Finite Group Databasebeta

← #27400969

Subgroups — #27400969

Order 13568.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811342C211yesmaximal, characteristic
2C2×#16810753C211yesmaximal, characteristic, complemented
3#20277667C211yesmaximal, characteristic, complemented
4#20275648C211yesmaximal, characteristic
5#20275646C211yesmaximal, characteristic
6#20275636C211yesmaximal, characteristic, complemented
7#20275533C211yesmaximal, characteristic, complemented
8#20275530C211yesmaximal, characteristic
9#20274727C211yesmaximal, characteristic, complemented
10#20274723C211yesmaximal, characteristic
11#20274718C211yesmaximal, characteristic, complemented
12#20274547C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1958C211yesmaximal, characteristic
16#16811342—24no—
17#16811342C418yescomplemented
18C2×#13133215C2211yescharacteristic
19C2×#13133210—21no—
20C2×#13133210—21no—
21C2×#13133204C2211yescharacteristic
22C2×#13133194—21no—
23C2×Q8×Dic53C2211yescharacteristic
24C2×C4×Dic106C2211yescharacteristic
25C2×C4×Dic106C2211yescharacteristic
26C2×D4×Dic53C2211yescharacteristic
27C2×#13133173—21no—
28#16810753—24no—
29C2×C4⋊#88322C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
36C2×#13133140C2211yescharacteristic
37C2×#13133140C2211yescharacteristic
38C2×#13133138C2211yescharacteristic
39C2×#13133138C2211yescharacteristic
40C2×#13133127C2211yescharacteristic
41C2×#13133127C2211yescharacteristic
42#16810527—21no—
43#16810508—21no—
44#16810505—21no—
45#16810492C2211yescharacteristic
46#16810488C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48C4⋊C4×Dic53C2211yescharacteristic
49C4×C4⋊Dic53C2211yescharacteristic
50#16810474—21no—
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#124×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C53×#382C2211yescharacteristic
62C22×Dic106D411yescharacteristic
63C22×Dic106C2311yescharacteristic
64C23×Dic53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133215—22no—
68#13133215C2×C414yes—
69#13133210—28no—
70#13133210—28no—
71#13133204—22no—
72#13133204C2×C414yes—
73#13133194—28no—
74Q8×Dic53—22no—
75Q8×Dic53C2×C414yes—
76C4×Dic106—22no—
77C4×Dic106C2×C414yes—
78C4×Dic106C2×C414yes—
79C4×Dic106D414yes—
80D4×Dic53—22no—
81D4×Dic53C2×C414yes—
82#13133173—28no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53D411yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919D411yescharacteristic
94C2×#11719919D411yescharacteristic
95C2×C22⋊Dic53—21no—
96C2×C22⋊Dic53—21no—
97C2×C22⋊Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53C2311yescharacteristic
104C2×C4×Dic53C2311yescharacteristic
105C4⋊#88322—22no—
106#13133146—22no—
107#13133146—22no—
108#13133142—22no—
109#13133142—22no—
110#13133141—22no—
111#13133141—41no—
112#13133140—41no—
113#13133140—41no—
114#13133138—22no—
115#13133138—41no—
116#13133127—22no—
117#13133127—22no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121#13133124—21no—
122#13133124—21no—
123C22×Q8×C53C2311yescharacteristic
124C22×D4×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127#124×C53—22no—
128#124×C53C2×C414yes—
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132C2×C22⋊C4×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#103×C53—22no—
135#103×C53—41no—
136#102×C53—22no—
137#102×C53—41no—
138C4×C8×C53—22no—
139#95×C53—21no—
140#95×C53—21no—
141C24×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53C22×C412yes—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53C22×C412yes—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—24no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—41no—
177C2×Dic106—41no—
178C2×Dic106C22⋊C414yes—
179C2×Dic106C22×C412yes—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53#4112yes—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C22⋊C4×C53—24no—
191C22⋊C4×C53—24no—
192C42×C53C22×C412yes—
193C42×C53C2×D412yes—
194C22⋊Dic53—24no—
195C22⋊Dic53—24no—
196C22⋊Dic53—24no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—24no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53C22⋊C414yes—
203C4⋊Dic53C22×C412yes—
204C4⋊Dic53#4112yes—
205#11719919—24no—
206#11719919—24no—
207#11719919—24no—
208#11719919—42no—
209#11719919—81no—
210#11719919C22⋊C414yes—
211#11719919C22⋊C414yes—
212C4×Dic53—24no—
213C4×Dic53—24no—
214C4×Dic53—42no—
215C4×Dic53—81no—
216C4×Dic53—81no—
217C4×Dic53C22×C412yes—
218C4×Dic53C22×C412yes—
219C4×Dic53C2×D412yes—
220C4×Dic53C2×D412yes—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C23×C53—22no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53C22×D411yescharacteristic
235Q8×C53—22no—
236Q8×C53—22no—
237Q8×C53—41no—
238Q8×C53—41no—
239D4×C53—22no—
240D4×C53—22no—
241D4×C53—41no—
242D4×C53—41no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
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×C53C2×C22⋊C412yes—
257C2×C4×C53C2×C22⋊C412yes—
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C532−1+411yescharacteristic
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—48no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53C2×C22⋊C412yes—
286C2×Dic53C2×C22⋊C412yes—
287C2×Dic53C2×C22⋊C412yes—
288C2×Dic53C2×C22⋊C412yes—
289Dic106—22no—
290Dic106—22no—
291Dic106—44no—
292Dic106—41no—
293Dic106—41no—
294C8×C53—22no—
295C8×C53—41no—
296#88322—41no—
297#88322—41no—
298#22548—531nomaximal, Sylow
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53C22×C22⋊C411yescharacteristic
307C22×C53#54811yescharacteristic
308C22×C53#55011yescharacteristic
309C22×C53#57511yescharacteristic
310C22×C53#57511yescharacteristic
311C22×C53#57711yescharacteristic
312C22×C53#57811yescharacteristic
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—42no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53C4×#4111yescharacteristic
319C4×C53#52011yescharacteristic
320C4×C53#54811yescharacteristic
321C4×C53#55011yescharacteristic
322Dic53—22no—
323Dic53—22no—
324Dic53—22no—
325Dic53—22no—
326Dic53—42no—
327Dic53—42no—
328Dic53—42no—
329Dic53—81no—
330C2×#556—531no—
331C2×#446—531no—
332#2355—531no—
333#1983—531no—
334#1983—531no—
335#1981—531no—
336#1960—531no—
337#1959—531no—
338#1958D5311yescharacteristic
339#1805—531no—
340#1804—531no—
341#1803—531no—
342#1763—531no—
343#1762—531no—
344C4×#103—531no—
345C2×C53—82no—
346C2×C53#234111yescharacteristic
347C2×C53#289611yescharacteristic
348C2×C53#289811yescharacteristic
349C2×C53#337611yescharacteristic
350C2×C53#337711yescharacteristic
351C2×C53#340111yescharacteristic
352C2×C53#340211yescharacteristic
353#556—538no—
354#556—1064no—
355C2×C4⋊Q8—531no—
356C2×#126—1061no—
357C2×#126—1061no—
358C2×#124—531no—
359C2×#124C2×D5311yescharacteristic
360C2×C22⋊Q8—1061no—
361C2×C4×Q8—531no—
362C2×C4×Q8—531no—
363C2×C4×Q8—531no—
364C2×C4×D4—531no—
365C2×#117—1061no—
366#446—1064no—
367C2×C4⋊C8—531no—
368C2×C4⋊C8—531no—
369C2×C4⋊C8—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103C2×D5311yescharacteristic
377C2×#103C2×D5311yescharacteristic
378C2×#102—531no—
379C2×#102—531no—
380C2×#102C2×D5311yescharacteristic
381C2×#102C2×D5311yescharacteristic
382C2×#97—531no—
383C2×#97—531no—
384C2×C4×C8C2×D5311yescharacteristic
385#391—1061no—
386#387—1061no—
387#387—1061no—
388#384—531no—
389#382—531no—
390#382C2×D5311yescharacteristic
391C4×C4⋊C4—531no—
392C4×C4⋊C4—531no—
393C4×C4⋊C4—531no—
394C4×C4⋊C4—1061no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399C53#2254811yescharacteristic, complemented, Sylow
400C22×Q8—531no—
401C22×Q8—531no—
402C22×Q8C22×D5311yescharacteristic
403C22×D4C22×D5311yescharacteristic
404C23×C4—1061no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409C4⋊Q8—534no—
410C4⋊Q8—1062no—
411#126—1068no—
412#126—1068no—
413#124—22no—
414#124—534no—
415#124—1062no—
416#124C4×D5314yes—
417C22⋊Q8—1068no—
418C4×Q8—534no—
419C4×Q8—534no—
420C4×Q8—534no—
421C4×Q8—534no—
422C4×Q8—1062no—
423C4×Q8—1062no—
424C4×D4—534no—
425C4×D4—1062no—
426#117—1068no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—1061no—
449C2×C42—1061no—
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42C22×D5311yescharacteristic
454C4⋊C8—1062no—
455C4⋊C8—1062no—
456C4⋊C8—1062no—
457#103—22no—
458#103—41no—
459#103—1062no—
460#103—1062no—
461#103—1062no—
462#103—2121no—
463#103—2121no—
464#103—2121no—
465#102—22no—
466#102—41no—
467#102—1062no—
468#102—2121no—
469#97—1062no—
470#97—1062no—
471C4×C8—22no—
472#95—21no—
473#95—21no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479C24—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—534no—
486C2×Q8—532no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8—1061no—
491C2×Q8—1061no—
492C2×Q8—2121no—
493C2×Q8—2121no—
494C2×Q8C2×C4×D5312yes—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C2×C4×D5312yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—531no—
508C22×C4—1064no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4D4×D5311yescharacteristic
516C22×C4D4×D5311yescharacteristic
517C22×C4C23×D5311yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8#1171994712yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—534no—
535C4⋊C4—534no—
536C4⋊C4—534no—
537C4⋊C4—532no—
538C4⋊C4—532no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1064no—
542C4⋊C4—1064no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—4241no—
548C4⋊C4—2122no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—1064no—
552C22⋊C4—1064no—
553C22⋊C4—1064no—
554C42—532no—
555C42—532no—
556C42—532no—
557C42—532no—
558C42—1064no—
559C42—1064no—
560C42—4241no—
561C42—4241no—
562C42—2122no—
563C42C2×C4×D5312yes—
564C42D4×D5312yes—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D5311yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—1062no—
578Q8—1062no—
579Q8—2124no—
580Q8—2121no—
581Q8—2121no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—532no—
600C2×C4—532no—
601C2×C4—532no—
602C2×C4—532no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—2128no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4C22⋊C4×D5312yes—
625C2×C4C22⋊C4×D5312yes—
626C2×C4C22×C4×D5311yescharacteristic
627C2×C4C2×D4×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4C2×#1171994711yescharacteristic
630C2×C4#1313326211yescharacteristic
631C2×C4#1313326411yescharacteristic
632C8—22no—
633C8—41no—
634C8—2121no—
635C8—2121no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×C22⋊C4×D5311yescharacteristic
644C22#1681128111yescharacteristic
645C22#1681129311yescharacteristic
646C22#1681142011yescharacteristic
647C22#1681142011yescharacteristic
648C22#1681143011yescharacteristic
649C22#1681143711yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—1062no—
656C4—1062no—
657C4—1062no—
658C4—1062no—
659C4—4241no—
660C4—2122no—
661C4—2122no—
662C4—2122no—
663C4C4×#1171994711yescharacteristic
664C4#1681110211yescharacteristic
665C4#1681127911yescharacteristic
666C4#1681129111yescharacteristic
667C2—82no—
668C2#2027755411yescharacteristic
669C2#2028323011yescharacteristic
670C2#2028324411yescharacteristic
671C2#2028754811yescharacteristic
672C2#2028756311yescharacteristic
673C2#2028778711yescharacteristic
674C2#2028779711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1958C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1958D53
hypocenterC53#22548
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16810753C2semidirect
3. #20277667C2semidirect
6. #20275636C2semidirect
7. #20275533C2semidirect
9. #20274727C2semidirect
11. #20274718C2semidirect
12. #20274547C2semidirect
14. C4×#13133142C2semidirect
17. #16811342C4semidirect
399. C53#22548semidirect

Maximal subgroups

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

subgroupindex
C2×#168113422
C2×#168107532
#202776672
#202756482
#202756462
#202756362
#202755332
#202755302
#202747272
#202747232
#202747182
#202745472
#202745422
C4×#131331422
C53×#19582
#2254853