Finite Group Databasebeta

← #27307472

Subgroups — #27307472

Order 13568.

1810 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×#16810530C211yesmaximal, characteristic
2C2×#16810530C211yesmaximal, characteristic
3C2×#16810519C211yesmaximal, characteristic
4C2×C4×#11719919C211yesmaximal, characteristic
5#20272978C212yesmaximal
6#20272974C212yesmaximal
7#20272948C212yesmaximal
8#20272948C212yesmaximal
9#20272945C211yesmaximal, characteristic
10#20272944C211yesmaximal, characteristic
11C22×C4×Q8×C53C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×#11719919C2212yes—
16C22×#11719919C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810530—24no—
20#16810530C418yescomplemented
21#16810519—24no—
22C4×#11719919—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C23×Q8×C53C2211yescharacteristic
35C2×C4×Q8×C53—24no—
36C2×C4×Q8×C53—24no—
37C2×C4×Q8×C53—24no—
38C22×C4⋊C4×C53C2212yes—
39C22×C4⋊C4×C53C2211yescharacteristic
40C22×C42×C53C2212yes—
41C22×C42×C53C2211yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53C2×C414yes—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—24no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—24no—
61C2×#11719919—42no—
62C2×#11719919Q814yescomplemented
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68C2×C4×Dic53C2×C414yes—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—22no—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—42no—
77#13133124—42no—
78#13133124C2×C418yes—
79#13133124C2×C418yes—
80C22×Q8×C53—22no—
81C22×Q8×C53—22no—
82C22×Q8×C53—22no—
83C22×Q8×C53—22no—
84C22×Q8×C53C2×C414yes—
85C22×Q8×C53D414yes—
86C22×Q8×C53Q814yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C4×Q8×C53—232no—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53D414yes—
99C2×C42×C53—24no—
100C2×C42×C53—24no—
101C2×C42×C53—22no—
102C2×C42×C53—22no—
103C2×C42×C53—24no—
104C2×C42×C53D414yes—
105C24×C53C2411yescharacteristic
106C2×Q8×C53—28no—
107C2×Q8×C53—28no—
108C2×Q8×C53—28no—
109C2×Q8×C53—28no—
110C2×Q8×C53—28no—
111C2×Q8×C53—28no—
112C2×Q8×C53C4⋊C4116yescomplemented
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D414yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D414yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×Q814yes—
143C22×C4×C53C2×Q812yes—
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53#4112yes—
147C22×C4×C53#4112yes—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—22no—
159C22×Dic53—21no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53—22no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—22no—
166C22×Dic53—21no—
167C22×Dic53—22no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—22no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53—21no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—22no—
179C22×Dic53—22no—
180C22×Dic53—22no—
181C22×Dic53—22no—
182C22×Dic53—22no—
183C22×Dic53C22×C412yes—
184C22×Dic53C22×C414yes—
185C22×Dic53C22×C412yes—
186C22×Dic53C2×Q812yes—
187C22×Dic53C2×Q812yes—
188C22×Dic53#4112yes—
189C22×Dic53#4112yes—
190C4⋊C4×C53—216no—
191C4⋊C4×C53—28no—
192C42×C53—216no—
193C42×C53—28no—
194C4⋊Dic53—28no—
195C4⋊Dic53—44no—
196#11719919—28no—
197#11719919—28no—
198#11719919—48no—
199C4×Dic53—28no—
200C4×Dic53—44no—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C22×D411yescharacteristic
205C23×C53C22×Q811yescharacteristic
206C23×C53C22×Q811yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53C2×#4111yescharacteristic
211C23×C53C2×#4111yescharacteristic
212C23×C53H(5,𝔽2)11yescharacteristic
213C23×C532−1+411yescharacteristic
214C23×C532−1+411yescharacteristic
215C23×C532−1+411yescharacteristic
216Q8×C53—232no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53C2×C4⋊C418yes—
244C2×C4×C53C2×C4⋊C414yes—
245C2×C4×C53C4×D414yes—
246C2×C4×C53C4×D414yes—
247C2×C4×C53#12014yes—
248C2×C4×C53C22⋊Q814yes—
249C2×C4×C53C22⋊Q814yes—
250C2×C4×C53C22⋊Q814yes—
251C2×C4×C53#12314yes—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—41no—
280C2×Dic53—42no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
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×Dic53C4×Q814yescomplemented
294#14017—531nomaximal, Sylow
295C22×C53C22×C4⋊C411yescharacteristic
296C22×C53C2×C4×D411yescharacteristic
297C22×C53C2×C4×D411yescharacteristic
298C22×C53C2×C4×Q811yescharacteristic
299C22×C53C4×#4111yescharacteristic
300C22×C53#51911yescharacteristic
301C22×C53#52011yescharacteristic
302C22×C53C2×#12011yescharacteristic
303C22×C53C2×C22⋊Q811yescharacteristic
304C22×C53C2×C22⋊Q811yescharacteristic
305C22×C53C2×C22⋊Q811yescharacteristic
306C22×C53C2×#12311yescharacteristic
307C22×C53#52511yescharacteristic
308C22×C53#52911yescharacteristic
309C22×C53#53311yescharacteristic
310C22×C53#53611yescharacteristic
311C22×C53#53711yescharacteristic
312C22×C53#54111yescharacteristic
313C22×C53#54211yescharacteristic
314C22×C53#54611yescharacteristic
315C22×C53#54711yescharacteristic
316C22×C53#54811yescharacteristic
317C22×C53#54811yescharacteristic
318C22×C53#54811yescharacteristic
319C22×C53D4×Q811yescharacteristic
320C22×C53D4×Q811yescharacteristic
321C22×C53#55011yescharacteristic
322C22×C53#55611yescharacteristic
323C22×C53#55611yescharacteristic
324C22×C53#55711yescharacteristic
325C22×C53#55711yescharacteristic
326C22×C53#55711yescharacteristic
327C22×C53#55711yescharacteristic
328C22×C53Q8211yescharacteristic
329C22×C53#55911yescharacteristic
330C4×C53—24no—
331C4×C53—28no—
332C4×C53—42no—
333C4×C53—44no—
334C4×C53#38518yes—
335Dic53—24no—
336Dic53—42no—
337Dic53—44no—
338Dic53—42no—
339Dic53—42no—
340Dic53—82no—
341C22×C4×Q8D5311yescharacteristic
342C2×#391—531no—
343C2×#391—531no—
344C2×#389—531no—
345C2×C4×C4⋊C4—531no—
346#1444—532no—
347#1442—532no—
348#1436—532no—
349#1436—532no—
350C4⋊#95—531no—
351C4⋊#95—531no—
352C2×C53C2×#38511yescharacteristic
353C2×C53C4×C22⋊Q811yescharacteristic
354C2×C53#232311yescharacteristic
355C2×C53Q8×C4⋊C411yescharacteristic
356C2×C53#235111yescharacteristic
357C2×C53#236511yescharacteristic
358C2×C53#236711yescharacteristic
359C2×C53#240711yescharacteristic
360C2×C53#240911yescharacteristic
361C2×C53#244611yescharacteristic
362C2×C53#244811yescharacteristic
363C2×C53#245111yescharacteristic
364C2×C53#245311yescharacteristic
365C2×C53#258111yescharacteristic
366C2×C53#258211yescharacteristic
367C23×Q8C2×D5311yescharacteristic
368C2×C4×Q8—24no—
369C2×C4×Q8—24no—
370C2×C4×Q8—24no—
371C22×C4⋊C4—531no—
372C22×C4⋊C4—531no—
373C22×C4⋊C4—531no—
374C22×C4⋊C4—531no—
375C22×C4⋊C4—532no—
376C22×C4⋊C4C2×D5312yes—
377C22×C4⋊C4C2×D5311yescharacteristic
378C22×C42—531no—
379C22×C42—531no—
380C22×C42C2×D5312yes—
381C22×C42C2×D5311yescharacteristic
382#391—538no—
383#391—1064no—
384#389—1064no—
385C4×C4⋊C4—1064no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—531no—
391C2×#95—532no—
392C2×#95—532no—
393C2×#95—532no—
394C2×#95—531no—
395C2×#95—532no—
396C2×#95—532no—
397C53#1401711yescharacteristic, complemented, Sylow
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—22no—
402C22×Q8Dic10614yes—
403C22×Q8C4×D5314yes—
404C22×Q8C22⋊D5314yes—
405C23×C4—531no—
406C23×C4—531no—
407C23×C4—532no—
408C23×C4—531no—
409C23×C4—531no—
410C23×C4—532no—
411C23×C4C22×D5312yes—
412C23×C4C22×D5311yescharacteristic
413C23×C4C22×D5312yes—
414C23×C4C22×D5311yescharacteristic
415C23×C4C22×D5311yescharacteristic
416C4×Q8—232no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—24no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—534no—
423C2×C4⋊C4—534no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1064no—
432C2×C4⋊C4—1062no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4—2122no—
437C2×C4⋊C4C22⋊D5314yes—
438C2×C42—24no—
439C2×C42—24no—
440C2×C42—24no—
441C2×C42—22no—
442C2×C42—22no—
443C2×C42—534no—
444C2×C42—1062no—
445C2×C42—1062no—
446C2×C42—1062no—
447C2×C42—1062no—
448C2×C42—1062no—
449C2×C42C22⋊D5314yes—
450#95—538no—
451#95—538no—
452#95—1064no—
453#95—1064no—
454#95—1062no—
455#95—1062no—
456#95—1064no—
457#95—1064no—
458#95—1064no—
459#95—2122no—
460#95—2122no—
461C24C23×D5311yescharacteristic
462C2×Q8—28no—
463C2×Q8—28no—
464C2×Q8—28no—
465C2×Q8—28no—
466C2×Q8—28no—
467C2×Q8—28no—
468C2×Q8#11719919116yescomplemented
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—532no—
495C22×C4—532no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—1062no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1062no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—1061no—
520C22×C4—1061no—
521C22×C4—1061no—
522C22×C4—1062no—
523C22×C4—1061no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4C2×Dic10614yes—
532C22×C4C2×Dic10612yes—
533C22×C4C2×C4×D5314yes—
534C22×C4C2×C4×D5312yes—
535C22×C4#1171994512yes—
536C22×C4#1171994512yes—
537C22×C4Q8×D5312yes—
538C22×C4#1171994912yes—
539C22×C4C2×C22⋊D5314yes—
540C22×C4C2×C22⋊D5312yes—
541C22×C4C2×C22⋊D5314yes—
542C22×C4C2×C22⋊D5312yes—
543C22×C4C2×C22⋊D5312yes—
544C22×C4C2×C22⋊D5312yes—
545C22×C4C2×C22⋊D5312yes—
546C4⋊C4—216no—
547C4⋊C4—28no—
548C4⋊C4—1068no—
549C4⋊C4—1068no—
550C4⋊C4—1068no—
551C4⋊C4—2128no—
552C4⋊C4—2124no—
553C42—216no—
554C42—28no—
555C42—1068no—
556C42—2124no—
557C23C22×C4×D5311yescharacteristic
558C23C22×C22⋊D5311yescharacteristic
559C23C22×C22⋊D5311yescharacteristic
560C23C22×C22⋊D5311yescharacteristic
561C23C2×Q8×D5311yescharacteristic
562C23C22×Dic10611yescharacteristic
563C23#41×D5311yescharacteristic
564C23#41×D5311yescharacteristic
565C23C2×#1171994511yescharacteristic
566C23C2×#1171994511yescharacteristic
567C23C2×#1171994911yescharacteristic
568C23#1313326311yescharacteristic
569C23#1313326411yescharacteristic
570C23#1313326511yescharacteristic
571C23#1313326511yescharacteristic
572Q8—232no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—534no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—1062no—
613C2×C4—1064no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2121no—
626C2×C4—2122no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2122no—
641C2×C4C2×#1171991918yes—
642C2×C4C2×#1171991914yes—
643C2×C4C4×C22⋊D5314yes—
644C2×C4C4×C22⋊D5314yes—
645C2×C4#1313318614yes—
646C2×C4#1313319314yes—
647C2×C4#1313319614yes—
648C2×C4#1313319614yes—
649C2×C4#1313320014yes—
650C22C22×#1171991911yescharacteristic
651C22C2×C4×C22⋊D5311yescharacteristic
652C22C2×C4×C22⋊D5311yescharacteristic
653C22C4×Q8×D5311yescharacteristic
654C22C4×#1171994911yescharacteristic
655C22#1681109911yescharacteristic
656C22#1681110511yescharacteristic
657C22C2×#1313318611yescharacteristic
658C22C2×#1313319311yescharacteristic
659C22C2×#1313319611yescharacteristic
660C22C2×#1313319611yescharacteristic
661C22C2×#1313320011yescharacteristic
662C22#1681113011yescharacteristic
663C22#1681116011yescharacteristic
664C22#1681117711yescharacteristic
665C22#1681119711yescharacteristic
666C22#1681120111yescharacteristic
667C22#1681123111yescharacteristic
668C22#1681124511yescharacteristic
669C22#1681126111yescharacteristic
670C22#1681127211yescharacteristic
671C22#1681128011yescharacteristic
672C22#1681128411yescharacteristic
673C22#1681128411yescharacteristic
674C22Q8×C22⋊D5311yescharacteristic
675C22Q8×C22⋊D5311yescharacteristic
676C22#1681129211yescharacteristic
677C22#1681134511yescharacteristic
678C22#1681134511yescharacteristic
679C22#1681134911yescharacteristic
680C22#1681134911yescharacteristic
681C22#1681135111yescharacteristic
682C22#1681135211yescharacteristic
683C22Q8×Dic10611yescharacteristic
684C22#1681135811yescharacteristic
685C4—24no—
686C4—28no—
687C4—44no—
688C4—42no—
689C4—1064no—
690C4—4242no—
691C4—2124no—
692C4—2122no—
693C4—2122no—
694C4—2122no—
695C4#1681049618yes—
696C2C2×#1681049611yescharacteristic
697C2C4×#1313319311yescharacteristic
698C2#2027739111yescharacteristic
699C2Q8×#1171991911yescharacteristic
700C2#2027763311yescharacteristic
701C2#2027776911yescharacteristic
702C2#2027778211yescharacteristic
703C2#2027805411yescharacteristic
704C2#2027807411yescharacteristic
705C2#2027848511yescharacteristic
706C2#2027850711yescharacteristic
707C2#2027854811yescharacteristic
708C2#2027857011yescharacteristic
709C2#2028007711yescharacteristic
710C2#2028008511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C4×Q8×C53C2
socle, prime radical, cosocleC24×C53C24
hypercenterC22×C4×Q8D53
hypocenterC53#14017
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
20. #16810530C4semidirect
62. C2×#11719919Q8semidirect
112. C2×Q8×C53C4⋊C4semidirect
293. C2×Dic53C4×Q8semidirect
397. C53#14017semidirect
468. C2×Q8#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168105302
C2×#168105192
C2×C4×#117199192
#202729782
#202729742
#202729482
#202729452
#202729442
C22×C4×Q8×C532
#1401753