Finite Group Databasebeta

← #27446569

Subgroups — #27446569

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810901C211yesmaximal, characteristic, complemented
2C22×#13133151C211yesmaximal, characteristic, complemented
3#20278418C211yesmaximal, characteristic, complemented
4#20277735C211yesmaximal, characteristic
5#20276241C211yesmaximal, characteristic, complemented
6#20276205C211yesmaximal, characteristic, complemented
7#20276101C211yesmaximal, characteristic, complemented
8#20276042C211yesmaximal, characteristic, complemented
9#20275472C211yesmaximal, characteristic, complemented
10#20275285C211yesmaximal, characteristic, complemented
11#20275033C211yesmaximal, characteristic
12#20274964C211yesmaximal, characteristic, complemented
13#20274902C211yesmaximal, characteristic, complemented
14#20274893C211yesmaximal, characteristic, complemented
15C53×#1893C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133194—21no—
18C2×D4×Dic53—21no—
19C2×D4×Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810901—24no—
23C2×#13133151—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133150C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35#16810557—21no—
36#16810544—21no—
37#16810526C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810519C2211yescharacteristic
41#16810516—21no—
42#16810503—21no—
43#16810498—21no—
44#16810495—21no—
45#16810492C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C4×C22⋊Dic53C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×#121×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C23×Dic53—21no—
63C23×Dic53—21no—
64C23×Dic53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216—41no—
69#13133194—44no—
70D4×Dic53—22no—
71D4×Dic53—41no—
72D4×Dic53—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53—41no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53D411yescharacteristic
93C2×C22⋊Dic53D411yescharacteristic
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133151—22no—
98#13133151—22no—
99#13133151—22no—
100#13133151D414yescomplemented
101#13133150—22no—
102#13133150—41no—
103#13133146—22no—
104#13133146—22no—
105#13133138—22no—
106#13133138—41no—
107C22⋊#88322—22no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124D411yescharacteristic
113#13133124D411yescharacteristic
114C22×D4×C53—21no—
115C22×D4×C53C2311yescharacteristic
116C23×C4×C53C2311yescharacteristic
117C22×C8×C53—22no—
118C22×C8×C53—22no—
119C22×C8×C53—22no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122#121×C53—22no—
123#121×C53Q814yescomplemented
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#102×C53—22no—
130#102×C53—22no—
131#95×C53—21no—
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—24no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—42no—
141C2×D4×C53—42no—
142C2×D4×C53C2×Q812yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×Q812yes—
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—81no—
165C22×Dic53—81no—
166C22×Dic53—81no—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C2×Dic106—42no—
172C2×Dic106—42no—
173C2×Dic106—81no—
174C2×C8×C53—28no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53C2×D412yes—
180C2×C8×C53#4112yes—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53C2×Q812yes—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—42no—
189C22⋊Dic53—22no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—81no—
193C22⋊Dic53D814yes—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53C2×D412yes—
205C4⋊Dic53C2×D412yes—
206#11719919—42no—
207#11719919—42no—
208#11719919—42no—
209#11719919—81no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—42no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—22no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53C22⋊Q812yes—
233C23×C53C2×D812yes—
234C23×C53#13712yes—
235C23×C53C22×D411yescharacteristic
236D4×C53—22no—
237D4×C53—22no—
238D4×C53—48no—
239D4×C53—41no—
240D4×C53—41no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
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×C53C22⋊Q812yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C22×Q811yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C532−1+411yescharacteristic
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—41no—
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×Dic53—84no—
286C2×Dic53—84no—
287C2×Dic53—84no—
288C2×Dic53C2×D812yes—
289C2×Dic53C2×D812yes—
290Dic106—84no—
291C8×C53—22no—
292C8×C53—22no—
293#88322—41no—
294#88322—41no—
295#26336—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53#47414yescomplemented
310C22×C53C2×C22⋊Q811yescharacteristic
311C22×C53D4211yescharacteristic
312C22×C53#54711yescharacteristic
313C22×C53C22×D811yescharacteristic
314C22×C53C2×#13711yescharacteristic
315C22×C53#57611yescharacteristic
316C22×C53#57811yescharacteristic
317C4×C53—22no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53#52511yescharacteristic
322C4×C53#53911yescharacteristic
323C4×C53D4×Q811yescharacteristic
324C4×C53#55411yescharacteristic
325Dic53—22no—
326Dic53—22no—
327Dic53—41no—
328Dic53—41no—
329Dic53—42no—
330Dic53—42no—
331Dic53—81no—
332Dic53—81no—
333C2×#474—531no—
334C22×#107—531no—
335#2440—531no—
336#2361—531no—
337#2093—531no—
338#2086—531no—
339#2067—531no—
340#2057—531no—
341#1947—531no—
342#1918—531no—
343#1893D5311yescharacteristic
344#1874—531no—
345#1862—531no—
346#1852—531no—
347#1850—531no—
348C2×C53—24no—
349C2×C53—82no—
350C2×C53#244911yescharacteristic
351C2×C53C2×#47411yescharacteristic
352C2×C53#308311yescharacteristic
353C2×C53#327011yescharacteristic
354C2×C53D4×D811yescharacteristic
355C2×C53#333411yescharacteristic
356C2×C53#334611yescharacteristic
357C23×C8C2×D5311yescharacteristic
358C2×C4⋊Q8—531no—
359C2×C22⋊Q8—1061no—
360C2×#121C2×D5311yescharacteristic
361C2×C4×D4—531no—
362C2×C4×D4—1061no—
363C22×C4⋊C4—531no—
364C22×C4⋊C4—531no—
365#474—1064no—
366C2×#107—531no—
367C2×#107—531no—
368C2×#107—531no—
369C2×#107—531no—
370C2×#107—531no—
371C2×#107—531no—
372C2×#107—1064no—
373C2×C4⋊C8—531no—
374C2×C4⋊C8—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×#102C2×D5311yescharacteristic
378C2×#102C2×D5311yescharacteristic
379C2×C22⋊C8—531no—
380#398—1061no—
381#395—1061no—
382#390—531no—
383#389—531no—
384#389—531no—
385#389—531no—
386#388—1061no—
387#386—1061no—
388#385—1061no—
389#385—1061no—
390#384—531no—
391#380C2×D5311yescharacteristic
392C4×C4⋊C4—531no—
393C4×C22⋊C4—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340C2×D5311yescharacteristic
399#340C2×D5311yescharacteristic
400#336—531no—
401#336—531no—
402C53#2633611yescharacteristic, complemented, Sylow
403C22×Q8—1061no—
404C22×D4—21no—
405C22×D4C22×D5311yescharacteristic
406C23×C4—1061no—
407C23×C4—1061no—
408C23×C4—1061no—
409C23×C4C22×D5311yescharacteristic
410C22×C8—22no—
411C22×C8—22no—
412C22×C8—22no—
413C22×C8—531no—
414C22×C8—531no—
415C22×C8C22×D5311yescharacteristic
416C22×C8C22×D5311yescharacteristic
417C4⋊Q8—1062no—
418C4⋊Q8—2121no—
419C22⋊Q8—2124no—
420#121—22no—
421#121Dic10614yescomplemented
422C4×D4—1062no—
423C4×D4—2121no—
424C4×D4—2124no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
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⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—2121no—
440C2×C4⋊C4—2121no—
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—531no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C42—531no—
452C2×C42—531no—
453C2×C42—1061no—
454#107—534no—
455#107—1062no—
456#107—1062no—
457#107—1062no—
458#107—1062no—
459#107—2121no—
460C4⋊C8—1062no—
461C4⋊C8—1062no—
462#102—22no—
463#102—22no—
464#102—1062no—
465#102—2121no—
466C22⋊C8—1062no—
467#95—21no—
468#95—531no—
469#95—531no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474C24—21no—
475C24D4×D5311yescharacteristic
476C2×Q8—4241no—
477C2×Q8—2122no—
478C2×Q8—2122no—
479C2×D4—24no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4C2×Dic10612yes—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—4241no—
506C22×C4—2122no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4—2122no—
511C22×C4—2122no—
512C22×C4C2×Dic10612yes—
513C22×C4D4×D5311yescharacteristic
514C22×C4#1171994712yes—
515C22×C4C23×D5311yescharacteristic
516C2×C8—28no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8D4×D5312yes—
528C2×C8#1171994712yes—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—532no—
535C4⋊C4—532no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—4241no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4—2122no—
550C4⋊C4C2×Dic10612yes—
551C22⋊C4—24no—
552C22⋊C4—42no—
553C22⋊C4—534no—
554C22⋊C4—1062no—
555C22⋊C4—4241no—
556C22⋊C4—2122no—
557C22⋊C4—2122no—
558C42—1061no—
559C42—1061no—
560C42—1061no—
561C42—1061no—
562C42—2122no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23#1313319412yes—
575C23D8×D5312yes—
576C23#1313324612yes—
577C23C2×D4×D5311yescharacteristic
578Q8—4244no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—41no—
583D4—41no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
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—1061no—
602C2×C4—1061no—
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—4244no—
614C2×C4—4244no—
615C2×C4—4244no—
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×C4—2121no—
625C2×C4—2121no—
626C2×C4#1313319412yes—
627C2×C4C2×D4×D5311yescharacteristic
628C2×C4C22×Dic10611yescharacteristic
629C2×C4C2×#1171994711yescharacteristic
630C2×C4C2×#1171994711yescharacteristic
631C2×C4#1313326311yescharacteristic
632C2×C4#1313326411yescharacteristic
633C8—22no—
634C8—22no—
635C8—2121no—
636C8—2121no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#1681090114yescomplemented
651C22C2×#1313319411yescharacteristic
652C22#1681125711yescharacteristic
653C22#1681127311yescharacteristic
654C22C2×D8×D5311yescharacteristic
655C22C2×#1313324611yescharacteristic
656C22#1681142511yescharacteristic
657C22#1681143711yescharacteristic
658C4—22no—
659C4—42no—
660C4—81no—
661C4—81no—
662C4—1062no—
663C4—1062no—
664C4—4241no—
665C4—4241no—
666C4—2121no—
667C4—2122no—
668C4—2122no—
669C4—2121no—
670C4#1681112911yescharacteristic
671C4#1681121711yescharacteristic
672C4D4×Dic10611yescharacteristic
673C4#1681133011yescharacteristic
674C2—24no—
675C2—82no—
676C2#2027853011yescharacteristic
677C2C2×#1681090111yescharacteristic
678C2#2028456411yescharacteristic
679C2#2028628211yescharacteristic
680C2#2028638911yescharacteristic
681C2#2028705911yescharacteristic
682C2#2028720611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1893C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1893D53
hypocenterC53#26336
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. C2×#16810901C2semidirect
2. C22×#13133151C2semidirect
3. #20278418C2semidirect
5. #20276241C2semidirect
6. #20276205C2semidirect
7. #20276101C2semidirect
8. #20276042C2semidirect
9. #20275472C2semidirect
10. #20275285C2semidirect
12. #20274964C2semidirect
13. #20274902C2semidirect
14. #20274893C2semidirect
100. #13133151D4semidirect
123. #121×C53Q8semidirect
309. C22×C53#474semidirect
402. C53#26336semidirect
421. #121Dic106semidirect
650. C22#16810901semidirect

Maximal subgroups

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

subgroupindex
C2×#168109012
C22×#131331512
#202784182
#202777352
#202762412
#202762052
#202761012
#202760422
#202754722
#202752852
#202750332
#202749642
#202749022
#202748932
C53×#18932
#2633653