Finite Group Databasebeta

← #27400823

Subgroups — #27400823

Order 13568.

1364 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719945C211yesmaximal, characteristic
2C2×#16810751C212yesmaximal, complemented
3#20275513C212yesmaximal, complemented
4#20275511C212yesmaximal, complemented
5#20275507C212yesmaximal, complemented
6#20274695C211yesmaximal, characteristic, complemented
7#20274694C211yesmaximal, characteristic, complemented
8#20274694C211yesmaximal, characteristic, complemented
9#20274476C211yesmaximal, characteristic, complemented
10#20274472C211yesmaximal, characteristic, complemented
11C53×#1749C211yesmaximal, characteristic, complemented
12C22×#11719945C411yescharacteristic
13C4×#11719945—28no—
14C2×C4×Dic106C411yescharacteristic
15C2×C4×Dic106C2211yescharacteristic
16C2×C4×D212C411yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×C22⋊D53C411yescharacteristic
19C2×C4×C22⋊D53C411yescharacteristic
20C2×C4×C22⋊D53C2211yescharacteristic
21C2×C4×C22⋊D53C2211yescharacteristic
22C2×#13133176C411yescharacteristic
23C2×#13133174C411yescharacteristic
24C2×#13133174C411yescharacteristic
25C2×C42×D53C2211yescharacteristic
26C2×C42×D53C2211yescharacteristic
27#16810751—28no—
28C2×C4⋊#88322C2212yes—
29C2×#13133146C2212yes—
30C2×#13133146C2212yes—
31C2×C22⋊#88322C2212yes—
32C2×#13133129C2212yes—
33C2×#13133129C2212yes—
34C2×#13133129C2212yes—
35C2×#13133129C2212yes—
36C2×#13133127C2212yes—
37C2×C8×Dic53C2212yes—
38#16810378C2211yescharacteristic
39#16810378C2211yescharacteristic
40C22×C42×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2212yes—
42C2×#97×C53C2212yes—
43C53×#336C2211yescharacteristic
44C53×#336C2211yescharacteristic
45C2×#11719945—24no—
46C2×#11719945—24no—
47C2×#11719945—24no—
48C22×Dic106C2×C411yescharacteristic
49C22×D212C2×C411yescharacteristic
50C22×C22⋊D53C2×C411yescharacteristic
51C22×C22⋊D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C22×C4×D53C2311yescharacteristic
58C22×#88322C2312yes—
59C22×#88322C2312yes—
60C4×Dic106—24no—
61C4×Dic106D418yes—
62C4×D212—24no—
63C4×D212—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66C4×C22⋊D53—24no—
67C4×C22⋊D53—24no—
68#13133176—24no—
69#13133174—24no—
70#13133174—24no—
71C2×C4⋊Dic53C2×C411yescharacteristic
72C2×C4⋊Dic53C2×C411yescharacteristic
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2×C411yescharacteristic
75C2×#11719919C2×C411yescharacteristic
76C2×#11719919C2×C411yescharacteristic
77C2×C22⋊Dic53C2×C411yescharacteristic
78C2×C22⋊Dic53C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×C4×Dic53C2×C411yescharacteristic
84C2×C4×Dic53C2×C411yescharacteristic
85C2×C4×Dic53C2×C411yescharacteristic
86C2×C4×Dic53C2×C411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C42×D53—24no—
90C42×D53—24no—
91C4⋊#88322—24no—
92#13133146—24no—
93#13133146—24no—
94C22⋊#88322—42no—
95#13133129—24no—
96#13133129—24no—
97#13133129—42no—
98#13133129—42no—
99#13133127—24no—
100C8×Dic53—24no—
101C23×C4×C53C2×C411yescharacteristic
102C23×C4×C53C2×C411yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2312yes—
105C22×C8×C53C2312yes—
106C2×C42×C53—24no—
107C2×C42×C53C2×C411yescharacteristic
108C2×C42×C53C2×C411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C22⋊C8×C53—24no—
112#97×C53—24no—
113C24×C53C22×C411yescharacteristic
114C23×D53C22×C411yescharacteristic
115C23×D53C22×C411yescharacteristic
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—41no—
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C2×C22⋊D53—22no—
133C2×C22⋊D53—22no—
134C2×C22⋊D53—22no—
135C2×C22⋊D53—22no—
136C2×C22⋊D53—22no—
137C2×C22⋊D53M1614yes—
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144#11719945—232no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—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—21no—
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—21no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—41no—
166C2×C4×D53—41no—
167C2×C4×D53M1614yes—
168C2×Dic106—22no—
169C2×Dic106—22no—
170C2×Dic106C22⋊C414yes—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53#4114yescomplemented
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53C2×D414yes—
181C22⋊Dic53—22no—
182C22⋊Dic53—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719921—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53C22⋊C414yes—
189#11719919—22no—
190#11719919—22no—
191#11719919C22⋊C414yes—
192#11719919C22⋊C414yes—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197C4×Dic53C2×D414yes—
198C4×Dic53C2×D414yes—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53C2×M1612yes—
212C23×C53C23×C411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53C2×M1612yes—
226C22×D53C2×M1612yes—
227C2×C4×C53—28no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—44no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×M1612yes—
245C2×C4×C53#13112yes—
246C2×C4×C53C22×D412yes—
247C2×C4×C53C2×#4112yes—
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C22⋊D53—28no—
251C22⋊D53—28no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53C2×C22⋊C412yes—
265C2×Dic53C2×C22⋊C412yes—
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53C2×C22⋊C412yes—
268C2×Dic53C2×M1612yes—
269C2×Dic53C2×M1612yes—
270D212—28no—
271C4×D53—28no—
272C4×D53—28no—
273C4×D53—44no—
274C4×D53—44no—
275C4×D53—42no—
276C4×D53—42no—
277C4×D53—42no—
278C4×D53—42no—
279Dic106—28no—
280C8×C53—24no—
281C8×C53—42no—
282#88322—42no—
283#88322—42no—
284#22532—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C22×C22⋊C411yescharacteristic
293C22×C53C4×#4111yescharacteristic
294C22×C53#52011yescharacteristic
295C22×C53C22×M1611yescharacteristic
296C22×C53C22×M1611yescharacteristic
297C22×C53C2×#13111yescharacteristic
298C22×C53#56811yescharacteristic
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—41no—
306C2×D53—41no—
307C2×D53—41no—
308C2×D53—41no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53—41no—
312C2×D53—41no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—41no—
318C4×C53#54812yes—
319C4×C53#55012yes—
320Dic53—22no—
321Dic53—22no—
322Dic53—22no—
323Dic53—41no—
324Dic53—41no—
325Dic53#40714yes—
326C2×C4×#41—531no—
327C2×#446—532no—
328#1955—532no—
329#1955—532no—
330#1954—532no—
331#1797—531no—
332#1797—531no—
333#1797—531no—
334#1749—531no—
335#1749D5311yescharacteristic, complemented
336C4×C22⋊C8—531no—
337C2×C53—42no—
338C2×C53#234111yescharacteristic
339C2×C53C2×#40711yescharacteristic
340C2×C53#288011yescharacteristic
341C2×C53#296511yescharacteristic
342C2×C53#297011yescharacteristic
343C2×C53#298911yescharacteristic
344C2×C53#299011yescharacteristic
345D53—42no—
346D53—42no—
347C22×#41—531no—
348C4×#41—1068no—
349C2×C4×Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×#117—531no—
358C2×#117—531no—
359C2×#117—531no—
360C22×C42—531no—
361C22×C42—531no—
362C22×C42C2×D5311yescharacteristic
363#446—1068no—
364C2×C4⋊C8—532no—
365C2×C4⋊C8—532no—
366C2×C4⋊C8—532no—
367C2×C22⋊C8—532no—
368C2×C22⋊C8—532no—
369C2×C22⋊C8—532no—
370C2×C22⋊C8—532no—
371C2×C22⋊C8—532no—
372C2×C22⋊C8C2×D5312yes—
373C2×#97—532no—
374C2×#97C2×D5312yes—
375C2×C4×C8—532no—
376#336—531no—
377#336—531no—
378#336C2×D5311yescharacteristic
379#336C2×D5311yescharacteristic
380C53#2253211yescharacteristic, complemented, Sylow
381C2×#41—1064no—
382C2×#41—1064no—
383C2×#41—1064no—
384C22×Q8—531no—
385C22×D4—531no—
386C22×D4—531no—
387C22×D4—531no—
388C23×C4—531no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4C4×D5311yescharacteristic
395C23×C4C4×D5311yescharacteristic
396C23×C4C22×D5311yescharacteristic
397C22×C8—532no—
398C22×C8—532no—
399C22×C8C22×D5312yes—
400C22×C8C22×D5312yes—
401C4×Q8—538no—
402C4×Q8—1064no—
403C4×D4—1064no—
404C4×D4—1064no—
405C4×D4—1064no—
406C4×D4—1064no—
407C4×D4—1064no—
408C4×D4—1064no—
409#117—1064no—
410#117—1064no—
411#117—1064no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—531no—
424C2×C42—24no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1064no—
432C2×C42—1064no—
433C2×C42C4×D5311yescharacteristic
434C2×C42C4×D5311yescharacteristic
435C2×C42C22×D5311yescharacteristic
436C2×C42C22×D5311yescharacteristic
437C4⋊C8—1064no—
438C4⋊C8—1064no—
439C4⋊C8—1064no—
440C22⋊C8—24no—
441C22⋊C8—1064no—
442C22⋊C8—1064no—
443C22⋊C8—2122no—
444C22⋊C8—2122no—
445C22⋊C8—2122no—
446#97—24no—
447#97—1064no—
448C4×C8—1064no—
449C24—531no—
450C24—531no—
451C24C2×C4×D5311yescharacteristic
452#41—10632no—
453C2×Q8—534no—
454C2×Q8—1062no—
455C2×Q8—1062no—
456C2×D4—534no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1062no—
464C2×D4—1062no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—41no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—534no—
480C22×C4—531no—
481C22×C4—531no—
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×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—2121no—
500C22×C4—2121no—
501C22×C4C2×C4×D5311yescharacteristic
502C22×C4C2×C4×D5311yescharacteristic
503C22×C4C2×C4×D5311yescharacteristic
504C22×C4C2×C4×D5311yescharacteristic
505C22×C4C2×C4×D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—1062no—
513C2×C8—1062no—
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—1062no—
518C2×C8#1171994514yescomplemented
519C4⋊C4—534no—
520C4⋊C4—534no—
521C4⋊C4—534no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1062no—
530C22⋊C4—1062no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—534no—
535C42—534no—
536C42—1062no—
537C42—1062no—
538C42—1062no—
539C42—1062no—
540C42D4×D5314yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—532no—
548C23—532no—
549C23—1061no—
550C23—1061no—
551C23—1061no—
552C23—1061no—
553C23—1061no—
554C23—1061no—
555C23—1061no—
556C23—1061no—
557C23—1061no—
558C23—1061no—
559C23—1061no—
560C23—1061no—
561C23M16×D5312yes—
562C23C22×C4×D5311yescharacteristic
563Q8—1068no—
564D4—1068no—
565D4—1068no—
566D4—1068no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—532no—
586C2×C4—532no—
587C2×C4—532no—
588C2×C4—1068no—
589C2×C4—1068no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—2124no—
603C2×C4—2124no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4C22⋊C4×D5312yes—
609C2×C4C22⋊C4×D5312yes—
610C2×C4M16×D5312yes—
611C2×C4#1313322512yes—
612C2×C4C2×D4×D5312yes—
613C2×C4C2×#1171994512yes—
614C2×C4#1313326311yescharacteristic
615C2×C4#1313326411yescharacteristic
616C8—24no—
617C8—42no—
618C8—2122no—
619C8—2122no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—1062no—
628C22—1062no—
629C22—1062no—
630C22—1062no—
631C22—1062no—
632C22—1062no—
633C22—2121no—
634C22—2121no—
635C22—2121no—
636C22—2121no—
637C22—2121no—
638C22—2121no—
639C22—2121no—
640C22—2121no—
641C22C2×C22⋊C4×D5311yescharacteristic
642C22C4×#1171994511yescharacteristic
643C22#1681110311yescharacteristic
644C22C2×M16×D5311yescharacteristic
645C22C2×M16×D5311yescharacteristic
646C22C2×#1313322511yescharacteristic
647C22#1681138911yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—534no—
654C4—1062no—
655C4—1062no—
656C4—1062no—
657C4—2121no—
658C4—2121no—
659C4#1681128212yes—
660C4#1681129412yes—
661C2—42no—
662C2—2122no—
663C2—2122no—
664C2#2027755511yescharacteristic
665C2#407×D5311yescharacteristic
666C2#2028316511yescharacteristic
667C2#2028367711yescharacteristic
668C2#2028372611yescharacteristic
669C2#2028383411yescharacteristic
670C2#2028384211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1749C2
prime radicalC22×#11719945C4
cosocleC22×C4×C53C24
hypercenter#1749D53
hypocenterC53#22532
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#16810751C2semidirect
3. #20275513C2semidirect
4. #20275511C2semidirect
5. #20275507C2semidirect
6. #20274695C2semidirect
7. #20274694C2semidirect
8. #20274694C2semidirect
9. #20274476C2semidirect
10. #20274472C2semidirect
11. C53×#1749C2semidirect
176. C2×C8×C53#41semidirect
335. #1749D53semidirect
380. C53#22532semidirect
518. C2×C8#11719945semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199452
C2×#168107512
#202755132
#202755112
#202755072
#202746952
#202746942
#202744762
#202744722
C53×#17492
#2253253