Finite Group Databasebeta

← #27350183

Subgroups — #27350183

Order 13568.

1386 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133196C211yesmaximal, characteristic
2C2×#16810587C211yesmaximal, characteristic
3#20273080C214yesmaximal, complemented
4#20273078C214yesmaximal, complemented
5#20273044C214yesmaximal, complemented
6C2×C53×#407C211yesmaximal, characteristic
7C23×Dic106C411yescharacteristic
8C22×#11719917—21no—
9C2×#13133196—21no—
10C2×#13133196—21no—
11C2×#13133196—21no—
12C2×#13133196—21no—
13C2×#13133196—21no—
14C2×#13133196—21no—
15C2×#13133196—21no—
16C2×#13133196—21no—
17C2×#13133196C412yes—
18C2×#13133196C412yes—
19C2×#13133196C2212yes—
20C2×#13133196C2212yes—
21C22×C4⋊Dic53C411yescharacteristic
22C22×#11719919C2212yes—
23C22×C22⋊Dic53C412yes—
24#16810587—21no—
25#16810587—21no—
26#16810587C2212yes—
27#16810587C2212yes—
28C2×C22⋊#88322—21no—
29C2×C22⋊#88322C2212yes—
30#16810348C22116yescomplemented
31C24×C4×C53C2211yescharacteristic
32C22×M16×C53—21no—
33C53×#407—21no—
34C53×#407—21no—
35C53×#407C2212yes—
36C53×#407C2212yes—
37C2×C22⋊C8×C53—21no—
38C2×C22⋊C8×C53C2212yes—
39C22×Dic106—21no—
40C22×Dic106—21no—
41C22×Dic106—21no—
42C22×Dic106—21no—
43C22×Dic106—44no—
44C22×Dic106C2×C412yes—
45C22×Dic106C2×C412yes—
46C23×Dic53C2×C412yes—
47C23×Dic53C2×C412yes—
48C2×#11719917—22no—
49C2×#11719917—22no—
50C2×#11719917—44no—
51C22×#88322—22no—
52#13133196—22no—
53#13133196—22no—
54#13133196—22no—
55#13133196—22no—
56#13133196—22no—
57#13133196—22no—
58#13133196—22no—
59#13133196—22no—
60#13133196—22no—
61#13133196—22no—
62#13133196—22no—
63#13133196—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—22no—
67#13133196—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2×C412yes—
73C2×C4⋊Dic53C2×C412yes—
74C2×#11719919—22no—
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919C2×C414yes—
79C2×#11719919C2314yes—
80C2×C22⋊Dic53—22no—
81C2×C22⋊Dic53—22no—
82C2×C22⋊Dic53—22no—
83C2×C22⋊Dic53—22no—
84C2×C22⋊Dic53C2×C414yes—
85C2×C22⋊Dic53C2×C414yes—
86C22⋊#88322—22no—
87C22⋊#88322—22no—
88C22⋊#88322—22no—
89C22⋊#88322C2314yes—
90C25×C53C2×C411yescharacteristic
91C23×C4×C53—44no—
92C23×C4×C53C2×C411yescharacteristic
93C23×C4×C53C2×C411yescharacteristic
94C23×C4×C53C2×C411yescharacteristic
95C23×C4×C53D411yescharacteristic
96C23×C4×C53D411yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99C23×C4×C53D411yescharacteristic
100C23×C4×C53D411yescharacteristic
101C23×C4×C53D411yescharacteristic
102C23×C4×C53D411yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53—44no—
109C22×C8×C53—22no—
110C22⋊C8×C53—22no—
111C22⋊C8×C53—22no—
112C22⋊C8×C53—22no—
113C22⋊C8×C53C2314yes—
114C24×C53—24no—
115C24×C53—42no—
116C24×C53—42no—
117C24×C53C22⋊C411yescharacteristic
118C24×C53C22⋊C411yescharacteristic
119C24×C53C22⋊C411yescharacteristic
120C24×C53C22⋊C411yescharacteristic
121C24×C53C22×C411yescharacteristic
122C24×C53C22×C411yescharacteristic
123C24×C53C22×C411yescharacteristic
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—24no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22⋊C411yescharacteristic
140C22×C4×C53C22⋊C411yescharacteristic
141C22×C4×C53C22⋊C411yescharacteristic
142C22×C4×C53C22⋊C411yescharacteristic
143C22×C4×C53C22⋊C411yescharacteristic
144C22×C4×C53C22⋊C411yescharacteristic
145C22×C4×C53C22⋊C411yescharacteristic
146C22×C4×C53C22⋊C411yescharacteristic
147C22×C4×C53C22⋊C411yescharacteristic
148C22×C4×C53C22⋊C411yescharacteristic
149C22×C4×C53C22⋊C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2411yescharacteristic
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—44no—
171C22×Dic53—44no—
172C22×Dic53C22×C414yes—
173C22×Dic53C22×C414yes—
174C2×Dic106—22no—
175C2×Dic106—22no—
176C2×Dic106—22no—
177C2×Dic106—22no—
178C2×Dic106—44no—
179C2×Dic106—44no—
180C2×Dic106—44no—
181C2×Dic106—44no—
182C2×Dic106—44no—
183C2×Dic106—44no—
184M16×C53—44no—
185M16×C53—44no—
186C2×C8×C53—24no—
187C2×C8×C53—44no—
188C22⋊Dic53—24no—
189C22⋊Dic53—24no—
190C22⋊Dic53—24no—
191C22⋊Dic53—24no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196#11719919—24no—
197#11719919—24no—
198#11719919—24no—
199#11719919—24no—
200#11719917—44no—
201#11719917—44no—
202C2×#88322—24no—
203C2×#88322—44no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—416no—
225C23×C53#10114yes—
226C23×C53C2×C22⋊C411yescharacteristic
227C23×C53C2×C22⋊C411yescharacteristic
228C23×C53C2×C22⋊C411yescharacteristic
229C23×C53C2×C22⋊C411yescharacteristic
230C23×C53C2×C22⋊C411yescharacteristic
231C23×C53C2×C22⋊C411yescharacteristic
232C23×C53C23×C411yescharacteristic
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53#9914yes—
261C2×C4×C53C2×C22⋊C411yescharacteristic
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C2×C22⋊C411yescharacteristic
264C2×C4×C53C2×C22⋊C411yescharacteristic
265C2×C4×C53C2×C22⋊C411yescharacteristic
266C2×C4×C53C2×C22⋊C411yescharacteristic
267C2×C4×C53C22×D411yescharacteristic
268C2×C4×C53C22×D411yescharacteristic
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—28no—
272C2×Dic53—44no—
273C2×Dic53—44no—
274C2×Dic53—44no—
275C2×Dic53—44no—
276C2×Dic53—44no—
277Dic106—44no—
278Dic106—44no—
279Dic106—44no—
280Dic106—44no—
281C8×C53—44no—
282#88322—44no—
283#17397—531nomaximal, Sylow
284C22×C53—28no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—44no—
302C22×C53—44no—
303C22×C53—44no—
304C22×C53—44no—
305C22×C53—44no—
306C22×C53—44no—
307C22×C53C2×#9912yes—
308C22×C53C2×#10112yes—
309C22×C53C2×#10112yes—
310C22×C53C2×#10112yes—
311C22×C53C22×C22⋊C411yescharacteristic
312C22×C53#56812yes—
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—21no—
319C4×C53C2×#9912yes—
320C4×C53C2×#9912yes—
321Dic53—44no—
322Dic53—44no—
323C22×C22⋊Q8—531no—
324C2×#407—531no—
325C2×#407D5311yescharacteristic
326#1472—534no—
327#1472—534no—
328#1463—534no—
329C2×C53—22no—
330C2×C53—22no—
331C2×C53—22no—
332C2×C53—22no—
333C2×C53—22no—
334C2×C53—41no—
335C2×C53—41no—
336C2×C53#287911yescharacteristic
337C2×C53C22×#9911yescharacteristic
338C2×C53C22×#10111yescharacteristic
339C23×Q8—531no—
340C24×C4C2×D5311yescharacteristic
341C22×M16—21no—
342C22×M16—1061no—
343C2×C22⋊Q8—532no—
344C2×C22⋊Q8—532no—
345C2×C22⋊Q8—532no—
346C2×C22⋊Q8—532no—
347C2×C22⋊Q8—1061no—
348C2×C22⋊Q8—1061no—
349C2×C22⋊Q8—1061no—
350C2×C22⋊Q8—1061no—
351C2×C22⋊Q8—1061no—
352C2×C22⋊Q8—1061no—
353C2×C22⋊Q8—1061no—
354C2×C22⋊Q8—1061no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—532no—
357C22×C22⋊C4—532no—
358#407—21no—
359#407—21no—
360#407—532no—
361#407—532no—
362#407—1061no—
363#407—1061no—
364#407C2×D5312yes—
365#407C2×D5312yes—
366C2×C22⋊C8—21no—
367C2×C22⋊C8—532no—
368C2×C22⋊C8—1061no—
369C2×C22⋊C8C2×D5312yes—
370#324—5316no—
371C53#1739711yescharacteristic, complemented, Sylow
372C25C4×D5311yescharacteristic
373C22×Q8—532no—
374C22×Q8—532no—
375C22×Q8—1061no—
376C22×Q8—1061no—
377C22×Q8—1061no—
378C22×Q8—1061no—
379C22×Q8—2124no—
380C23×C4—44no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4C4×D5311yescharacteristic
384C23×C4C4×D5311yescharacteristic
385C23×C4C4×D5311yescharacteristic
386C23×C4D21211yescharacteristic
387C23×C4D21211yescharacteristic
388C23×C4D21211yescharacteristic
389C23×C4D21211yescharacteristic
390C23×C4C22⋊D5311yescharacteristic
391C23×C4C22⋊D5311yescharacteristic
392C23×C4C22⋊D5311yescharacteristic
393C23×C4C22⋊D5311yescharacteristic
394C23×C4C22×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C23×C4C22×D5311yescharacteristic
397C2×M16—22no—
398C2×M16—22no—
399C2×M16—44no—
400C2×M16—1062no—
401C2×M16—1062no—
402C2×M16—2124no—
403C22×C8—22no—
404C22×C8—1062no—
405C22⋊Q8—1062no—
406C22⋊Q8—1062no—
407C22⋊Q8—1062no—
408C22⋊Q8—1062no—
409C22⋊Q8—1062no—
410C22⋊Q8—1062no—
411C22⋊Q8—1062no—
412C22⋊Q8—1062no—
413C22⋊Q8—1062no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—1062no—
419C22⋊Q8—1062no—
420C22⋊Q8—1062no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—534no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C22⋊C4—534no—
434C2×C22⋊C4—534no—
435C2×C22⋊C4—1062no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1062no—
438C2×C22⋊C4—1062no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—22no—
442C22⋊C8—534no—
443C22⋊C8—1062no—
444C22⋊C8—1062no—
445C22⋊C8—1062no—
446C22⋊C8C22×D5314yes—
447C24—24no—
448C24—42no—
449C24—42no—
450C24#1171992111yescharacteristic
451C24#1171992111yescharacteristic
452C24#1171992111yescharacteristic
453C24#1171992111yescharacteristic
454C24C2×C4×D5311yescharacteristic
455C24C2×C4×D5311yescharacteristic
456C24C2×C4×D5311yescharacteristic
457C2×Q8—1062no—
458C2×Q8—1062no—
459C2×Q8—1062no—
460C2×Q8—1062no—
461C2×Q8—2124no—
462C2×Q8—2124no—
463C2×Q8—2124no—
464C2×Q8—2124no—
465C2×Q8—2124no—
466C2×Q8—2124no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—534no—
482C22×C4—534no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—2124no—
488C22×C4—2124no—
489C22×C4#1171992111yescharacteristic
490C22×C4#1171992111yescharacteristic
491C22×C4#1171992111yescharacteristic
492C22×C4#1171992111yescharacteristic
493C22×C4#1171992111yescharacteristic
494C22×C4#1171992111yescharacteristic
495C22×C4#1171992111yescharacteristic
496C22×C4#1171992111yescharacteristic
497C22×C4#1171992111yescharacteristic
498C22×C4#1171992111yescharacteristic
499C22×C4#1171992111yescharacteristic
500C22×C4#1171992111yescharacteristic
501C22×C4C2×C4×D5311yescharacteristic
502C22×C4C2×C4×D5311yescharacteristic
503C22×C4C2×C4×D5311yescharacteristic
504C22×C4C2×D21211yescharacteristic
505C22×C4C2×D21211yescharacteristic
506C22×C4C2×D21211yescharacteristic
507C22×C4C2×D21211yescharacteristic
508C22×C4C2×D21211yescharacteristic
509C22×C4C2×D21211yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C23×D5311yescharacteristic
517M16—44no—
518M16—44no—
519M16—2124no—
520M16—2124no—
521C2×C8—24no—
522C2×C8—44no—
523C2×C8—1064no—
524C2×C8—2124no—
525C4⋊C4—1064no—
526C4⋊C4—1064no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1064no—
532C4⋊C4—1064no—
533C22⋊C4—1064no—
534C22⋊C4—1064no—
535C22⋊C4—1064no—
536C22⋊C4—1064no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—416no—
558C23#1313313514yes—
559C23C2×#1171992111yescharacteristic
560C23C2×#1171992111yescharacteristic
561C23C2×#1171992111yescharacteristic
562C23C2×#1171992111yescharacteristic
563C23C2×#1171992111yescharacteristic
564C23C2×#1171992111yescharacteristic
565C23C22×C4×D5311yescharacteristic
566Q8—2124no—
567Q8—2124no—
568Q8—2124no—
569Q8—2124no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—1068no—
600C2×C4—2124no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2124no—
604C2×C4—2124no—
605C2×C4#1313313114yes—
606C2×C4C2×#1171992111yescharacteristic
607C2×C4C2×#1171992111yescharacteristic
608C2×C4C2×#1171992111yescharacteristic
609C2×C4C2×#1171992111yescharacteristic
610C2×C4C2×#1171992111yescharacteristic
611C2×C4C2×#1171992111yescharacteristic
612C2×C4C22×C22⋊D5311yescharacteristic
613C2×C4C22×D21211yescharacteristic
614C8—44no—
615C8—2124no—
616C22—28no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22C2×#1313313112yes—
640C22C2×#1313313512yes—
641C22C2×#1313313512yes—
642C22C2×#1313313512yes—
643C22C22×#1171992111yescharacteristic
644C22#1681138912yes—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—2124no—
652C4—2124no—
653C4C2×#1313313112yes—
654C4C2×#1313313112yes—
655C2—22no—
656C2—22no—
657C2—22no—
658C2—22no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2#2028315911yescharacteristic
663C2C22×#1313313111yescharacteristic
664C2C22×#1313313511yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#407C2
socleC22×C53C22×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenterC2×#407D53
hypocenterC53#17397
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
3. #20273080C2semidirect
4. #20273078C2semidirect
5. #20273044C2semidirect
30. #16810348C22semidirect
371. C53#17397semidirect

Maximal subgroups

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

subgroupindex
C22×#131331962
C2×#168105872
#202730802
#202730782
#202730442
C2×C53×#4072
#1739753