Finite Group Databasebeta

← #27367932

Subgroups — #27367932

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285290C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic, complemented
3C2×Q8×#88322C211yesmaximal, characteristic, complemented
4#20274161C212yesmaximal, complemented
5#20274157C212yesmaximal, complemented
6#20274107C212yesmaximal, complemented
7#20274073C212yesmaximal, complemented
8#20273473C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10#20273392C211yesmaximal, characteristic, complemented
11C53×#3037C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13#16811173C2211yescharacteristic
14#16811026C2211yescharacteristic
15#16810999C2211yescharacteristic
16#16810971C2211yescharacteristic
17#16810969C2211yescharacteristic
18#16810857—21no—
19#16810857—21no—
20#16810845—22no—
21#16810833C2212yes—
22#16810823C2212yes—
23Q8×#88322—21no—
24Q8×#88322—21no—
25Q8×#88322C2212yes—
26Q8×#88322C2212yes—
27C4×#11719922—21no—
28C4×#11719922—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32#16810622—21no—
33C2×C4×#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yes—
36#16810375C2212yes—
37#16810370C2212yes—
38#16810369C2212yes—
39#16810367C2212yes—
40#16810366C2212yes—
41C2×#137×C53—21no—
42C53×#532C2211yescharacteristic
43C2×C4×Q8×C53C2211yescharacteristic
44C53×#464—22no—
45C53×#462C2212yes—
46C53×#460C2212yes—
47C53×#423C2211yescharacteristic
48C53×#417—21no—
49C2×#11719945—21no—
50#13133248—44no—
51C2×#11719925—22no—
52C2×#11719924—22no—
53C2×#11719922—21no—
54C2×#11719922—21no—
55C2×#11719917—21no—
56C22×#88322—21no—
57C22×#88322D411yescharacteristic
58C22×#88322D411yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133150—21no—
67#13133150—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—22no—
71C4⋊#88322—22no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74#13133140—22no—
75#13133138—21no—
76#13133138—21no—
77#13133137—22no—
78#13133137—21no—
79#13133137—21no—
80C4×#88322—22no—
81C4×#88322—22no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C2×#41×C53—21no—
87C22×Q8×C53D411yescharacteristic
88#137×C53—44no—
89C2×Q16×C53—22no—
90C2×SD16×C53—22no—
91C2×M16×C53—21no—
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53C2311yescharacteristic
94#124×C53—21no—
95#121×C53—21no—
96#121×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53D412yes—
100C4×Q8×C53C2312yes—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#103×C53—22no—
110#102×C53—22no—
111#41×C53—42no—
112#41×C53—42no—
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—21no—
116C2×Q8×C53D814yescomplemented
117C2×Q8×C53C2×D412yes—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×C22⋊D53—41no—
129#11719945—42no—
130#11719945—42no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×C4×D53—41no—
135C2×Dic106—21no—
136Q16×C53—44no—
137SD16×C53—44no—
138M16×C53—42no—
139C2×C8×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D412yes—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C42×C53—22no—
149C42×C53—22no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719925—44no—
155#11719924—44no—
156#11719922—42no—
157#11719922—42no—
158#11719921—41no—
159#11719921—41no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162#11719917—42no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—41no—
167C2×#88322—42no—
168C2×#88322C2×D411yescharacteristic
169C2×#88322C2×D411yescharacteristic
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172C23×C53—41no—
173C23×C53#12111yescharacteristic
174C22×D53—41no—
175Q8×C53—22no—
176Q8×C53—24no—
177Q8×C53—41no—
178Q8×C53—41no—
179Q8×C53C2×D814yes—
180D4×C53—41no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—81no—
185D4×C53—82no—
186C2×C4×C53—22no—
187C2×C4×C53—24no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—41no—
194C2×C4×C53—82no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53C2×D812yes—
199C2×C4×C53#13512yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C22⋊D53—82no—
204C2×Dic53—41no—
205D212—41no—
206D212—41no—
207D212—41no—
208D212—41no—
209D212—81no—
210C4×D53—82no—
211Dic106—41no—
212Dic106—41no—
213C8×C53—42no—
214#88322—21no—
215#88322—21no—
216#88322—21no—
217#88322—42no—
218#88322—42no—
219#88322#13712yescomplemented
220#19261—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53#46612yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C4×C53—21no—
232C4×C53—24no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—81no—
237C4×C53#46612yes—
238C4×C53C22×D811yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53C2×#13711yescharacteristic
241C4×C53#57511yescharacteristic
242Dic53—81no—
243#3169—531no—
244#3037—531no—
245#3037D5311yescharacteristic, complemented
246C2×C8×Q8—531no—
247#1682—532no—
248#1682—532no—
249#1674—532no—
250#1668—532no—
251#1567—531no—
252#1547—531no—
253#1547—531no—
254C2×C53—22no—
255C2×C53—161no—
256C2×C53#303711yescharacteristic
257C2×C53C2×#46611yescharacteristic
258C2×C53#306711yescharacteristic
259D53—161no—
260C2×#137—21no—
261C2×#137—1061no—
262#532—531no—
263#532C2×D5311yescharacteristic
264C2×C4×Q8C2×D5311yescharacteristic
265#500—531no—
266#493—531no—
267#486—531no—
268#486—531no—
269#466—1061no—
270#466—1061no—
271#464—22no—
272#464—1062no—
273#462—532no—
274#462C2×D5312yes—
275#460—532no—
276#460C2×D5312yes—
277C8×Q8—532no—
278C8×Q8—532no—
279C8×Q8—1061no—
280C8×Q8—1061no—
281C4×D8—1061no—
282C4×D8—1061no—
283#423—531no—
284#423C2×D5311yescharacteristic
285C2×C4⋊C8—531no—
286C2×C4⋊C8—1061no—
287#417—21no—
288#417—1061no—
289C2×C4×C8—531no—
290C2×C4×C8—1061no—
291#335—532no—
292#335—532no—
293#332—532no—
294#332—532no—
295#331—532no—
296#331—532no—
297C53#1926111yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—1061no—
300C22×Q8C22⋊D5311yescharacteristic
301#137—44no—
302#137—2124no—
303C2×Q16—22no—
304C2×Q16—1062no—
305C2×SD16—22no—
306C2×SD16—1062no—
307C2×D8—1061no—
308C2×D8—1061no—
309C2×M16—21no—
310C2×M16—1061no—
311C22×C8—531no—
312C22×C8—531no—
313C22×C8—1061no—
314C4⋊Q8—531no—
315C4⋊Q8C22×D5311yescharacteristic
316#127—531no—
317#127C22×D5311yescharacteristic
318#124—21no—
319#124—1061no—
320#121—21no—
321#121—21no—
322#121—1061no—
323#121—1061no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22⋊D5312yes—
327C4×Q8C22×D5312yes—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1061no—
331C4×D4—1061no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C22⋊D5311yescharacteristic
334C2×C42—21no—
335C2×C42C22×D5311yescharacteristic
336#107—1061no—
337#107—1061no—
338C4⋊C8—21no—
339C4⋊C8—532no—
340C4⋊C8—532no—
341C4⋊C8—1062no—
342C4⋊C8—1062no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8C22×D5312yes—
346#103—22no—
347#103—1062no—
348#102—22no—
349#102—1061no—
350#102—1061no—
351#102—1061no—
352#102—1061no—
353#102—1062no—
354C4×C8—531no—
355C4×C8—531no—
356C4×C8—531no—
357C4×C8—531no—
358C4×C8—1062no—
359C4×C8—1062no—
360#41—42no—
361#41—42no—
362#41—2122no—
363#41—2122no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—1061no—
368C2×Q8#1171992214yescomplemented
369C2×Q8C2×C22⋊D5312yes—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1061no—
375C2×D4—1061no—
376C2×D4—1061no—
377C2×D4—2121no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—2121no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—44no—
386Q16—2124no—
387SD16—44no—
388SD16—2124no—
389D8—2122no—
390D8—2122no—
391M16—42no—
392M16—2122no—
393C2×C8—22no—
394C2×C8—531no—
395C2×C8—531no—
396C2×C8—531no—
397C2×C8—531no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—2121no—
402C2×C8—2122no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1061no—
409C4⋊C4—1061no—
410C4⋊C4C2×C22⋊D5312yes—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C42—22no—
416C42—22no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—41no—
422C23—2121no—
423C23#1313318911yescharacteristic
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—2121no—
429Q8—2121no—
430Q8C2×#1171992214yes—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—82no—
437D4—4241no—
438D4—4242no—
439D4—2121no—
440D4—2121no—
441D4—2121no—
442D4—2121no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—4242no—
453C2×C4—2121no—
454C2×C4#1313318911yescharacteristic
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4C2×#1171992212yes—
458C2×C4#1313323712yes—
459C2×C4C2×D4×D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C2×#1171994711yescharacteristic
462C8—42no—
463C8—532no—
464C8—1061no—
465C8—1061no—
466C8—1061no—
467C8—2122no—
468C8—2122no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4241no—
475C22—4241no—
476C22—4241no—
477C22#1681085712yes—
478C22C2×#1313318911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—24no—
483C4—42no—
484C4—81no—
485C4—4241no—
486C4#1681085712yes—
487C4C22×#1171992211yescharacteristic
488C4C2×#1313323711yescharacteristic
489C4#137×D5311yescharacteristic
490C4#1681142211yescharacteristic
491C2—22no—
492C2—161no—
493C2—8481no—
494C2#2028418411yescharacteristic
495C2C2×#1681085711yescharacteristic
496C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3037C2
socleC22×C53C2×#121
prime radical#20285290C2
hypercenter#3037D53
hypocenterC53#19261
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. #20284187C2semidirect
3. C2×Q8×#88322C2semidirect
4. #20274161C2semidirect
5. #20274157C2semidirect
6. #20274107C2semidirect
7. #20274073C2semidirect
8. #20273473C2semidirect
9. #20273393C2semidirect
10. #20273392C2semidirect
11. C53×#3037C2semidirect
116. C2×Q8×C53D8semidirect
219. #88322#137semidirect
245. #3037D53semidirect
297. C53#19261semidirect
368. C2×Q8#11719922semidirect

Maximal subgroups

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

subgroupindex
#202852902
#202841872
C2×Q8×#883222
#202741612
#202741572
#202741072
#202740732
#202734732
#202733932
#202733922
C53×#30372
#1926153