Finite Group Databasebeta

← #27305918

Subgroups — #27305918

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
3C2×#16810474C212yesmaximal, complemented
4C2×C22⋊C4×Dic53C212yesmaximal
5C2×C22⋊C4×Dic53C211yesmaximal, characteristic
6#20272982C211yesmaximal, characteristic, complemented
7#20272981C212yesmaximal, complemented
8#20272978C211yesmaximal, characteristic, complemented
9#20272958C212yesmaximal
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C22⋊Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22#16810559C418yescomplemented
23C4⋊C4×Dic53—24no—
24#16810474—28no—
25C22⋊C4×Dic53—28no—
26C22⋊C4×Dic53—24no—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C23×Dic53—24no—
41C23×Dic53—22no—
42C23×Dic53—24no—
43C23×Dic53—22no—
44C23×Dic53Q814yes—
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53C2×C414yescomplemented
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919C2×C418yescomplemented
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53C2×C418yes—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53D418yes—
75C2×C4×Dic53D414yes—
76#13133124—24no—
77#13133124—22no—
78#13133124—24no—
79#13133124—22no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124C2×C414yescomplemented
84C25×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—24no—
91C2×C22⋊C4×C53—24no—
92C2×C22⋊C4×C53—24no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—24no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53C2×C414yes—
97#95×C53—24no—
98#95×C53—22no—
99C24×C53—22no—
100C24×C53—21no—
101C24×C53—21no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—21no—
105C24×C53—22no—
106C24×C53—21no—
107C24×C53C22×C412yes—
108C24×C53C2×Q812yes—
109C24×C53#4112yes—
110C24×C53C2411yescharacteristic
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D414yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53#4114yes—
134C22×C4×C53#4112yes—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—28no—
158C22×Dic53—216no—
159C22×Dic53—48no—
160C22×Dic53—44no—
161C22×Dic53—44no—
162C22×Dic53C22×C414yes—
163C22×Dic53C22×C414yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×D414yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×Q812yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53#4112yes—
171C22×Dic53#4114yes—
172C22×Dic53#4114yes—
173C22×Dic53#4112yes—
174C4⋊C4×C53—28no—
175C22⋊C4×C53—216no—
176C22⋊C4×C53—28no—
177C22⋊Dic53—28no—
178C22⋊Dic53—216no—
179C4⋊Dic53—28no—
180#11719919—216no—
181#11719919—216no—
182C4×Dic53—216no—
183C4×Dic53—28no—
184C4×Dic53—44no—
185C4×Dic53—48no—
186C23×C53—24no—
187C23×C53—22no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—24no—
192C23×C53—22no—
193C23×C53—24no—
194C23×C53—24no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—24no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—41no—
206C23×C53—42no—
207C23×C53—41no—
208C23×C53C4×Q814yes—
209C23×C53C23×C411yescharacteristic
210C23×C53C22×D412yes—
211C23×C53C22×D411yescharacteristic
212C23×C53C22×Q811yescharacteristic
213C23×C53C2×#4112yes—
214C23×C53C2×#4112yes—
215C23×C53C2×#4111yescharacteristic
216C23×C53C2×#4111yescharacteristic
217C23×C53C2×#4111yescharacteristic
218C23×C53C2×#4111yescharacteristic
219C23×C53C2×#4112yes—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53C4×D414yes—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—416no—
273C2×Dic53C4×D418yes—
274C2×Dic53C22⋊Q814yes—
275C2×Dic53C22⋊Q818yes—
276C2×Dic53#12318yes—
277C2×Dic53#12314yes—
278C2×Dic53#12714yes—
279C4×#399—531nomaximal, Sylow
280C22×C53—24no—
281C22×C53—28no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—44no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—44no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—44no—
291C22×C53C2×C4×D412yes—
292C22×C53C2×C4×D411yescharacteristic
293C22×C53C2×C4×Q811yescharacteristic
294C22×C53C4×#4112yes—
295C22×C53C4×#4111yescharacteristic
296C22×C53C2×C22⋊Q811yescharacteristic
297C22×C53C2×C22⋊Q812yes—
298C22×C53C2×#12312yes—
299C22×C53C2×#12311yescharacteristic
300C22×C53#52512yes—
301C22×C53#52511yescharacteristic
302C22×C53#52511yescharacteristic
303C22×C53#52512yes—
304C22×C53#52912yes—
305C22×C53#52912yes—
306C22×C53#52911yescharacteristic
307C22×C53#52912yes—
308C22×C53#52912yes—
309C22×C53C2×#12711yescharacteristic
310C22×C53#53212yes—
311C22×C53#53211yescharacteristic
312C22×C53#53312yes—
313C22×C53#53311yescharacteristic
314C4×C53—24no—
315C4×C53—42no—
316C4×C53—44no—
317C4×C53—44no—
318Dic53—28no—
319Dic53—44no—
320Dic53—42no—
321Dic53—42no—
322Dic53—42no—
323Dic53#39918yescomplemented
324C2×#399—531no—
325C2×C4×C4⋊C4—532no—
326C2×C4×C4⋊C4—531no—
327C2×C4×C22⋊C4—532no—
328C2×C4×C22⋊C4—531no—
329#1445—532no—
330#1445—531no—
331#1444—531no—
332#1439—532no—
333#1439D5311yescharacteristic
334C4×#95—531no—
335C2×C53—44no—
336C2×C53C4×#12311yescharacteristic
337C2×C53C4×#12312yes—
338C2×C53C4×C22⋊Q811yescharacteristic
339C2×C53C4×C22⋊Q812yes—
340C2×C53C4×#12711yescharacteristic
341C2×C53C2×#39911yescharacteristic
342C2×C53#239512yes—
343C2×C53#239511yescharacteristic
344C2×C53#239611yescharacteristic
345C2×C53#239612yes—
346C2×C53#239711yescharacteristic
347C24×C4—531no—
348C22×C4⋊C4—532no—
349C22×C4⋊C4—531no—
350C22×C4⋊C4—532no—
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—532no—
353C22×C22⋊C4—531no—
354C22×C22⋊C4C2×D5312yes—
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C42—532no—
357C22×C42—531no—
358C22×C42—531no—
359C22×C42—532no—
360#399—538no—
361C4×C4⋊C4—1068no—
362C4×C4⋊C4—1064no—
363C4×C22⋊C4—1068no—
364C4×C22⋊C4—1064no—
365C2×#95—531no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—531no—
371C2×#95—531no—
372C2×#95—532no—
373C2×#95C2×D5311yescharacteristic
374C2×#95C2×D5312yes—
375C53C4×#39911yescharacteristic, complemented, Sylow
376C25C22×D5311yescharacteristic
377C23×C4—531no—
378C23×C4—532no—
379C23×C4—534no—
380C23×C4—532no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—1064no—
385C23×C4—1062no—
386C23×C4—1064no—
387C23×C4—1062no—
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5312yes—
390C23×C4C22×D5312yes—
391C23×C4C22×D5311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—538no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1062no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—538no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1062no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4C4×D5314yes—
415C2×C42—538no—
416C2×C42—534no—
417C2×C42—1064no—
418C2×C42—1064no—
419C2×C42—1064no—
420C2×C42—1062no—
421C2×C42—1064no—
422C2×C42—1064no—
423C2×C42—1064no—
424C2×C42—1064no—
425#95—24no—
426#95—22no—
427#95—534no—
428#95—1064no—
429#95—1064no—
430#95—1062no—
431#95—1064no—
432#95—1064no—
433#95—1062no—
434#95—1064no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—22no—
443C24C2×C4×D5312yes—
444C24#1171994712yes—
445C24Q8×D5312yes—
446C24C23×D5311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—534no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—534no—
469C22×C4—532no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—534no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—1068no—
478C22×C4—10616no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—2128no—
502C22×C4—2124no—
503C22×C4—2124no—
504C22×C4C2×C4×D5312yes—
505C22×C4C2×C4×D5312yes—
506C22×C4D4×D5314yes—
507C22×C4D4×D5312yes—
508C22×C4#1171994714yes—
509C22×C4#1171994712yes—
510C4⋊C4—28no—
511C4⋊C4—10616no—
512C4⋊C4—10616no—
513C4⋊C4—1068no—
514C22⋊C4—216no—
515C22⋊C4—28no—
516C22⋊C4—10616no—
517C22⋊C4—1068no—
518C42—1068no—
519C42—10616no—
520C42—2124no—
521C42—2128no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23#1313318214yes—
545C23C22×C4×D5311yescharacteristic
546C23C2×D4×D5312yes—
547C23C2×D4×D5311yescharacteristic
548C23C2×Q8×D5311yescharacteristic
549C23#41×D5312yes—
550C23#41×D5312yes—
551C23#41×D5311yescharacteristic
552C23#41×D5311yescharacteristic
553C23C2×#1171994711yescharacteristic
554C23C2×#1171994711yescharacteristic
555C23C2×#1171994712yes—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—534no—
580C2×C4—538no—
581C2×C4—538no—
582C2×C4—534no—
583C2×C4—534no—
584C2×C4—538no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—21216no—
602C2×C4—2122no—
603C2×C4—2121no—
604C2×C4—2122no—
605C2×C4—2121no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4#1313317714yes—
615C22—24no—
616C22—28no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22C4×D4×D5312yes—
627C22C2×#1313317711yescharacteristic
628C22C2×#1313318211yescharacteristic
629C22C4×#1171994712yes—
630C22#1681109111yescharacteristic
631C22C22⋊Q8×D5311yescharacteristic
632C22C22⋊Q8×D5312yes—
633C22#123×D5312yes—
634C22#123×D5311yescharacteristic
635C22#1681112912yes—
636C22#1681112911yescharacteristic
637C22#1681113311yescharacteristic
638C22#1681113312yes—
639C22#1681115012yes—
640C22#1681115212yes—
641C22#1681115411yescharacteristic
642C22#1681115412yes—
643C22#1681115612yes—
644C22#127×D5311yescharacteristic
645C22#1681116812yes—
646C22#1681117211yescharacteristic
647C22#1681117512yes—
648C22#1681117511yescharacteristic
649C4—24no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—538no—
654C4—1068no—
655C4—2124no—
656C4—2122no—
657C4—2122no—
658C4—2122no—
659C2—44no—
660C2#2027724011yescharacteristic
661C2#2027724212yes—
662C2#2027724311yescharacteristic
663C2#2027724712yes—
664C2#2027727911yescharacteristic
665C2#399×D5311yescharacteristic
666C2#2027797112yes—
667C2#2027797711yescharacteristic
668C2#2027798211yescharacteristic
669C2#2027799012yes—
670C2#2027799311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53C4×#399
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic53C2semidirect
3. C2×#16810474C2semidirect
6. #20272982C2semidirect
7. #20272981C2semidirect
8. #20272978C2semidirect
10. #95×Dic53C2semidirect
22. #16810559C4semidirect
53. C2×C4⋊Dic53C2×C4semidirect
59. C2×#11719919C2×C4semidirect
83. #13133124C2×C4semidirect
323. Dic53#399semidirect
375. C53C4×#399semidirect

Maximal subgroups

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

subgroupindex
C2×#168105592
C2×C4⋊C4×Dic532
C2×#168104742
C2×C22⋊C4×Dic532
#202729822
#202729812
#202729782
#202729582
#95×Dic532
C53×#14392
C4×#39953