×D53—82no—
175Q8×C53—21no—
176Q8×C53—21no—
177D4×C53—21no—
178D4×C53—42no—
179D4×C53AGL(1,ℤ/8)12yes—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—81no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×Dic53—41no—
197C2×Dic53—41no—
198D212—42no—
199D212—44no—
200D212—42no—
201D212—82no—
202D212—82no—
203D212—82no—
204D212—82no—
205Dic106—41no—
206Dic106—41no—
207Dic106—81no—
208C8×C53—22no—
209C8×C53—22no—
210C8×C53—42no—
211#88322—42no—
212#88322—41no—
213#88322—41no—
214#19316—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—22no—
218C22×C53—21no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53C2×#12111yescharacteristic
222C2×D53—82no—
223C2×D53—82no—
224C2×D53—82no—
225C2×D53—164no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
234C4×C53#57512yes—
235C4×C53#57511yescharacteristic
236Dic53—81no—
237Dic53—81no—
238#3138—531no—
239#3116—531no—
240#3038—531no—
241#2970D5311yescharacteristic, complemented
242#1702—532no—
243#1698—532no—
244#1681—532no—
245#1677—532no—
246#1569—531no—
247#1559—531no—
248#1555—531no—
249C2×C53—41no—
250C2×C53—42no—
251C2×C53#304011yescharacteristic
252C2×C53#305511yescharacteristic
253C2×C53#306811yescharacteristic
254D53—162no—
255C2×AGL(1,ℤ/8)—1061no—
256#533—531no—
257C2×#127—531no—
258C4×#41C2×D5311yescharacteristic
259#490—1061no—
260#490—1061no—
261#486—532no—
262#486—532no—
263#485—1061no—
264#477—531no—
265#476—531no—
266#462—531no—
267#461—531no—
268#460—532no—
269#460—1061no—
270#459—532no—
271#459—1061no—
272#452—1061no—
273#446C2×D5312yescomplemented
274#441—1061no—
275#440—1061no—
276#435—22no—
277C8×D4C2×D5312yescomplemented
278#432—21no—
279#432—531no—
280#432C2×D5311yescharacteristic
281#423—531no—
282#418—1062no—
283#418—1061no—
284C4×M16C2×D5311yescharacteristic
285C2×#97—21no—
286#335—532no—
287#334—532no—
288#333—532no—
289#332—532no—
290#331—532no—
291#331—532no—
292C53#1931611yescharacteristic, complemented, Sylow
293C2×#41D21211yescharacteristic
294C22×D4—1061no—
295C22×D4—1061no—
296C22×D4—1061no—
297AGL(1,ℤ/8)—2124no—
298C2×Q16—1061no—
299C2×SD16—1062no—
300C2×SD16—1061no—
301C2×D8—1062no—
302C2×M16—1061no—
303C2×M16C22⋊D5312yes—
304C22×C8—22no—
305C4⋊Q8—531no—
306C4⋊Q8—531no—
307#127—532no—
308#127—1061no—
309#127—1061no—
310#127—1061no—
311#125—1061no—
312C22⋊Q8—1061no—
313C22⋊Q8—1061no—
314C4×Q8—1061no—
315C4×Q8—1061no—
316C4×Q8D21211yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4D21211yescharacteristic
321C4×D4C22×D5311yescharacteristic
322#117—21no—
323#117—1061no—
324#117D21211yescharacteristic
325C2×C42—21no—
326C2×C42C22×D5311yescharacteristic
327#107—1061no—
328#106—1061no—
329C4⋊C8—22no—
330C4⋊C8—531no—
331C4⋊C8—531no—
332C4⋊C8—532no—
333C4⋊C8—1061no—
334C4⋊C8C22×D5312yes—
335#103—1061no—
336#103—1061no—
337#103—1061no—
338#102—1061no—
339#102—1062no—
340#102—1062no—
341#102—1062no—
342#102—1062no—
343#102—1062no—
344#102—1062no—
345C22⋊C8—21no—
346C22⋊C8—22no—
347C22⋊C8—1061no—
348#97—22no—
349#97—21no—
350#97—1061no—
351#97C22⋊D5311yescharacteristic
352#97C22⋊D5311yescharacteristic
353C4×C8C22×D5312yes—
354C24—2121no—
355#41—22no—
356#41—41no—
357C2×Q8—1061no—
358C2×Q8—1061no—
359C2×Q8C2×D21211yescharacteristic
360C2×D4—21no—
361C2×D4—1062no—
362C2×D4—1062no—
363C2×D4—1062no—
364C2×D4—4242no—
365C2×D4—2122no—
366C2×D4—2124no—
367C2×D4—2122no—
368C2×D4—2121no—
369C2×D4—2121no—
370C2×D4—2121no—
371C2×D4C2×D21211yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C2×D21211yescharacteristic
376C22×C4#1171994511yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—2122no—
379SD16—2124no—
380SD16—2122no—
381D8—2124no—
382M16—42no—
383M16—2122no—
384C2×C8—22no—
385C2×C8—44no—
386C2×C8—1061no—
387C2×C8—1061no—
388C2×C8—1062no—
389C2×C8C2×C22⋊D5312yes—
390C2×C8C2×C22⋊D5312yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—1061no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—2121no—
398C4⋊C4—2121no—
399C4⋊C4C2×D21211yescharacteristic
400C4⋊C4C2×D21211yescharacteristic
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—2121no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—2121no—
409C42C2×D21211yescharacteristic
410C42#1171994511yescharacteristic
411C42C2×C22⋊D5311yescharacteristic
412C42C23×D5311yescharacteristic
413C23—21no—
414C23—4242no—
415C23—4242no—
416C23—4242no—
417C23—2122no—
418C23#1313319111yescharacteristic
419Q8—21no—
420Q8—21no—
421Q8—4241no—
422Q8—2121no—
423Q8—2121no—
424D4—21no—
425D4—42no—
426D4—4242no—
427D4—4242no—
428D4—4242no—
429D4—4242no—
430D4—2122no—
431D4—2124no—
432D4—2122no—
433D4#1313324512yes—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4#1313319111yescharacteristic
447C2×C4#1313319111yescharacteristic
448C2×C4#1313319111yescharacteristic
449C2×C4C22×C22⋊D5311yescharacteristic
450C2×C4C22×D21211yescharacteristic
451C2×C4C2×#1171994511yescharacteristic
452C8—22no—
453C8—22no—
454C8—42no—
455C8—2122no—
456C8—2121no—
457C8—2121no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—8484no—
465C22—4242no—
466C22—4242no—
467C22—4242no—
468C22C2×#1313319111yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—4241no—
477C4—4241no—
478C4C2×#1313324511yescharacteristic
479C4#1681142112yes—
480C4#1681142311yescharacteristic
481C2—41no—
482C2—42no—
483C2—8482no—
484C2#2028420911yescharacteristic
485C2#2028433411yescharacteristic
486C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2970C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#2970D53
hypocenterC53#19316
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2970

Complements

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

NH ≅ G/Ntype
1. #20285090C2semidirect
2. #20284877C2semidirect
4. #20274283C2semidirect
5. #20274255C2semidirect
6. #20274152C2semidirect
7. #20274124C2semidirect
8. #20273482C2semidirect
9. #20273446C2semidirect
10. #20273428C2semidirect
11. C53×#2970C2semidirect
20. #16810918C22semidirect
21. #16810913C22semidirect
22. #16810833C22semidirect
23. #16810828C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810368C22semidirect
42. C53×#446C22semidirect
44. C8×D4×C53C22semidirect
241. #2970D53semidirect
273. #446C2×D53semidirect
277. C8×D4C2×D53semidirect
292. C53#19316semidirect

Maximal subgroups

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

subgroupindex
#202850902
#202848772
#202841952
#202742832
#202742552
#202741522
#202741242
#202734822
#202734462
#202734282
C53×#29702
#1931653