Finite Group Databasebeta

← #27361438

Subgroups — #27361438

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2C2×#16810971C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic
4#20273818C214yesmaximal, complemented
5#20273811C214yesmaximal, complemented
6#20273400C211yesmaximal, characteristic, complemented
7#20273170C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3503C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133214C2211yescharacteristic
12#16810997—21no—
13#16810971—22no—
14#16810971—21no—
15#16810971C2214yes—
16#16810823—21no—
17#16810823C2212yes—
18#16810823C2214yes—
19#16810818—21no—
20#16810818C2212yes—
21#16810773—22no—
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810627—21no—
25C2×#13133137—22no—
26C2×C4×#88322C2211yescharacteristic
27#16810369C2214yescomplemented
28Q8⋊#88322C2214yescomplemented
29Q8⋊#88322C2212yescomplemented
30D4⋊#88322C2212yescomplemented
31Q82×C53—21no—
32C53×#557C2212yescomplemented
33C53×#554—21no—
34C53×#554—22no—
35D4×Q8×C53—22no—
36D4×Q8×C53C2212yescomplemented
37C53×#544—22no—
38C53×#544—21no—
39C53×#533—21no—
40C53×#533C2211yescharacteristic
41C2×C4⋊Q8×C53—21no—
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#520—21no—
44C4×#41×C53C2211yescharacteristic
45C22×D212—21no—
46C22×D212—21no—
47C22×D212—21no—
48#13133242—44no—
49C2×#11719924—22no—
50C2×#11719924—24no—
51C2×#11719922—22no—
52C2×#11719917—21no—
53C22×#88322—21no—
54#13133214—21no—
55#13133214—21no—
56#13133214—21no—
57#13133214C2312yes—
58C4⋊#88322—21no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133137—24no—
62#13133137—22no—
63#13133137—22no—
64#13133137—28no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322—21no—
68C4×#88322C2312yes—
69C2×#41×C53D411yescharacteristic
70C22×Q8×C53—22no—
71C22×Q8×C53—21no—
72C4⋊Q8×C53—22no—
73C4⋊Q8×C53—22no—
74C4⋊Q8×C53—24no—
75C4⋊Q8×C53—22no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53—22no—
79C4⋊Q8×C53D412yes—
80C4⋊Q8×C53D411yescharacteristic
81C4⋊Q8×C53D411yescharacteristic
82C4⋊Q8×C53C2312yes—
83#125×C53—22no—
84#125×C53—24no—
85#125×C53—22no—
86#125×C53—21no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—24no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—22no—
91C22⋊Q8×C53—24no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53D412yes—
94C4×Q8×C53—22no—
95C4×Q8×C53—22no—
96C4×Q8×C53—21no—
97C4×Q8×C53—22no—
98C4×Q8×C53D411yescharacteristic
99C4×Q8×C53C2311yescharacteristic
100C4×Q8×C53C2312yes—
101C4×D4×C53—24no—
102C4×D4×C53—22no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53D411yescharacteristic
106C4×D4×C53C2311yescharacteristic
107#117×C53—22no—
108#117×C53—21no—
109#117×C53—21no—
110#117×C53D411yescharacteristic
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D412yes—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C23×D53—41no—
118#41×C53—22no—
119#41×C53SD1614yescomplemented
120C2×Q8×C53—24no—
121C2×Q8×C53—21no—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—82no—
125C2×Q8×C53—84no—
126C2×Q8×C53C2×D412yes—
127C2×Q8×C53C2×D411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—24no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—44no—
142C2×D212—42no—
143C2×D212—42no—
144C2×D212—41no—
145C2×D212—41no—
146C2×D212—41no—
147C2×D212—82no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C2×D414yes—
163C4⋊C4×C53C2×D411yescharacteristic
164C4⋊C4×C53C2×D411yescharacteristic
165C4⋊C4×C53C2×D412yes—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C42×C53—24no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—22no—
176C42×C53C2×D411yescharacteristic
177C42×C53C2×D411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53C2411yescharacteristic
180#11719924—44no—
181#11719924—84no—
182#11719922—44no—
183#11719917—42no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—42no—
188C23×C53—21no—
189C23×C53#12011yescharacteristic
190C22×D53—42no—
191C22×D53—82no—
192C22×D53—82no—
193C22×D53—82no—
194Q8×C53—21no—
195Q8×C53—24no—
196Q8×C53—44no—
197Q8×C53—42no—
198Q8×C53—82no—
199Q8×C53—84no—
200Q8×C53C2×SD1612yes—
201D4×C53—24no—
202D4×C53—21no—
203D4×C53C2×SD1612yes—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—82no—
216C2×C4×C53—84no—
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53C2×SD1612yes—
221C2×C4×C53#13712yes—
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225D212—44no—
226D212—42no—
227D212—42no—
228D212—44no—
229D212—82no—
230D212—82no—
231D212—82no—
232#88322—42no—
233#88322—42no—
234#88322—42no—
235#18642—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—22no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53#44812yes—
242C22×C53C2×#12011yescharacteristic
243C2×D53—82no—
244C2×D53—82no—
245C2×D53—82no—
246C2×D53—164no—
247C4×C53—22no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—41no—
251C4×C53—42no—
252C4×C53—44no—
253C4×C53—41no—
254C4×C53—42no—
255C4×C53#44812yes—
256C4×C53#45412yes—
257C4×C53C22×SD1611yescharacteristic
258C4×C53C2×#13711yescharacteristic
259C4×C53#57612yes—
260#3503D5311yescharacteristic, complemented
261#3208—531no—
262C2×#486—531no—
263#3038—531no—
264#1627—534no—
265#1626—534no—
266#1549—531no—
267#1493—531no—
268#1476—531no—
269C2×C53—22no—
270C2×C53—42no—
271C2×C53C2×#44811yescharacteristic
272C2×C53#300811yescharacteristic
273C2×C53C2×#45411yescharacteristic
274D53—162no—
275C2×AGL(1,ℤ/8)—1061no—
276Q82—21no—
277#557C2×D5312yescomplemented
278#554—21no—
279#554—22no—
280D4×Q8—22no—
281D4×Q8C2×D5312yescomplemented
282#544—22no—
283#544—21no—
284#533—21no—
285#533C2×D5311yescharacteristic
286C2×C4⋊Q8—21no—
287C2×C4⋊Q8C2×D5311yescharacteristic
288C2×#127—531no—
289#520—21no—
290C4×#41C2×D5311yescharacteristic
291#492—1061no—
292#486—534no—
293#486—1062no—
294#486—1061no—
295#460—532no—
296#460—534no—
297#460—1061no—
298#459—532no—
299#459—1061no—
300#450—1062no—
301#431—531no—
302#423—531no—
303#418—1061no—
304C2×#102—1062no—
305C2×C4×C8—531no—
306#332—534no—
307Q8⋊C8—532no—
308Q8⋊C8—534no—
309D4⋊C8—532no—
310C53#1864211yescharacteristic, complemented, Sylow
311C2×#41C22⋊D5311yescharacteristic
312C22×Q8—22no—
313C22×Q8—21no—
314C22×D4—1061no—
315C22×D4—1061no—
316C22×D4—1061no—
317AGL(1,ℤ/8)—2124no—
318C2×SD16—1064no—
319C2×SD16—1062no—
320C2×D8—1062no—
321C2×M16—1061no—
322C22×C8—1061no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—24no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8C22⋊D5311yescharacteristic
331C4⋊Q8C22⋊D5312yes—
332C4⋊Q8C22⋊D5311yescharacteristic
333C4⋊Q8C22×D5312yes—
334#127—532no—
335#127—1061no—
336#127—1061no—
337#127—1061no—
338#125—22no—
339#125—24no—
340#125—22no—
341#125—21no—
342C22⋊Q8—22no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8—24no—
348C22⋊Q8C22⋊D5312yes—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—21no—
353C4×Q8C22⋊D5311yescharacteristic
354C4×Q8C22×D5311yescharacteristic
355C4×Q8C22×D5312yes—
356C4×D4—24no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4C22⋊D5311yescharacteristic
361C4×D4C22×D5311yescharacteristic
362#117—22no—
363#117—21no—
364#117—21no—
365#117C22⋊D5311yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C22⋊D5312yes—
370C2×C42—21no—
371C2×C42C22×D5311yescharacteristic
372C4⋊C8—532no—
373C4⋊C8—532no—
374C4⋊C8—1061no—
375#102—1068no—
376#102—1062no—
377#102—1062no—
378#102—1064no—
379C22⋊C8—1061no—
380C4×C8—532no—
381C4×C8—1061no—
382C4×C8—1061no—
383C24—2121no—
384#41—22no—
385#41#1171992414yescomplemented
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—84no—
392C2×Q8C2×C22⋊D5312yes—
393C2×Q8C2×C22⋊D5311yescharacteristic
394C2×D4—21no—
395C2×D4—1062no—
396C2×D4—1062no—
397C2×D4—1062no—
398C2×D4—4242no—
399C2×D4—2124no—
400C2×D4—2121no—
401C2×D4—2121no—
402C2×D4—2121no—
403C2×D4—2122no—
404C2×D4—2122no—
405C2×D4C2×C22⋊D5311yescharacteristic
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414SD16—4244no—
415SD16—2124no—
416D8—2124no—
417M16—2122no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—1062no—
421C2×C8—2122no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4C2×C22⋊D5314yes—
437C4⋊C4C2×C22⋊D5311yescharacteristic
438C4⋊C4C2×C22⋊D5311yescharacteristic
439C4⋊C4C2×C22⋊D5312yes—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C42—24no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—22no—
450C42C2×C22⋊D5311yescharacteristic
451C42C2×C22⋊D5311yescharacteristic
452C42C2×C22⋊D5311yescharacteristic
453C42C23×D5311yescharacteristic
454C23—21no—
455C23—4242no—
456C23—4242no—
457C23—4242no—
458C23—2122no—
459C23#1313318611yescharacteristic
460Q8—21no—
461Q8—24no—
462Q8—42no—
463Q8—44no—
464Q8—82no—
465Q8—84no—
466Q8C2×#1171992412yes—
467D4—24no—
468D4—21no—
469D4—4242no—
470D4—4242no—
471D4—4242no—
472D4—2124no—
473D4—2122no—
474D4—2122no—
475D4—2124no—
476D4C2×#1171992412yes—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4#1313318611yescharacteristic
491C2×C4#1313318611yescharacteristic
492C2×C4#1313318611yescharacteristic
493C2×C4C2×#1171992412yes—
494C2×C4#1313324812yes—
495C2×C4C22×C22⋊D5311yescharacteristic
496C2×C4C22×C22⋊D5311yescharacteristic
497C2×C4C22×C22⋊D5311yescharacteristic
498C8—2122no—
499C8—2122no—
500C8—2122no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—8484no—
507C22—4242no—
508C22—4242no—
509C22—4242no—
510C22#1681076212yes—
511C22C2×#1313318611yescharacteristic
512C4—21no—
513C4—21no—
514C4—22no—
515C4—41no—
516C4—42no—
517C4—44no—
518C4—41no—
519C4—42no—
520C4#1681076212yes—
521C4#1681079612yes—
522C4C22×#1171992411yescharacteristic
523C4C2×#1313324811yescharacteristic
524C4#1681142612yes—
525C2—22no—
526C2—42no—
527C2—8482no—
528C2C2×#1681076211yescharacteristic
529C2#2028398311yescharacteristic
530C2C2×#1681079611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3503C2
socleC22×C53C2×#120
prime radical#20284195C2
hypercenter#3503D53
hypocenterC53#18642
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
2. C2×#16810971C2semidirect
4. #20273818C2semidirect
5. #20273811C2semidirect
6. #20273400C2semidirect
7. #20273170C2semidirect
8. #20273097C2semidirect
9. C53×#3503C2semidirect
27. #16810369C22semidirect
28. Q8⋊#88322C22semidirect
29. Q8⋊#88322C22semidirect
30. D4⋊#88322C22semidirect
32. C53×#557C22semidirect
36. D4×Q8×C53C22semidirect
119. #41×C53SD16semidirect
260. #3503D53semidirect
277. #557C2×D53semidirect
281. D4×Q8C2×D53semidirect
310. C53#18642semidirect
385. #41#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168109712
#202841952
#202738182
#202738112
#202734002
#202731702
#202730972
C53×#35032
#1864253