Finite Group Databasebeta

← #27393536

Subgroups — #27393536

Order 13568.

1474 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133194C211yesmaximal, characteristic
2#20277093C212yesmaximal
3#20275698C218yesmaximal, complemented
4#20274789C211yesmaximal, characteristic
5#20274787C212yesmaximal
6C53×#1819C211yesmaximal, characteristic
7C23×Dic106C411yescharacteristic, complemented
8C24×Dic53C2211yescharacteristic
9C2×#13133194—22no—
10C2×#13133194—22no—
11C2×#13133194—22no—
12C2×#13133194—22no—
13C2×#13133194C414yescomplemented
14C2×#13133194C2214yes—
15C2×#13133173—22no—
16C22×C4⋊Dic53C411yescharacteristic, complemented
17C22×#11719919C411yescharacteristic, complemented
18C22×#11719919C411yescharacteristic, complemented
19C22×C22⋊Dic53C2211yescharacteristic
20#16810484—22no—
21#16810481—22no—
22#16810481C2214yes—
23#16810479C2212yes—
24C22⋊C4×Dic53—22no—
25C22⋊C4×Dic53C2214yes—
26#16810390C2214yes—
27#16810389C2218yes—
28C22×C22⋊C4×C53C2211yescharacteristic
29C53×#379C2212yes—
30C53×#342C2214yes—
31C22×Dic106—21no—
32C22×Dic106—22no—
33C22×Dic106—21no—
34C22×Dic106—44no—
35C22×Dic106C2×C414yes—
36C23×Dic53—44no—
37C23×Dic53C2×C411yescharacteristic
38C23×Dic53C2×C411yescharacteristic
39C23×Dic53C2×C411yescharacteristic
40C23×Dic53C2×C411yescharacteristic
41C23×Dic53C2×C412yes—
42C23×Dic53D412yes—
43C23×Dic53D412yes—
44C23×Dic53D412yes—
45C23×Dic53D412yes—
46C23×Dic53C2312yes—
47#13133194—24no—
48#13133194—24no—
49#13133194—24no—
50#13133194—24no—
51#13133194—24no—
52#13133194—24no—
53#13133194—24no—
54#13133194—24no—
55#13133173—48no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53C2×C414yes—
60C2×#11719919—22no—
61C2×#11719919—21no—
62C2×#11719919—22no—
63C2×#11719919—21no—
64C2×#11719919—21no—
65C2×#11719919—21no—
66C2×#11719919—24no—
67C2×#11719919C2×C414yes—
68C2×#11719919C2×C414yes—
69C2×C22⋊Dic53—22no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—22no—
72C2×C22⋊Dic53—22no—
73C2×C22⋊Dic53C2×C412yes—
74C2×C22⋊Dic53C2314yes—
75C2×C22⋊Dic53C2312yes—
76C2×C4×Dic53—24no—
77#13133124—24no—
78#13133124—24no—
79C25×C53C2311yescharacteristic
80C23×C4×C53C2×C411yescharacteristic
81C23×C4×C53C2×C411yescharacteristic
82C2×C22⋊C4×C53—22no—
83C2×C22⋊C4×C53—22no—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53C2×C412yes—
87C2×C22⋊C4×C53C2314yes—
88C2×C22⋊C4×C53C2312yes—
89C24×C53—24no—
90C24×C53—42no—
91C24×C53—42no—
92C24×C53C22×C411yescharacteristic
93C24×C53C22×C411yescharacteristic
94C24×C53C2×D412yes—
95C24×C53C2×D412yes—
96C24×C53C2411yescharacteristic
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—24no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C22×C412yes—
105C22×C4×C53C22×C412yes—
106C22×Dic53—28no—
107C22×Dic53—24no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—24no—
113C22×Dic53—44no—
114C22×Dic53—44no—
115C22×Dic53—44no—
116C22×Dic53—44no—
117C22×Dic53—42no—
118C22×Dic53—44no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C22×Dic53C22⋊C412yes—
122C22×Dic53C22⋊C411yescharacteristic
123C22×Dic53C22⋊C412yes—
124C22×Dic53C22⋊C412yes—
125C22×Dic53C22⋊C412yes—
126C22×Dic53C22⋊C411yescharacteristic
127C22×Dic53C22⋊C411yescharacteristic
128C22×Dic53C22⋊C412yes—
129C22×Dic53C22⋊C412yes—
130C22×Dic53C22⋊C411yescharacteristic
131C22×Dic53C22×C412yes—
132C22×Dic53C22×C412yes—
133C22×Dic53C22×C412yes—
134C22×Dic53C22×C412yes—
135C22×Dic53C2×D412yes—
136C22×Dic53C2×D412yes—
137C22×Dic53C2×D412yes—
138C22×Dic53C2×D412yes—
139C2×Dic106—24no—
140C2×Dic106—24no—
141C2×Dic106—44no—
142C2×Dic106—48no—
143C2×Dic106—44no—
144C2×Dic106—48no—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—24no—
147C22⋊C4×C53—28no—
148C22⋊C4×C53—44no—
149C22⋊C4×C53—44no—
150C22⋊Dic53—24no—
151C22⋊Dic53—24no—
152C22⋊Dic53—28no—
153C22⋊Dic53—44no—
154C22⋊Dic53—44no—
155C4⋊Dic53—24no—
156C4⋊Dic53—24no—
157#11719919—24no—
158#11719919—24no—
159#11719919—24no—
160#11719919—24no—
161#11719919—48no—
162C4×Dic53—48no—
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—24no—
169C23×C53—24no—
170C23×C53—24no—
171C23×C53—41no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—42no—
176C23×C53—41no—
177C23×C53—416no—
178C23×C53C2×C22⋊C412yes—
179C23×C53C2×C22⋊C411yescharacteristic
180C23×C53C2×C22⋊C411yescharacteristic
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D412yes—
183C23×C532−1+414yes—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—44no—
193C2×C4×C53—44no—
194C2×C4×C53—44no—
195C2×Dic53—22no—
196C2×Dic53—24no—
197C2×Dic53—24no—
198C2×Dic53—24no—
199C2×Dic53—24no—
200C2×Dic53—24no—
201C2×Dic53—24no—
202C2×Dic53—24no—
203C2×Dic53—22no—
204C2×Dic53—22no—
205C2×Dic53—42no—
206C2×Dic53—42no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—44no—
219C2×Dic53—44no—
220C2×Dic53—44no—
221C2×Dic53#9914yes—
222C2×Dic53C2×C22⋊C412yes—
223C2×Dic53C2×C22⋊C412yes—
224C2×Dic53C2×C22⋊C412yes—
225C2×Dic53C2×C22⋊C412yes—
226Dic106—48no—
227Dic106—48no—
228#21825—531nomaximal, Sylow
229C22×C53—28no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—22no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—44no—
242C22×C53—48no—
243C22×C53—48no—
244C22×C53—44no—
245C22×C53C2×#9912yes—
246C22×C53#41012yes—
247C22×C53C22×C22⋊C411yescharacteristic
248C22×C53#51912yes—
249C22×C53#53614yes—
250C4×C53—42no—
251C4×C53—42no—
252C4×C53—84no—
253Dic53—22no—
254Dic53—22no—
255Dic53—22no—
256Dic53—42no—
257Dic53—42no—
258Dic53—84no—
259Dic53C2×#9914yes—
260C22×C22⋊Q8—531no—
261#2280—532no—
262#1996—538no—
263#1819—532no—
264#1819—531no—
265#1819D5311yescharacteristic
266C2×C53—24no—
267C2×C53—22no—
268C2×C53—22no—
269C2×C53—22no—
270C2×C53—42no—
271C2×C53#231011yescharacteristic
272C2×C53C22×#9911yescharacteristic
273C2×C53C2×#41011yescharacteristic
274C23×Q8—531no—
275C24×C4—531no—
276C2×C22⋊Q8—534no—
277C2×C22⋊Q8—534no—
278C2×C22⋊Q8—1062no—
279C2×C22⋊Q8—1062no—
280C2×C22⋊Q8—1062no—
281C2×C22⋊Q8—1062no—
282C2×#117—1062no—
283C22×C4⋊C4—531no—
284C22×C4⋊C4—531no—
285C22×C4⋊C4—531no—
286C22×C22⋊C4—531no—
287C22×C22⋊C4C2×D5311yescharacteristic
288#381—1062no—
289#380—534no—
290#380—1062no—
291#379—532no—
292#379C2×D5312yes—
293C4×C22⋊C4—534no—
294C4×C22⋊C4—1062no—
295#342—538no—
296#342—534no—
297#342C2×D5314yes—
298C53#2182511yescharacteristic, complemented, Sylow
299C25C22×D5311yescharacteristic
300C22×Q8—534no—
301C22×Q8—1061no—
302C22×Q8—1062no—
303C22×Q8—1061no—
304C22×Q8—2124no—
305C23×C4—531no—
306C23×C4—532no—
307C23×C4—531no—
308C23×C4—532no—
309C23×C4—532no—
310C23×C4—532no—
311C23×C4—532no—
312C23×C4—532no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—2124no—
316C23×C4C4×D5311yescharacteristic
317C23×C4C4×D5311yescharacteristic
318C22⋊Q8—1064no—
319C22⋊Q8—1064no—
320C22⋊Q8—1064no—
321C22⋊Q8—1064no—
322C22⋊Q8—1064no—
323C22⋊Q8—1064no—
324C22⋊Q8—1064no—
325C22⋊Q8—1064no—
326#117—2128no—
327C2×C4⋊C4—534no—
328C2×C4⋊C4—534no—
329C2×C4⋊C4—534no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4—1062no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4—1062no—
334C2×C4⋊C4—1061no—
335C2×C4⋊C4—1061no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4—1061no—
339C2×C4⋊C4—1064no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—532no—
345C2×C22⋊C4—532no—
346C2×C22⋊C4—534no—
347C2×C22⋊C4—1062no—
348C2×C22⋊C4—1062no—
349C2×C22⋊C4—1062no—
350C2×C22⋊C4—1062no—
351C2×C22⋊C4C4×D5312yes—
352C2×C22⋊C4C22×D5314yes—
353C2×C22⋊C4C22×D5312yes—
354C2×C42—1064no—
355#95—1064no—
356#95—1064no—
357C24—24no—
358C24—42no—
359C24—42no—
360C24C2×C4×D5311yescharacteristic
361C24C2×C4×D5311yescharacteristic
362C24D4×D5312yes—
363C24D4×D5312yes—
364C24C23×D5311yescharacteristic
365C2×Q8—1064no—
366C2×Q8—1064no—
367C2×Q8—2124no—
368C2×Q8—2128no—
369C2×Q8—2128no—
370C2×Q8—2124no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—531no—
379C22×C4—532no—
380C22×C4—532no—
381C22×C4—531no—
382C22×C4—532no—
383C22×C4—532no—
384C22×C4—532no—
385C22×C4—532no—
386C22×C4—532no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—532no—
390C22×C4—532no—
391C22×C4—532no—
392C22×C4—532no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—532no—
396C22×C4—1068no—
397C22×C4—1064no—
398C22×C4—1061no—
399C22×C4—1061no—
400C22×C4—1061no—
401C22×C4—1061no—
402C22×C4—1064no—
403C22×C4—2124no—
404C22×C4—2124no—
405C22×C4—2124no—
406C22×C4—2124no—
407C22×C4—2122no—
408C22×C4—2124no—
409C22×C4—2122no—
410C22×C4—2122no—
411C22×C4C2×C4×D5312yes—
412C22×C4C2×C4×D5312yes—
413C4⋊C4—1064no—
414C4⋊C4—1064no—
415C4⋊C4—1064no—
416C4⋊C4—1064no—
417C4⋊C4—1064no—
418C4⋊C4—1064no—
419C4⋊C4—2128no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—1064no—
426C22⋊C4—1064no—
427C22⋊C4—1068no—
428C22⋊C4—2124no—
429C22⋊C4—2124no—
430C42—2128no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—416no—
446C23C22⋊C4×D5312yes—
447C23C22⋊C4×D5311yescharacteristic
448C23C22⋊C4×D5311yescharacteristic
449C23C22×C4×D5311yescharacteristic
450C23C2×D4×D5312yes—
451C23#1313326414yes—
452Q8—2128no—
453Q8—2128no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—532no—
466C2×C4—534no—
467C2×C4—532no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—1064no—
471C2×C4—1064no—
472C2×C4—1064no—
473C2×C4—1064no—
474C2×C4—1064no—
475C2×C4—1064no—
476C2×C4—1064no—
477C2×C4—1062no—
478C2×C4—1062no—
479C2×C4—1062no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4—2122no—
486C2×C4—2122no—
487C2×C4—2124no—
488C2×C4—2124no—
489C2×C4—2124no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C22—28no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—48no—
510C22—48no—
511C22—44no—
512C22#99×D5312yes—
513C22#1681059512yes—
514C22C2×C22⋊C4×D5311yescharacteristic
515C22#1681110012yes—
516C22#1681119514yes—
517C4—42no—
518C4—42no—
519C4—84no—
520C4—534no—
521C4—1062no—
522C4—1062no—
523C4—1062no—
524C4—4244no—
525C4—2122no—
526C4—2122no—
527C2—24no—
528C2—22no—
529C2—22no—
530C2—22no—
531C2—42no—
532C2#2027730011yescharacteristic
533C2C2×#99×D5311yescharacteristic
534C2C2×#1681059511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1819C2
socleC22×C53C22×C22⋊C4
prime radicalC25×C53C23
cosocleC24×C53C24
hypercenter#1819D53
hypocenterC53#21825
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20275698C2semidirect
7. C23×Dic106C4semidirect
13. C2×#13133194C4semidirect
16. C22×C4⋊Dic53C4semidirect
17. C22×#11719919C4semidirect
18. C22×#11719919C4semidirect
298. C53#21825semidirect

Maximal subgroups

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

subgroupindex
C22×#131331942
#202770932
#202756982
#202747892
#202747872
C53×#18192
#2182553