Finite Group Databasebeta

← #27362885

Subgroups — #27362885

Order 13568.

932 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#20285843C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3#20284187C211yesmaximal, characteristic
4#20273992C214yesmaximal, complemented
5#20273811C214yesmaximal, complemented
6#20273400C211yesmaximal, characteristic, complemented
7#20273144C211yesmaximal, characteristic, complemented
8#20273110C211yesmaximal, characteristic, complemented
9C2×C53×#557C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16811019C2212yes—
13#16810971C2212yes—
14#16810941—21no—
15#16810941—22no—
16#16810910—22no—
17#16810908C2212yes—
18#16810845—22no—
19#16810833C2212yes—
20#16810823C2212yes—
21#16810823C2212yes—
22#16810762—21no—
23C4×#11719924—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810622—21no—
28#16810369C2214yes—
29Q8⋊#88322C2214yes—
30Q8⋊#88322C2214yes—
31C53×#557—22no—
32C53×#557—22no—
33C53×#557—22no—
34C53×#557C2214yes—
35C2×C4⋊Q8×C53—21no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#125×C53—22no—
38C2×#125×C53—21no—
39C2×C4×Q8×C53—21no—
40C2×C4×Q8×C53—21no—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×#11719945—21no—
44#13133248—44no—
45C2×#11719925—22no—
46C2×#11719924—22no—
47C2×#11719924—22no—
48C2×#11719917—21no—
49#13133217C2311yescharacteristic
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56#13133148—24no—
57#13133148—22no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133140—22no—
64#13133137—22no—
65#13133137—22no—
66#13133137—24no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69C4×#88322—21no—
70C4×#88322C2312yes—
71C22×Q8×C53—21no—
72C22×Q8×C53D411yescharacteristic
73C22×Q8×C53D411yescharacteristic
74C4⋊Q8×C53—24no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—24no—
78C4⋊Q8×C53D412yes—
79C4⋊Q8×C53C2312yes—
80#125×C53—28no—
81#125×C53—24no—
82#125×C53—28no—
83#125×C53—24no—
84C4×Q8×C53—24no—
85C4×Q8×C53—24no—
86C4×Q8×C53—24no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53—24no—
92C4×Q8×C53D412yes—
93C4×Q8×C53D412yes—
94C4×Q8×C53C2312yes—
95C4×Q8×C53C2312yes—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D412yes—
105C2×C42×C53—22no—
106C2×C42×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—42no—
115C2×Q8×C53—44no—
116C2×Q8×C53SD1614yescomplemented
117C2×Q8×C53C2×D412yes—
118C2×Q8×C53C2×D412yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C2×C22⋊D53—41no—
128#11719945—42no—
129#11719945—42no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×C4×D53—41no—
134C2×Dic106—21no—
135C4⋊C4×C53—28no—
136C4⋊C4×C53—28no—
137C4⋊C4×C53—28no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53C2×D412yes—
148C4⋊C4×C53C2×D412yes—
149C4⋊C4×C53C2×D414yes—
150C42×C53—28no—
151C42×C53—22no—
152C42×C53—22no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2411yescharacteristic
159#11719925—44no—
160#11719924—44no—
161#11719924—44no—
162#11719921—41no—
163#11719921—41no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166#11719917—42no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C23×C53#12011yescharacteristic
171C22×D53—41no—
172Q8×C53—22no—
173Q8×C53—24no—
174Q8×C53—22no—
175Q8×C53—22no—
176Q8×C53—42no—
177Q8×C53—44no—
178Q8×C53—44no—
179Q8×C53C2×SD1614yes—
180C2×C4×C53—22no—
181C2×C4×C53—24no—
182C2×C4×C53—22no—
183C2×C4×C53—24no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—44no—
192C2×C4×C53#12011yescharacteristic
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53#12011yescharacteristic
195C2×C4×C53C2×SD1612yes—
196C2×C4×C53#13712yes—
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C22⋊D53—82no—
201C2×Dic53—41no—
202D212—42no—
203D212—41no—
204D212—41no—
205D212—42no—
206C4×D53—82no—
207Dic106—41no—
208Dic106—41no—
209#88322—42no—
210#88322—42no—
211#88322—42no—
212#18778—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53#44812yes—
216C22×C53C2×#12011yescharacteristic
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C4×C53—24no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—42no—
225C4×C53—44no—
226C4×C53—42no—
227C4×C53—42no—
228C4×C53#44812yes—
229C4×C53C22×SD1611yescharacteristic
230C4×C53C2×#13711yescharacteristic
231C4×C53#57512yes—
232Dic53—81no—
233C2×#557D5311yescharacteristic, complemented
234#3227—531no—
235#3103—531no—
236#3037—531no—
237#1653—534no—
238#1626—534no—
239#1549—531no—
240#1487—531no—
241#1479—531no—
242C2×C53—22no—
243C2×C53C2×#44811yescharacteristic
244C2×C53#300511yescharacteristic
245C2×C53#302011yescharacteristic
246D53—161no—
247C2×#137—1061no—
248#557—22no—
249#557—22no—
250#557—22no—
251#557C2×D5314yes—
252#532—531no—
253C2×C4⋊Q8—21no—
254C2×C4⋊Q8C2×D5311yescharacteristic
255C2×#125—22no—
256C2×#125—21no—
257C2×C4×Q8—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8C2×D5311yescharacteristic
260C2×C4×Q8C2×D5311yescharacteristic
261#498—532no—
262#486—532no—
263#481—1061no—
264#481—1062no—
265#476—1062no—
266#475—532no—
267#464—1062no—
268#462—532no—
269#460—532no—
270#460—532no—
271#448—1061no—
272C4×SD16—1062no—
273#431—531no—
274#431—531no—
275#423—531no—
276#417—1061no—
277#332—534no—
278Q8⋊C8—534no—
279Q8⋊C8—534no—
280C53#1877811yescharacteristic, complemented, Sylow
281C2×#41—1061no—
282C22×Q8—21no—
283C22×Q8C22⋊D5311yescharacteristic
284C22×Q8C22⋊D5311yescharacteristic
285#137—2124no—
286C2×Q16—1062no—
287C2×SD16—1062no—
288C2×SD16—1062no—
289C2×M16—1061no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—531no—
295C4⋊Q8C22⋊D5312yes—
296C4⋊Q8C22×D5312yes—
297#127—531no—
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—1061no—
303#121—1061no—
304#121—1061no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8C22⋊D5312yes—
314C4×Q8C22⋊D5312yes—
315C4×Q8C22×D5312yes—
316C4×Q8C22×D5312yes—
317C4×D4—1061no—
318C4×D4—1061no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D5311yescharacteristic
326C2×C4⋊C4C22⋊D5311yescharacteristic
327C2×C4⋊C4C22⋊D5312yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#106—1062no—
333#106—1064no—
334C4⋊C8—532no—
335C4⋊C8—532no—
336C4⋊C8—1061no—
337C4⋊C8—1061no—
338#103—1062no—
339#103—1062no—
340#102—1062no—
341#102—1062no—
342#102—1064no—
343C22⋊C8—1061no—
344C22⋊C8—1061no—
345C4×C8—532no—
346C4×C8—1061no—
347#41—2122no—
348#41—2122no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—1061no—
357C2×Q8#1171992414yescomplemented
358C2×Q8C2×C22⋊D5312yes—
359C2×Q8C2×C22⋊D5312yes—
360C2×D4—1061no—
361C2×D4—1061no—
362C2×D4—1061no—
363C2×D4—2121no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—2121no—
370C22×C4C2×C22⋊D5311yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C2×C22⋊D5311yescharacteristic
373Q16—2124no—
374SD16—2124no—
375SD16—2124no—
376M16—2122no—
377C2×C8—1062no—
378C2×C8—1062no—
379C2×C8—1062no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4C2×C22⋊D5312yes—
395C4⋊C4C2×C22⋊D5312yes—
396C4⋊C4C2×C22⋊D5314yes—
397C22⋊C4—2121no—
398C22⋊C4—2121no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×C22⋊D5311yescharacteristic
405C42C2×C22⋊D5311yescharacteristic
406C42C2×C22⋊D5311yescharacteristic
407C42C23×D5311yescharacteristic
408C23—2121no—
409C23#1313318611yescharacteristic
410Q8—24no—
411Q8—22no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—2121no—
418Q8—2121no—
419Q8C2×#1171992414yes—
420D4—4242no—
421D4—2122no—
422D4—2121no—
423D4—2121no—
424D4—2122no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—4242no—
438C2×C4—2121no—
439C2×C4#1313318611yescharacteristic
440C2×C4#1313318611yescharacteristic
441C2×C4#1313318611yescharacteristic
442C2×C4C2×#1171992412yes—
443C2×C4#1313324812yes—
444C2×C4C22×C22⋊D5311yescharacteristic
445C2×C4C22×C22⋊D5311yescharacteristic
446C2×C4C22×C22⋊D5311yescharacteristic
447C8—2122no—
448C8—2122no—
449C8—2122no—
450C22—21no—
451C22—21no—
452C22—4241no—
453C22—4241no—
454C22—4241no—
455C22#1681076212yes—
456C22C2×#1313318611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—24no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—4241no—
466C4#1681076212yes—
467C4C22×#1171992411yescharacteristic
468C4C2×#1313324811yescharacteristic
469C4#1681142112yes—
470C2—22no—
471C2—8481no—
472C2C2×#1681076211yescharacteristic
473C2#2028395711yescharacteristic
474C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#557C2
socleC22×C53C2×#120
prime radical#16811173C22
hypercenterC2×#557D53
hypocenterC53#18778
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #20273992C2semidirect
5. #20273811C2semidirect
6. #20273400C2semidirect
7. #20273144C2semidirect
8. #20273110C2semidirect
9. C2×C53×#557C2semidirect
116. C2×Q8×C53SD16semidirect
233. C2×#557D53semidirect
280. C53#18778semidirect
357. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202847462
#202841872
#202739922
#202738112
#202734002
#202731442
#202731102
C2×C53×#5572
#1877853