Finite Group Databasebeta

← #27400720

Subgroups — #27400720

Order 13568.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811329C211yesmaximal, characteristic
2C2×#16810752C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic106C211yesmaximal, characteristic, complemented
4#20275685C211yesmaximal, characteristic, complemented
5#20275682C211yesmaximal, characteristic, complemented
6#20275674C211yesmaximal, characteristic, complemented
7#20275669C211yesmaximal, characteristic, complemented
8#20275661C211yesmaximal, characteristic, complemented
9#20274716C211yesmaximal, characteristic
10#20274714C211yesmaximal, characteristic
11#20274714C211yesmaximal, characteristic
12#20274543C211yesmaximal, characteristic
13#20274539C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1989C211yesmaximal, characteristic, complemented
16#16811329—24no—
17#16811329C418yescomplemented
18C2×#13133216C2211yescharacteristic
19C2×#13133208C2211yescharacteristic
20C2×#13133197—21no—
21C2×#13133195—21no—
22C2×#13133195—21no—
23C2×C4×Dic106—21no—
24C2×C4×Dic106C2211yescharacteristic
25C2×C4×D212C2211yescharacteristic
26C2×#13133179C2211yescharacteristic
27C2×#13133179C2211yescharacteristic
28C2×C4⋊C4×D53—21no—
29#16810752—24no—
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133139C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2211yescharacteristic
36C2×#13133128C2211yescharacteristic
37C2×#13133127C2211yescharacteristic
38C2×C8×Dic53C2211yescharacteristic
39#16810521—21no—
40#16810520—21no—
41#16810520—21no—
42#16810493C2211yescharacteristic
43#16810492C2211yescharacteristic
44#16810492C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#125×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106D411yescharacteristic
64C22×D212C2311yescharacteristic
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53—21no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133216—22no—
71#13133216C2×C414yes—
72#13133208—22no—
73#13133208C2×C414yes—
74#13133197—28no—
75#13133195—28no—
76#13133195—28no—
77C4×Dic106—44no—
78C4×Dic106C2×C414yes—
79C4×Dic106D414yes—
80C4×D212—22no—
81C4×D212C2×C414yes—
82#13133179—22no—
83#13133179—22no—
84#13133179C2×C414yes—
85#13133179C2×C414yes—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53D411yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919D411yescharacteristic
100C2×#11719919D411yescharacteristic
101C4⋊C4×D53—28no—
102C2×#11719921—21no—
103C2×#11719921—21no—
104C2×#11719921—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53—21no—
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109#13133146—22no—
110#13133146—22no—
111#13133139—22no—
112#13133139—22no—
113#13133137—22no—
114#13133137Q814yescomplemented
115#13133128—22no—
116#13133127—22no—
117C8×Dic53—22no—
118#13133124—21no—
119#13133124—21no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122#125×C53—22no—
123#125×C53C2×C414yes—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C42×C53—21no—
131C2×C42×C53C2311yescharacteristic
132C4⋊C8×C53—22no—
133#95×C53—21no—
134C23×D53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212C22×C412yes—
153C2×D212C2×Q812yes—
154C2×D212#4112yes—
155C2×C4×D53—24no—
156C2×C4×D53—24no—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106—24no—
167C2×Dic106—81no—
168C2×Dic106—81no—
169C2×Dic106C22⋊C414yes—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C22×C412yes—
184C4⋊C4×C53C22×C412yes—
185C4⋊C4×C53C2×Q812yes—
186C42×C53—42no—
187C42×C53C22×C412yes—
188C42×C53C2×D412yes—
189#11719921—24no—
190#11719921—24no—
191#11719921—24no—
192C4⋊Dic53—21no—
193C4⋊Dic53—24no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200C4⋊Dic53C22⋊C414yes—
201C4⋊Dic53C22×C412yes—
202#11719919—24no—
203#11719919—24no—
204#11719919—42no—
205#11719919—42no—
206#11719919—81no—
207#11719919—81no—
208#11719919C22⋊C414yes—
209#11719919C22⋊C414yes—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53C22×C412yes—
213C4×Dic53C22×C412yes—
214C4×Dic53C2×D412yes—
215C4×Dic53C2×D412yes—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322C2×Q812yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C23×C411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
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×Dic53C2×C22⋊C412yes—
270C2×Dic53C2×C22⋊C412yes—
271C2×Dic53C2×C22⋊C412yes—
272C2×Dic53C2×C22⋊C412yes—
273C2×Dic53AGL(1,ℤ/8)12yes—
274C2×Dic53#13712yes—
275D212—22no—
276D212—22no—
277D212—22no—
278D212C4×Q814yescomplemented
279C4×D53—48no—
280C4×D53—84no—
281C4×D53—84no—
282Dic106—28no—
283Dic106—84no—
284C8×C53—41no—
285C8×C53—41no—
286#88322—22no—
287#88322—41no—
288#22522—531nomaximal, Sylow
289C22×C53C22×C22⋊C411yescharacteristic
290C22×C53#54811yescharacteristic
291C22×C53D4×Q811yescharacteristic
292C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C53C2×#13711yescharacteristic
294C22×C53#57611yescharacteristic
295C22×C53#57711yescharacteristic
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—81no—
309C4×C53C2×C4×Q811yescharacteristic
310C4×C53#51911yescharacteristic
311C4×C53#54811yescharacteristic
312C4×C53D4×Q811yescharacteristic
313Dic53—22no—
314Dic53—22no—
315Dic53—22no—
316Dic53—42no—
317Dic53—81no—
318Dic53—81no—
319Dic53#41714yes—
320C2×#554—531no—
321C2×#446—531no—
322Q8×C4⋊C4—531no—
323#1992—531no—
324#1992—531no—
325#1990—531no—
326#1989—531no—
327#1989D5311yescharacteristic, complemented
328#1987—531no—
329#1802—531no—
330#1802—531no—
331#1802—531no—
332#1762—531no—
333#1762—531no—
334C4×#102—531no—
335C2×C53Q8×C22⋊C411yescharacteristic
336C2×C53C2×#41711yescharacteristic
337C2×C53#289711yescharacteristic
338C2×C53#336611yescharacteristic
339C2×C53#337211yescharacteristic
340C2×C53#338411yescharacteristic
341C2×C53#338511yescharacteristic
342D53—44no—
343#554—538no—
344#554—1064no—
345C2×C4⋊Q8—531no—
346C2×#125—531no—
347C2×#125C2×D5311yescharacteristic
348C2×C22⋊Q8—1061no—
349C2×C22⋊Q8—1061no—
350C2×C22⋊Q8—1061no—
351C2×C4×Q8—531no—
352C2×C4×Q8—1061no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×C4×D4—531no—
356C22×C4⋊C4—1061no—
357#446—1064no—
358C2×C4⋊C8—531no—
359C2×C4⋊C8—531no—
360C2×C4⋊C8C2×D5311yescharacteristic
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#97—531no—
366C2×#97—531no—
367C2×C4×C8—531no—
368#389—1061no—
369#389—1061no—
370#389—1061no—
371#384—531no—
372#384—531no—
373#384—531no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4C2×D5311yescharacteristic
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#340C2×D5311yescharacteristic
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388#340C2×D5311yescharacteristic
389C53#2252211yescharacteristic, complemented, Sylow
390C22×Q8—531no—
391C22×Q8—1061no—
392C22×D4—531no—
393C23×C4—1061no—
394C23×C4—1061no—
395C23×C4—1061no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C4⋊Q8—534no—
401C4⋊Q8—1062no—
402#125—22no—
403#125—534no—
404#125—1062no—
405#125C4×D5314yes—
406C22⋊Q8—1068no—
407C22⋊Q8—1068no—
408C22⋊Q8—1068no—
409C4×Q8—534no—
410C4×Q8—534no—
411C4×Q8—2124no—
412C4×D4—534no—
413C4×D4—534no—
414C4×D4—534no—
415C4×D4—1062no—
416C4×D4—1062no—
417C4×D4—1062no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1068no—
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⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C42—21no—
444C2×C42—531no—
445C2×C42—531no—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449C4⋊C8—22no—
450C4⋊C8—1062no—
451C4⋊C8—1062no—
452#102—534no—
453#102—1062no—
454#102—1062no—
455#102—1062no—
456#97—1062no—
457#97—1062no—
458C4×C8—1062no—
459#95—21no—
460#95—1061no—
461#95—1061no—
462C24—1061no—
463C2×Q8—534no—
464C2×Q8—1062no—
465C2×Q8—1062no—
466C2×Q8—1064no—
467C2×Q8—4241no—
468C2×Q8—4241no—
469C2×D4—532no—
470C2×D4—532no—
471C2×D4—532no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1061no—
486C22×C4—1064no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—4241no—
490C22×C4—4241no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4D4×D5311yescharacteristic
496C22×C4D4×D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—532no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—534no—
518C4⋊C4—534no—
519C4⋊C4—534no—
520C4⋊C4—532no—
521C4⋊C4—1064no—
522C4⋊C4—1064no—
523C4⋊C4—1064no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—4241no—
530C4⋊C4—4241no—
531C4⋊C4—4241no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4C2×C4×D5312yes—
536C4⋊C4C2×C4×D5312yes—
537C4⋊C4Q8×D5312yes—
538C22⋊C4—1064no—
539C22⋊C4—1064no—
540C22⋊C4—1064no—
541C42—42no—
542C42—532no—
543C42—532no—
544C42—532no—
545C42—532no—
546C42—2122no—
547C42—2122no—
548C42C2×C4×D5312yes—
549C42D4×D5312yes—
550C23—1062no—
551C23—1062no—
552C23—1062no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23C2×D4×D5311yescharacteristic
558Q8—1068no—
559Q8—4244no—
560D4—534no—
561D4—1062no—
562D4—1062no—
563D4—1062no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—532no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1062no—
593C2×C4—4244no—
594C2×C4—4244no—
595C2×C4—2128no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4C22⋊C4×D5312yes—
605C2×C4C22⋊C4×D5312yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4C2×D4×D5311yescharacteristic
608C2×C4C2×Q8×D5311yescharacteristic
609C2×C4#41×D5311yescharacteristic
610C2×C4#1313326411yescharacteristic
611C2×C4#1313326511yescharacteristic
612C8—41no—
613C8—41no—
614C8—1062no—
615C8—2121no—
616C22—1064no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22—2122no—
622C22—2122no—
623C22C2×C22⋊C4×D5311yescharacteristic
624C22#1681128311yescharacteristic
625C22#1681128611yescharacteristic
626C22AGL(1,ℤ/8)×D5311yescharacteristic
627C22#137×D5311yescharacteristic
628C22#1681142511yescharacteristic
629C22#1681143011yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—534no—
637C4—1062no—
638C4—1062no—
639C4—1062no—
640C4—4241no—
641C4—4241no—
642C4—2122no—
643C4C4×Q8×D5311yescharacteristic
644C4#1681109911yescharacteristic
645C4#1681128111yescharacteristic
646C4#1681128811yescharacteristic
647C2—2124no—
648C2#2027754911yescharacteristic
649C2#417×D5311yescharacteristic
650C2#2028323711yescharacteristic
651C2#2028742511yescharacteristic
652C2#2028749911yescharacteristic
653C2#2028761711yescharacteristic
654C2#2028762811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1989C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1989D53
hypocenterC53#22522
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#16810752C2semidirect
3. C4⋊C4×Dic106C2semidirect
4. #20275685C2semidirect
5. #20275682C2semidirect
6. #20275674C2semidirect
7. #20275669C2semidirect
8. #20275661C2semidirect
15. C53×#1989C2semidirect
17. #16811329C4semidirect
114. #13133137Q8semidirect
278. D212C4×Q8semidirect
327. #1989D53semidirect
389. C53#22522semidirect

Maximal subgroups

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

subgroupindex
C2×#168113292
C2×#168107522
C4⋊C4×Dic1062
#202756852
#202756822
#202756742
#202756692
#202756612
#202747162
#202747142
#202745432
#202745392
#202745292
C53×#19892
#2252253