Finite Group Databasebeta

← #27312508

Subgroups — #27312508

Order 13568.

2118 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810524C211yesmaximal, characteristic
2C2×#16810501C212yesmaximal
3C2×#16810497C212yesmaximal, complemented
4C2×#16810494C212yesmaximal, complemented
5#20272968C211yesmaximal, characteristic, complemented
6#20272959C212yesmaximal, complemented
7#20272959C212yesmaximal, complemented
8#20272943C212yesmaximal, complemented
9C53×#1441C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C4×D53C2212yes—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810524C418yescomplemented
22#16810501—28no—
23#16810497—28no—
24#16810494—28no—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C2×#95×C53C2212yes—
33C2×#95×C53C2212yes—
34C2×#95×C53C2211yescharacteristic
35C24×D53C2312yes—
36C22×D212—41no—
37C22×D212—41no—
38C22×D212—41no—
39C22×D212—41no—
40C22×D212C2×C414yes—
41C22×D212D414yes—
42C22×D212Q814yes—
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C2×C4⋊Dic53—24no—
54C2×C4⋊Dic53—41no—
55C2×#11719919—24no—
56C2×#11719919—42no—
57C2×#11719919—42no—
58C2×#11719919—42no—
59C2×#11719919—42no—
60C2×#11719919D418yescomplemented
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—24no—
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921—42no—
71C2×#11719921—42no—
72C2×#11719921—42no—
73C2×#11719921—42no—
74C2×#11719921C2×C418yescomplemented
75C2×#11719921C2×C418yescomplemented
76C2×C4×Dic53—41no—
77C2×C4×Dic53—41no—
78C2×C4×Dic53C2×C414yescomplemented
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—41no—
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—24no—
90C2×C4⋊C4×C53—41no—
91C2×C4⋊C4×C53—41no—
92C2×C4⋊C4×C53—41no—
93C2×C4⋊C4×C53C2×C414yescomplemented
94#95×C53—24no—
95#95×C53—24no—
96#95×C53—41no—
97C24×C53C2411yescharacteristic
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53—22no—
110C23×D53C22×C414yes—
111C23×D53C2×D414yes—
112C23×D53C2×Q814yes—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D414yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×Q812yes—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C22×C414yes—
162C22×Dic53C2×D414yes—
163C22×Dic53C2×D414yes—
164C22×Dic53#4114yes—
165C2×D212—44no—
166C2×D212—44no—
167C2×D212—44no—
168C2×D212—44no—
169C2×D212—44no—
170C2×D212—44no—
171C2×D212C4⋊C4116yescomplemented
172C2×C4×D53—48no—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C4⋊C4×C53—44no—
180C4⋊C4×C53—44no—
181#11719921—216no—
182#11719921—216no—
183#11719921—48no—
184#11719921—48no—
185#11719921—48no—
186C4⋊Dic53—44no—
187#11719919—48no—
188#11719919—48no—
189C4×Dic53—44no—
190C23×C53C23×C411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D412yes—
193C23×C53C22×Q811yescharacteristic
194C23×C53C2×#4112yes—
195C23×C53H(5,𝔽2)11yescharacteristic
196C23×C53H(5,𝔽2)11yescharacteristic
197C23×C53H(5,𝔽2)11yescharacteristic
198C23×C53H(5,𝔽2)11yescharacteristic
199C23×C53H(5,𝔽2)12yes—
200C23×C532−1+412yes—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—24no—
215C22×D53—24no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—42no—
231C22×D53—42no—
232C22×D53—42no—
233C22×D53—42no—
234C22×D53—42no—
235C22×D53C2×C4⋊C418yes—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53—42no—
266C2×C4×C53C2×C4⋊C414yes—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53C4×D418yes—
288D212—416no—
289C4×D53—432no—
290#14440—531nomaximal, Sylow
291C22×C53C22×C4⋊C411yescharacteristic
292C22×C53C2×C4×D412yes—
293C22×C53#51811yescharacteristic
294C22×C53#51811yescharacteristic
295C22×C53#52012yes—
296C22×C53#53411yescharacteristic
297C22×C53#53411yescharacteristic
298C22×C53#53412yes—
299C22×C53#53712yes—
300C22×C53#53812yes—
301C22×C53#53912yes—
302C22×C53#54311yescharacteristic
303C22×C53#54311yescharacteristic
304C22×C53D4211yescharacteristic
305C22×C53D4212yes—
306C22×C53#54712yes—
307C22×C53D4×Q812yes—
308C22×C53#55012yes—
309C22×C53#55412yes—
310C22×C53#55911yescharacteristic
311C22×C53H(3,𝔽4)11yescharacteristic
312C22×C53#56212yes—
313C22×C53#56311yescharacteristic
314C2×D53—28no—
315C2×D53—28no—
316C2×D53—28no—
317C2×D53—44no—
318C2×D53—44no—
319C2×D53—44no—
320C2×D53—44no—
321C2×D53—44no—
322C2×D53—44no—
323C2×D53—44no—
324C2×D53—44no—
325C2×D53—44no—
326C2×D53—44no—
327C2×D53—44no—
328C2×D53—44no—
329C4×C53—24no—
330C4×C53—42no—
331C4×C53—42no—
332C4×C53—44no—
333C4×C53—82no—
334Dic53—28no—
335Dic53—44no—
336Dic53—82no—
337C2×#390—531no—
338C2×#386—532no—
339C2×#385—532no—
340C2×#385—532no—
341#1441—531no—
342#1441D5311yescharacteristic, complemented
343#1439—532no—
344#1439—532no—
345C4⋊#95—532no—
346C2×C53#231511yescharacteristic
347C2×C53#232512yes—
348C2×C53D4×C4⋊C412yes—
349C2×C53#236611yescharacteristic
350C2×C53#237911yescharacteristic
351C2×C53#267911yescharacteristic
352C2×C53#268212yes—
353C2×C53#268311yescharacteristic
354C2×C53#272411yescharacteristic
355C2×C53#273212yes—
356C2×C53#280411yescharacteristic
357D53—48no—
358C23×D4—531no—
359C24×C4—532no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4—531no—
362C22×C4⋊C4—532no—
363C22×C4⋊C4C2×D5311yescharacteristic
364C22×C4⋊C4C2×D5311yescharacteristic
365C22×C22⋊C4—532no—
366C22×C22⋊C4—532no—
367C22×C22⋊C4—532no—
368C22×C22⋊C4—532no—
369C22×C22⋊C4—532no—
370C22×C42—531no—
371#390—538no—
372#386—1068no—
373#385—1068no—
374#385—1068no—
375C2×#95—532no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—531no—
380C2×#95C2×D5312yes—
381C2×#95C2×D5312yes—
382C2×#95C2×D5311yescharacteristic
383C53#1444011yescharacteristic, complemented, Sylow
384C25—532no—
385C22×D4—534no—
386C22×D4—534no—
387C22×D4—534no—
388C22×D4—2121no—
389C22×D4—2121no—
390C22×D4—2121no—
391C22×D4—2121no—
392C23×C4—532no—
393C23×C4—532no—
394C23×C4—532no—
395C23×C4—1064no—
396C23×C4—1064no—
397C23×C4—1064no—
398C23×C4—2122no—
399C23×C4—2122no—
400C23×C4—2122no—
401C23×C4—2122no—
402C23×C4C22×D5312yes—
403C23×C4C22×D5311yescharacteristic
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5312yes—
406C23×C4C22×D5311yescharacteristic
407C2×C4⋊C4—24no—
408C2×C4⋊C4—41no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—41no—
411C2×C4⋊C4—538no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—2122no—
415C2×C4⋊C4—2122no—
416C2×C4⋊C4—2122no—
417C2×C4⋊C4—2122no—
418C2×C4⋊C4—2121no—
419C2×C4⋊C4C4×D5314yescomplemented
420C2×C22⋊C4—538no—
421C2×C22⋊C4—538no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1064no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—1064no—
430C2×C22⋊C4—1064no—
431C2×C22⋊C4—2122no—
432C2×C22⋊C4—2122no—
433C2×C22⋊C4—2122no—
434C2×C22⋊C4—2122no—
435C2×C42—534no—
436C2×C42—2121no—
437C2×C42—2121no—
438#95—24no—
439#95—24no—
440#95—41no—
441#95—1064no—
442#95—1064no—
443#95—1064no—
444#95—1064no—
445#95—2121no—
446C24—534no—
447C24—534no—
448C24—534no—
449C24—1062no—
450C24—1062no—
451C24—1062no—
452C24—1062no—
453C24—1062no—
454C24—1062no—
455C24—1062no—
456C24—1062no—
457C24—1062no—
458C24—1062no—
459C24—1062no—
460C24—1062no—
461C24C23×D5311yescharacteristic
462C2×D4—5316no—
463C2×D4—2124no—
464C2×D4—2124no—
465C2×D4—2124no—
466C2×D4—2124no—
467C2×D4—2124no—
468C2×D4—2124no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—534no—
497C22×C4—534no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—2128no—
516C22×C4—2128no—
517C22×C4—2128no—
518C22×C4—2128no—
519C22×C4—2128no—
520C22×C4—2128no—
521C22×C4—2128no—
522C22×C4C2×C4×D5312yes—
523C22×C4C2×C4×D5312yes—
524C22×C4C2×C4×D5312yes—
525C22×C4D4×D5314yes—
526C22×C4D4×D5312yes—
527C22×C4Q8×D5312yes—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—2128no—
531C4⋊C4—2124no—
532C4⋊C4—2128no—
533C22⋊C4—10616no—
534C22⋊C4—10616no—
535C22⋊C4—2128no—
536C22⋊C4—2128no—
537C22⋊C4—2128no—
538C42—2124no—
539C23—538no—
540C23—1064no—
541C23—1064no—
542C23—1064no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1064no—
547C23—1064no—
548C23—1064no—
549C23—1064no—
550C23—1064no—
551C23—1064no—
552C23—1064no—
553C23—1064no—
554C23—1064no—
555C23—1064no—
556C23—1064no—
557C23—1064no—
558C23—2122no—
559C23—2122no—
560C23—2122no—
561C23—2122no—
562C23—2122no—
563C23—2122no—
564C23—2122no—
565C23—2122no—
566C23—2122no—
567C23—2122no—
568C23—2122no—
569C23—2122no—
570C23—2122no—
571C23—2122no—
572C23—2122no—
573C23—2122no—
574C23C22×C4×D5311yescharacteristic
575C23C2×D4×D5311yescharacteristic
576C23C2×D4×D5312yes—
577C23C2×Q8×D5311yescharacteristic
578C23#41×D5312yes—
579C23#1313326211yescharacteristic
580C23#1313326211yescharacteristic
581C23#1313326312yes—
582C23#1313326311yescharacteristic
583C23#1313326311yescharacteristic
584C23#1313326512yes—
585D4—21216no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—538no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1064no—
620C2×C4—1064no—
621C2×C4—1064no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—1064no—
625C2×C4—1064no—
626C2×C4—21232no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4C4⋊C4×D5314yes—
639C22—1068no—
640C22—1068no—
641C22—1068no—
642C22—2124no—
643C22—2124no—
644C22—2124no—
645C22—2124no—
646C22—2124no—
647C22—2124no—
648C22—2124no—
649C22—2124no—
650C22—2124no—
651C22—2124no—
652C22—2124no—
653C22—2124no—
654C22C2×C4⋊C4×D5311yescharacteristic
655C22C4×D4×D5312yes—
656C22#1681109611yescharacteristic
657C22#1681109711yescharacteristic
658C22#1681110512yes—
659C22#1681118111yescharacteristic
660C22#1681118111yescharacteristic
661C22#1681118412yes—
662C22#1681120312yes—
663C22#1681121212yes—
664C22#1681122112yes—
665C22#1681124811yescharacteristic
666C22#1681124811yescharacteristic
667C22#1681125511yescharacteristic
668C22#1681125712yes—
669C22#1681127512yes—
670C22#1681128812yes—
671C22#1681129112yes—
672C22#1681132912yes—
673C22#1681135711yescharacteristic
674C22#1681136511yescharacteristic
675C22#1681136912yes—
676C22#1681137211yescharacteristic
677C4—24no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—82no—
682C4—1068no—
683C4—4242no—
684C4—2124no—
685C2—2128no—
686C2#2027732811yescharacteristic
687C2#2027741412yes—
688C2#2027761612yes—
689C2#2027777811yescharacteristic
690C2#2027788711yescharacteristic
691C2#2028104811yescharacteristic
692C2#2028107512yes—
693C2#2028108911yescharacteristic
694C2#2028164211yescharacteristic
695C2#2028173612yes—
696C2#2028266011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenter#1441D53
hypocenterC53#14440
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#16810497C2semidirect
4. C2×#16810494C2semidirect
5. #20272968C2semidirect
6. #20272959C2semidirect
7. #20272959C2semidirect
8. #20272943C2semidirect
9. C53×#1441C2semidirect
21. #16810524C4semidirect
60. C2×#11719919D4semidirect
74. C2×#11719921C2×C4semidirect
75. C2×#11719921C2×C4semidirect
78. C2×C4×Dic53C2×C4semidirect
93. C2×C4⋊C4×C53C2×C4semidirect
171. C2×D212C4⋊C4semidirect
342. #1441D53semidirect
383. C53#14440semidirect
419. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105242
C2×#168105012
C2×#168104972
C2×#168104942
#202729682
#202729592
#202729432
C53×#14412
#1444053