Finite Group Databasebeta

← #28319215

Subgroups — #28319215

Order 13568.

2038 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811642C212yesmaximal, complemented
2#20290607C212yesmaximal, complemented
3#20290570C212yesmaximal, complemented
4#20290569C212yesmaximal, complemented
5#20290538C212yesmaximal
6C4×#13133274C212yesmaximal, complemented
7C4×C22⋊#88324C212yesmaximal, complemented
8C22×C42×D53C211yesmaximal, characteristic
9#16811642—28no—
10C2×C4⋊#88324C2214yescomplemented
11C2×#13133274C2214yes—
12C2×#13133274C2214yes—
13C2×C22⋊#88324C2214yes—
14C2×C4×#88324C2214yescomplemented
15#16811489C2214yescomplemented
16#16811489C2214yescomplemented
17C23×C4×D53C411yescharacteristic
18C23×C4×D53C411yescharacteristic
19C23×C4×D53C2211yescharacteristic
20C22×C4×Dic53C411yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C2×C42×D53—28no—
24C2×C42×D53C411yescharacteristic
25C2×C42×D53C411yescharacteristic
26C2×C42×D53C411yescharacteristic
27C2×C42×D53C411yescharacteristic
28C2×C42×D53C2211yescharacteristic
29C2×C42×D53C2211yescharacteristic
30C2×C42×D53C2211yescharacteristic
31C2×C42×D53C2211yescharacteristic
32C22×C42×C53C411yescharacteristic
33C22×#88324C2318yes—
34C4⋊#88324—28no—
35#13133274—28no—
36#13133274—28no—
37C22⋊#88324—28no—
38C4×#88324—28no—
39C24×D53C2×C411yescharacteristic
40C23×Dic53C2×C411yescharacteristic
41C23×Dic53C2×C411yescharacteristic
42C23×Dic53C2311yescharacteristic
43C22×C4×D53—24no—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53C818yescomplemented
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2×C411yescharacteristic
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53C2×C411yescharacteristic
60C22×C4×D53C2×C411yescharacteristic
61C22×C4×D53C2×C411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×C4×D53C2311yescharacteristic
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53C2×C411yescharacteristic
68C2×C4×Dic53C2×C411yescharacteristic
69C2×C4×Dic53C2×C411yescharacteristic
70C2×C4×Dic53C2×C411yescharacteristic
71C2×C4×Dic53C2×C411yescharacteristic
72C2×C4×Dic53C2×C411yescharacteristic
73C2×C4×Dic53C2×C411yescharacteristic
74C2×C4×Dic53C2×C411yescharacteristic
75C2×C4×Dic53C2311yescharacteristic
76C2×C4×Dic53C2311yescharacteristic
77C2×C4×Dic53C2311yescharacteristic
78C2×C4×Dic53C2311yescharacteristic
79C42×D53—24no—
80C42×D53—24no—
81C42×D53—24no—
82C42×D53—24no—
83C42×D53—24no—
84C42×D53—24no—
85C42×D53—24no—
86C42×D53D418yes—
87C23×C4×C53C2×C411yescharacteristic
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C2×C42×C53—24no—
91C2×C42×C53C2×C411yescharacteristic
92C2×C42×C53C2×C411yescharacteristic
93C2×C42×C53C2×C411yescharacteristic
94C2×C42×C53C2×C411yescharacteristic
95C24×C53C22×C411yescharacteristic
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53C2×C814yes—
103C23×D53C22×C411yescharacteristic
104C23×D53C22×C411yescharacteristic
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53C2×C814yes—
113C22×C4×C53M1614yes—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—22no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53C2×C814yes—
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53#4114yes—
134C22×Dic53C2411yescharacteristic
135C2×C4×D53—232no—
136C2×C4×D53—232no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
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—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—232no—
164C2×C4×D53C22⋊C414yes—
165C2×C4×D53C22⋊C414yes—
166C2×C4×D53C2×C814yes—
167C2×C4×D53C2×C814yes—
168C2×C4×D53C2×C814yes—
169C2×C4×D53C2×C814yes—
170C2×C4×D53M1614yes—
171C2×C4×D53M1614yes—
172C2×C4×D53C2×D414yes—
173C2×C4×D53#4114yes—
174C2×#88324—28no—
175C2×#88324—28no—
176C2×#88324—28no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53C22⋊C414yes—
181C4×Dic53—22no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—22no—
189C4×Dic53—22no—
190C4×Dic53C22⋊C414yes—
191C4×Dic53C2×D414yes—
192C4×Dic53C2×D414yes—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53#11712yes—
198C23×C53C22×C812yes—
199C23×C53C2×M1612yes—
200C23×C53C23×C411yescharacteristic
201C22×D53—28no—
202C22×D53—28no—
203C22×D53—28no—
204C22×D53—28no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—28no—
214C22×D53—28no—
215C22×D53—28no—
216C22×D53C2×C22⋊C412yes—
217C22×D53#11712yes—
218C22×D53C22×C812yes—
219C22×D53C22×C812yes—
220C22×D53C2×M1612yes—
221C22×D53C2×M1612yes—
222C2×C4×C53—28no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—28no—
232C2×C4×C53—28no—
233C2×C4×C53C2×C22⋊C412yes—
234C2×C4×C53C2×C22⋊C412yes—
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53#11712yes—
237C2×C4×C53C22×C812yes—
238C2×C4×C53C22×C812yes—
239C2×C4×C53C2×M1612yes—
240C2×C4×C53C2×M1612yes—
241C2×C4×C53C2×M1612yes—
242C2×C4×C53C2×M1612yes—
243C2×Dic53—28no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—28no—
253C2×Dic53—28no—
254C2×Dic53C2×C22⋊C412yes—
255C2×Dic53C2×C22⋊C412yes—
256C2×Dic53#11712yes—
257C2×Dic53C22×C812yes—
258C2×Dic53C22×C812yes—
259C2×Dic53C2×M1612yes—
260C2×Dic53C2×M1612yes—
261C2×Dic53C22×D412yes—
262C2×Dic53C2×#4112yes—
263C2×Dic53C2×#4112yes—
264C4×D53—28no—
265C4×D53—28no—
266C4×D53—28no—
267C4×D53—28no—
268C4×D53—28no—
269C4×D53—28no—
270C4×D53—28no—
271C4×D53—28no—
272C4×D53—28no—
273C4×D53—28no—
274C4×D53—28no—
275C4×D53C22⋊C8116yescomplemented
276#88324—48no—
277#20714—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53#43114yescomplemented
285C22×C53C22×C22⋊C411yescharacteristic
286C22×C53C2×#11711yescharacteristic
287C22×C53C2×#11711yescharacteristic
288C22×C53C23×C811yescharacteristic
289C22×C53C22×M1611yescharacteristic
290C22×C53C22×M1611yescharacteristic
291C22×C53C22×M1611yescharacteristic
292C2×D53—232no—
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53C2×C22⋊C814yes—
306C2×D53#43114yescomplemented
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—22no—
311C4×C53C2×C22⋊C814yes—
312C4×C53#40714yes—
313Dic53—22no—
314Dic53—22no—
315Dic53—22no—
316Dic53—22no—
317Dic53C2×C22⋊C814yes—
318Dic53#52514yes—
319C23×C42—531no—
320C2×#431—532no—
321#1817—532no—
322#1797—532no—
323#1797—532no—
324#1779—532no—
325C4×C22⋊C8—532no—
326C4×C22⋊C8—532no—
327C2×C53—24no—
328C2×C53#228011yescharacteristic
329C2×C53C22×C22⋊C811yescharacteristic
330C2×C53C2×#40711yescharacteristic
331C2×C53C2×#43111yescharacteristic
332C2×C53C2×#43111yescharacteristic
333C2×C53#292011yescharacteristic
334C2×C53#292011yescharacteristic
335D53—24no—
336D53—24no—
337C24×C4—531no—
338C24×C4—531no—
339C24×C4—531no—
340C22×C42—531no—
341C22×C42—531no—
342C22×C42—531no—
343C22×C42—531no—
344C22×C42—531no—
345C22×C42—531no—
346C22×C42—531no—
347C22×C42—531no—
348C22×C42—531no—
349C22×C42—531no—
350C22×C42—531no—
351C22×C42—1068no—
352C22×C42#615711yescharacteristic
353#431—1068no—
354C2×C4⋊C8—534no—
355C2×C22⋊C8—534no—
356C2×C22⋊C8—534no—
357C2×C22⋊C8—534no—
358C2×C4×C8—534no—
359#336—534no—
360#336—534no—
361C53#2071411yescharacteristic, complemented, Sylow
362C25—531no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—538no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4—1064no—
382C23×C4—1064no—
383C23×C4—1064no—
384C23×C4—1064no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4C2×#615711yescharacteristic
388C23×C4C2×#615711yescharacteristic
389C23×C4C2×#615711yescharacteristic
390C22×C8—538no—
391C2×C42—24no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—538no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—1064no—
406C2×C42—1064no—
407C2×C42—1064no—
408C2×C42—1064no—
409C2×C42—1064no—
410C2×C42—1064no—
411C2×C42—1064no—
412C2×C42—1064no—
413C2×C42—1064no—
414C2×C42—1064no—
415C2×C42C2×#615711yescharacteristic
416C2×C42C2×#615711yescharacteristic
417C2×C42C2×#615711yescharacteristic
418C2×C42C2×#615711yescharacteristic
419C4⋊C8—1068no—
420C22⋊C8—1068no—
421C22⋊C8—1068no—
422C22⋊C8—1068no—
423C4×C8—1068no—
424C24—531no—
425C24—531no—
426C24—534no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24C22×#615711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—534no—
446C22×C4—534no—
447C22×C4—534no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—534no—
453C22×C4—534no—
454C22×C4—534no—
455C22×C4—534no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—10632no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—10632no—
477C22×C4—10632no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4D53⋊C814yes—
496C22×C4#1171993614yes—
497C22×C4C22×#615711yescharacteristic
498C22×C4C22×#615711yescharacteristic
499C22×C4C22×#615711yescharacteristic
500C22×C4C22×#615711yescharacteristic
501C22×C4C22×#615711yescharacteristic
502C22×C4C22×#615711yescharacteristic
503C2×C8—1068no—
504C2×C8—1068no—
505C2×C8—1068no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—534no—
510C42—534no—
511C42—534no—
512C42—1062no—
513C42—1062no—
514C42—1062no—
515C42—1062no—
516C42—1062no—
517C42—1062no—
518C42—1062no—
519C42—1062no—
520C42—1062no—
521C42C22⋊#615714yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—532no—
527C23—532no—
528C23—532no—
529C23—532no—
530C23—532no—
531C23—532no—
532C23—1068no—
533C23—1068no—
534C23—1068no—
535C23—1068no—
536C23—1068no—
537C23—1068no—
538C23—1068no—
539C23—1061no—
540C23—1061no—
541C23—1061no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23—1061no—
547C23#1313329612yes—
548C23C2×D53⋊C812yes—
549C23C2×#1171993612yes—
550C23C23×#615711yescharacteristic
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—532no—
563C2×C4—532no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—532no—
567C2×C4—532no—
568C2×C4—5316no—
569C2×C4—532no—
570C2×C4—532no—
571C2×C4—532no—
572C2×C4—532no—
573C2×C4—1068no—
574C2×C4—1068no—
575C2×C4—1068no—
576C2×C4—1068no—
577C2×C4—1068no—
578C2×C4—1068no—
579C2×C4—1068no—
580C2×C4—1068no—
581C2×C4—1068no—
582C2×C4—1068no—
583C2×C4—1068no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1068no—
593C2×C4—1068no—
594C2×C4—1068no—
595C2×C4C2×C22⋊#615712yes—
596C2×C4C2×C22⋊#615712yes—
597C2×C4C2×C22⋊#615712yes—
598C2×C4#1313329612yes—
599C2×C4C2×D53⋊C812yes—
600C2×C4C2×D53⋊C812yes—
601C2×C4D53⋊M1612yes—
602C2×C4D53⋊M1612yes—
603C2×C4C2×#1171993612yes—
604C2×C4C2×#1171993612yes—
605C8—2128no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—534no—
613C22—534no—
614C22—10632no—
615C22—1062no—
616C22—1062no—
617C22—1062no—
618C22—1062no—
619C22—1062no—
620C22—1062no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—1062no—
625C22—1062no—
626C22—1062no—
627C22#1681164214yescomplemented
628C22C22×C22⋊#615711yescharacteristic
629C22C2×#1313329611yescharacteristic
630C22C2×#1313329611yescharacteristic
631C22C22×D53⋊C811yescharacteristic
632C22C2×D53⋊M1611yescharacteristic
633C22C2×D53⋊M1611yescharacteristic
634C22C22×#1171993611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—534no—
640C4—534no—
641C4—1062no—
642C4—1062no—
643C4—1062no—
644C4—1062no—
645C4D53⋊(C22⋊C8)14yes—
646C4#1681159414yes—
647C2—24no—
648C2—1064no—
649C2—1064no—
650C2#2029115911yescharacteristic
651C2C2×D53⋊(C22⋊C8)11yescharacteristic
652C2C2×#1681159411yescharacteristic
653C2D53⋊#43111yescharacteristic
654C2C2×#1681164211yescharacteristic
655C2#2029139211yescharacteristic
656C2#2029139311yescharacteristic
subgroupG/H
centerC22×C4C22×#6157
commutatorC22×C53C23×C8
FrattiniC23C23×#6157
FittingC22×C42×C53C4
socleC23×C53C23×C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C42#6157
hypocenterC53#20714
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C2×#16811642C2semidirect
2. #20290607C2semidirect
3. #20290570C2semidirect
4. #20290569C2semidirect
6. C4×#13133274C2semidirect
7. C4×C22⋊#88324C2semidirect
10. C2×C4⋊#88324C22semidirect
14. C2×C4×#88324C22semidirect
15. #16811489C22semidirect
16. #16811489C22semidirect
51. C22×C4×D53C8semidirect
275. C4×D53C22⋊C8semidirect
284. C22×C53#431semidirect
306. C2×D53#431semidirect
361. C53#20714semidirect
627. C22#16811642semidirect

Maximal subgroups

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

subgroupindex
C2×#168116422
#202906072
#202905702
#202905692
#202905382
C4×#131332742
C4×C22⋊#883242
C22×C42×D532
#2071453