Finite Group Databasebeta

← #27385886

Subgroups — #27385886

Order 13568.

1994 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2C2×#16810596C212yesmaximal, complemented
3#20277098C211yesmaximal, characteristic, complemented
4#20275173C214yesmaximal, complemented
5#20275168C214yesmaximal, complemented
6#20274447C212yesmaximal, complemented
7C22×#117×C53C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9#16811134—22no—
10#16811134—22no—
11#16811134—21no—
12#16811134—21no—
13#16811134—21no—
14#16811134—21no—
15C2×#13133200C2212yes—
16C2×#13133196C412yescomplemented
17C2×#13133191C412yescomplemented
18C2×#13133186C2212yes—
19C2×C4×C22⋊D53C414yescomplemented
20C2×#13133176C2212yes—
21C2×#13133176C2211yescharacteristic
22#16810596—28no—
23C2×#13133131C2218yes—
24#16810486C2212yes—
25#16810483—22no—
26#16810479C2212yes—
27C4×C22⋊Dic53—22no—
28#16810389C2218yes—
29C24×C4×C53C2211yescharacteristic
30C2×#117×C53—21no—
31C2×#117×C53—21no—
32C2×#117×C53—28no—
33C2×#117×C53C2212yes—
34C2×#117×C53C2212yes—
35C22×C4⋊C4×C53—21no—
36C22×C22⋊C4×C53C2212yes—
37C22×C42×C53—21no—
38C2×#11719945—24no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—24no—
44C22×Dic106C2×C411yescharacteristic, complemented
45C22×D212C2×C411yescharacteristic, complemented
46C22×C22⋊D53C2×C414yes—
47C22×C22⋊D53C2312yes—
48C22×C4×D53C2×C412yescomplemented
49#13133200—24no—
50#13133200—22no—
51#13133200—22no—
52#13133196—22no—
53#13133196—22no—
54#13133196—22no—
55#13133196—22no—
56#13133191—22no—
57#13133191—22no—
58#13133191—22no—
59#13133191—22no—
60#13133186—22no—
61#13133186—24no—
62#13133186—22no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66C4×C22⋊D53—24no—
67#13133176—21no—
68#13133176—21no—
69#13133176—21no—
70#13133176—21no—
71#13133176—24no—
72#13133176—24no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53C2×C412yescomplemented
75C2×#11719919C2×C414yescomplemented
76C2×C22⋊Dic53—24no—
77C2×C22⋊Dic53C2×C414yes—
78C2×C22⋊Dic53C2314yes—
79C2×C22⋊Dic53C2312yes—
80C2×#11719921C2×C414yescomplemented
81C2×C4×Dic53—22no—
82C2×C4×Dic53C2×C412yescomplemented
83#13133131—216no—
84#13133124—24no—
85C25×C53C2311yescharacteristic
86C23×C4×C53—24no—
87C23×C4×C53—24no—
88C23×C4×C53—22no—
89C23×C4×C53C2×C412yes—
90C23×C4×C53C2×C412yes—
91C23×C4×C53D412yes—
92C23×C4×C53D412yes—
93C23×C4×C53D411yescharacteristic
94C23×C4×C53D411yescharacteristic
95C23×C4×C53D411yescharacteristic
96C23×C4×C53D411yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99#117×C53—28no—
100#117×C53—28no—
101#117×C53—28no—
102#117×C53—24no—
103#117×C53D418yescomplemented
104C2×C4⋊C4×C53—28no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53—22no—
107C2×C22⋊C4×C53—28no—
108C2×C22⋊C4×C53—22no—
109C2×C22⋊C4×C53C2314yes—
110C2×C42×C53—28no—
111C2×C42×C53—22no—
112C2×C42×C53—22no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—44no—
116C24×C53C22×C412yes—
117C24×C53C2×D411yescharacteristic
118C24×C53C2×D412yes—
119C24×C53C2×D411yescharacteristic
120C24×C53C2411yescharacteristic
121C23×D53C22×C412yes—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—28no—
125C22×C4×C53—28no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—28no—
131C22×C4×C53—28no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—28no—
137C22×C4×C53—24no—
138C22×C4×C53—48no—
139C22×C4×C53C22⋊C412yes—
140C22×C4×C53C22⋊C412yes—
141C22×C4×C53C22⋊C412yes—
142C22×C4×C53C22⋊C412yes—
143C22×C4×C53C22⋊C412yes—
144C22×C4×C53C22⋊C412yes—
145C22×C4×C53C22⋊C412yes—
146C22×C4×C53C22⋊C412yes—
147C22×C4×C53C22×C412yes—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D414yes—
158C22×C4×C53#4114yes—
159C2×C22⋊D53—24no—
160C2×C22⋊D53—24no—
161C2×C22⋊D53—24no—
162C2×C22⋊D53—22no—
163C2×C22⋊D53—22no—
164C2×C22⋊D53—24no—
165C2×C22⋊D53—48no—
166C22×Dic53—24no—
167C22×Dic53C22×C412yes—
168C22×Dic53C22×C414yes—
169#11719945—42no—
170#11719945—42no—
171#11719945—44no—
172#11719945—44no—
173#11719945—42no—
174#11719945—42no—
175C2×D212—21no—
176C2×D212—21no—
177C2×D212—24no—
178C2×C4×D53—22no—
179C2×C4×D53—22no—
180C2×C4×D53—28no—
181C2×Dic106—21no—
182C2×Dic106—21no—
183C2×Dic106—24no—
184C4⋊C4×C53—28no—
185C4⋊C4×C53—28no—
186C22⋊C4×C53—28no—
187C22⋊C4×C53—24no—
188C22⋊C4×C53C2×D418yes—
189C42×C53—28no—
190C42×C53—28no—
191C22⋊Dic53—24no—
192C22⋊Dic53—24no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53—24no—
196C22⋊Dic53—24no—
197C22⋊Dic53—48no—
198#11719921—24no—
199#11719921—24no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—44no—
203#11719919—24no—
204#11719919—24no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—44no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—21no—
213C23×C53—24no—
214C23×C53—21no—
215C23×C53—24no—
216C23×C53—24no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—48no—
222C23×C53—48no—
223C23×C53C2×C22⋊C412yes—
224C23×C53C2×C22⋊C412yes—
225C23×C53C23×C411yescharacteristic
226C23×C53C22×D411yescharacteristic
227C23×C53C22×D412yes—
228C23×C53C22×D411yescharacteristic
229C23×C53C2×#4112yes—
230C22×D53—22no—
231C22×D53—24no—
232C22×D53—44no—
233C2×C4×C53—24no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—24no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—24no—
246C2×C4×C53—24no—
247C2×C4×C53—28no—
248C2×C4×C53—416no—
249C2×C4×C53—48no—
250C2×C4×C53C2×C22⋊C412yes—
251C2×C4×C53C2×C22⋊C412yes—
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C4×D414yes—
257C2×C4×C53C4×D414yes—
258C2×C4×C53#12012yes—
259C2×C4×C53#12012yes—
260C2×C4×C53#12114yes—
261C2×C4×C53#12114yes—
262C2×C4×C53#12312yes—
263C2×C4×C53#12312yes—
264C22⋊D53—48no—
265C22⋊D53—42no—
266C22⋊D53—44no—
267C22⋊D53—42no—
268C22⋊D53—48no—
269C2×Dic53—22no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—44no—
273C2×Dic53—48no—
274C2×Dic53—44no—
275D212—42no—
276D212—42no—
277C4×D53—44no—
278C4×D53—44no—
279Dic106—42no—
280Dic106—42no—
281#20983—531nomaximal, Sylow
282C22×C53—24no—
283C22×C53—24no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—48no—
295C22×C53—48no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53#41014yescomplemented
299C22×C53C22×C22⋊C411yescharacteristic
300C22×C53C2×C4×D412yes—
301C22×C53C2×#12011yescharacteristic
302C22×C53C2×#12112yes—
303C22×C53C2×#12311yescharacteristic
304C2×D53—44no—
305C2×D53—42no—
306C2×D53—84no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—48no—
310C4×C53#38612yes—
311C4×C53#38612yes—
312C4×C53#38612yes—
313C4×C53#38612yes—
314Dic53—44no—
315Dic53—42no—
316Dic53—84no—
317C2×#525—531no—
318C22×#117D5311yescharacteristic, complemented
319C2×#410—532no—
320#2280—531no—
321#1899—534no—
322#1898—534no—
323#1740—532no—
324C2×C53—22no—
325C2×C53—21no—
326C2×C53—22no—
327C2×C53—21no—
328C2×C53—24no—
329C2×C53—42no—
330C2×C53C2×#38611yescharacteristic
331C2×C53C2×#41012yes—
332D53—82no—
333C22×#41—531no—
334C24×C4C2×D5311yescharacteristic
335#525—1062no—
336#525—1062no—
337#525—1061no—
338#525—1061no—
339#525—1061no—
340#525—1061no—
341C2×#123—532no—
342C2×C22⋊Q8—532no—
343C2×#121—532no—
344C2×#120—532no—
345C2×C4×D4—534no—
346C2×#117—21no—
347C2×#117—21no—
348C2×#117—28no—
349C2×#117—531no—
350C2×#117—532no—
351C2×#117C2×D5312yes—
352C2×#117C2×D5312yes—
353C22×C4⋊C4—21no—
354C22×C22⋊C4C2×D5312yes—
355C22×C42—21no—
356#410—1068no—
357C2×#99—538no—
358#381—532no—
359#380—1062no—
360#379—532no—
361C4×C22⋊C4—1062no—
362#342—538no—
363C53#2098311yescharacteristic, complemented, Sylow
364C25C22×D5311yescharacteristic
365C2×#41—1061no—
366C2×#41—1061no—
367C2×#41—1064no—
368C2×#41—1061no—
369C2×#41—1061no—
370C2×#41—1064no—
371C22×Q8—531no—
372C22×D4—532no—
373C22×D4—534no—
374C22×D4—531no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—22no—
378C23×C4—532no—
379C23×C4C4×D5312yes—
380C23×C4C4×D5312yes—
381C23×C4D21212yes—
382C23×C4D21212yes—
383C23×C4C22⋊D5311yescharacteristic
384C23×C4C22⋊D5311yescharacteristic
385C23×C4C22⋊D5311yescharacteristic
386C23×C4C22⋊D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389#123—1064no—
390#123—1062no—
391#123—1062no—
392C22⋊Q8—1062no—
393C22⋊Q8—1062no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396#121—1062no—
397#121—1062no—
398#121—1062no—
399#121—1062no—
400#120—1064no—
401#120—1062no—
402#120—1062no—
403C4×D4—1064no—
404C4×D4—1064no—
405C4×D4—1064no—
406C4×D4—1064no—
407#117—28no—
408#117—28no—
409#117—24no—
410#117—28no—
411#117—1061no—
412#117—1061no—
413#117—1061no—
414#117—1061no—
415#117—1064no—
416#117—1064no—
417#117C22⋊D5318yescomplemented
418C2×C4⋊C4—28no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—532no—
422C2×C4⋊C4—534no—
423C2×C4⋊C4—1062no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—28no—
426C2×C22⋊C4—534no—
427C2×C22⋊C4—532no—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—534no—
430C2×C22⋊C4—1064no—
431C2×C22⋊C4C22×D5314yes—
432C2×C42—28no—
433C2×C42—22no—
434C2×C42—22no—
435C2×C42—532no—
436C2×C42—1062no—
437#99—10616no—
438#95—1064no—
439C24—22no—
440C24—22no—
441C24—44no—
442C24—532no—
443C24C2×C4×D5312yes—
444C24C2×D21212yes—
445C24C2×C22⋊D5311yescharacteristic
446C24C2×C22⋊D5311yescharacteristic
447C24C23×D5311yescharacteristic
448#41—2122no—
449#41—2122no—
450#41—2124no—
451#41—2124no—
452#41—2122no—
453#41—2122no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1064no—
457C2×D4—1064no—
458C2×D4—1064no—
459C2×D4—1064no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1064no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1064no—
466C2×D4—2128no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—28no—
482C22×C4—24no—
483C22×C4—48no—
484C22×C4—532no—
485C22×C4—534no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1068no—
489C22×C4—1064no—
490C22×C4#1171992112yes—
491C22×C4#1171992112yes—
492C22×C4#1171992112yes—
493C22×C4#1171992112yes—
494C22×C4#1171992112yes—
495C22×C4#1171992112yes—
496C22×C4#1171992112yes—
497C22×C4#1171992112yes—
498C22×C4C2×C4×D5312yes—
499C22×C4C2×C4×D5311yescharacteristic
500C22×C4C2×C4×D5311yescharacteristic
501C22×C4C2×D21212yes—
502C22×C4C2×D21211yescharacteristic
503C22×C4C2×D21211yescharacteristic
504C22×C4#1171994514yes—
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5314yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1064no—
515C4⋊C4—1064no—
516C4⋊C4—2124no—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—1064no—
520C22⋊C4—1064no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—1064no—
524C22⋊C4—1064no—
525C22⋊C4—1064no—
526C22⋊C4—1064no—
527C22⋊C4—2128no—
528C22⋊C4C2×C22⋊D5318yes—
529C42—28no—
530C42—28no—
531C42—1062no—
532C42—1062no—
533C42—2124no—
534C23—24no—
535C23—21no—
536C23—21no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—21no—
541C23—24no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—48no—
548C23—48no—
549C23—1064no—
550C23—1062no—
551C23—2124no—
552C23C2×#1171992112yes—
553C23C2×#1171992112yes—
554C23C22×C4×D5311yescharacteristic
555C23C22×C22⋊D5311yescharacteristic
556C23C22×C22⋊D5312yes—
557C23C22×D21211yescharacteristic
558C23C2×#1171994512yes—
559Q8—2122no—
560Q8—2122no—
561D4—2128no—
562D4—2128no—
563D4—2122no—
564D4—2124no—
565D4—2122no—
566D4—2122no—
567D4—2122no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—1064no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—2128no—
589C2×C4—2124no—
590C2×C4—2124no—
591C2×C4—2124no—
592C2×C4—2124no—
593C2×C4C2×#1171992112yes—
594C2×C4C2×#1171992112yes—
595C2×C4C2×#1171992111yescharacteristic
596C2×C4C2×#1171992111yescharacteristic
597C2×C4C2×#1171992111yescharacteristic
598C2×C4C2×#1171992111yescharacteristic
599C2×C4C4×C22⋊D5314yes—
600C2×C4C4×C22⋊D5314yes—
601C2×C4#1313318612yes—
602C2×C4#1313318612yes—
603C2×C4#1313319114yes—
604C2×C4#1313319114yes—
605C2×C4#1313320012yes—
606C2×C4#1313320012yes—
607C22—24no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—48no—
620C22—48no—
621C22—42no—
622C22—42no—
623C22—4244no—
624C22—2124no—
625C22—2122no—
626C22#1681059614yescomplemented
627C22C22×#1171992111yescharacteristic
628C22C2×C4×C22⋊D5312yes—
629C22C2×#1313318611yescharacteristic
630C22C2×#1313319112yes—
631C22C2×#1313320011yescharacteristic
632C4—22no—
633C4—22no—
634C4—48no—
635C4—4244no—
636C4—2124no—
637C4—2122no—
638C4#1681050212yes—
639C4#1681050212yes—
640C4#1681050212yes—
641C4#1681050212yes—
642C2—21no—
643C2—22no—
644C2—24no—
645C2—21no—
646C2—22no—
647C2—42no—
648C2—4242no—
649C2C2×#1681050211yescharacteristic
650C2C2×#1681059612yes—
subgroupG/H
centerC2×C4C2×#11719921
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC22×#117×C53C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811134C2
cosocleC24×C53C24
hypercenterC22×#117D53
hypocenterC53#20983
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16810596C2semidirect
3. #20277098C2semidirect
4. #20275173C2semidirect
5. #20275168C2semidirect
6. #20274447C2semidirect
7. C22×#117×C53C2semidirect
16. C2×#13133196C4semidirect
17. C2×#13133191C4semidirect
19. C2×C4×C22⋊D53C4semidirect
44. C22×Dic106C2×C4semidirect
45. C22×D212C2×C4semidirect
48. C22×C4×D53C2×C4semidirect
74. C2×C4⋊Dic53C2×C4semidirect
75. C2×#11719919C2×C4semidirect
80. C2×#11719921C2×C4semidirect
82. C2×C4×Dic53C2×C4semidirect
103. #117×C53D4semidirect
298. C22×C53#410semidirect
318. C22×#117D53semidirect
363. C53#20983semidirect
417. #117C22⋊D53semidirect
626. C22#16810596semidirect

Maximal subgroups

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

subgroupindex
C2×#168111342
C2×#168105962
#202770982
#202751732
#202751682
#202744472
C22×#117×C532
#2098353