Finite Group Databasebeta

← #27388744

Subgroups — #27388744

Order 13568.

1142 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
1C2×#16811252C211yesmaximal, characteristic
2#20277332C211yesmaximal, characteristic
3#20277070C211yesmaximal, characteristic
4#20275701C211yesmaximal, characteristic
5#20275700C211yesmaximal, characteristic
6#20275699C211yesmaximal, characteristic
7#20275698C214yesmaximal
8#20274446C214yesmaximal
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—
21C2×#13133173C2212yes—
22C2×#13133173C2212yes—
23C22×#11719919C2211yescharacteristic
24#16810520—22no—
25#16810518—22no—
26#16810483C2211yescharacteristic
27#16810481C2212yes—
28C4×#11719919—22no—
29#16810474—22no—
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—
59#13133173—24no—
60#13133173—28no—
61#13133173Q818yescomplemented
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C412yes—
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×#11719919—22no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919C2×C412yes—
69C2×#11719919C2×C412yes—
70C2×#11719919C2×C412yes—
71C2×#11719919C2×C412yes—
72C2×#11719919C2×C412yes—
73C2×#11719919C2×C412yes—
74C2×#11719919C2×C412yes—
75C2×#11719919D412yes—
76C2×#11719919D412yes—
77C2×#11719919D412yes—
78C2×#11719919D412yes—
79C2×C22⋊Dic53C2312yes—
80C2×C22⋊Dic53C2312yes—
81C2×C22⋊Dic53C2312yes—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53C2×C412yes—
87C2×C4×Dic53C2×C412yes—
88#13133124—21no—
89#13133124—21no—
90#13133124—22no—
91#13133124—22no—
92C22×Q8×C53C2×C411yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2×C411yescharacteristic
100C2×C4⋊C4×C53C2×C412yes—
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107C24×C53C2411yescharacteristic
108C2×Q8×C53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—81no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—81no—
114C22×C4×C53C22×C412yes—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×Dic53—22no—
123C22×Dic53—22no—
124C22×Dic53—81no—
125C22×Dic53C22×C412yes—
126C22×Dic53C22×C412yes—
127C22×Dic53C22×C412yes—
128C22×Dic53C22×C412yes—
129C22×Dic53C2×D412yes—
130C22×Dic53C2×D412yes—
131C22×Dic53C2×D412yes—
132C22×Dic53C2×D412yes—
133C22×Dic53C2×Q814yes—
134C22×Dic53#4114yes—
135C2×Dic106—21no—
136C2×Dic106—21no—
137C2×Dic106—24no—
138C2×Dic106—82no—
139C2×Dic106—81no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—81no—
145C22⋊C4×C53—22no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—24no—
149C22⋊C4×C53C2×Q814yes—
150C42×C53—81no—
151C22⋊Dic53—22no—
152C22⋊Dic53—22no—
153C22⋊Dic53—22no—
154C22⋊Dic53—24no—
155C22⋊Dic53C2×Q814yes—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—24no—
159C4⋊Dic53—81no—
160#11719919—22no—
161#11719919—22no—
162#11719919—24no—
163#11719919—24no—
164#11719919—24no—
165#11719919—24no—
166#11719919—28no—
167#11719919—44no—
168#11719919—82no—
169#11719919C22⋊C414yes—
170#11719919C22⋊C414yes—
171#11719919C22⋊C414yes—
172#11719919C22⋊C414yes—
173C4×Dic53—24no—
174C4×Dic53—24no—
175C4×Dic53—28no—
176C4×Dic53—44no—
177C4×Dic53—82no—
178C4×Dic53—81no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C22×Q812yes—
185C23×C53C2×#4112yes—
186Q8×C53—81no—
187Q8×C53—81no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53C2×C22⋊C411yescharacteristic
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53C2×C22⋊C411yescharacteristic
207C2×C4×C53C2×C22⋊C411yescharacteristic
208C2×Dic53—22no—
209C2×Dic53—22no—
210C2×Dic53—24no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53C2×C22⋊C414yes—
222C2×Dic53C2×C22⋊C414yes—
223C2×Dic53C4×Q818yes—
224C2×Dic53C22⋊Q814yes—
225C2×Dic53C22⋊Q814yes—
226C2×Dic53#12414yes—
227C2×Dic53C4⋊Q814yes—
228Dic106—84no—
229Dic106—81no—
230Dic106—81no—
231#21334—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—42no—
239C22×C53—42no—
240C22×C53C22×C22⋊C411yescharacteristic
241C22×C53C2×C4×Q812yes—
242C22×C53C2×C22⋊Q812yes—
243C22×C53C2×#12411yescharacteristic
244C22×C53C2×C4⋊Q811yescharacteristic
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—44no—
250C4×C53—42no—
251C4×C53—42no—
252C4×C53—82no—
253Dic53—24no—
254Dic53—44no—
255Dic53—42no—
256Dic53—42no—
257Dic53—82no—
258Dic53#39118yes—
259C2×#544—531no—
260#2316—531no—
261#2274—531no—
262#1996—531no—
263#1996—531no—
264#1996—534no—
265#1996—531no—
266#1996D5311yescharacteristic
267#1740—534no—
268C2×C53—22no—
269C2×C53—21no—
270C2×C53—22no—
271C2×C53—21no—
272C2×C53C2×#39111yescharacteristic
273C2×C53#288311yescharacteristic
274C2×C53#288411yescharacteristic
275#544—1062no—
276#544—1062no—
277#544—1062no—
278#544—1062no—
279C2×C4⋊Q8—532no—
280C2×C4⋊Q8—532no—
281C2×#125—532no—
282C2×#125—532no—
283C2×C22⋊Q8—531no—
284C2×C22⋊Q8—532no—
285C2×C22⋊Q8C2×D5311yescharacteristic
286C2×#117—532no—
287C2×#117—532no—
288C2×#117—532no—
289C22×C4⋊C4—531no—
290#389—1062no—
291#389—1062no—
292#380—531no—
293#380—532no—
294#380C2×D5311yescharacteristic
295C4×C4⋊C4—1062no—
296C4×C4⋊C4—1062no—
297C4×C22⋊C4—531no—
298C4×C22⋊C4—532no—
299C4×C22⋊C4C2×D5311yescharacteristic
300#342—534no—
301#342—534no—
302#342—534no—
303#342C2×D5314yes—
304C53#2133411yescharacteristic, complemented, Sylow
305C22×Q8—532no—
306C22×Q8—531no—
307C22×Q8C4×D5311yescharacteristic
308C23×C4—531no—
309C23×C4—531no—
310C23×C4C22×D5311yescharacteristic
311C4⋊Q8—1064no—
312C4⋊Q8—1064no—
313C4⋊Q8—1064no—
314C4⋊Q8—1064no—
315#125—1064no—
316#125—1064no—
317#125—1064no—
318#125—1064no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—1064no—
324C22⋊Q8—1064no—
325C22⋊Q8—1061no—
326C22⋊Q8—1061no—
327C22⋊Q8—1061no—
328C22⋊Q8—1061no—
329#117—538no—
330#117—1064no—
331#117—1064no—
332#117—1064no—
333#117—1068no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—532no—
336C2×C4⋊C4—532no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—532no—
339C2×C4⋊C4—532no—
340C2×C4⋊C4—532no—
341C2×C4⋊C4—532no—
342C2×C4⋊C4—532no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—531no—
345C2×C4⋊C4—532no—
346C2×C4⋊C4—532no—
347C2×C4⋊C4—532no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1062no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4C4×D5311yescharacteristic
353C2×C4⋊C4C4×D5312yes—
354C2×C22⋊C4—532no—
355C2×C22⋊C4—532no—
356C2×C22⋊C4—532no—
357C2×C22⋊C4C22×D5312yes—
358C2×C22⋊C4C22×D5312yes—
359C2×C22⋊C4C22×D5312yes—
360C2×C42—21no—
361C2×C42—532no—
362C2×C42—532no—
363C2×C42—1062no—
364C2×C42—1062no—
365C2×C42—1061no—
366C2×C42—1062no—
367#95—21no—
368#95—21no—
369#95—1062no—
370#95—1062no—
371#95—1061no—
372#95—1061no—
373C24C23×D5311yescharacteristic
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—81no—
377C2×Q8—1064no—
378C2×Q8—1061no—
379C2×Q8—1061no—
380C2×Q8—4241no—
381C2×Q8—4242no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—81no—
385C22×C4—532no—
386C22×C4—532no—
387C22×C4—532no—
388C22×C4—534no—
389C22×C4—534no—
390C22×C4—532no—
391C22×C4—532no—
392C22×C4—532no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—1062no—
396C22×C4—1062no—
397C22×C4—4241no—
398C22×C4C2×C4×D5312yes—
399C22×C4C2×C4×D5312yes—
400C22×C4C2×C4×D5311yescharacteristic
401C22×C4C2×C4×D5311yescharacteristic
402C22×C4D4×D5311yescharacteristic
403C22×C4D4×D5311yescharacteristic
404C22×C4D4×D5311yescharacteristic
405C22×C4D4×D5311yescharacteristic
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—81no—
411C4⋊C4—534no—
412C4⋊C4—534no—
413C4⋊C4—534no—
414C4⋊C4—534no—
415C4⋊C4—1064no—
416C4⋊C4—1064no—
417C4⋊C4—1064no—
418C4⋊C4—1064no—
419C4⋊C4—1064no—
420C4⋊C4—1068no—
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—1062no—
424C4⋊C4—1062no—
425C4⋊C4—4242no—
426C4⋊C4—4241no—
427C4⋊C4—2124no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—534no—
433C22⋊C4—1062no—
434C22⋊C4—1062no—
435C22⋊C4—1062no—
436C22⋊C4—1064no—
437C22⋊C4Q8×D5314yes—
438C42—81no—
439C42—1064no—
440C42—1064no—
441C42—1068no—
442C42—4241no—
443C42—4242no—
444C42—2124no—
445C23—41no—
446C23—41no—
447C23C22×C4×D5311yescharacteristic
448C23C2×D4×D5311yescharacteristic
449C23C2×D4×D5311yescharacteristic
450C23C2×Q8×D5312yes—
451C23#41×D5312yes—
452Q8—81no—
453Q8—81no—
454Q8—4241no—
455Q8—4241no—
456Q8—4244no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—534no—
474C2×C4—538no—
475C2×C4—534no—
476C2×C4—534no—
477C2×C4—534no—
478C2×C4—534no—
479C2×C4—534no—
480C2×C4—1062no—
481C2×C4—1062no—
482C2×C4—1064no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4—2122no—
486C2×C4—2122no—
487C2×C4—2122no—
488C2×C4—2122no—
489C2×C4—2122no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4C22⋊C4×D5311yescharacteristic
494C2×C4C22⋊C4×D5311yescharacteristic
495C2×C4C22⋊C4×D5311yescharacteristic
496C2×C4C22⋊C4×D5311yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22C2×C22⋊C4×D5311yescharacteristic
506C22C4×Q8×D5312yes—
507C22C22⋊Q8×D5312yes—
508C22#124×D5311yescharacteristic
509C22C4⋊Q8×D5311yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—538no—
519C4—1064no—
520C4—4242no—
521C4—2124no—
522C4—2122no—
523C4—2122no—
524C2—22no—
525C2—21no—
526C2—22no—
527C2—21no—
528C2#391×D5311yescharacteristic
529C2#2028317311yescharacteristic
530C2#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#21334
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
61. #13133173Q8semidirect
304. C53#21334semidirect

Maximal subgroups

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

subgroupindex
C2×#168112522
#202773322
#202770702
#202757012
#202757002
#202756992
#202756982
#202744462
C53×#19962
#2133453