Finite Group Databasebeta

← #27546992

Subgroups — #27546992

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2#20277877C214yesmaximal
3#20277741C212yesmaximal
4#20277738C212yesmaximal
5#20277674C211yesmaximal, characteristic, complemented
6#20277674C212yesmaximal, complemented
7#20277673C212yesmaximal, complemented
8#20277672C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic106C211yesmaximal, characteristic, complemented
10#20277513C214yesmaximal, complemented
11#20277462C212yesmaximal, complemented
12#20277431C212yesmaximal, complemented
13#20277323C212yesmaximal
14#20277320C212yesmaximal
15#20277120C211yesmaximal, characteristic
16#20277111C211yesmaximal, characteristic
17Q8×C4⋊C4×C53C211yesmaximal, characteristic, complemented
18#16811177C4116yescomplemented
19C2×#13133216C2211yescharacteristic
20C2×#13133215C2211yescharacteristic
21C2×#13133207C2212yes—
22C2×#13133197C2211yescharacteristic
23C2×#13133195C2212yes—
24C2×#13133193C2211yescharacteristic
25C2×C4×Dic106C2212yes—
26C2×C4×Dic106C2211yescharacteristic
27C2×C4×Dic106C2212yes—
28C2×#13133175C2212yes—
29C2×#13133174C2212yes—
30C2×C42×D53C2211yescharacteristic
31#16810530C2212yes—
32#16810528C2214yes—
33#16810528C2212yes—
34#16810527C2214yes—
35#16810521C2212yes—
36#16810520C2214yes—
37#16810514C2214yes—
38#16810513C2214yes—
39#16810513C2214yes—
40#16810512C2214yes—
41#16810509C2214yes—
42#16810507C2214yes—
43#16810507C2214yes—
44#16810506C2214yes—
45#16810505C2214yes—
46#16810504C2214yes—
47#16810504C2214yes—
48#16810504C2214yes—
49#16810504C2214yes—
50#16810504C2214yes—
51#16810493C2211yescharacteristic
52#16810492C2212yes—
53#16810490C2212yes—
54#16810489C2212yes—
55#16810488C2212yes—
56#16810488C2211yescharacteristic
57#16810488C2211yescharacteristic
58#16810488C2212yes—
59#16810485C2212yes—
60#16810480C2211yescharacteristic
61#16810480C2212yes—
62#16810480C2211yescharacteristic
63#16810480C2212yes—
64C4⋊C4×Dic53C2212yes—
65C4⋊C4×Dic53C2214yes—
66C4⋊C4×Dic53C2212yes—
67C4⋊C4×Dic53C2212yes—
68C4⋊C4×Dic53C2212yes—
69C4⋊C4×Dic53C2212yes—
70C4⋊C4×Dic53C2212yes—
71C4×#11719919C2214yes—
72C4×C4⋊Dic53C2212yes—
73C4×C4⋊Dic53C2211yescharacteristic
74C4×#11719921C2212yes—
75#16810469C2212yes—
76C2×C4×Q8×C53C2212yes—
77C2×C4×Q8×C53C2211yescharacteristic
78C53×#389C2214yes—
79C53×#389C2212yes—
80C53×#384C2212yes—
81C53×#384C2211yescharacteristic
82C4×C4⋊C4×C53C2212yes—
83C4×C4⋊C4×C53C2211yescharacteristic
84C22×Dic106C2311yescharacteristic
85C22×Dic106C2312yes—
86C22×C4×D53C2311yescharacteristic
87C22×C4×D53C2311yescharacteristic
88C22×C4×D53C2311yescharacteristic
89#13133216C2×C418yes—
90#13133215C2×C418yes—
91#13133207C2×C4116yes—
92#13133197C2×C418yes—
93#13133195C2×C4116yes—
94#13133193C2×C418yes—
95C4×Dic106—44no—
96C4×Dic106C2×C418yes—
97C4×Dic106C2×C4116yes—
98#13133175—44no—
99#13133174C2×C4116yes—
100C2×C4⋊Dic53C2312yes—
101C2×C4⋊Dic53C2311yescharacteristic
102C2×C4⋊Dic53C2311yescharacteristic
103C2×C4⋊Dic53C2312yes—
104C2×C4⋊Dic53C2312yes—
105C2×C4⋊Dic53C2312yes—
106C2×C4⋊Dic53C2311yescharacteristic
107C2×#11719919C2314yes—
108C2×#11719919C2314yes—
109C2×#11719919C2312yes—
110C2×#11719919C2312yes—
111C2×#11719919C2312yes—
112C2×#11719919C2312yes—
113C2×#11719919C2312yes—
114C2×#11719921C2312yes—
115C2×#11719921C2314yes—
116C2×#11719921C2312yes—
117C2×#11719921C2312yes—
118C2×#11719921C2312yes—
119C2×C4×Dic53C2312yes—
120C2×C4×Dic53C2314yes—
121C2×C4×Dic53C2314yes—
122C2×C4×Dic53C2314yes—
123C2×C4×Dic53C2314yes—
124C2×C4×Dic53C2311yescharacteristic
125C2×C4×Dic53C2311yescharacteristic
126C2×C4×Dic53C2311yescharacteristic
127C2×C4×Dic53C2312yes—
128C42×D53C2×C418yes—
129#13133124C2312yes—
130#13133124C2314yes—
131#13133124C2314yes—
132#13133124C2314yes—
133#13133124C2314yes—
134#13133124C2314yes—
135#13133124C2312yes—
136#13133124C2314yes—
137#13133124C2312yes—
138#13133124C2312yes—
139#13133124C2314yes—
140#13133124C2314yes—
141#13133124C2314yes—
142#13133124C2312yes—
143#13133124C2312yes—
144#13133124C2312yes—
145C22×Q8×C53C2311yescharacteristic
146C4×Q8×C53—44no—
147C4×Q8×C53C2×C418yes—
148C2×C4⋊C4×C53C2312yes—
149C2×C4⋊C4×C53C2311yescharacteristic
150C2×C4⋊C4×C53C2312yes—
151C2×C4⋊C4×C53C2311yescharacteristic
152C2×C4⋊C4×C53C2314yes—
153C2×C4⋊C4×C53C2312yes—
154C2×C4⋊C4×C53C2314yes—
155C2×C4⋊C4×C53C2312yes—
156C2×C4⋊C4×C53C2311yescharacteristic
157C2×C42×C53C2312yes—
158C2×C42×C53C2314yes—
159C2×C42×C53C2311yescharacteristic
160C2×C42×C53C2312yes—
161#95×C53C2314yes—
162#95×C53C2312yes—
163C23×D53C2411yescharacteristic
164C2×Q8×C53—22no—
165C2×Q8×C53—22no—
166C2×Q8×C53C22×C414yes—
167C22×C4×C53C2412yes—
168C22×C4×C53C2412yes—
169C22×C4×C53C2414yes—
170C22×C4×C53C2412yes—
171C22×C4×C53C2411yescharacteristic
172C22×C4×C53C2412yes—
173C22×C4×C53C2411yescharacteristic
174C22×C4×C53C2411yescharacteristic
175C22×Dic53C2412yes—
176C22×Dic53C2412yes—
177C22×Dic53C2414yes—
178C22×Dic53C2411yescharacteristic
179C22×Dic53C2411yescharacteristic
180C22×Dic53C2411yescharacteristic
181C22×Dic53C2412yes—
182C22×Dic53C2412yes—
183C2×C4×D53—22no—
184C2×C4×D53—22no—
185C2×C4×D53—22no—
186C2×C4×D53—22no—
187C2×C4×D53—41no—
188C2×C4×D53—41no—
189C2×C4×D53C22×C414yes—
190C2×C4×D53C22×C414yes—
191C2×C4×D53C22×C414yes—
192C2×Dic106—22no—
193C2×Dic106—22no—
194C2×Dic106—42no—
195C2×Dic106—42no—
196C2×Dic106C22×C414yes—
197C2×Dic106C22×C418yes—
198C4⋊C4×C53—24no—
199C4⋊C4×C53—24no—
200C4⋊C4×C53—22no—
201C4⋊C4×C53—22no—
202C4⋊C4×C53—44no—
203C4⋊C4×C53—42no—
204C4⋊C4×C53—44no—
205C4⋊C4×C53C22×C418yes—
206C4⋊C4×C53C22×C414yes—
207C42×C53—24no—
208C42×C53—44no—
209C42×C53C22×C414yes—
210C42×C53C22×C418yes—
211#11719921—42no—
212#11719921—42no—
213#11719921—44no—
214#11719921C22×C418yes—
215#11719921C22×C418yes—
216C4⋊Dic53—24no—
217C4⋊Dic53—22no—
218C4⋊Dic53—22no—
219C4⋊Dic53—24no—
220C4⋊Dic53—42no—
221C4⋊Dic53C22×C414yes—
222C4⋊Dic53C22×C418yes—
223#11719919—44no—
224#11719919—44no—
225#11719919C22×C418yes—
226#11719919C22×C418yes—
227#11719919C22×C418yes—
228#11719919C22×C418yes—
229#11719919C22×C418yes—
230C4×Dic53—24no—
231C4×Dic53—44no—
232C4×Dic53—44no—
233C4×Dic53—44no—
234C4×Dic53—44no—
235C4×Dic53C22×C414yes—
236C4×Dic53C22×C414yes—
237C4×Dic53C22×C414yes—
238C4×Dic53C22×C418yes—
239C23×C53C2511yescharacteristic
240C22×D53—21no—
241C22×D53—21no—
242C22×D53—21no—
243C22×D53—21no—
244C22×D53—21no—
245C22×D53—21no—
246C22×D53C23×C412yes—
247Q8×C53—28no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—22no—
255C2×C4×C53—24no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—24no—
259C2×C4×C53—22no—
260C2×C4×C53—24no—
261C2×C4×C53—22no—
262C2×C4×C53—22no—
263C2×C4×C53C23×C414yes—
264C2×C4×C53C23×C412yes—
265C2×C4×C53C23×C414yes—
266C2×C4×C53C23×C412yes—
267C2×C4×C53C23×C412yes—
268C2×C4×C53H(5,𝔽2)12yes—
269C2×C4×C532−1+412yes—
270C2×C4×C532−1+412yes—
271C2×C4×C532−1+412yes—
272C2×Dic53—22no—
273C2×Dic53—21no—
274C2×Dic53—22no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—22no—
281C2×Dic53—22no—
282C2×Dic53—24no—
283C2×Dic53—22no—
284C2×Dic53—22no—
285C2×Dic53—22no—
286C2×Dic53—24no—
287C2×Dic53—22no—
288C2×Dic53—22no—
289C2×Dic53—22no—
290C2×Dic53—24no—
291C2×Dic53C23×C412yes—
292C2×Dic53C23×C414yes—
293C2×Dic53C23×C414yes—
294C2×Dic53C23×C412yes—
295C2×Dic53C23×C412yes—
296C4×D53—28no—
297C4×D53—28no—
298C4×D53—44no—
299Dic106—28no—
300Dic106—48no—
301#34373—531nomaximal, Sylow
302C22×C53C24×C411yescharacteristic
303C22×C53C2×H(5,𝔽2)11yescharacteristic
304C22×C53C2×2−1+411yescharacteristic
305C22×C53C2×2−1+411yescharacteristic
306C22×C53C2×2−1+411yescharacteristic
307C22×C53#58511yescharacteristic
308C22×C53#58511yescharacteristic
309C2×D53—22no—
310C2×D53—22no—
311C2×D53—22no—
312C2×D53—41no—
313C2×D53—41no—
314C2×D53—41no—
315C2×D53—41no—
316C4×C53—24no—
317C4×C53—24no—
318C4×C53—22no—
319C4×C53—42no—
320C4×C53—44no—
321C4×C53—42no—
322C4×C53#51914yes—
323C4×C53#52014yes—
324Dic53—22no—
325Dic53—24no—
326Dic53—22no—
327Dic53—22no—
328Dic53—24no—
329Dic53—44no—
330Dic53—42no—
331Dic53—42no—
332C2×#533—531no—
333#2378—534no—
334#2362—532no—
335#2362—532no—
336#2355—531no—
337#2355—532no—
338#2355—532no—
339#2355—531no—
340Q8×C4⋊C4—531no—
341Q8×C4⋊C4D5311yescharacteristic, complemented
342#2336—534no—
343#2331—532no—
344#2327—532no—
345#2314—532no—
346#2314—532no—
347#2285—531no—
348#2283—531no—
349C2×C53C2×#51911yescharacteristic
350C2×C53C2×#52011yescharacteristic
351C2×C53#342811yescharacteristic
352C2×C53#355411yescharacteristic
353C2×C53#355411yescharacteristic
354C2×C53#355511yescharacteristic
355C2×C53#355511yescharacteristic
356D53—42no—
357#533—5316no—
358C2×C4⋊Q8—531no—
359C2×C4⋊Q8—531no—
360C2×#125—532no—
361C2×C22⋊Q8—531no—
362C2×C22⋊Q8—531no—
363C2×C22⋊Q8—532no—
364C2×C4×Q8—531no—
365C2×C4×Q8—532no—
366C2×C4×Q8—532no—
367C2×C4×Q8C2×D5312yes—
368C2×C4×Q8C2×D5311yescharacteristic
369C2×#117—532no—
370C2×#117—532no—
371C22×C42—531no—
372#391—532no—
373#391—532no—
374#391—534no—
375#391—534no—
376#389—532no—
377#389—534no—
378#389C2×D5312yes—
379#389C2×D5314yes—
380#388—534no—
381#388—534no—
382#388—534no—
383#388—534no—
384#387—534no—
385#387—534no—
386#387—534no—
387#387—534no—
388#387—534no—
389#387—534no—
390#387—534no—
391#387—534no—
392#387—534no—
393#387—534no—
394#384—532no—
395#384—531no—
396#384C2×D5312yes—
397#384C2×D5311yescharacteristic
398#383—532no—
399#382—532no—
400#382—532no—
401#382—531no—
402#382—531no—
403#382—532no—
404#381—532no—
405#380—531no—
406#380—531no—
407#380—532no—
408#380—532no—
409C4×C4⋊C4—532no—
410C4×C4⋊C4—532no—
411C4×C4⋊C4—532no—
412C4×C4⋊C4—532no—
413C4×C4⋊C4—532no—
414C4×C4⋊C4—534no—
415C4×C4⋊C4—532no—
416C4×C4⋊C4—532no—
417C4×C4⋊C4—534no—
418C4×C4⋊C4—531no—
419C4×C4⋊C4C2×D5312yes—
420C4×C4⋊C4C2×D5311yescharacteristic
421C4×C22⋊C4—532no—
422#376—532no—
423C53#3437311yescharacteristic, complemented, Sylow
424C22×Q8—531no—
425C22×Q8—532no—
426C22×Q8C22×D5311yescharacteristic
427C23×C4—531no—
428C23×C4—531no—
429C23×C4—531no—
430C4⋊Q8—538no—
431C4⋊Q8—538no—
432#125—5316no—
433C22⋊Q8—538no—
434C22⋊Q8—5316no—
435C22⋊Q8—538no—
436C4×Q8—44no—
437C4×Q8—5316no—
438C4×Q8—538no—
439C4×Q8—2124no—
440C4×Q8C4×D5318yes—
441#117—5316no—
442#117—2124no—
443C2×C4⋊C4—532no—
444C2×C4⋊C4—532no—
445C2×C4⋊C4—532no—
446C2×C4⋊C4—532no—
447C2×C4⋊C4—532no—
448C2×C4⋊C4—532no—
449C2×C4⋊C4—531no—
450C2×C4⋊C4—532no—
451C2×C4⋊C4—531no—
452C2×C4⋊C4—531no—
453C2×C4⋊C4—532no—
454C2×C4⋊C4—534no—
455C2×C4⋊C4—534no—
456C2×C4⋊C4—532no—
457C2×C4⋊C4C22×D5312yes—
458C2×C4⋊C4C22×D5311yescharacteristic
459C2×C4⋊C4C22×D5312yes—
460C2×C4⋊C4C22×D5311yescharacteristic
461C2×C4⋊C4C22×D5314yes—
462C2×C4⋊C4C22×D5312yes—
463C2×C4⋊C4C22×D5314yes—
464C2×C4⋊C4C22×D5312yes—
465C2×C4⋊C4C22×D5311yescharacteristic
466C2×C22⋊C4—532no—
467C2×C22⋊C4—532no—
468C2×C22⋊C4—534no—
469C2×C22⋊C4—532no—
470C2×C22⋊C4—532no—
471C2×C42—532no—
472C2×C42—531no—
473C2×C42—531no—
474C2×C42—531no—
475C2×C42—538no—
476C2×C42—532no—
477C2×C42—534no—
478C2×C42—534no—
479C2×C42—534no—
480C2×C42—534no—
481C2×C42C22×D5312yes—
482C2×C42C22×D5314yes—
483C2×C42C22×D5312yes—
484C2×C42C22×D5311yescharacteristic
485#95—532no—
486#95—532no—
487#95—532no—
488#95—532no—
489#95—532no—
490#95—532no—
491#95—534no—
492#95—534no—
493#95—532no—
494#95—534no—
495#95—534no—
496#95—534no—
497#95—534no—
498#95—534no—
499#95—534no—
500#95—534no—
501#95C22×D5312yes—
502#95C22×D5314yes—
503C24—531no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—534no—
507C2×Q8—538no—
508C2×Q8—1062no—
509C2×Q8—1062no—
510C2×Q8—2122no—
511C2×Q8—2122no—
512C2×Q8C2×C4×D5314yes—
513C22×C4—531no—
514C22×C4—531no—
515C22×C4—531no—
516C22×C4—532no—
517C22×C4—532no—
518C22×C4—534no—
519C22×C4—534no—
520C22×C4—534no—
521C22×C4—532no—
522C22×C4—532no—
523C22×C4—534no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—2121no—
529C22×C4—2121no—
530C22×C4C23×D5312yes—
531C22×C4C23×D5314yes—
532C22×C4C23×D5312yes—
533C22×C4C23×D5312yes—
534C22×C4C23×D5312yes—
535C22×C4C23×D5311yescharacteristic
536C22×C4C23×D5311yescharacteristic
537C22×C4C23×D5311yescharacteristic
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—538no—
546C4⋊C4—538no—
547C4⋊C4—538no—
548C4⋊C4—538no—
549C4⋊C4—538no—
550C4⋊C4—538no—
551C4⋊C4—534no—
552C4⋊C4—1064no—
553C4⋊C4—1062no—
554C4⋊C4—1062no—
555C4⋊C4—1064no—
556C4⋊C4—2124no—
557C4⋊C4—2124no—
558C4⋊C4—2122no—
559C4⋊C4C2×C4×D5318yes—
560C4⋊C4C2×C4×D5314yes—
561C22⋊C4—538no—
562C22⋊C4—538no—
563C22⋊C4—2124no—
564C22⋊C4—2122no—
565C22⋊C4—2122no—
566C42—24no—
567C42—44no—
568C42—534no—
569C42—534no—
570C42—534no—
571C42—538no—
572C42—1064no—
573C42—2124no—
574C42—2124no—
575C42—2124no—
576C42—2124no—
577C42C2×C4×D5314yes—
578C42C2×C4×D5318yes—
579C23—532no—
580C23—1061no—
581C23—1061no—
582C23—1061no—
583C23—1061no—
584C23—1061no—
585C23—1061no—
586C23C24×D5311yescharacteristic
587Q8—28no—
588Q8—1068no—
589Q8—2128no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—532no—
606C2×C4—532no—
607C2×C4—532no—
608C2×C4—534no—
609C2×C4—534no—
610C2×C4—1061no—
611C2×C4—1062no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1061no—
618C2×C4—1062no—
619C2×C4—1061no—
620C2×C4—1068no—
621C2×C4—1068no—
622C2×C4—1064no—
623C2×C4—1062no—
624C2×C4—1062no—
625C2×C4—1062no—
626C2×C4—1064no—
627C2×C4—1062no—
628C2×C4—1062no—
629C2×C4—1062no—
630C2×C4—1064no—
631C2×C4—2124no—
632C2×C4C22×C4×D5312yes—
633C2×C4C22×C4×D5314yes—
634C2×C4C22×C4×D5314yes—
635C2×C4C22×C4×D5312yes—
636C2×C4C22×C4×D5312yes—
637C2×C4#1313326212yes—
638C2×C4#1313326412yes—
639C2×C4#1313326512yes—
640C2×C4#1313326512yes—
641C22—1062no—
642C22—1062no—
643C22—1062no—
644C22—2121no—
645C22—2121no—
646C22—2121no—
647C22—2121no—
648C22C23×C4×D5311yescharacteristic
649C22C2×#1313326211yescharacteristic
650C22C2×#1313326411yescharacteristic
651C22C2×#1313326511yescharacteristic
652C22C2×#1313326511yescharacteristic
653C22#1681145811yescharacteristic
654C22#1681146011yescharacteristic
655C4—24no—
656C4—22no—
657C4—24no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—1062no—
662C4—1062no—
663C4—1064no—
664C4—1062no—
665C4—1064no—
666C4—2124no—
667C4—2122no—
668C4—2122no—
669C4#1681109914yes—
670C4#1681110414yes—
671C2—2122no—
672C2C2×#1681109911yescharacteristic
673C2C2×#1681110411yescharacteristic
674C2#2028791511yescharacteristic
675C2#2028918211yescharacteristic
676C2#2028918711yescharacteristic
677C2#2028919911yescharacteristic
678C2#2028920411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingQ8×C4⋊C4×C53C2
socle, cosocleC23×C53C25
prime radicalC23×D53C24
hypercenterQ8×C4⋊C4D53
hypocenterC53#34373
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #20277674C2semidirect
6. #20277674C2semidirect
7. #20277673C2semidirect
8. #20277672C2semidirect
9. C4⋊C4×Dic106C2semidirect
10. #20277513C2semidirect
11. #20277462C2semidirect
12. #20277431C2semidirect
17. Q8×C4⋊C4×C53C2semidirect
18. #16811177C4semidirect
341. Q8×C4⋊C4D53semidirect
423. C53#34373semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
#202778772
#202777412
#202777382
#202776742
#202776732
#202776722
C4⋊C4×Dic1062
#202775132
#202774622
#202774312
#202773232
#202773202
#202771202
#202771112
Q8×C4⋊C4×C532
#3437353