Finite Group Databasebeta

← #27314765

Subgroups — #27314765

Order 13568.

1586 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810526C211yesmaximal, characteristic
2C2×#16810509C211yesmaximal, characteristic, complemented
3#20272980C211yesmaximal, characteristic, complemented
4#20272968C211yesmaximal, characteristic, complemented
5#20272964C212yesmaximal, complemented
6#20272960C212yesmaximal, complemented
7#20272960C212yesmaximal, complemented
8#20272960C212yesmaximal, complemented
9#20272956C211yesmaximal, characteristic
10#20272940C211yesmaximal, characteristic, complemented
11C2×C53×#394C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20#16810526—24no—
21#16810509—24no—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C23×D4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2212yes—
35C22×C22⋊C4×C53C2212yes—
36C53×#394C418yescomplemented
37C2×#95×C53C2211yescharacteristic
38C23×Dic53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—42no—
47C2×C4⋊Dic53—42no—
48C2×C4⋊Dic53—42no—
49C2×C4⋊Dic53—42no—
50C2×C4⋊Dic53—42no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—42no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53—42no—
57C2×C22⋊Dic53—42no—
58C2×C22⋊Dic53—42no—
59C2×C22⋊Dic53—42no—
60C2×C22⋊Dic53—42no—
61C2×C22⋊Dic53—42no—
62C2×C22⋊Dic53—42no—
63C2×C4×Dic53—42no—
64C2×C4×Dic53D414yes—
65#13133124—22no—
66#13133124—22no—
67#13133124—22no—
68#13133124—22no—
69#13133124—22no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75C25×C53C2312yes—
76C22×D4×C53—22no—
77C22×D4×C53—22no—
78C22×D4×C53—42no—
79C22×D4×C53—42no—
80C22×D4×C53C2×C414yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53C2×C414yes—
88C2×C22⋊C4×C53—42no—
89C2×C22⋊C4×C53—42no—
90C2×C22⋊C4×C53—42no—
91C2×C22⋊C4×C53—42no—
92C2×C22⋊C4×C53C2×C418yes—
93C2×C22⋊C4×C53C2×C418yes—
94#95×C53C2×C414yes—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53C22×C414yes—
110C24×C53C2411yescharacteristic
111C2×D4×C53—28no—
112C2×D4×C53—48no—
113C2×D4×C53—44no—
114C2×D4×C53—44no—
115C2×D4×C53—48no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53C22×C414yes—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
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—21no—
161C22×Dic53—21no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—21no—
169C22×Dic53—22no—
170C22×Dic53—21no—
171C22×Dic53—22no—
172C22×Dic53C2×D412yes—
173C22×Dic53C2×D412yes—
174C4⋊C4×C53—28no—
175C22⋊C4×C53—48no—
176C22⋊C4×C53—48no—
177C22⋊Dic53—48no—
178C22⋊Dic53—48no—
179C22⋊Dic53—48no—
180C22⋊Dic53—48no—
181C4⋊Dic53—44no—
182C4⋊Dic53—44no—
183C4⋊Dic53—48no—
184C4×Dic53—44no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53C23×C411yescharacteristic
221C23×C53C22×D411yescharacteristic
222C23×C53C2×#4111yescharacteristic
223C23×C53H(5,𝔽2)12yes—
224C23×C53H(5,𝔽2)12yes—
225C23×C53H(5,𝔽2)11yescharacteristic
226C23×C53H(5,𝔽2)11yescharacteristic
227C23×C53H(5,𝔽2)11yescharacteristic
228C23×C53H(5,𝔽2)12yes—
229C23×C532−1+412yes—
230C23×C532−1+411yescharacteristic
231D4×C53—416no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53C4×D414yes—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—41no—
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—
290#14624—531nomaximal, Sylow
291C22×C53—44no—
292C22×C53—44no—
293C22×C53—44no—
294C22×C53—44no—
295C22×C53—44no—
296C22×C53—44no—
297C22×C53—44no—
298C22×C53—82no—
299C22×C53—82no—
300C22×C53—82no—
301C22×C53—82no—
302C22×C53—82no—
303C22×C53—82no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53C2×C4×D411yescharacteristic
307C22×C53#51812yes—
308C22×C53#51811yescharacteristic
309C22×C53#52011yescharacteristic
310C22×C53#52012yes—
311C22×C53#53411yescharacteristic
312C22×C53#53511yescharacteristic
313C22×C53#53511yescharacteristic
314C22×C53#53511yescharacteristic
315C22×C53#53611yescharacteristic
316C22×C53#53711yescharacteristic
317C22×C53#53812yes—
318C22×C53#54011yescharacteristic
319C22×C53#54211yescharacteristic
320C22×C53#54212yes—
321C22×C53#56012yes—
322C22×C53#56012yes—
323C22×C53#56012yes—
324C22×C53#56212yes—
325C22×C53#56212yes—
326C22×C53#56212yes—
327C22×C53#56212yes—
328C22×C53#56312yes—
329C4×C53—24no—
330C4×C53—42no—
331C4×C53—42no—
332C4×C53—44no—
333Dic53—42no—
334Dic53—42no—
335Dic53—81no—
336Dic53—82no—
337Dic53—82no—
338Dic53—81no—
339C2×#394D5311yescharacteristic
340C2×#390—531no—
341C2×#387—531no—
342#1444—531no—
343#1441—531no—
344#1440—532no—
345#1439—532no—
346#1439—532no—
347#1439—532no—
348#1438—531no—
349#1434—531no—
350C2×C53—84no—
351C2×C53#232111yescharacteristic
352C2×C53#232411yescharacteristic
353C2×C53#232611yescharacteristic
354C2×C53#237612yes—
355C2×C53#237712yes—
356C2×C53#281311yescharacteristic
357C2×C53#281611yescharacteristic
358C2×C53#281811yescharacteristic
359C2×C53#282011yescharacteristic
360C2×C53#282912yes—
361C2×C53#283012yes—
362C23×D4C2×D5311yescharacteristic
363C22×C4⋊C4—531no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4—532no—
366C22×C4⋊C4C2×D5311yescharacteristic
367C22×C22⋊C4—532no—
368C22×C22⋊C4—532no—
369C22×C22⋊C4—532no—
370C22×C22⋊C4—532no—
371C22×C22⋊C4C2×D5312yes—
372C22×C22⋊C4C2×D5312yes—
373C22×C42—531no—
374#394Dic5318yescomplemented
375#390—1064no—
376#387—1064no—
377C2×#95—531no—
378C2×#95—531no—
379C2×#95—531no—
380C2×#95—531no—
381C2×#95—531no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95—532no—
385C2×#95—532no—
386C2×#95—532no—
387C2×#95C2×D5311yescharacteristic
388C53#1462411yescharacteristic, complemented, Sylow
389C25C22×D5312yes—
390C22×D4—22no—
391C22×D4—22no—
392C22×D4—42no—
393C22×D4—42no—
394C22×D4C2×Dic5314yes—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—532no—
399C23×C4—531no—
400C23×C4—532no—
401C23×C4C22×D5311yescharacteristic
402C23×C4C22×D5312yes—
403C23×C4C22×D5311yescharacteristic
404C23×C4C22×D5311yescharacteristic
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—2122no—
410C2×C4⋊C4—2122no—
411C2×C4⋊C4—2122no—
412C2×C4⋊C4—2122no—
413C2×C4⋊C4—2122no—
414C2×C4⋊C4C2×Dic5314yes—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—42no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—2122no—
422C2×C22⋊C4—2122no—
423C2×C22⋊C4—2122no—
424C2×C22⋊C4—2122no—
425C2×C22⋊C4—2122no—
426C2×C22⋊C4—2122no—
427C2×C22⋊C4—2122no—
428C2×C22⋊C4—2122no—
429C2×C22⋊C4—2122no—
430C2×C22⋊C4—2122no—
431C2×C22⋊C4C2×Dic5318yes—
432C2×C22⋊C4C2×Dic5318yes—
433C2×C42—534no—
434C2×C42—2122no—
435#95—1062no—
436#95—1062no—
437#95—1062no—
438#95—1062no—
439#95—1062no—
440#95—2122no—
441#95—2122no—
442#95—2122no—
443#95—2122no—
444#95—2122no—
445#95C2×Dic5314yes—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24C22×Dic5314yes—
461C24C23×D5311yescharacteristic
462C2×D4—28no—
463C2×D4—48no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—48no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1061no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1061no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1061no—
519C22×C4D4×D5312yes—
520C22×C4#1171994712yes—
521C22×C4C22×Dic5314yes—
522C22×C4C22×Dic5312yes—
523C22×C4C22×Dic5312yes—
524C22×C4C22×Dic5312yes—
525C4⋊C4—28no—
526C4⋊C4—2124no—
527C4⋊C4—2124no—
528C4⋊C4—2128no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—2128no—
532C22⋊C4—2128no—
533C22⋊C4—2128no—
534C22⋊C4—2128no—
535C42—2124no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C23×Dic5311yescharacteristic
572C23C2×D4×D5311yescharacteristic
573C23C2×#1171994711yescharacteristic
574C23#1313326211yescharacteristic
575C23#1313326211yescharacteristic
576C23#1313326212yes—
577C23#1313326212yes—
578C23#1313326312yes—
579C23#1313326311yescharacteristic
580C23#1313326412yes—
581C23#1313326411yescharacteristic
582D4—416no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2121no—
625C2×C4—2122no—
626C2×C4—2121no—
627C2×C4—2122no—
628C2×C4—2121no—
629C2×C4—2122no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4—2122no—
639C2×C4—2122no—
640C2×C4D4×Dic5314yes—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×D4×Dic5311yescharacteristic
657C22#1681109812yes—
658C22#1681109811yescharacteristic
659C22#1681110611yescharacteristic
660C22#1681110612yes—
661C22#1681118311yescharacteristic
662C22#1681118811yescharacteristic
663C22#1681118911yescharacteristic
664C22#1681119111yescharacteristic
665C22#1681119311yescharacteristic
666C22#1681120211yescharacteristic
667C22#1681120812yes—
668C22#1681122611yescharacteristic
669C22#1681124011yescharacteristic
670C22#1681124212yes—
671C22#1681136112yes—
672C22#1681136112yes—
673C22#1681136312yes—
674C22#1681136812yes—
675C22#1681136812yes—
676C22#1681137012yes—
677C22#1681137012yes—
678C22#1681137612yes—
679C4—24no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—4241no—
684C4—4241no—
685C4—4242no—
686C4—4242no—
687C4—2122no—
688C4—2122no—
689C2—84no—
690C2#2027737611yescharacteristic
691C2#2027741011yescharacteristic
692C2#2027742811yescharacteristic
693C2#2027786312yes—
694C2#2027787212yes—
695C2#2028271611yescharacteristic
696C2#2028274611yescharacteristic
697C2#2028275711yescharacteristic
698C2#2028276811yescharacteristic
699C2#2028285112yes—
700C2#2028286112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#394C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#394D53
hypocenterC53#14624
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C2×#16810509C2semidirect
3. #20272980C2semidirect
4. #20272968C2semidirect
5. #20272964C2semidirect
6. #20272960C2semidirect
7. #20272960C2semidirect
8. #20272960C2semidirect
10. #20272940C2semidirect
36. C53×#394C4semidirect
374. #394Dic53semidirect
388. C53#14624semidirect

Maximal subgroups

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

subgroupindex
C2×#168105262
C2×#168105092
#202729802
#202729682
#202729642
#202729602
#202729562
#202729402
C2×C53×#3942
#1462453