Finite Group Databasebeta

← #27320442

Subgroups — #27320442

Order 13568.

1678 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810562C211yesmaximal, characteristic, complemented
2C2×#16810540C211yesmaximal, characteristic
3C2×#16810525C211yesmaximal, characteristic
4#20272984C211yesmaximal, characteristic, complemented
5#20272963C212yesmaximal, complemented
6#20272957C212yesmaximal, complemented
7#20272955C211yesmaximal, characteristic
8#20272953C212yesmaximal, complemented
9#20272953C212yesmaximal, complemented
10#20272942C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20#16810562—24no—
21#16810540—24no—
22#16810525—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C22×C42×C53C2211yescharacteristic
36C4×C4⋊C4×C53—24no—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C24×D53C2312yes—
40C22×D212—24no—
41C22×D212—22no—
42C22×D212—22no—
43C22×D212—22no—
44C22×D212—42no—
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47C23×Dic53C2312yes—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×#11719921—24no—
52C2×#11719921—24no—
53C2×#11719921—24no—
54C2×#11719921—24no—
55C2×#11719921—24no—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—42no—
64C2×#11719921—42no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69#13133124—22no—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53D414yes—
90C2×C42×C53—22no—
91C2×C42×C53—22no—
92C2×C42×C53—22no—
93C2×C42×C53—22no—
94C2×C42×C53—42no—
95C2×C42×C53—42no—
96C2×C42×C53D414yes—
97#95×C53—22no—
98#95×C53—22no—
99C24×C53C2411yescharacteristic
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×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—22no—
114C23×D53#4114yes—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
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×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53#4112yes—
147C22×C4×C53#4112yes—
148C22×C4×C53#4112yes—
149C22×C4×C53#4112yes—
150C22×C4×C53#4112yes—
151C22×C4×C53#4114yes—
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×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C2×D212—28no—
174C2×D212—48no—
175C2×D212—48no—
176C2×D212—44no—
177C2×D212—44no—
178C4⋊C4×C53—28no—
179C4⋊C4×C53—28no—
180C42×C53—28no—
181C42×C53—44no—
182C42×C53—44no—
183#11719921—48no—
184#11719921—48no—
185#11719921—48no—
186#11719921—48no—
187#11719921—48no—
188#11719921—48no—
189C4⋊Dic53—44no—
190C23×C53C22×D411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C2×#4112yes—
193C23×C53C2×#4111yescharacteristic
194C23×C53C2×#4111yescharacteristic
195C23×C53C2×#4111yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C23×C53C2×#4111yescharacteristic
198C23×C53C2×#4112yes—
199C23×C53H(5,𝔽2)12yes—
200C23×C53H(5,𝔽2)12yes—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
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×D53—42no—
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—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—24no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—24no—
257C2×C4×C53—24no—
258C2×C4×C53—24no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—42no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—42no—
270C2×C4×C53#12114yes—
271C2×C4×C53#12414yes—
272C2×C4×C53#12414yes—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
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×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53—42no—
293C2×Dic53—42no—
294D212—416no—
295#15107—531nomaximal, Sylow
296C22×C53C2×#12111yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53#52511yescharacteristic
299C22×C53#52511yescharacteristic
300C22×C53C2×#12411yescharacteristic
301C22×C53C2×#12411yescharacteristic
302C22×C53#52911yescharacteristic
303C22×C53#52911yescharacteristic
304C22×C53#52912yes—
305C22×C53#52912yes—
306C22×C53#53211yescharacteristic
307C22×C53#53411yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#53812yes—
310C22×C53#54012yes—
311C22×C53#54612yes—
312C22×C53#55012yes—
313C22×C53#55112yes—
314C22×C53#55112yes—
315C22×C53#55312yes—
316C22×C53#55312yes—
317C22×C53#55312yes—
318C22×C53#55912yes—
319C2×D53—44no—
320C2×D53—44no—
321C2×D53—44no—
322C2×D53—44no—
323C2×D53—44no—
324C2×D53—44no—
325C2×D53—44no—
326C2×D53—82no—
327C2×D53—82no—
328C2×D53—82no—
329C2×D53—82no—
330C2×D53—82no—
331C2×D53—82no—
332C2×D53—82no—
333C2×D53—82no—
334C4×C53—24no—
335C4×C53—24no—
336C4×C53—24no—
337C4×C53—42no—
338C4×C53—42no—
339C4×C53—42no—
340C4×C53—42no—
341Dic53—82no—
342Dic53—82no—
343Dic53—82no—
344C2×#400—531no—
345C2×#394—531no—
346C2×#390—531no—
347C2×C4×C4⋊C4D5311yescharacteristic, complemented
348#1445—531no—
349#1440—532no—
350#1439—532no—
351#1438—531no—
352#1437—532no—
353#1437—532no—
354#1434—531no—
355C2×C53#239811yescharacteristic
356C2×C53#249012yes—
357C2×C53#251612yes—
358C2×C53#253411yescharacteristic
359C2×C53#254811yescharacteristic
360C2×C53#255412yes—
361C2×C53#259111yescharacteristic
362C2×C53#259512yes—
363C2×C53#259811yescharacteristic
364C2×C53#263511yescharacteristic
365C2×C53#285011yescharacteristic
366D53—84no—
367C23×D4—531no—
368C22×C4⋊C4—531no—
369C22×C4⋊C4C2×D5311yescharacteristic
370C22×C4⋊C4C2×D5311yescharacteristic
371C22×C22⋊C4—532no—
372C22×C22⋊C4—532no—
373C22×C22⋊C4—532no—
374C22×C22⋊C4—532no—
375C22×C22⋊C4—532no—
376C22×C22⋊C4—532no—
377C22×C42C2×D5311yescharacteristic
378C22×C42C2×D5311yescharacteristic
379C22×C42C2×D5311yescharacteristic
380#400—1064no—
381#394—1064no—
382#390—1064no—
383C4×C4⋊C4—24no—
384C2×#95—531no—
385C2×#95—531no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—532no—
391C2×#95—532no—
392C2×#95C2×D5311yescharacteristic
393C2×#95C2×D5311yescharacteristic
394C53#1510711yescharacteristic, complemented, Sylow
395C25—532no—
396C22×D4—1064no—
397C22×D4—1062no—
398C22×D4—1062no—
399C22×D4—1062no—
400C22×D4—2122no—
401C23×C4—532no—
402C23×C4—532no—
403C23×C4—532no—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408C23×C4C22×D5311yescharacteristic
409C23×C4C22×D5311yescharacteristic
410C23×C4C22×D5311yescharacteristic
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4D21214yes—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
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—1064no—
432C2×C22⋊C4—2122no—
433C2×C22⋊C4—2122no—
434C2×C22⋊C4—2122no—
435C2×C22⋊C4—2122no—
436C2×C22⋊C4—2122no—
437C2×C22⋊C4—2122no—
438C2×C42—22no—
439C2×C42—22no—
440C2×C42—22no—
441C2×C42—22no—
442C2×C42—42no—
443C2×C42—42no—
444C2×C42C22⋊D5314yes—
445#95—22no—
446#95—22no—
447#95—1062no—
448#95—1062no—
449#95—1064no—
450#95—1064no—
451#95—1064no—
452#95—2122no—
453#95—2122no—
454#95—2122no—
455C24—534no—
456C24—1062no—
457C24—1062no—
458C24—1062no—
459C24—1062no—
460C24—1062no—
461C24—1062no—
462C24—1062no—
463C24—1062no—
464C24—1062no—
465C24—1062no—
466C24—1062no—
467C24—1062no—
468C24—1062no—
469C24—1062no—
470C24C23×D5311yescharacteristic
471C2×D4—1068no—
472C2×D4—2128no—
473C2×D4—2128no—
474C2×D4—2124no—
475C2×D4—2124no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
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—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4C2×D21212yes—
523C22×C4C2×D21212yes—
524C22×C4C2×D21212yes—
525C22×C4#1171994512yes—
526C22×C4#1171994512yes—
527C22×C4#1171994512yes—
528C22×C4#1171994512yes—
529C22×C4#1171994512yes—
530C22×C4#1171994914yes—
531C22×C4C2×C22⋊D5312yes—
532C22×C4C2×C22⋊D5312yes—
533C22×C4C2×C22⋊D5312yes—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—2124no—
537C22⋊C4—2128no—
538C22⋊C4—2128no—
539C22⋊C4—2128no—
540C22⋊C4—2128no—
541C22⋊C4—2128no—
542C22⋊C4—2128no—
543C42—28no—
544C42—44no—
545C42—44no—
546C23—1064no—
547C23—1064no—
548C23—1064no—
549C23—1064no—
550C23—1064no—
551C23—1064no—
552C23—1064no—
553C23—2122no—
554C23—2122no—
555C23—2122no—
556C23—2122no—
557C23—2122no—
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—
574C23—2122no—
575C23—2122no—
576C23—2122no—
577C23—2122no—
578C23—2122no—
579C23—2122no—
580C23—2122no—
581C23C22×C22⋊D5311yescharacteristic
582C23C22×D21211yescharacteristic
583C23#41×D5312yes—
584C23C2×#1171994511yescharacteristic
585C23C2×#1171994511yescharacteristic
586C23C2×#1171994511yescharacteristic
587C23C2×#1171994511yescharacteristic
588C23C2×#1171994511yescharacteristic
589C23C2×#1171994912yes—
590C23#1313326212yes—
591C23#1313326312yes—
592D4—21216no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
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×C4—2122no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4—2122no—
643C2×C4—2122no—
644C2×C4—2122no—
645C2×C4—2122no—
646C2×C4—2122no—
647C2×C4—2122no—
648C2×C4#1313319114yes—
649C2×C4#1313320514yes—
650C2×C4#1313320614yes—
651C22—4242no—
652C22—4242no—
653C22—4242no—
654C22—4242no—
655C22—4242no—
656C22—4242no—
657C22—4242no—
658C22—4242no—
659C22—2124no—
660C22—2124no—
661C22—2124no—
662C22—2124no—
663C22—2124no—
664C22—2124no—
665C22—2124no—
666C22C2×#1313319111yescharacteristic
667C22#1681113011yescharacteristic
668C22#1681113411yescharacteristic
669C22#1681113411yescharacteristic
670C22C2×#1313320511yescharacteristic
671C22C2×#1313320611yescharacteristic
672C22#1681115711yescharacteristic
673C22#1681115711yescharacteristic
674C22#1681116012yes—
675C22#1681116012yes—
676C22#1681117311yescharacteristic
677C22#1681118211yescharacteristic
678C22#1681118711yescharacteristic
679C22#1681120612yes—
680C22#1681122412yes—
681C22#1681126812yes—
682C22#1681129512yes—
683C22#1681129712yes—
684C22#1681130212yes—
685C22#1681131712yes—
686C22#1681132112yes—
687C22#1681132112yes—
688C22#1681135812yes—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—4242no—
697C4—4242no—
698C4—4242no—
699C2—4244no—
700C2#2027799811yescharacteristic
701C2#2027906112yes—
702C2#2027934312yes—
703C2#2027960111yescharacteristic
704C2#2027971411yescharacteristic
705C2#2027976912yes—
706C2#2028019511yescharacteristic
707C2#2028023612yes—
708C2#2028028111yescharacteristic
709C2#2028059911yescharacteristic
710C2#2028301611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC2×C4×C4⋊C4×C53C2
prime radicalC23×D212C22
hypercenterC2×C4×C4⋊C4D53
hypocenterC53#15107
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16810562C2semidirect
4. #20272984C2semidirect
5. #20272963C2semidirect
6. #20272957C2semidirect
8. #20272953C2semidirect
9. #20272953C2semidirect
10. #20272942C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
347. C2×C4×C4⋊C4D53semidirect
394. C53#15107semidirect

Maximal subgroups

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

subgroupindex
C2×#168105622
C2×#168105402
C2×#168105252
#202729842
#202729632
#202729572
#202729552
#202729532
#202729422
C2×C4×C4⋊C4×C532
#1510753