Finite Group Databasebeta

← #27439939

Subgroups — #27439939

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811251C211yesmaximal, characteristic
2C2×#16811032C211yesmaximal, characteristic, complemented
3C2×#16811025C211yesmaximal, characteristic, complemented
4#20280545C211yesmaximal, characteristic
5#20276054C211yesmaximal, characteristic, complemented
6#20276054C211yesmaximal, characteristic, complemented
7#20276049C211yesmaximal, characteristic, complemented
8#20276044C211yesmaximal, characteristic, complemented
9#20275684C211yesmaximal, characteristic, complemented
10#20275676C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274916C211yesmaximal, characteristic
13#20274910C211yesmaximal, characteristic
14#20274905C211yesmaximal, characteristic
15C53×#1750C211yesmaximal, characteristic
16#16811251—24no—
17#16811251—24no—
18C2×#13133217C2211yescharacteristic
19C2×#13133217C2211yescharacteristic
20C2×#13133216—21no—
21C2×#13133209C2211yescharacteristic
22C2×#13133209C2211yescharacteristic
23C2×#13133209C2211yescharacteristic
24C2×#13133208—21no—
25C2×#13133194—21no—
26C2×#13133194—21no—
27C2×#13133176C2211yescharacteristic
28C22×C4⋊Dic53C2211yescharacteristic
29#16811032—24no—
30#16811025—24no—
31C2×#13133151C2211yescharacteristic
32C2×#13133151C2211yescharacteristic
33C2×#13133151C2211yescharacteristic
34C2×#13133151C2211yescharacteristic
35C2×#13133149C2211yescharacteristic
36C2×#13133149C2211yescharacteristic
37C2×#13133149C2211yescharacteristic
38C2×#13133149C2211yescharacteristic
39#16810556—21no—
40#16810556—21no—
41#16810546—21no—
42#16810546—21no—
43#16810521C2211yescharacteristic
44#16810521C2211yescharacteristic
45#16810493C2211yescharacteristic
46#16810483C2211yescharacteristic
47#16810483C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54#16810386C2211yescharacteristic
55#16810386C2211yescharacteristic
56C2×#117×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C53×#336C2211yescharacteristic
62C53×#336C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C23×Dic53—21no—
66#13133217—22no—
67#13133217—22no—
68#13133217—22no—
69#13133217D414yes—
70#13133216—28no—
71#13133209—22no—
72#13133209—22no—
73#13133209—22no—
74#13133209—22no—
75#13133209—22no—
76#13133209D414yes—
77#13133208—28no—
78#13133194—28no—
79#13133194—28no—
80#13133176—22no—
81#13133176—22no—
82C2×C4⋊Dic53—22no—
83C2×C4⋊Dic53—22no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×C4⋊Dic53C2311yescharacteristic
94C2×C4⋊Dic53C2311yescharacteristic
95C2×C4⋊Dic53C2311yescharacteristic
96C2×C4⋊Dic53C2311yescharacteristic
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919—21no—
100C2×#11719919—21no—
101C2×#11719919—21no—
102C2×#11719919—21no—
103C2×#11719919—21no—
104C2×#11719919—21no—
105C2×C22⋊Dic53—21no—
106C2×C22⋊Dic53—21no—
107C2×C22⋊Dic53—21no—
108C2×C4×Dic53—21no—
109#13133151—22no—
110#13133151—22no—
111#13133151—22no—
112#13133151—22no—
113#13133149—22no—
114#13133149—22no—
115#13133149—22no—
116#13133149—22no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121C23×C4×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126#117×C53—22no—
127#117×C53—22no—
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C4⋊C4×C53D411yescharacteristic
130C2×C22⋊C4×C53D411yescharacteristic
131C2×C22⋊C4×C53D411yescharacteristic
132C2×C42×C53C2311yescharacteristic
133C2×C42×C53C2311yescharacteristic
134C22⋊C8×C53Q814yescomplemented
135C22⋊C8×C53Q814yescomplemented
136#97×C53—22no—
137C4×C8×C53—22no—
138C24×C53C2×D411yescharacteristic
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×Q812yes—
149C22×C4×C53C2×Q812yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—24no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—81no—
161C22×Dic53—81no—
162C2×Dic106—24no—
163C2×Dic106—24no—
164C2×Dic106—81no—
165C2×Dic106—81no—
166C2×Dic106—81no—
167C2×Dic106—81no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53C2×Q812yes—
177C2×C8×C53C2×Q812yes—
178C2×C8×C53C2×Q812yes—
179C2×C8×C53C2×Q812yes—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53C2×D412yes—
187C42×C53C2×D412yes—
188C22⋊Dic53—24no—
189C22⋊Dic53—81no—
190C22⋊Dic53—81no—
191C4⋊Dic53—24no—
192C4⋊Dic53—24no—
193C4⋊Dic53—24no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—81no—
207C4⋊Dic53—81no—
208C4⋊Dic53C2×D412yes—
209C4⋊Dic53C2×D412yes—
210C4⋊Dic53C2×D412yes—
211C4⋊Dic53C2×D412yes—
212#11719919—24no—
213#11719919—24no—
214#11719919—24no—
215#11719919—24no—
216#11719919—42no—
217#11719919—42no—
218#11719919—81no—
219#11719919—81no—
220C4×Dic53—24no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53C4⋊Q812yes—
228C23×C53C22×D411yescharacteristic
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—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53#12012yes—
246C2×C4×C53#12012yes—
247C2×C4×C53C4⋊Q812yes—
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C22×Q811yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—84no—
279Dic106—84no—
280Dic106—84no—
281C8×C53—22no—
282C8×C53—22no—
283C8×C53—22no—
284C8×C53—22no—
285#25822—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×#12011yescharacteristic
294C22×C53C2×C4⋊Q811yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#57611yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57811yescharacteristic
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53D4×Q811yescharacteristic
306C4×C53D4×Q811yescharacteristic
307C4×C53D4×Q811yescharacteristic
308C4×C53D4×Q811yescharacteristic
309Dic53—42no—
310Dic53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#544—531no—
318C2×#501—531no—
319C2×#499—531no—
320#2629—531no—
321#2059—531no—
322#2059—531no—
323#2058—531no—
324#2057—531no—
325#1992—531no—
326#1990—531no—
327#1854—531no—
328#1854—531no—
329#1853—531no—
330#1852—531no—
331#1750D5311yescharacteristic
332C2×C53—42no—
333C2×C53#243411yescharacteristic
334C2×C53#301211yescharacteristic
335C2×C53#301311yescharacteristic
336C2×C53#316311yescharacteristic
337C2×C53#316411yescharacteristic
338C2×C53#324111yescharacteristic
339C2×C53#324211yescharacteristic
340#544—1064no—
341#544—1064no—
342C2×C4⋊Q8—531no—
343C2×C4⋊Q8—531no—
344C2×C4⋊Q8—1061no—
345C2×#125—531no—
346C2×#125—531no—
347C2×#125—531no—
348C2×#125—1061no—
349C2×C22⋊Q8—1061no—
350C2×C22⋊Q8—1061no—
351C2×#117—531no—
352C2×#117C2×D5311yescharacteristic
353C22×C4⋊C4—531no—
354#501—1064no—
355#499—1064no—
356C2×#107—531no—
357C2×#107—531no—
358C2×#107—531no—
359C2×#107—531no—
360C2×#106—531no—
361C2×#106—531no—
362C2×#106—531no—
363C2×#106—531no—
364C2×C22⋊C8C2×D5311yescharacteristic
365C2×C22⋊C8C2×D5311yescharacteristic
366C2×#97C2×D5311yescharacteristic
367C2×C4×C8C2×D5311yescharacteristic
368#398—1061no—
369#398—1061no—
370#396—1061no—
371#396—1061no—
372#389—531no—
373#389—531no—
374#384—531no—
375#380—531no—
376#380—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#336C2×D5311yescharacteristic
386#336C2×D5311yescharacteristic
387C53#2582211yescharacteristic, complemented, Sylow
388C22×Q8—1061no—
389C22×Q8—1061no—
390C23×C4—1061no—
391C23×C4C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396C4⋊Q8—534no—
397C4⋊Q8—1062no—
398C4⋊Q8—1062no—
399C4⋊Q8—1068no—
400C4⋊Q8—1062no—
401#125—534no—
402#125—1062no—
403#125—1062no—
404#125—1068no—
405#125—1062no—
406#125—1062no—
407#125—1062no—
408C22⋊Q8—1068no—
409C22⋊Q8—1068no—
410#117—22no—
411#117—22no—
412#117—1062no—
413#117—1062no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4D21211yescharacteristic
438C2×C4⋊C4D21211yescharacteristic
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4D21211yescharacteristic
443C2×C22⋊C4D21211yescharacteristic
444C2×C42—1061no—
445C2×C42C22×D5311yescharacteristic
446C2×C42C22×D5311yescharacteristic
447#107—1062no—
448#107—1062no—
449#107—1062no—
450#107—1062no—
451#106—1062no—
452#106—1062no—
453#106—1062no—
454#106—1062no—
455C22⋊C8Dic10614yescomplemented
456C22⋊C8Dic10614yescomplemented
457#97—22no—
458C4×C8—22no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463C24C2×D21211yescharacteristic
464C2×Q8—1064no—
465C2×Q8—1064no—
466C2×Q8—4241no—
467C2×Q8—4241no—
468C2×Q8—4241no—
469C2×Q8—4241no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—1064no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—4241no—
484C22×C4—4241no—
485C22×C4C2×Dic10612yes—
486C22×C4C2×Dic10612yes—
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4C2×D21211yescharacteristic
492C22×C4C23×D5311yescharacteristic
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8C2×Dic10612yes—
502C2×C8C2×Dic10612yes—
503C2×C8C2×Dic10612yes—
504C2×C8C2×Dic10612yes—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—532no—
508C4⋊C4—532no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—1064no—
512C4⋊C4—1064no—
513C4⋊C4—1064no—
514C4⋊C4—1064no—
515C4⋊C4—1064no—
516C4⋊C4—1064no—
517C4⋊C4—1064no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—4241no—
531C4⋊C4—4241no—
532C4⋊C4—4241no—
533C4⋊C4—4241no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—1064no—
539C22⋊C4—4241no—
540C22⋊C4—4241no—
541C42—21no—
542C42—21no—
543C42—1064no—
544C42D4×D5312yes—
545C42D4×D5312yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#1313321712yes—
553C23C22×D21211yescharacteristic
554Q8—4244no—
555Q8—4244no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—4244no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313318712yes—
598C2×C4#1313318712yes—
599C2×C4#1313321712yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C2×D4×D5311yescharacteristic
602C2×C4C22×Dic10611yescharacteristic
603C2×C4C22×Dic10611yescharacteristic
604C2×C4#1313326411yescharacteristic
605C2×C4#1313326411yescharacteristic
606C8—22no—
607C8—22no—
608C8—22no—
609C8—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22C2×#1313318711yescharacteristic
618C22C2×#1313321711yescharacteristic
619C22#1681119911yescharacteristic
620C22#1681142811yescharacteristic
621C22#1681143511yescharacteristic
622C22#1681143511yescharacteristic
623C22#1681144011yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—4241no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—2122no—
634C4—2122no—
635C4—2122no—
636C4—2122no—
637C4D4×Dic10611yescharacteristic
638C4D4×Dic10611yescharacteristic
639C4D4×Dic10611yescharacteristic
640C4D4×Dic10611yescharacteristic
641C2—42no—
642C2#2027835911yescharacteristic
643C2#2028402211yescharacteristic
644C2#2028403311yescharacteristic
645C2#2028524611yescharacteristic
646C2#2028525411yescharacteristic
647C2#2028597011yescharacteristic
648C2#2028598511yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1750C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1750D53
hypocenterC53#25822
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#16811032C2semidirect
3. C2×#16811025C2semidirect
5. #20276054C2semidirect
6. #20276054C2semidirect
7. #20276049C2semidirect
8. #20276044C2semidirect
9. #20275684C2semidirect
10. #20275676C2semidirect
134. C22⋊C8×C53Q8semidirect
135. C22⋊C8×C53Q8semidirect
387. C53#25822semidirect
455. C22⋊C8Dic106semidirect
456. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168112512
C2×#168110322
C2×#168110252
#202805452
#202760542
#202760492
#202760442
#202756842
#202756762
#202749162
#202749102
#202749052
C53×#17502
#2582253