Finite Group Databasebeta

← #27370173

Subgroups — #27370173

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285642C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3C2×#16810908C211yesmaximal, characteristic, complemented
4#20274297C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274173C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273433C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic
11C53×#2984C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16810996C2212yes—
15#16810974C2212yes—
16#16810936—21no—
17#16810918C2211yescharacteristic
18#16810918C2211yescharacteristic
19#16810908—21no—
20#16810908—21no—
21#16810908C2212yes—
22#16810908C2212yes—
23#16810829—22no—
24#16810823C2211yescharacteristic
25#16810823C2211yescharacteristic
26#16810764—21no—
27#16810764—21no—
28#16810762—21no—
29#16810721—21no—
30#16810721—21no—
31#16810680C2211yescharacteristic
32C2×#13133148—21no—
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133141—21no—
35#16810376C2212yes—
36#16810374C2212yes—
37#16810371C2212yes—
38#16810370C2212yes—
39#16810370C2212yes—
40#16810366C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yes—
44C8×Q8×C53C2212yes—
45C53×#432—21no—
46C53×#432C2211yescharacteristic
47C53×#431—21no—
48C53×#431C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719913—22no—
52C2×#11719913—22no—
53C2×#11719924—21no—
54C2×#11719924—21no—
55C22×#88322—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217C2312yes—
60#13133217C2311yescharacteristic
61#13133214C2311yescharacteristic
62#13133205—21no—
63#13133191—21no—
64#13133191—21no—
65C4×D212—21no—
66C4×D212—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69#13133150—21no—
70#13133150—21no—
71#13133148—22no—
72#13133148—22no—
73C4⋊#88322—21no—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133142—22no—
78#13133142—22no—
79#13133141—21no—
80#13133141—21no—
81#13133141—22no—
82#13133141—22no—
83#13133137—21no—
84#13133137—21no—
85C22⋊#88322—21no—
86#13133128—21no—
87C22×Q8×C53D411yescharacteristic
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53D411yescharacteristic
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C42×C53—21no—
97C2×C42×C53C2311yescharacteristic
98C4⋊C8×C53—22no—
99C4⋊C8×C53—22no—
100C4⋊C8×C53D412yes—
101C4⋊C8×C53C2312yes—
102C22⋊C8×C53—21no—
103C22⋊C8×C53D411yescharacteristic
104C22⋊C8×C53D411yescharacteristic
105#97×C53—22no—
106#97×C53—21no—
107C4×C8×C53—22no—
108C4×C8×C53C2312yes—
109C2×Q8×C53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53—41no—
112C2×Q8×C53C2×D411yescharacteristic
113C2×Q8×C53C2×D411yescharacteristic
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—41no—
120C22×Dic53—41no—
121#11719945—82no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—22no—
127C2×Dic106—21no—
128C2×Dic106—41no—
129C2×Dic106—82no—
130C2×C8×C53—22no—
131C2×C8×C53C2×D412yes—
132C2×C8×C53C2×D412yes—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53C2×D411yescharacteristic
139C4⋊C4×C53C2×D411yescharacteristic
140C42×C53—21no—
141C42×C53—21no—
142C42×C53—21no—
143C42×C53—21no—
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53#4111yescharacteristic
147C42×C53C2411yescharacteristic
148#11719924—42no—
149#11719924—42no—
150#11719921—41no—
151#11719921—41no—
152C4⋊Dic53—22no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—22no—
156C4⋊Dic53—41no—
157C4⋊Dic53—41no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—22no—
161C2×#88322—42no—
162#11719913—82no—
163#11719913—82no—
164C23×C53#12111yescharacteristic
165C22×D53—41no—
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—42no—
170Q8×C53#13712yes—
171C2×C4×C53—21no—
172C2×C4×C53—22no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—22no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C22×D411yescharacteristic
186C2×C4×C53C2×#4111yescharacteristic
187C22⋊D53—82no—
188C2×Dic53—42no—
189C2×Dic53—41no—
190C2×Dic53—82no—
191D212—41no—
192D212—41no—
193D212—42no—
194D212—81no—
195C4×D53—82no—
196Dic106—42no—
197Dic106—42no—
198Dic106—82no—
199Dic106—81no—
200C8×C53—22no—
201C8×C53—22no—
202C8×C53—42no—
203#88322—41no—
204#88322—42no—
205#88322—41no—
206#19469—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53#47112yes—
210C22×C53C2×#12111yescharacteristic
211C2×D53—81no—
212C2×D53—81no—
213C2×D53—81no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—22no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—41no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53C2×#13711yescharacteristic
223C4×C53#57511yescharacteristic
224C4×C53#57712yes—
225Dic53—82no—
226Dic53—81no—
227#3206—531no—
228#3112—531no—
229C2×#475—531no—
230#2984D5311yescharacteristic, complemented
231#1704—532no—
232#1686—532no—
233#1684—532no—
234#1666—532no—
235#1573—531no—
236#1556—531no—
237#1551—531no—
238C2×C53—22no—
239C2×C53#304311yescharacteristic
240C2×C53#305611yescharacteristic
241C2×C53C2×#47111yescharacteristic
242D53—161no—
243#532—531no—
244C2×C4⋊Q8—531no—
245C2×C4×Q8C2×D5311yescharacteristic
246#492—532no—
247#487—532no—
248#480—1061no—
249#477—531no—
250#477—531no—
251#475—532no—
252#475—532no—
253#475—1061no—
254#475—1061no—
255#461—1062no—
256#460—531no—
257#460—531no—
258#448—1061no—
259#448—1061no—
260#448—1061no—
261#446—22no—
262#446C2×D5312yes—
263C8×Q8C2×D5312yes—
264#440—1061no—
265#440—1061no—
266#432—21no—
267#432—531no—
268#432C2×D5311yescharacteristic
269#431—21no—
270#431C2×D5311yescharacteristic
271C2×#106—1061no—
272C2×C4⋊C8—531no—
273C2×#103—1061no—
274#335—532no—
275#334—532no—
276#333—532no—
277#332—532no—
278#332—532no—
279#331—532no—
280C53#1946911yescharacteristic, complemented, Sylow
281C2×#41—1061no—
282C22×Q8—1061no—
283C22×Q8D21211yescharacteristic
284C2×SD16—1061no—
285C2×SD16—1061no—
286C2×SD16—1062no—
287C2×SD16—1062no—
288C22×C8—1061no—
289C4⋊Q8—532no—
290C4⋊Q8—531no—
291C4⋊Q8—1061no—
292C4⋊Q8—1061no—
293C4⋊Q8—1061no—
294#127—531no—
295#124—1061no—
296#121—1061no—
297#121—1061no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8D21211yescharacteristic
301C4×Q8D21211yescharacteristic
302C4×Q8C22×D5311yescharacteristic
303C4×Q8C22×D5311yescharacteristic
304C4×D4—1061no—
305C4×D4—1061no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4D21211yescharacteristic
310C2×C42—21no—
311C2×C42C22×D5311yescharacteristic
312#107—1061no—
313#107—1061no—
314#106—1062no—
315#106—1062no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—532no—
319C4⋊C8—531no—
320C4⋊C8—531no—
321C4⋊C8—1061no—
322C4⋊C8C22⋊D5312yes—
323C4⋊C8C22×D5312yes—
324#103—1062no—
325#103—1062no—
326#103—1061no—
327#103—1061no—
328#103—1062no—
329#103—1062no—
330#102—1061no—
331#102—1061no—
332C22⋊C8—21no—
333C22⋊C8—1061no—
334C22⋊C8C22⋊D5311yescharacteristic
335C22⋊C8C22⋊D5311yescharacteristic
336#97—22no—
337#97—21no—
338#97—1061no—
339C4×C8—22no—
340C4×C8C22×D5312yes—
341#41—4242no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—41no—
345C2×Q8—1062no—
346C2×Q8—1061no—
347C2×Q8—4242no—
348C2×Q8—2121no—
349C2×Q8C2×D21211yescharacteristic
350C2×Q8C2×D21211yescharacteristic
351C2×D4—1061no—
352C2×D4—1061no—
353C2×D4—1061no—
354C2×D4—2121no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—2121no—
358C22×C4—2121no—
359C22×C4C2×D21211yescharacteristic
360C22×C4#1171994511yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362SD16—4242no—
363SD16—4242no—
364SD16—2122no—
365SD16—2122no—
366C2×C8—22no—
367C2×C8—1061no—
368C2×C8—1061no—
369C2×C8—1062no—
370C2×C8—2122no—
371C2×C8C2×C22⋊D5312yes—
372C2×C8C2×C22⋊D5312yes—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—1061no—
379C4⋊C4—1062no—
380C4⋊C4—1061no—
381C4⋊C4—1062no—
382C4⋊C4—2121no—
383C4⋊C4—2121no—
384C4⋊C4C2×D21211yescharacteristic
385C4⋊C4C2×D21211yescharacteristic
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42C2×D21211yescharacteristic
393C42#1171994511yescharacteristic
394C42C2×C22⋊D5311yescharacteristic
395C42C23×D5311yescharacteristic
396C23—2121no—
397C23#1313319111yescharacteristic
398Q8—21no—
399Q8—21no—
400Q8—21no—
401Q8—42no—
402Q8—4241no—
403Q8—4242no—
404Q8—2122no—
405Q8—2122no—
406Q8#1313325012yes—
407D4—4241no—
408D4—4242no—
409D4—2121no—
410D4—2121no—
411D4—2122no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—4242no—
423C2×C4—4242no—
424C2×C4—2121no—
425C2×C4—2122no—
426C2×C4#1313319111yescharacteristic
427C2×C4#1313319111yescharacteristic
428C2×C4#1313319111yescharacteristic
429C2×C4C22×C22⋊D5311yescharacteristic
430C2×C4C22×D21211yescharacteristic
431C2×C4C2×#1171994511yescharacteristic
432C8—22no—
433C8—22no—
434C8—42no—
435C8—2121no—
436C8—2122no—
437C8—2121no—
438C22—21no—
439C22—21no—
440C22—4241no—
441C22—4241no—
442C22—4241no—
443C22#1681088612yes—
444C22C2×#1313319111yescharacteristic
445C4—21no—
446C4—21no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—4242no—
454C4—4241no—
455C4C2×#1313325011yescharacteristic
456C4#1681142311yescharacteristic
457C4#1681143112yes—
458C2—22no—
459C2—8481no—
460C2#2028423611yescharacteristic
461C2#2028434511yescharacteristic
462C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2984C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2984D53
hypocenterC53#19469
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
3. C2×#16810908C2semidirect
4. #20274297C2semidirect
5. #20274187C2semidirect
6. #20274173C2semidirect
7. #20274063C2semidirect
8. #20273500C2semidirect
9. #20273433C2semidirect
11. C53×#2984C2semidirect
230. #2984D53semidirect
280. C53#19469semidirect

Maximal subgroups

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

subgroupindex
#202856422
#202848372
C2×#168109082
#202742972
#202741872
#202741732
#202740632
#202735002
#202734332
#202734112
C53×#29842
#1946953