Finite Group Databasebeta

← #27312588

Subgroups — #27312588

Order 13568.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810509C212yesmaximal, complemented
2#20272986C212yesmaximal, complemented
3#20272983C212yesmaximal, complemented
4#20272960C212yesmaximal
5#20272952C212yesmaximal
6#20272944C212yesmaximal, complemented
7#20272938C211yesmaximal, characteristic
8#20272938C211yesmaximal, characteristic
9C22×C22⋊Q8×C53C211yesmaximal, characteristic
10C22×C4⋊Dic53C2212yes—
11C22×C22⋊Dic53C2212yes—
12C22×C22⋊Dic53C2211yescharacteristic
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C4×Dic53C2212yes—
15#16810509—28no—
16C2×#13133124C2212yes—
17C2×#13133124C2212yes—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C23×Q8×C53C2211yescharacteristic
28C24×C4×C53C2211yescharacteristic
29C2×C22⋊Q8×C53—28no—
30C2×C22⋊Q8×C53C418yescomplemented
31C22×C4⋊C4×C53C2212yes—
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2212yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—24no—
40C2×C4⋊Dic53—42no—
41C2×C22⋊Dic53—24no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—24no—
44C2×C22⋊Dic53—41no—
45C2×C22⋊Dic53—42no—
46C2×C22⋊Dic53—41no—
47C2×C22⋊Dic53—41no—
48C2×C22⋊Dic53—41no—
49C2×C22⋊Dic53Q814yescomplemented
50C2×C4×Dic53—24no—
51C2×C4×Dic53—24no—
52C2×C4×Dic53—42no—
53#13133124—24no—
54#13133124—24no—
55#13133124—24no—
56#13133124—24no—
57#13133124—41no—
58#13133124—42no—
59#13133124—42no—
60#13133124—42no—
61#13133124—42no—
62#13133124—41no—
63#13133124—42no—
64C25×C53C2311yescharacteristic
65C22×Q8×C53—24no—
66C22×Q8×C53—41no—
67C22×Q8×C53—41no—
68C22×Q8×C53—42no—
69C22×Q8×C53C2×C414yes—
70C23×C4×C53—24no—
71C23×C4×C53—41no—
72C23×C4×C53—41no—
73C23×C4×C53—42no—
74C23×C4×C53C2×C414yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2312yes—
78C23×C4×C53C2311yescharacteristic
79C22⋊Q8×C53—232no—
80C2×C4⋊C4×C53—24no—
81C2×C4⋊C4×C53—24no—
82C2×C4⋊C4×C53C2×C418yes—
83C2×C4⋊C4×C53C2×C414yes—
84C2×C4⋊C4×C53D418yes—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53C2×C418yes—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—22no—
93C24×C53—21no—
94C24×C53—22no—
95C24×C53—21no—
96C24×C53C22×C412yes—
97C24×C53C2×Q812yes—
98C24×C53#4112yes—
99C24×C53C2411yescharacteristic
100C2×Q8×C53—28no—
101C2×Q8×C53—44no—
102C2×Q8×C53—44no—
103C2×Q8×C53—48no—
104C2×Q8×C53—48no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
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—28no—
123C22×C4×C53—44no—
124C22×C4×C53—44no—
125C22×C4×C53—48no—
126C22×C4×C53—48no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C414yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D414yes—
132C22×C4×C53C2×D414yes—
133C22×C4×C53C2×D414yes—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
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×Dic53C2×Q814yes—
161C22×Dic53#4114yes—
162C4⋊C4×C53—216no—
163C4⋊C4×C53C22⋊C4116yescomplemented
164C22⋊C4×C53—216no—
165C22⋊Dic53—44no—
166C22⋊Dic53—48no—
167C22⋊Dic53—44no—
168C4⋊Dic53—48no—
169C4×Dic53—48no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53C4×Q814yes—
193C23×C53C23×C411yescharacteristic
194C23×C53C22×D412yes—
195C23×C53C22×Q811yescharacteristic
196C23×C53C2×#4112yes—
197C23×C53C2×#4111yescharacteristic
198C23×C53H(5,𝔽2)11yescharacteristic
199C23×C53H(5,𝔽2)11yescharacteristic
200C23×C53H(5,𝔽2)12yes—
201C23×C53H(5,𝔽2)11yescharacteristic
202C23×C532−1+411yescharacteristic
203C23×C532−1+412yes—
204Q8×C53—416no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—416no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53C2×C22⋊C414yes—
229C2×C4×C53C2×C22⋊C418yes—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258#14448—531nomaximal, Sylow
259C22×C53—24no—
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—44no—
267C22×C53—44no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53C22×C22⋊C411yescharacteristic
271C22×C53C2×C4×Q811yescharacteristic
272C22×C53C4×#4111yescharacteristic
273C22×C53#51811yescharacteristic
274C22×C53#51811yescharacteristic
275C22×C53#51911yescharacteristic
276C22×C53#52011yescharacteristic
277C22×C53#53512yes—
278C22×C53#53512yes—
279C22×C53#53811yescharacteristic
280C22×C53#53912yes—
281C22×C53#53911yescharacteristic
282C22×C53#54212yes—
283C22×C53#54311yescharacteristic
284C22×C53#54411yescharacteristic
285C22×C53#54612yes—
286C22×C53#54712yes—
287C22×C53#54812yes—
288C22×C53D4×Q812yes—
289C22×C53#55112yes—
290C22×C53#55412yes—
291C22×C53#56212yes—
292C22×C53#56312yes—
293C4×C53—24no—
294C4×C53—28no—
295C4×C53—42no—
296C4×C53—44no—
297Dic53—44no—
298Dic53—44no—
299Dic53—82no—
300Dic53—82no—
301C22×C22⋊Q8D5311yescharacteristic
302C2×#387—532no—
303#1446—532no—
304#1445—532no—
305#1439—532no—
306#1437—532no—
307C4⋊#95—532no—
308C22⋊#95—531no—
309C22⋊#95—531no—
310C2×C53—44no—
311C2×C53#230911yescharacteristic
312C2×C53#232911yescharacteristic
313C2×C53#233211yescharacteristic
314C2×C53#233911yescharacteristic
315C2×C53Q8×C22⋊C411yescharacteristic
316C2×C53#236811yescharacteristic
317C2×C53#237811yescharacteristic
318C2×C53#269212yes—
319C2×C53#269612yes—
320C2×C53#272012yes—
321C2×C53#272312yes—
322C23×Q8C2×D5311yescharacteristic
323C24×C4C2×D5311yescharacteristic
324C2×C22⋊Q8—28no—
325C2×C22⋊Q8Dic5318yescomplemented
326C22×C4⋊C4—532no—
327C22×C4⋊C4C2×D5312yes—
328C22×C4⋊C4C2×D5311yescharacteristic
329C22×C22⋊C4—531no—
330C22×C22⋊C4—532no—
331C22×C22⋊C4—531no—
332C22×C22⋊C4C2×D5312yes—
333C22×C42—532no—
334#387—1068no—
335C2×#95—532no—
336C2×#95—532no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—531no—
340C2×#95—532no—
341C2×#95—531no—
342C2×#95—532no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95—532no—
346C53#1444811yescharacteristic, complemented, Sylow
347C25C22×D5311yescharacteristic
348C22×Q8—24no—
349C22×Q8—42no—
350C22×Q8—41no—
351C22×Q8—41no—
352C22×Q8C2×Dic5314yes—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4—41no—
356C23×C4—41no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—532no—
360C23×C4—532no—
361C23×C4C2×Dic5314yes—
362C23×C4C22×D5312yes—
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5312yes—
365C23×C4C22×D5311yescharacteristic
366C22⋊Q8—232no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—2122no—
372C2×C4⋊C4C2×Dic5318yes—
373C2×C4⋊C4C2×Dic5314yes—
374C2×C4⋊C4C22⋊D5318yes—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—534no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—2121no—
382C2×C22⋊C4—2121no—
383C2×C22⋊C4—2122no—
384C2×C22⋊C4—2121no—
385C2×C22⋊C4—2121no—
386C2×C22⋊C4C2×Dic5318yes—
387C2×C42—1064no—
388C2×C42—1064no—
389C2×C42—2122no—
390#95—1064no—
391#95—1064no—
392#95—1064no—
393#95—1064no—
394#95—2121no—
395#95—2122no—
396#95—2121no—
397#95—2122no—
398#95—2122no—
399#95—2122no—
400#95—2122no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24Q8×D5312yes—
410C24#1171994912yes—
411C24C22×Dic5312yes—
412C24C23×D5311yescharacteristic
413C2×Q8—28no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—48no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4C22×Dic5314yes—
469C22×C4C22×Dic5312yes—
470C22×C4C22×Dic5314yes—
471C22×C4C22×Dic5312yes—
472C22×C4C2×C22⋊D5314yes—
473C22×C4C2×C22⋊D5314yes—
474C22×C4C2×C22⋊D5314yes—
475C4⋊C4—216no—
476C4⋊C4—2128no—
477C4⋊C4C22⋊Dic53116yescomplemented
478C22⋊C4—216no—
479C22⋊C4—2124no—
480C22⋊C4—2128no—
481C22⋊C4—2124no—
482C42—2128no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23Q8×Dic5314yes—
506C23C23×Dic5311yescharacteristic
507C23C22×C22⋊D5312yes—
508C23C2×Q8×D5311yescharacteristic
509C23#41×D5312yes—
510C23C2×#1171994911yescharacteristic
511C23#1313326212yes—
512C23#1313326211yescharacteristic
513C23#1313326211yescharacteristic
514C23#1313326311yescharacteristic
515C23#1313326411yescharacteristic
516C23#1313326512yes—
517Q8—416no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4C2×C22⋊Dic5318yes—
570C2×C4C2×C22⋊Dic5314yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22C22×C22⋊Dic5311yescharacteristic
583C22C2×Q8×Dic5311yescharacteristic
584C22#41×Dic5311yescharacteristic
585C22#1681109811yescharacteristic
586C22#1681109811yescharacteristic
587C22#1681110111yescharacteristic
588C22#1681110611yescharacteristic
589C22#1681119012yes—
590C22#1681119012yes—
591C22#1681121311yescharacteristic
592C22#1681121912yes—
593C22#1681122211yescharacteristic
594C22#1681124312yes—
595C22#1681124811yescharacteristic
596C22#1681125211yescharacteristic
597C22#1681126112yes—
598C22#1681127212yes—
599C22#1681128412yes—
600C22Q8×C22⋊D5312yes—
601C22#1681130012yes—
602C22#1681132712yes—
603C22#1681136912yes—
604C22#1681137412yes—
605C4—28no—
606C4—24no—
607C4—42no—
608C4—44no—
609C4—4242no—
610C4—4242no—
611C4—2124no—
612C4—2124no—
613C2—44no—
614C2#2027729711yescharacteristic
615C2#2027745011yescharacteristic
616C2#2027747311yescharacteristic
617C2#2027754511yescharacteristic
618C2Q8×C22⋊Dic5311yescharacteristic
619C2#2027780011yescharacteristic
620C2#2027788111yescharacteristic
621C2#2028120512yes—
622C2#2028125912yes—
623C2#2028158912yes—
624C2#2028162812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C22⋊Q8×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22×C22⋊Q8D53
hypocenterC53#14448
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810509C2semidirect
2. #20272986C2semidirect
3. #20272983C2semidirect
6. #20272944C2semidirect
30. C2×C22⋊Q8×C53C4semidirect
49. C2×C22⋊Dic53Q8semidirect
163. C4⋊C4×C53C22⋊C4semidirect
325. C2×C22⋊Q8Dic53semidirect
346. C53#14448semidirect
477. C4⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105092
#202729862
#202729832
#202729602
#202729522
#202729442
#202729382
C22×C22⋊Q8×C532
#1444853