Finite Group Databasebeta

← #27306880

Subgroups — #27306880

Order 13568.

2266 conjugacy classes of subgroups.

The subgroup lattice has 854 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133193C211yesmaximal, characteristic
2C2×#16810530C211yesmaximal, characteristic, complemented
3C2×#16810480C211yesmaximal, characteristic
4C2×C4×#11719921C211yesmaximal, characteristic
5#20272965C212yesmaximal, complemented
6#20272957C212yesmaximal
7#20272953C212yesmaximal
8#20272948C212yesmaximal, complemented
9#20272945C211yesmaximal, characteristic, complemented
10#20272944C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C53C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133193—24no—
14C2×#13133193—24no—
15C2×#13133193C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2212yes—
21C22×#11719921C2212yes—
22C22×#11719921C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810530—24no—
25#16810480—24no—
26C4×#11719921—24no—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C23×Q8×C53C2211yescharacteristic
37C2×C4×Q8×C53—24no—
38C2×C4×Q8×C53—24no—
39C2×C4×Q8×C53—24no—
40C22×C4⋊C4×C53C2212yes—
41C22×C4⋊C4×C53C2211yescharacteristic
42C22×C42×C53C2212yes—
43C22×C42×C53C2211yescharacteristic
44C24×D53C2311yescharacteristic
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53C2×C414yes—
57#13133193—232no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53C2×C414yes—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919C2×C418yes—
67C2×#11719921—22no—
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921—24no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—42no—
77C2×#11719921C2×C418yes—
78C2×#11719921Q814yescomplemented
79C2×C4×Dic53—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—22no—
82#13133124—24no—
83#13133124—24no—
84#13133124—22no—
85#13133124—22no—
86#13133124—24no—
87#13133124—24no—
88#13133124—24no—
89#13133124—42no—
90#13133124—42no—
91C22×Q8×C53—22no—
92C22×Q8×C53—22no—
93C22×Q8×C53—22no—
94C22×Q8×C53—22no—
95C22×Q8×C53C2×C414yes—
96C22×Q8×C53D414yes—
97C22×Q8×C53D414yes—
98C23×C4×C53C2312yes—
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2312yes—
101C23×C4×C53C2311yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C4×Q8×C53—232no—
104C2×C4⋊C4×C53—24no—
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53—24no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53D414yes—
110C2×C42×C53—24no—
111C2×C42×C53—24no—
112C2×C42×C53—22no—
113C2×C42×C53—22no—
114C2×C42×C53—24no—
115C2×C42×C53D414yes—
116C24×C53C2411yescharacteristic
117C23×D53—21no—
118C23×D53—21no—
119C23×D53—21no—
120C23×D53—21no—
121C23×D53—21no—
122C23×D53—21no—
123C23×D53—21no—
124C23×D53—21no—
125C23×D53—21no—
126C23×D53—21no—
127C23×D53—21no—
128C23×D53—21no—
129C23×D53C22×C412yes—
130C23×D53C2×Q812yes—
131C23×D53#4112yes—
132C2×Q8×C53—28no—
133C2×Q8×C53—28no—
134C2×Q8×C53—28no—
135C2×Q8×C53—28no—
136C2×Q8×C53—28no—
137C2×Q8×C53—28no—
138C2×Q8×C53C22⋊C4116yescomplemented
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—21no—
149C22×C4×C53—22no—
150C22×C4×C53—21no—
151C22×C4×C53—22no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—22no—
155C22×C4×C53—21no—
156C22×C4×C53—22no—
157C22×C4×C53—21no—
158C22×C4×C53—22no—
159C22×C4×C53C22×C414yes—
160C22×C4×C53C22×C412yes—
161C22×C4×C53C2×D414yes—
162C22×C4×C53C2×D412yes—
163C22×C4×C53C2×D414yes—
164C22×C4×C53C2×D414yes—
165C22×C4×C53C2×D412yes—
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2×D412yes—
168C22×C4×C53C2×D412yes—
169C22×C4×C53C2×D412yes—
170C22×C4×C53C2×Q812yes—
171C22×C4×C53#4112yes—
172C22×C4×C53#4112yes—
173C22×C4×C53#4112yes—
174C22×Dic53—21no—
175C22×Dic53—21no—
176C22×Dic53—21no—
177C22×Dic53—21no—
178C22×Dic53—21no—
179C22×Dic53—21no—
180C22×Dic53—22no—
181C22×Dic53—22no—
182C22×Dic53—21no—
183C22×Dic53—21no—
184C22×Dic53—22no—
185C22×Dic53—22no—
186C22×Dic53—21no—
187C22×Dic53—21no—
188C22×Dic53—21no—
189C22×Dic53—21no—
190C22×Dic53—21no—
191C22×Dic53—21no—
192C22×Dic53—22no—
193C22×Dic53—21no—
194C22×Dic53—22no—
195C22×Dic53—21no—
196C22×Dic53—21no—
197C22×Dic53—21no—
198C22×Dic53—22no—
199C22×Dic53—22no—
200C22×Dic53—22no—
201C22×Dic53—22no—
202C22×Dic53—22no—
203C22×Dic53—22no—
204C22×Dic53—22no—
205C22×Dic53C22×C414yes—
206C22×Dic53C22×C412yes—
207C22×Dic53C2×Q812yes—
208C22×Dic53#4112yes—
209C2×C4×D53—28no—
210C2×C4×D53—28no—
211C2×C4×D53—28no—
212C2×C4×D53—44no—
213C2×C4×D53—44no—
214C2×C4×D53—44no—
215C2×C4×D53—44no—
216C4⋊C4×C53—216no—
217C4⋊C4×C53—28no—
218C42×C53—216no—
219C42×C53—28no—
220#11719921—28no—
221#11719921—216no—
222#11719921—28no—
223#11719921—48no—
224C4⋊Dic53—28no—
225C4⋊Dic53—44no—
226#11719919—216no—
227C4×Dic53—44no—
228C23×C53C23×C411yescharacteristic
229C23×C53C22×D411yescharacteristic
230C23×C53C22×D411yescharacteristic
231C23×C53C22×D411yescharacteristic
232C23×C53C22×D411yescharacteristic
233C23×C53C22×Q811yescharacteristic
234C23×C53C2×#4111yescharacteristic
235C23×C53C2×#4111yescharacteristic
236C23×C53C2×#4111yescharacteristic
237C23×C53C2×#4111yescharacteristic
238C23×C53C2×#4111yescharacteristic
239C23×C53H(5,𝔽2)11yescharacteristic
240C23×C532−1+411yescharacteristic
241C23×C532−1+411yescharacteristic
242C23×C532−1+411yescharacteristic
243C22×D53—22no—
244C22×D53—22no—
245C22×D53—22no—
246C22×D53—22no—
247C22×D53—22no—
248C22×D53—22no—
249C22×D53—22no—
250C22×D53—22no—
251C22×D53—22no—
252C22×D53—22no—
253C22×D53—22no—
254C22×D53—22no—
255C22×D53—22no—
256C22×D53—22no—
257C22×D53—22no—
258C22×D53—22no—
259C22×D53—22no—
260C22×D53—22no—
261C22×D53—41no—
262C22×D53—41no—
263C22×D53—41no—
264C22×D53—41no—
265C22×D53—41no—
266C22×D53—41no—
267C22×D53—41no—
268C22×D53—41no—
269C22×D53—41no—
270C22×D53—41no—
271C22×D53—41no—
272C22×D53—41no—
273C22×D53—41no—
274C22×D53—41no—
275C22×D53—41no—
276C22×D53—41no—
277C22×D53C4×Q814yescomplemented
278Q8×C53—232no—
279C2×C4×C53—24no—
280C2×C4×C53—24no—
281C2×C4×C53—24no—
282C2×C4×C53—22no—
283C2×C4×C53—24no—
284C2×C4×C53—24no—
285C2×C4×C53—24no—
286C2×C4×C53—22no—
287C2×C4×C53—22no—
288C2×C4×C53—22no—
289C2×C4×C53—22no—
290C2×C4×C53—22no—
291C2×C4×C53—24no—
292C2×C4×C53—22no—
293C2×C4×C53—22no—
294C2×C4×C53—22no—
295C2×C4×C53—24no—
296C2×C4×C53—24no—
297C2×C4×C53—42no—
298C2×C4×C53—42no—
299C2×C4×C53—41no—
300C2×C4×C53—41no—
301C2×C4×C53—41no—
302C2×C4×C53—42no—
303C2×C4×C53—42no—
304C2×C4×C53—41no—
305C2×C4×C53C2×C22⋊C418yes—
306C2×C4×C53C2×C22⋊C414yes—
307C2×C4×C53C4×D414yes—
308C2×C4×C53C4×D414yes—
309C2×C4×C53#12014yes—
310C2×C4×C53#12114yes—
311C2×C4×C53#12114yes—
312C2×C4×C53C22⋊Q814yescomplemented
313C2×C4×C53#12314yes—
314C2×Dic53—22no—
315C2×Dic53—24no—
316C2×Dic53—22no—
317C2×Dic53—22no—
318C2×Dic53—24no—
319C2×Dic53—24no—
320C2×Dic53—22no—
321C2×Dic53—22no—
322C2×Dic53—22no—
323C2×Dic53—22no—
324C2×Dic53—22no—
325C2×Dic53—22no—
326C2×Dic53—41no—
327C2×Dic53—41no—
328C2×Dic53—41no—
329C2×Dic53—41no—
330C2×Dic53—41no—
331C2×Dic53—41no—
332C2×Dic53—41no—
333C2×Dic53—41no—
334C2×Dic53—42no—
335C2×Dic53—42no—
336C2×Dic53—41no—
337C2×Dic53—42no—
338C2×Dic53—42no—
339C2×Dic53—41no—
340C2×Dic53—41no—
341C2×Dic53—41no—
342C2×Dic53—42no—
343C2×Dic53—42no—
344C2×Dic53—42no—
345C2×Dic53—42no—
346C2×Dic53—42no—
347C2×Dic53—42no—
348C2×Dic53—42no—
349C4×D53—416no—
350#13969—531nomaximal, Sylow
351C22×C53C22×C22⋊C411yescharacteristic
352C22×C53C2×C4×D411yescharacteristic
353C22×C53C2×C4×D411yescharacteristic
354C22×C53C2×C4×Q811yescharacteristic
355C22×C53C4×#4111yescharacteristic
356C22×C53#51911yescharacteristic
357C22×C53#52011yescharacteristic
358C22×C53C2×#12011yescharacteristic
359C22×C53C2×#12111yescharacteristic
360C22×C53C2×#12111yescharacteristic
361C22×C53C2×C22⋊Q811yescharacteristic
362C22×C53C2×#12311yescharacteristic
363C22×C53#52511yescharacteristic
364C22×C53#52911yescharacteristic
365C22×C53#53311yescharacteristic
366C22×C53#53611yescharacteristic
367C22×C53#53711yescharacteristic
368C22×C53#54111yescharacteristic
369C22×C53#54211yescharacteristic
370C22×C53#54611yescharacteristic
371C22×C53#54711yescharacteristic
372C22×C53#54811yescharacteristic
373C22×C53#54811yescharacteristic
374C22×C53#54811yescharacteristic
375C22×C53#54811yescharacteristic
376C22×C53#54811yescharacteristic
377C22×C53D4×Q811yescharacteristic
378C22×C53D4×Q811yescharacteristic
379C22×C53D4×Q811yescharacteristic
380C22×C53#55011yescharacteristic
381C22×C53#55011yescharacteristic
382C22×C53#55211yescharacteristic
383C22×C53#55211yescharacteristic
384C22×C53#55411yescharacteristic
385C22×C53#55611yescharacteristic
386C2×D53—24no—
387C2×D53—24no—
388C2×D53—24no—
389C2×D53—42no—
390C2×D53—42no—
391C2×D53—42no—
392C2×D53—42no—
393C2×D53—42no—
394C2×D53—42no—
395C2×D53—42no—
396C2×D53—42no—
397C2×D53—42no—
398C2×D53—42no—
399C2×D53—42no—
400C2×D53—42no—
401C4×C53—24no—
402C4×C53—28no—
403C4×C53—42no—
404C4×C53—44no—
405C4×C53#38618yes—
406Dic53—44no—
407Dic53—42no—
408Dic53—42no—
409Dic53—42no—
410Dic53—82no—
411C22×C22⋊Q8—531no—
412C22×C4×Q8D5311yescharacteristic, complemented
413C2×#391—531no—
414C2×#380—531no—
415C2×C4×C22⋊C4—531no—
416#1441—532no—
417#1439—532no—
418#1437—532no—
419#1436—532no—
420C4⋊#95—531no—
421C4⋊#95—531no—
422C2×C53C2×#38611yescharacteristic
423C2×C53C4×C22⋊Q811yescharacteristic
424C2×C53#232311yescharacteristic
425C2×C53Q8×C22⋊C411yescharacteristic
426C2×C53#234111yescharacteristic
427C2×C53#236011yescharacteristic
428C2×C53#236211yescharacteristic
429C2×C53#240711yescharacteristic
430C2×C53#240911yescharacteristic
431C2×C53#242111yescharacteristic
432C2×C53#242311yescharacteristic
433C2×C53#242611yescharacteristic
434C2×C53#242911yescharacteristic
435C2×C53#253611yescharacteristic
436C2×C53#253811yescharacteristic
437D53—44no—
438C23×Q8C2×D5311yescharacteristic
439C24×C4—531no—
440C2×C22⋊Q8—538no—
441C2×C22⋊Q8—1064no—
442C2×C22⋊Q8—1064no—
443C2×C4×Q8—24no—
444C2×C4×Q8—24no—
445C2×C4×Q8—24no—
446C22×C4⋊C4—532no—
447C22×C4⋊C4—531no—
448C22×C4⋊C4—531no—
449C22×C4⋊C4C2×D5312yes—
450C22×C4⋊C4C2×D5311yescharacteristic
451C22×C22⋊C4—532no—
452C22×C22⋊C4—531no—
453C22×C22⋊C4—531no—
454C22×C22⋊C4—532no—
455C22×C42—531no—
456C22×C42C2×D5312yes—
457C22×C42C2×D5311yescharacteristic
458#391—1064no—
459#380—1064no—
460C4×C22⋊C4—1064no—
461C2×#95—532no—
462C2×#95—532no—
463C2×#95—531no—
464C2×#95—532no—
465C2×#95—532no—
466C2×#95—532no—
467C2×#95—531no—
468C2×#95—532no—
469C2×#95—532no—
470C53#1396911yescharacteristic, complemented, Sylow
471C25—531no—
472C22×Q8—22no—
473C22×Q8—22no—
474C22×Q8—22no—
475C22×Q8—22no—
476C22×Q8C4×D5314yes—
477C22×Q8D21214yes—
478C22×Q8C22⋊D5314yes—
479C23×C4—532no—
480C23×C4—531no—
481C23×C4—531no—
482C23×C4—531no—
483C23×C4—534no—
484C23×C4—532no—
485C23×C4—1062no—
486C23×C4—1062no—
487C23×C4—1062no—
488C23×C4—1062no—
489C23×C4—1062no—
490C23×C4—1062no—
491C23×C4C22×D5312yes—
492C23×C4C22×D5311yescharacteristic
493C23×C4C22×D5312yes—
494C23×C4C22×D5311yescharacteristic
495C23×C4C22×D5311yescharacteristic
496C22⋊Q8—10632no—
497C4×Q8—232no—
498C2×C4⋊C4—24no—
499C2×C4⋊C4—24no—
500C2×C4⋊C4—24no—
501C2×C4⋊C4—22no—
502C2×C4⋊C4—22no—
503C2×C4⋊C4—538no—
504C2×C4⋊C4—534no—
505C2×C4⋊C4—1064no—
506C2×C4⋊C4—1062no—
507C2×C4⋊C4—1064no—
508C2×C4⋊C4—1062no—
509C2×C4⋊C4—1062no—
510C2×C4⋊C4—1062no—
511C2×C4⋊C4—1062no—
512C2×C4⋊C4C22⋊D5314yes—
513C2×C22⋊C4—538no—
514C2×C22⋊C4—534no—
515C2×C22⋊C4—1062no—
516C2×C22⋊C4—1062no—
517C2×C22⋊C4—1064no—
518C2×C22⋊C4—1062no—
519C2×C22⋊C4—1064no—
520C2×C22⋊C4—1062no—
521C2×C22⋊C4—1062no—
522C2×C22⋊C4—1064no—
523C2×C22⋊C4—1064no—
524C2×C22⋊C4—2122no—
525C2×C42—24no—
526C2×C42—24no—
527C2×C42—24no—
528C2×C42—22no—
529C2×C42—22no—
530C2×C42—1062no—
531C2×C42—1062no—
532C2×C42—1062no—
533C2×C42C22⋊D5314yes—
534#95—1064no—
535#95—1064no—
536#95—1064no—
537#95—1064no—
538#95—1062no—
539#95—1062no—
540#95—1064no—
541#95—2122no—
542#95—2122no—
543C24—532no—
544C24—532no—
545C24—532no—
546C24—1061no—
547C24—1061no—
548C24—1061no—
549C24—1061no—
550C24—1061no—
551C24—1061no—
552C24—1061no—
553C24—1061no—
554C24—1061no—
555C24—1061no—
556C24—1061no—
557C24—1061no—
558C24C23×D5311yescharacteristic
559C2×Q8—28no—
560C2×Q8—28no—
561C2×Q8—28no—
562C2×Q8—28no—
563C2×Q8—28no—
564C2×Q8—28no—
565C2×Q8#11719921116yescomplemented
566C22×C4—21no—
567C22×C4—21no—
568C22×C4—22no—
569C22×C4—22no—
570C22×C4—22no—
571C22×C4—22no—
572C22×C4—21no—
573C22×C4—21no—
574C22×C4—21no—
575C22×C4—22no—
576C22×C4—21no—
577C22×C4—22no—
578C22×C4—22no—
579C22×C4—21no—
580C22×C4—21no—
581C22×C4—22no—
582C22×C4—21no—
583C22×C4—22no—
584C22×C4—22no—
585C22×C4—21no—
586C22×C4—534no—
587C22×C4—532no—
588C22×C4—532no—
589C22×C4—532no—
590C22×C4—1061no—
591C22×C4—1061no—
592C22×C4—1061no—
593C22×C4—1061no—
594C22×C4—1062no—
595C22×C4—1061no—
596C22×C4—1061no—
597C22×C4—1061no—
598C22×C4—1062no—
599C22×C4—1061no—
600C22×C4—1062no—
601C22×C4—1061no—
602C22×C4—1061no—
603C22×C4—1061no—
604C22×C4—1061no—
605C22×C4—1061no—
606C22×C4—1062no—
607C22×C4—1061no—
608C22×C4—1061no—
609C22×C4—1062no—
610C22×C4—1061no—
611C22×C4—1061no—
612C22×C4—1061no—
613C22×C4—1062no—
614C22×C4—1068no—
615C22×C4—1068no—
616C22×C4—1068no—
617C22×C4—1062no—
618C22×C4—1062no—
619C22×C4—1062no—
620C22×C4—1062no—
621C22×C4—1062no—
622C22×C4—1062no—
623C22×C4—1062no—
624C22×C4—2124no—
625C22×C4—2124no—
626C22×C4—2124no—
627C22×C4—2124no—
628C22×C4C2×C4×D5314yes—
629C22×C4C2×C4×D5312yes—
630C22×C4C2×D21214yes—
631C22×C4C2×D21212yes—
632C22×C4#1171994512yes—
633C22×C4#1171994512yes—
634C22×C4Q8×D5312yes—
635C22×C4#1171994912yes—
636C22×C4C2×C22⋊D5314yes—
637C22×C4C2×C22⋊D5312yes—
638C22×C4C2×C22⋊D5314yes—
639C22×C4C2×C22⋊D5312yes—
640C22×C4C2×C22⋊D5312yes—
641C22×C4C2×C22⋊D5312yes—
642C22×C4C2×C22⋊D5312yes—
643C4⋊C4—216no—
644C4⋊C4—28no—
645C4⋊C4—10616no—
646C4⋊C4—1068no—
647C4⋊C4—2124no—
648C22⋊C4—1068no—
649C22⋊C4—10616no—
650C22⋊C4—1068no—
651C22⋊C4—2128no—
652C42—216no—
653C42—28no—
654C42—2124no—
655C23—534no—
656C23—1062no—
657C23—1062no—
658C23—1062no—
659C23—1062no—
660C23—1062no—
661C23—1062no—
662C23—1062no—
663C23—1062no—
664C23—1062no—
665C23—1062no—
666C23—1062no—
667C23—1062no—
668C23—1062no—
669C23—1062no—
670C23—1062no—
671C23—1062no—
672C23—1062no—
673C23—1062no—
674C23—2121no—
675C23—2121no—
676C23—2121no—
677C23—2121no—
678C23—2121no—
679C23—2121no—
680C23—2121no—
681C23—2121no—
682C23—2121no—
683C23—2121no—
684C23—2121no—
685C23—2121no—
686C23—2121no—
687C23—2121no—
688C23—2121no—
689C23—2121no—
690C23C22×C4×D5311yescharacteristic
691C23C22×C22⋊D5311yescharacteristic
692C23C22×C22⋊D5311yescharacteristic
693C23C22×C22⋊D5311yescharacteristic
694C23C22×D21211yescharacteristic
695C23C2×Q8×D5311yescharacteristic
696C23#41×D5311yescharacteristic
697C23#41×D5311yescharacteristic
698C23C2×#1171994511yescharacteristic
699C23C2×#1171994511yescharacteristic
700C23C2×#1171994911yescharacteristic
701C23#1313326311yescharacteristic
702C23#1313326411yescharacteristic
703C23#1313326511yescharacteristic
704C23#1313326511yescharacteristic
705Q8—232no—
706C2×C4—22no—
707C2×C4—22no—
708C2×C4—22no—
709C2×C4—22no—
710C2×C4—24no—
711C2×C4—24no—
712C2×C4—22no—
713C2×C4—24no—
714C2×C4—24no—
715C2×C4—24no—
716C2×C4—24no—
717C2×C4—22no—
718C2×C4—24no—
719C2×C4—24no—
720C2×C4—22no—
721C2×C4—22no—
722C2×C4—22no—
723C2×C4—24no—
724C2×C4—41no—
725C2×C4—42no—
726C2×C4—41no—
727C2×C4—42no—
728C2×C4—41no—
729C2×C4—42no—
730C2×C4—42no—
731C2×C4—41no—
732C2×C4—1062no—
733C2×C4—1062no—
734C2×C4—1062no—
735C2×C4—1064no—
736C2×C4—1062no—
737C2×C4—1062no—
738C2×C4—1062no—
739C2×C4—1062no—
740C2×C4—1064no—
741C2×C4—1064no—
742C2×C4—1062no—
743C2×C4—1062no—
744C2×C4—2122no—
745C2×C4—2122no—
746C2×C4—2122no—
747C2×C4—2122no—
748C2×C4—2122no—
749C2×C4—2122no—
750C2×C4—2122no—
751C2×C4—2122no—
752C2×C4—2122no—
753C2×C4—2121no—
754C2×C4—2122no—
755C2×C4—2122no—
756C2×C4—2121no—
757C2×C4—2121no—
758C2×C4—2121no—
759C2×C4—2121no—
760C2×C4—2121no—
761C2×C4—2121no—
762C2×C4—2121no—
763C2×C4—2121no—
764C2×C4—2121no—
765C2×C4—2121no—
766C2×C4—2121no—
767C2×C4—21216no—
768C2×C4C2×#1171992118yes—
769C2×C4C2×#1171992114yes—
770C2×C4C4×C22⋊D5314yes—
771C2×C4C4×C22⋊D5314yes—
772C2×C4#1313318614yes—
773C2×C4#1313319114yes—
774C2×C4#1313319114yes—
775C2×C4#1313319314yescomplemented
776C2×C4#1313320014yes—
777C22—1064no—
778C22—1064no—
779C22—1064no—
780C22—2122no—
781C22—2122no—
782C22—2122no—
783C22—2122no—
784C22—2122no—
785C22—2122no—
786C22—2122no—
787C22—2122no—
788C22—2122no—
789C22—2122no—
790C22—2122no—
791C22—2122no—
792C22C22×#1171992111yescharacteristic
793C22C2×C4×C22⋊D5311yescharacteristic
794C22C2×C4×C22⋊D5311yescharacteristic
795C22C4×Q8×D5311yescharacteristic
796C22C4×#1171994911yescharacteristic
797C22#1681109911yescharacteristic
798C22#1681110511yescharacteristic
799C22C2×#1313318611yescharacteristic
800C22C2×#1313319111yescharacteristic
801C22C2×#1313319111yescharacteristic
802C22C2×#1313319311yescharacteristic
803C22C2×#1313320011yescharacteristic
804C22#1681113011yescharacteristic
805C22#1681116011yescharacteristic
806C22#1681117711yescharacteristic
807C22#1681119711yescharacteristic
808C22#1681120111yescharacteristic
809C22#1681123111yescharacteristic
810C22#1681124511yescharacteristic
811C22#1681126111yescharacteristic
812C22#1681127211yescharacteristic
813C22#1681128011yescharacteristic
814C22#1681128411yescharacteristic
815C22#1681128411yescharacteristic
816C22#1681128511yescharacteristic
817C22#1681128511yescharacteristic
818C22Q8×C22⋊D5311yescharacteristic
819C22Q8×C22⋊D5311yescharacteristic
820C22Q8×D21211yescharacteristic
821C22#1681129211yescharacteristic
822C22#1681129511yescharacteristic
823C22#1681131411yescharacteristic
824C22#1681131411yescharacteristic
825C22#1681133311yescharacteristic
826C22#1681134611yescharacteristic
827C4—24no—
828C4—28no—
829C4—44no—
830C4—42no—
831C4—4242no—
832C4—2124no—
833C4—2122no—
834C4—2122no—
835C4—2122no—
836C4#1681050218yes—
837C2—2124no—
838C2C2×#1681050211yescharacteristic
839C2C4×#1313319311yescharacteristic
840C2#2027739111yescharacteristic
841C2Q8×#1171992111yescharacteristic
842C2#2027755311yescharacteristic
843C2#2027771611yescharacteristic
844C2#2027773811yescharacteristic
845C2#2027805411yescharacteristic
846C2#2027807411yescharacteristic
847C2#2027821211yescharacteristic
848C2#2027823411yescharacteristic
849C2#2027827511yescharacteristic
850C2#2027830811yescharacteristic
851C2#2027962011yescharacteristic
852C2#2027963311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C4×Q8×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC22×C4×Q8D53
hypocenterC53#13969
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16810530C2semidirect
5. #20272965C2semidirect
8. #20272948C2semidirect
9. #20272945C2semidirect
10. #20272944C2semidirect
11. C22×C4×Q8×C53C2semidirect
15. C2×#13133193C4semidirect
78. C2×#11719921Q8semidirect
138. C2×Q8×C53C22⋊C4semidirect
277. C22×D53C4×Q8semidirect
312. C2×C4×C53C22⋊Q8semidirect
412. C22×C4×Q8D53semidirect
470. C53#13969semidirect
565. C2×Q8#11719921semidirect
775. C2×C4#13133193semidirect

Maximal subgroups

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

subgroupindex
C22×#131331932
C2×#168105302
C2×#168104802
C2×C4×#117199212
#202729652
#202729572
#202729532
#202729482
#202729452
#202729442
C22×C4×Q8×C532
#1396953