Finite Group Databasebeta

← #27393546

Subgroups — #27393546

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811252C211yesmaximal, characteristic
2#20277317C211yesmaximal, characteristic
3#20277315C211yesmaximal, characteristic
4#20275701C211yesmaximal, characteristic
5#20275700C211yesmaximal, characteristic
6#20275699C211yesmaximal, characteristic
7#20275698C214yesmaximal
8#20274791C214yesmaximal
9C53×#1996C211yesmaximal, characteristic
10#16811252—22no—
11#16811252—22no—
12#16811252—22no—
13#16811252—22no—
14C2×#13133216C412yescomplemented
15C2×#13133215C412yescomplemented
16C2×#13133207C412yescomplemented
17C2×#13133207C412yescomplemented
18C2×#13133196C2211yescharacteristic
19C2×#13133194C2212yes—
20C2×#13133173C2212yes—
21C22×#11719919C2211yescharacteristic
22#16810510—22no—
23#16810507—22no—
24#16810505—22no—
25#16810504—22no—
26#16810484C2212yes—
27#16810484C2212yes—
28#16810483C2211yescharacteristic
29#16810481C2212yes—
30C22⋊C4×Dic53C2212yes—
31C4×C22⋊Dic53C2211yescharacteristic
32#16810390C2214yes—
33#16810389C2214yes—
34#16810389C2214yes—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C4×C22⋊C4×C53C2211yescharacteristic
38C53×#342C2214yes—
39C22×Dic106C2×C412yes—
40C22×Dic106C2×C411yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43#13133216—24no—
44#13133216—24no—
45#13133215—24no—
46#13133215—24no—
47#13133207—24no—
48#13133207—24no—
49#13133207—24no—
50#13133207—24no—
51#13133196—21no—
52#13133196—21no—
53#13133196—21no—
54#13133196—21no—
55#13133194—24no—
56#13133194—24no—
57#13133173—24no—
58#13133173—24no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53C2×C412yes—
61C2×C4⋊Dic53C2×C411yescharacteristic
62C2×#11719919—22no—
63C2×#11719919C2×C412yes—
64C2×#11719919C2×C412yes—
65C2×#11719919C2×C412yes—
66C2×#11719919C2×C412yes—
67C2×#11719919C2×C412yes—
68C2×#11719919C2×C412yes—
69C2×#11719919C2×C412yes—
70C2×#11719919D412yes—
71C2×#11719919D412yes—
72C2×#11719919D412yes—
73C2×#11719919D412yes—
74C2×C22⋊Dic53C2312yes—
75C2×C22⋊Dic53C2312yes—
76C2×C22⋊Dic53C2312yes—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—22no—
79C2×C4×Dic53C2×C412yes—
80C2×C4×Dic53C2×C412yes—
81#13133124—21no—
82#13133124—21no—
83#13133124—22no—
84#13133124—24no—
85#13133124—22no—
86#13133124—24no—
87C22×Q8×C53C2×C411yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53C2×C411yescharacteristic
95C2×C4⋊C4×C53C2×C412yes—
96C2×C22⋊C4×C53C2312yes—
97C2×C22⋊C4×C53C2312yes—
98C2×C22⋊C4×C53C2312yes—
99C2×C42×C53—21no—
100#95×C53—21no—
101#95×C53—21no—
102C24×C53C2411yescharacteristic
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—81no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—81no—
109C22×C4×C53C22×C412yes—
110C22×C4×C53C22×C412yes—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—81no—
122C22×Dic53C22×C412yes—
123C22×Dic53C22×C412yes—
124C22×Dic53C22×C412yes—
125C22×Dic53C22×C412yes—
126C22×Dic53C2×D412yes—
127C22×Dic53C2×D412yes—
128C22×Dic53C2×D412yes—
129C22×Dic53C2×D412yes—
130C2×Dic106—21no—
131C2×Dic106—21no—
132C2×Dic106—24no—
133C2×Dic106—82no—
134C2×Dic106—81no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—81no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—22no—
145C42×C53—81no—
146C22⋊Dic53—22no—
147C22⋊Dic53—22no—
148C22⋊Dic53—22no—
149C22⋊Dic53—22no—
150C22⋊Dic53—24no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—24no—
154C4⋊Dic53—81no—
155#11719919—22no—
156#11719919—22no—
157#11719919—24no—
158#11719919—24no—
159#11719919—24no—
160#11719919—24no—
161#11719919—82no—
162#11719919C22⋊C414yes—
163#11719919C22⋊C414yes—
164#11719919C22⋊C414yes—
165#11719919C22⋊C414yes—
166C4×Dic53—24no—
167C4×Dic53—24no—
168C4×Dic53—82no—
169C4×Dic53—81no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53C23×C411yescharacteristic
173C23×C53C22×D411yescharacteristic
174C23×C53C22×D411yescharacteristic
175C23×C532−1+412yes—
176C23×C532−1+412yes—
177Q8×C53—81no—
178Q8×C53—81no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53C2×C22⋊C411yescharacteristic
196C2×C4×C53C2×C22⋊C411yescharacteristic
197C2×C4×C53C2×C22⋊C411yescharacteristic
198C2×C4×C53C2×C22⋊C411yescharacteristic
199C2×Dic53—24no—
200C2×Dic53—22no—
201C2×Dic53—24no—
202C2×Dic53—22no—
203C2×Dic53—22no—
204C2×Dic53—22no—
205C2×Dic53—42no—
206C2×Dic53—42no—
207C2×Dic53—42no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53C2×C22⋊C414yes—
217C2×Dic53C2×C22⋊C414yes—
218Dic106—84no—
219Dic106—81no—
220Dic106—81no—
221#21828—531nomaximal, Sylow
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53C22×C22⋊C411yescharacteristic
231C22×C53#51912yes—
232C22×C53#53611yescharacteristic
233C22×C53#53612yes—
234C22×C53#53611yescharacteristic
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—42no—
240C4×C53—42no—
241C4×C53—82no—
242C4×C53—82no—
243Dic53—24no—
244Dic53—24no—
245Dic53—42no—
246Dic53—42no—
247Dic53—82no—
248Dic53—82no—
249C2×#544—531no—
250#2313—531no—
251#2313—531no—
252#1996—531no—
253#1996—531no—
254#1996—534no—
255#1996—531no—
256#1996D5311yescharacteristic
257#1820—534no—
258C2×C53—21no—
259C2×C53—22no—
260C2×C53—21no—
261C2×C53—22no—
262C2×C53#231011yescharacteristic
263C2×C53#288311yescharacteristic
264C2×C53#288411yescharacteristic
265#544—1062no—
266#544—1062no—
267#544—1062no—
268#544—1062no—
269C2×C4⋊Q8—532no—
270C2×C4⋊Q8—532no—
271C2×#125—532no—
272C2×#125—532no—
273C2×C22⋊Q8—531no—
274C2×C22⋊Q8—532no—
275C2×C22⋊Q8C2×D5311yescharacteristic
276C2×#117—532no—
277C22×C4⋊C4—531no—
278#387—1062no—
279#387—1062no—
280#387—1062no—
281#387—1062no—
282#381—532no—
283#381—532no—
284#380—531no—
285#380—532no—
286#380C2×D5311yescharacteristic
287C4×C22⋊C4—531no—
288C4×C22⋊C4—532no—
289C4×C22⋊C4C2×D5311yescharacteristic
290#342—534no—
291#342—534no—
292#342—534no—
293#342C2×D5314yes—
294C53#2182811yescharacteristic, complemented, Sylow
295C22×Q8—532no—
296C22×Q8—531no—
297C22×Q8C4×D5311yescharacteristic
298C23×C4—531no—
299C23×C4—531no—
300C23×C4C22×D5311yescharacteristic
301C4⋊Q8—1064no—
302C4⋊Q8—1064no—
303C4⋊Q8—1064no—
304C4⋊Q8—1064no—
305#125—1064no—
306#125—1064no—
307#125—1064no—
308#125—1064no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—1064no—
314C22⋊Q8—1064no—
315C22⋊Q8—1061no—
316C22⋊Q8—1061no—
317C22⋊Q8—1061no—
318C22⋊Q8—1061no—
319#117—1064no—
320#117—1064no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—532no—
323C2×C4⋊C4—532no—
324C2×C4⋊C4—532no—
325C2×C4⋊C4—532no—
326C2×C4⋊C4—532no—
327C2×C4⋊C4—532no—
328C2×C4⋊C4—532no—
329C2×C4⋊C4—532no—
330C2×C4⋊C4—532no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—532no—
333C2×C4⋊C4—532no—
334C2×C4⋊C4—532no—
335C2×C4⋊C4—1062no—
336C2×C4⋊C4—1061no—
337C2×C4⋊C4C4×D5311yescharacteristic
338C2×C4⋊C4C4×D5312yes—
339C2×C22⋊C4—532no—
340C2×C22⋊C4—532no—
341C2×C22⋊C4—532no—
342C2×C22⋊C4C22×D5312yes—
343C2×C22⋊C4C22×D5312yes—
344C2×C22⋊C4C22×D5312yes—
345C2×C42—21no—
346C2×C42—532no—
347C2×C42—532no—
348C2×C42—1062no—
349C2×C42—1061no—
350#95—21no—
351#95—21no—
352#95—1062no—
353#95—1062no—
354#95—1064no—
355#95—1064no—
356#95—1061no—
357#95—1061no—
358C24C23×D5311yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—81no—
362C2×Q8—1064no—
363C2×Q8—1061no—
364C2×Q8—1061no—
365C2×Q8—4241no—
366C2×Q8—4242no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—81no—
370C22×C4—532no—
371C22×C4—532no—
372C22×C4—532no—
373C22×C4—532no—
374C22×C4—532no—
375C22×C4—532no—
376C22×C4—532no—
377C22×C4—532no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—4241no—
383C22×C4C2×C4×D5312yes—
384C22×C4C2×C4×D5312yes—
385C22×C4C2×C4×D5311yescharacteristic
386C22×C4C2×C4×D5311yescharacteristic
387C22×C4D4×D5311yescharacteristic
388C22×C4D4×D5311yescharacteristic
389C22×C4D4×D5311yescharacteristic
390C22×C4D4×D5311yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—81no—
396C4⋊C4—534no—
397C4⋊C4—534no—
398C4⋊C4—534no—
399C4⋊C4—534no—
400C4⋊C4—1064no—
401C4⋊C4—1064no—
402C4⋊C4—1064no—
403C4⋊C4—1064no—
404C4⋊C4—1064no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—1062no—
408C4⋊C4—1062no—
409C4⋊C4—4242no—
410C4⋊C4—4241no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—1062no—
417C22⋊C4—1062no—
418C22⋊C4—1062no—
419C22⋊C4—1062no—
420C22⋊C4—1064no—
421C42—81no—
422C42—1064no—
423C42—1064no—
424C42—4241no—
425C42—4242no—
426C23—41no—
427C23—41no—
428C23C22×C4×D5311yescharacteristic
429C23C2×D4×D5311yescharacteristic
430C23C2×D4×D5311yescharacteristic
431C23#1313326412yes—
432C23#1313326512yes—
433Q8—81no—
434Q8—81no—
435Q8—4241no—
436Q8—4241no—
437Q8—4244no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—534no—
455C2×C4—534no—
456C2×C4—1064no—
457C2×C4—1062no—
458C2×C4—1064no—
459C2×C4—1062no—
460C2×C4—1062no—
461C2×C4—1062no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4—2122no—
467C2×C4—2122no—
468C2×C4—2122no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4—2122no—
473C2×C4C22⋊C4×D5311yescharacteristic
474C2×C4C22⋊C4×D5311yescharacteristic
475C2×C4C22⋊C4×D5311yescharacteristic
476C2×C4C22⋊C4×D5311yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D5311yescharacteristic
486C22#1681109912yes—
487C22#1681119311yescharacteristic
488C22#1681119612yes—
489C22#1681119811yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—82no—
498C4—1064no—
499C4—1064no—
500C4—4242no—
501C4—4242no—
502C4—2122no—
503C4—2122no—
504C2—21no—
505C2—22no—
506C2—21no—
507C2—22no—
508C2#2027729811yescharacteristic
509C2#2028317311yescharacteristic
510C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1996C2
socleC22×C53C22×C22⋊C4
prime radical, cosocleC24×C53C24
hypercenter#1996D53
hypocenterC53#21828
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#13133216C4semidirect
15. C2×#13133215C4semidirect
16. C2×#13133207C4semidirect
17. C2×#13133207C4semidirect
294. C53#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#168112522
#202773172
#202773152
#202757012
#202757002
#202756992
#202756982
#202747912
C53×#19962
#2182853