Finite Group Databasebeta

← #28312889

Subgroups — #28312889

Order 13568.

2438 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133296C211yesmaximal, characteristic
2C2×#16811566C211yesmaximal, characteristic
3C2×#16811566C212yesmaximal
4#20289900C214yesmaximal, complemented
5#20289894C214yesmaximal, complemented
6#20289889C212yesmaximal
7C22×C4⋊C4×D53C211yesmaximal, characteristic
8C2×#13133296—28no—
9C2×#13133296—24no—
10C22×C4⋊#6157C2212yes—
11C22×C22⋊#6157C2212yes—
12C22×C22⋊#6157C2212yes—
13C22×C4×#6157C2212yes—
14#16811566—28no—
15#16811566—24no—
16C2×(C2×#6157)⋊C4C2212yes—
17C2×(C2×#6157)⋊C4C2218yes—
18C2×(C2×#6157)⋊C4C2212yes—
19C2×(C2×#6157)⋊C4C2218yes—
20C23×C4×D53C2211yescharacteristic
21C23×C4×D53C2211yescharacteristic
22C23×C4×D53C2211yescharacteristic
23C22×C4⋊Dic53C411yescharacteristic, complemented
24C22×#11719919C411yescharacteristic, complemented
25C22×#11719919C411yescharacteristic, complemented
26C2×C4⋊C4×D53—24no—
27C2×C4⋊C4×D53—24no—
28C2×C4⋊C4×D53C412yescomplemented
29C2×C4⋊C4×D53C412yescomplemented
30C2×C4⋊C4×D53C2212yes—
31C2×C4⋊C4×D53C2212yes—
32C22×C4⋊C4×C53C411yescharacteristic, complemented
33C23×#6157C2314yes—
34C23×#6157C2314yes—
35#13133296—232no—
36C2×C4⋊#6157—28no—
37C2×C4⋊#6157—24no—
38C2×C22⋊#6157—24no—
39C2×C22⋊#6157—24no—
40C2×C22⋊#6157—24no—
41C2×C22⋊#6157—24no—
42C2×C22⋊#6157—24no—
43C2×C22⋊#6157D418yes—
44C2×C4×#6157—28no—
45C2×C4×#6157—24no—
46(C2×#6157)⋊C4—216no—
47(C2×#6157)⋊C4—24no—
48(C2×#6157)⋊C4—24no—
49(C2×#6157)⋊C4—216no—
50C24×D53C2311yescharacteristic
51C23×Dic53C2×C411yescharacteristic
52C23×Dic53C2×C411yescharacteristic
53C23×Dic53C2×C411yescharacteristic
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—24no—
57C22×C4×D53—24no—
58C22×C4×D53—28no—
59C22×C4×D53—24no—
60C22×C4×D53—22no—
61C22×C4×D53—22no—
62C22×C4×D53—24no—
63C22×C4×D53C2×C412yes—
64C22×C4×D53C2×C412yes—
65C22×C4×D53C2×C411yescharacteristic
66C22×C4×D53C2×C411yescharacteristic
67C22×C4×D53D414yes—
68C22×C4×D53D414yes—
69C22×C4×D53C2312yes—
70C22×C4×D53C2311yescharacteristic
71C22×C4×D53C2311yescharacteristic
72C22×C4×D53C2312yes—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53C2×C412yes—
75C2×C4⋊Dic53C2×C412yes—
76C2×#11719919—24no—
77C2×#11719919—24no—
78C2×#11719919C2×C412yes—
79C2×#11719919C2×C412yes—
80C2×#11719919C2×C412yes—
81C2×#11719919C2×C412yes—
82C4⋊C4×D53—28no—
83C4⋊C4×D53—28no—
84C4⋊C4×D53—24no—
85C4⋊C4×D53—24no—
86C4⋊C4×D53—24no—
87C4⋊C4×D53—24no—
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53C2×C412yes—
93C2×C4⋊C4×C53C2×C412yes—
94C24×C53C22×C411yescharacteristic
95C23×D53—24no—
96C23×D53—24no—
97C23×D53C22×C411yescharacteristic
98C23×D53C2×D412yes—
99C23×D53C2×D412yes—
100C23×D53C2×D414yes—
101C23×D53#4114yes—
102C23×D53C2411yescharacteristic
103C22×#6157—24no—
104C22×#6157—24no—
105C22×#6157—28no—
106C22×#6157—28no—
107C22×#6157—24no—
108C22×#6157—24no—
109C22×#6157—28no—
110C22×#6157—24no—
111C22×#6157—28no—
112C22×#6157C2×D418yes—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—24no—
118C22×C4×C53C22⋊C414yes—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C412yes—
123C22×Dic53—22no—
124C22×Dic53—24no—
125C22×Dic53—24no—
126C22×Dic53—22no—
127C22×Dic53—24no—
128C22×Dic53C22⋊C414yes—
129C22×Dic53C22×C412yes—
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C412yes—
133C2×C4×D53—216no—
134C2×C4×D53—216no—
135C2×C4×D53—24no—
136C2×C4×D53—24no—
137C2×C4×D53—24no—
138C2×C4×D53—28no—
139C2×C4×D53—216no—
140C2×C4×D53—28no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—24no—
150C2×C4×D53—24no—
151C2×C4×D53—28no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—24no—
155C2×C4×D53—24no—
156C2×C4×D53—44no—
157C2×C4×D53—48no—
158C2×C4×D53—44no—
159C2×C4×D53C22⋊C414yes—
160C2×C4×D53C22⋊C414yes—
161C2×C4×D53C2×D412yes—
162C2×C4×D53C2×D412yes—
163C2×C4×D53C2×D412yes—
164C2×C4×D53C2×D412yes—
165C2×C4×D53#4118yes—
166C22⋊#6157—216no—
167C22⋊#6157—48no—
168C4⋊#6157—216no—
169C4×#6157—216no—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—24no—
172C4⋊Dic53—24no—
173C4⋊Dic53—24no—
174#11719919—24no—
175#11719919—24no—
176#11719919—24no—
177#11719919—24no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53C2×C22⋊C412yes—
181C23×C53C4×D414yes—
182C23×C53C23×C411yescharacteristic
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53—216no—
190C22×D53—216no—
191C22×D53C2×C22⋊C412yes—
192C22×D53C4×D414yes—
193C22×D53#12014yes—
194C22×D53#12118yes—
195C22×D53#12314yes—
196C22×D53C22×D412yes—
197C22×D53C22×D411yescharacteristic
198C22×D53C22×D411yescharacteristic
199C22×D53C2×#4112yes—
200C22×D53C2×#4114yes—
201C22×D532−1+414yes—
202C2×#6157—28no—
203C2×#6157—216no—
204C2×#6157—44no—
205C2×#6157—44no—
206C2×#6157—48no—
207C2×#6157—44no—
208C2×#6157—44no—
209C2×#6157—44no—
210C2×#6157—48no—
211C2×#6157—48no—
212C2×C4×C53—28no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—28no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—44no—
221C2×C4×C53C2×C22⋊C412yes—
222C2×C4×C53C2×C22⋊C412yes—
223C2×C4×C53#11714yescomplemented
224C2×Dic53—28no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—28no—
230C2×Dic53—22no—
231C2×Dic53—24no—
232C2×Dic53—44no—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53#11714yescomplemented
236C4×D53—216no—
237C4×D53—22no—
238C4×D53—22no—
239C4×D53—24no—
240C4×D53—24no—
241C4×D53—22no—
242C4×D53—22no—
243C4×D53—28no—
244C4×D53—28no—
245C4×D53—28no—
246C4×D53—28no—
247C4×D53—42no—
248C4×D53—44no—
249C4×D53—42no—
250C4×D53—48no—
251#13870—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—28no—
254C22×C53—22no—
255C22×C53#38614yes—
256C22×C53C22×C22⋊C411yescharacteristic
257C22×C53C2×#11712yes—
258C22×C53C2×C4×D412yes—
259C22×C53#51912yes—
260C2×D53—22no—
261C2×D53—28no—
262C2×D53—22no—
263C2×D53—28no—
264C2×D53—216no—
265C2×D53—28no—
266C2×D53#38614yes—
267C2×D53C2×#12011yescharacteristic
268C2×D53C2×#12112yes—
269C2×D53C2×#12311yescharacteristic
270C2×D53#52512yes—
271C2×D53#52512yes—
272C2×D53#53612yes—
273C2×D53#53612yes—
274C2×D53#54718yes—
275C2×D53#55218yes—
276#6157—48no—
277#6157—84no—
278C4×C53—24no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—42no—
282Dic53—24no—
283Dic53—22no—
284Dic53—22no—
285Dic53—42no—
286C22×#117—531no—
287C23×C4⋊C4—531no—
288C2×#381—532no—
289C2×#381—531no—
290#1441—534no—
291#1436—534no—
292C22⋊#95—532no—
293C2×C53—24no—
294C2×C53#228011yescharacteristic
295C2×C53C2×#38611yescharacteristic
296C2×C53#231011yescharacteristic
297C2×C53#234414yes—
298D53—24no—
299D53#240612yes—
300D53#241314yes—
301D53#248412yes—
302C24×C4—531no—
303C24×C4—531no—
304C24×C4—531no—
305C2×#117—1068no—
306C2×#117—1064no—
307C22×C4⋊C4—531no—
308C22×C4⋊C4—531no—
309C22×C4⋊C4—531no—
310C22×C4⋊C4—532no—
311C22×C4⋊C4—532no—
312C22×C4⋊C4—532no—
313C22×C4⋊C4—532no—
314C22×C4⋊C4—532no—
315C22×C4⋊C4—1064no—
316C22×C4⋊C4—1064no—
317C22×C4⋊C4#615711yescharacteristic, complemented
318C22×C22⋊C4—532no—
319C22×C22⋊C4—532no—
320C22×C42—532no—
321#381—1068no—
322#381—1064no—
323C2×#95—538no—
324C2×#95—532no—
325C2×#95—538no—
326C2×#95—532no—
327C53#1387011yescharacteristic, complemented, Sylow
328C25—531no—
329C23×C4—531no—
330C23×C4—534no—
331C23×C4—534no—
332C23×C4—531no—
333C23×C4—532no—
334C23×C4—531no—
335C23×C4—531no—
336C23×C4—531no—
337C23×C4—534no—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4—532no—
341C23×C4—532no—
342C23×C4—532no—
343C23×C4—534no—
344C23×C4—1068no—
345C23×C4—1062no—
346C23×C4—1062no—
347C23×C4—1064no—
348C23×C4—1064no—
349C23×C4—1062no—
350C23×C4—1062no—
351C23×C4—1064no—
352C23×C4—1064no—
353C23×C4C2×#615711yescharacteristic
354C23×C4C2×#615711yescharacteristic
355C23×C4C2×#615711yescharacteristic
356#117—10632no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—532no—
361C2×C4⋊C4—532no—
362C2×C4⋊C4—532no—
363C2×C4⋊C4—532no—
364C2×C4⋊C4—1068no—
365C2×C4⋊C4—1068no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1068no—
374C2×C4⋊C4—1064no—
375C2×C4⋊C4C2×#615712yes—
376C2×C4⋊C4C2×#615712yes—
377C2×C22⋊C4—538no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—1064no—
382C2×C22⋊C4—1064no—
383C2×C42—1064no—
384C2×C42—1068no—
385#95—1064no—
386#95—1064no—
387#95—10616no—
388#95—10616no—
389C24—531no—
390C24—532no—
391C24—532no—
392C24—534no—
393C24—534no—
394C24—531no—
395C24—1064no—
396C24—1064no—
397C24C22×#615711yescharacteristic
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—534no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—532no—
407C22×C4—534no—
408C22×C4—534no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—538no—
414C22×C4—538no—
415C22×C4—532no—
416C22×C4—1064no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1064no—
421C22×C4—1062no—
422C22×C4—1068no—
423C22×C4—10616no—
424C22×C4—1068no—
425C22×C4—1064no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1064no—
429C22×C4—1064no—
430C22×C4—10616no—
431C22×C4—10616no—
432C22×C4—1064no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1064no—
436C22×C4—1064no—
437C22×C4—1068no—
438C22×C4—1064no—
439C22×C4—1062no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1068no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1068no—
446C22×C4—1062no—
447C22×C4—1064no—
448C22×C4—1064no—
449C22×C4—1064no—
450C22×C4—1068no—
451C22×C4—1064no—
452C22×C4—1068no—
453C22×C4—2124no—
454C22×C4—2128no—
455C22×C4—2124no—
456C22×C4C22⋊#615714yes—
457C22×C4C22×#615712yes—
458C22×C4C22×#615711yescharacteristic
459C22×C4C22×#615711yescharacteristic
460C22×C4C22×#615712yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—1064no—
464C4⋊C4—1064no—
465C4⋊C4—1064no—
466C4⋊C4—1064no—
467C4⋊C4—10616no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C22⋊C4—10616no—
471C22⋊C4—2128no—
472C42—10616no—
473C23—22no—
474C23—22no—
475C23—531no—
476C23—531no—
477C23—534no—
478C23—532no—
479C23—532no—
480C23—534no—
481C23—534no—
482C23—538no—
483C23—534no—
484C23—532no—
485C23—534no—
486C23—1064no—
487C23—1064no—
488C23—1064no—
489C23—1064no—
490C23—10616no—
491C23—10616no—
492C23—1062no—
493C23—1062no—
494C23C2×C22⋊#615712yes—
495C23D4×#615714yes—
496C23C23×#615711yescharacteristic
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—532no—
507C2×C4—532no—
508C2×C4—534no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1068no—
512C2×C4—1068no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1068no—
518C2×C4—1068no—
519C2×C4—10616no—
520C2×C4—1068no—
521C2×C4—1068no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1068no—
527C2×C4—10616no—
528C2×C4—1062no—
529C2×C4—1064no—
530C2×C4—2122no—
531C2×C4—2124no—
532C2×C4—2122no—
533C2×C4—2128no—
534C2×C4—2124no—
535C2×C4—2124no—
536C2×C4—2124no—
537C2×C4—2128no—
538C2×C4—2128no—
539C2×C4—2124no—
540C2×C4—2124no—
541C2×C4—2128no—
542C2×C4—2124no—
543C2×C4C2×C22⋊#615712yes—
544C2×C4C2×C22⋊#615712yes—
545C2×C4#1313329614yescomplemented
546C22—22no—
547C22—22no—
548C22—28no—
549C22—532no—
550C22—532no—
551C22—531no—
552C22—532no—
553C22—531no—
554C22—532no—
555C22—532no—
556C22—538no—
557C22—538no—
558C22—534no—
559C22—10616no—
560C22—1068no—
561C22—1068no—
562C22—1062no—
563C22—1062no—
564C22—1068no—
565C22#1681157114yes—
566C22C22×C22⋊#615711yescharacteristic
567C22C2×#1313329612yes—
568C22C2×D4×#615712yes—
569C22#1681169112yes—
570C4—24no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—1064no—
575C4—1062no—
576C4—1062no—
577C4—4244no—
578C4—2128no—
579C4—2122no—
580C2—24no—
581C2—534no—
582C2—532no—
583C2—532no—
584C2—1064no—
585C2#2029115911yescharacteristic
586C2C2×#1681157111yescharacteristic
587C2#2029118911yescharacteristic
588C2#2029122314yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC22×C4⋊C4×C53C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#13870
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20289900C2semidirect
5. #20289894C2semidirect
23. C22×C4⋊Dic53C4semidirect
24. C22×#11719919C4semidirect
25. C22×#11719919C4semidirect
28. C2×C4⋊C4×D53C4semidirect
29. C2×C4⋊C4×D53C4semidirect
32. C22×C4⋊C4×C53C4semidirect
223. C2×C4×C53#117semidirect
235. C2×Dic53#117semidirect
317. C22×C4⋊C4#6157semidirect
327. C53#13870semidirect
545. C2×C4#13133296semidirect

Maximal subgroups

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

subgroupindex
C22×#131332962
C2×#168115662
#202899002
#202898942
#202898892
C22×C4⋊C4×D532
#1387053