Finite Group Databasebeta

← #27418112

Subgroups — #27418112

Order 13568.

1088 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810927C211yesmaximal, characteristic, complemented
2C2×C4×#11719925C211yesmaximal, characteristic, complemented
3#20279624C211yesmaximal, characteristic
4#20275915C211yesmaximal, characteristic, complemented
5#20275908C211yesmaximal, characteristic, complemented
6#20275723C211yesmaximal, characteristic, complemented
7#20275723C211yesmaximal, characteristic, complemented
8#20275681C211yesmaximal, characteristic, complemented
9#20275510C211yesmaximal, characteristic
10#20275027C211yesmaximal, characteristic, complemented
11#20275022C211yesmaximal, characteristic, complemented
12#20274914C211yesmaximal, characteristic, complemented
13#20274903C211yesmaximal, characteristic, complemented
14#20274730C211yesmaximal, characteristic, complemented
15C2×C53×#552C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810927—24no—
21C4×#11719925—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×C22⋊#88322C2211yescharacteristic
36C2×C4×#88322C2211yescharacteristic
37#16810554—21no—
38#16810537—21no—
39#16810511—21no—
40#16810511—21no—
41#16810489C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C53×#552—24no—
51C53×#552—24no—
52C2×#126×C53—21no—
53C2×#125×C53—21no—
54C2×#125×C53C2211yescharacteristic
55C2×#123×C53—21no—
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×C22⋊Q8×C53C2211yescharacteristic
58C2×C4×Q8×C53C2211yescharacteristic
59C2×C4×D4×C53C2211yescharacteristic
60C2×#117×C53—21no—
61C2×#117×C53C2211yescharacteristic
62C22×C4⋊C4×C53C2211yescharacteristic
63C22×Dic106C2311yescharacteristic
64C2×#11719925—22no—
65C2×#11719925—22no—
66C2×#11719925—22no—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133196—22no—
72#13133196—22no—
73#13133196—41no—
74#13133196—41no—
75C4×Dic106—22no—
76C4×Dic106—22no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88#13133150—22no—
89#13133150—22no—
90#13133148—22no—
91C4⋊#88322—22no—
92#13133141—22no—
93#13133141—41no—
94#13133141—41no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98C22⋊#88322—22no—
99C22⋊#88322—22no—
100C22⋊#88322—41no—
101C22⋊#88322—41no—
102C4×#88322—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×Q8×C53C2311yescharacteristic
108C22×D4×C53D411yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112#126×C53—28no—
113#125×C53—28no—
114#125×C53—22no—
115#125×C53D414yes—
116#123×C53—28no—
117C22⋊Q8×C53—22no—
118C22⋊Q8×C53—22no—
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C4×Q8×C53—22no—
122C4×Q8×C53D414yes—
123C4×D4×C53—22no—
124C4×D4×C53—22no—
125#117×C53—28no—
126#117×C53—22no—
127#117×C53—22no—
128C2×C4⋊C4×C53—22no—
129C2×C4⋊C4×C53—22no—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53—21no—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53—21no—
134C2×C4⋊C4×C53—21no—
135C2×C4⋊C4×C53—21no—
136C2×C4⋊C4×C53D411yescharacteristic
137C2×C4⋊C4×C53C2311yescharacteristic
138C2×C4⋊C4×C53C2311yescharacteristic
139C2×C4⋊C4×C53C2311yescharacteristic
140C2×C22⋊C4×C53—21no—
141C2×C22⋊C4×C53—21no—
142C2×C22⋊C4×C53—21no—
143C2×C22⋊C4×C53D411yescharacteristic
144C2×C22⋊C4×C53D411yescharacteristic
145C2×C42×C53—21no—
146C2×C42×C53—21no—
147C2×C42×C53C2311yescharacteristic
148C24×C53C2×D411yescharacteristic
149C24×C53C2×D411yescharacteristic
150C2×Q8×C53—21no—
151C2×Q8×C53—21no—
152C2×Q8×C53—21no—
153C2×Q8×C53—21no—
154C2×Q8×C53—21no—
155C2×Q8×C53C2×D412yes—
156C2×D4×C53—22no—
157C2×D4×C53—22no—
158C2×D4×C53—41no—
159C22×C4×C53—24no—
160C22×C4×C53—21no—
161C22×C4×C53—21no—
162C22×C4×C53—21no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—21no—
166C22×C4×C53—21no—
167C22×C4×C53—21no—
168C22×C4×C53—21no—
169C22×C4×C53—21no—
170C22×C4×C53—21no—
171C22×C4×C53—21no—
172C22×C4×C53—21no—
173C22×C4×C53—21no—
174C22×C4×C53—21no—
175C22×C4×C53—21no—
176C22×C4×C53—42no—
177C22×C4×C53—42no—
178C22×C4×C53C2×D411yescharacteristic
179C22×C4×C53C2×D411yescharacteristic
180C22×C4×C53C2×D411yescharacteristic
181C22×C4×C53C2×D411yescharacteristic
182C22×C4×C53C2411yescharacteristic
183C22×Dic53—21no—
184C22×Dic53—21no—
185C22×Dic53—21no—
186C22×Dic53—21no—
187C2×Dic106—21no—
188C2×Dic106—21no—
189C2×Dic106—21no—
190C2×Dic106—21no—
191C2×Dic106—21no—
192C2×Dic106#4112yes—
193C4⋊C4×C53—24no—
194C4⋊C4×C53—24no—
195C4⋊C4×C53—24no—
196C4⋊C4×C53—24no—
197C4⋊C4×C53—24no—
198C4⋊C4×C53—24no—
199C4⋊C4×C53—22no—
200C4⋊C4×C53—21no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53—21no—
203C4⋊C4×C53C2×D412yes—
204C4⋊C4×C53C2×D412yes—
205C4⋊C4×C53C2×D412yes—
206C22⋊C4×C53—24no—
207C22⋊C4×C53—24no—
208C22⋊C4×C53—24no—
209C22⋊C4×C53—22no—
210C22⋊C4×C53—22no—
211C42×C53—24no—
212C42×C53—24no—
213C42×C53C2×D412yes—
214C42×C53C2×D412yes—
215C22⋊Dic53—42no—
216C22⋊Dic53—42no—
217C22⋊Dic53—42no—
218C22⋊Dic53—42no—
219#11719925—28no—
220C4⋊Dic53—21no—
221C4⋊Dic53—21no—
222C4⋊Dic53—21no—
223C4⋊Dic53—21no—
224C4⋊Dic53—21no—
225C4⋊Dic53—21no—
226#11719919—42no—
227#11719919—42no—
228#11719919—42no—
229C4×Dic53—42no—
230C2×#88322—21no—
231C2×#88322—21no—
232C2×#88322—21no—
233C2×#88322—21no—
234C2×#88322—21no—
235C2×#88322—21no—
236C2×#88322—21no—
237C2×#88322—21no—
238C2×#88322—21no—
239C2×#88322—21no—
240C2×#88322—21no—
241C2×#88322#4112yes—
242C23×C53—21no—
243C23×C53—21no—
244C23×C53—21no—
245C23×C53—21no—
246C23×C53—21no—
247C23×C53—21no—
248C23×C53—21no—
249C23×C53—21no—
250C23×C53—21no—
251C23×C53—21no—
252C23×C53—21no—
253C23×C53—21no—
254C23×C53—21no—
255C23×C53#13512yes—
256C23×C53C22×D411yescharacteristic
257Q8×C53—22no—
258Q8×C53—22no—
259Q8×C53—41no—
260Q8×C53—41no—
261D4×C53—44no—
262C2×C4×C53—21no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—22no—
267C2×C4×C53—22no—
268C2×C4×C53—21no—
269C2×C4×C53—21no—
270C2×C4×C53—22no—
271C2×C4×C53—22no—
272C2×C4×C53—21no—
273C2×C4×C53—21no—
274C2×C4×C53—21no—
275C2×C4×C53—21no—
276C2×C4×C53—48no—
277C2×C4×C53—41no—
278C2×C4×C53—41no—
279C2×C4×C53—41no—
280C2×C4×C53—41no—
281C2×C4×C53—41no—
282C2×C4×C53—41no—
283C2×C4×C53—41no—
284C2×C4×C53—41no—
285C2×C4×C53—41no—
286C2×C4×C53—41no—
287C2×C4×C53—41no—
288C2×C4×C53—41no—
289C2×C4×C53—41no—
290C2×C4×C53—41no—
291C2×C4×C53—41no—
292C2×C4×C53—41no—
293C2×C4×C53#12012yes—
294C2×C4×C53#12012yes—
295C2×C4×C53C22×D411yescharacteristic
296C2×C4×C53C22×D411yescharacteristic
297C2×C4×C53C2×#4111yescharacteristic
298C2×C4×C53C2×#4111yescharacteristic
299C2×C4×C53H(5,𝔽2)11yescharacteristic
300C2×C4×C532−1+411yescharacteristic
301C2×Dic53—22no—
302C2×Dic53—41no—
303C2×Dic53—41no—
304C2×Dic53—41no—
305C2×Dic53—41no—
306C2×Dic53—41no—
307C2×Dic53—41no—
308C2×Dic53—41no—
309C2×Dic53—41no—
310C2×Dic53—41no—
311C2×Dic53—41no—
312C2×Dic53—41no—
313Dic106—22no—
314Dic106—22no—
315Dic106—41no—
316Dic106—41no—
317#88322—22no—
318#88322—41no—
319#88322—41no—
320#88322—41no—
321#24127—531nomaximal, Sylow
322C22×C53—22no—
323C22×C53—22no—
324C22×C53—22no—
325C22×C53—41no—
326C22×C53—41no—
327C22×C53—41no—
328C22×C53—41no—
329C22×C53—41no—
330C22×C53—41no—
331C22×C53—41no—
332C22×C53—41no—
333C22×C53—41no—
334C22×C53—41no—
335C22×C53—41no—
336C22×C53C2×#12011yescharacteristic
337C22×C53#54611yescharacteristic
338C22×C53#54711yescharacteristic
339C22×C53C2×#13511yescharacteristic
340C22×C53#57511yescharacteristic
341C22×C53#57811yescharacteristic
342C22×C53#57811yescharacteristic
343C4×C53—22no—
344C4×C53—22no—
345C4×C53—41no—
346C4×C53—41no—
347C4×C53—42no—
348C4×C53—42no—
349C4×C53—42no—
350C4×C53—42no—
351C4×C53#52511yescharacteristic
352C4×C53#53711yescharacteristic
353C4×C53#54811yescharacteristic
354C4×C53#55011yescharacteristic
355Dic53—42no—
356Dic53—81no—
357Dic53—81no—
358Dic53—81no—
359C2×#552D5311yescharacteristic
360C2×#479—531no—
361C2×C4×Q16—531no—
362#2536—531no—
363#2036—531no—
364#2035—531no—
365#2001—531no—
366#2001—531no—
367#1991—531no—
368#1954—531no—
369#1873—531no—
370#1872—531no—
371#1854—531no—
372#1852—531no—
373#1805—531no—
374C2×C53—42no—
375C2×C53—81no—
376C2×C53#240711yescharacteristic
377C2×C53#300211yescharacteristic
378C2×C53#301111yescharacteristic
379C2×C53#330011yescharacteristic
380C2×C53#332711yescharacteristic
381C2×C53#333611yescharacteristic
382C2×C53#335211yescharacteristic
383C22×Q16—531no—
384#552—24no—
385#552—24no—
386C2×#126—21no—
387C2×#125—21no—
388C2×#125C2×D5311yescharacteristic
389C2×#123—21no—
390C2×C22⋊Q8—531no—
391C2×C22⋊Q8—531no—
392C2×C22⋊Q8C2×D5311yescharacteristic
393C2×C22⋊Q8C2×D5311yescharacteristic
394C2×C4×Q8—531no—
395C2×C4×Q8C2×D5311yescharacteristic
396C2×C4×D4C2×D5311yescharacteristic
397C2×#117—21no—
398C2×#117C2×D5311yescharacteristic
399C22×C4⋊C4C2×D5311yescharacteristic
400#479—1064no—
401C4×Q16—1064no—
402C2×#107—531no—
403C2×#107—531no—
404C2×#106—531no—
405C2×C4⋊C8—531no—
406C2×#103—531no—
407C2×#103—531no—
408C2×#103—531no—
409C2×#103—531no—
410C2×#103—531no—
411C2×#103—531no—
412C2×C22⋊C8—531no—
413C2×C22⋊C8—531no—
414C2×C22⋊C8—531no—
415C2×C22⋊C8—531no—
416C2×C4×C8—531no—
417#398—1061no—
418#394—1061no—
419#388—1061no—
420#388—1061no—
421#382—531no—
422#380—531no—
423#380—531no—
424#340—531no—
425#340—531no—
426#340—531no—
427#340—531no—
428#340—531no—
429#340—531no—
430C53#2412711yescharacteristic, complemented, Sylow
431C22×Q8—531no—
432C22×Q8C22×D5311yescharacteristic
433C22×D4C22⋊D5311yescharacteristic
434C23×C4—21no—
435C23×C4C22×D5311yescharacteristic
436C23×C4C22×D5311yescharacteristic
437C2×Q16—1062no—
438C2×Q16—1062no—
439C2×Q16—1062no—
440C22×C8—531no—
441C22×C8—531no—
442C22×C8—531no—
443C22×C8—531no—
444#126—28no—
445#125—28no—
446#125—22no—
447#125C22⋊D5314yes—
448#123—28no—
449C22⋊Q8—22no—
450C22⋊Q8—22no—
451C22⋊Q8—22no—
452C22⋊Q8—22no—
453C22⋊Q8—1062no—
454C22⋊Q8—1062no—
455C22⋊Q8—2121no—
456C22⋊Q8—2121no—
457C4×Q8—22no—
458C4×Q8—1062no—
459C4×Q8—1062no—
460C4×Q8C22⋊D5314yes—
461C4×D4—22no—
462C4×D4—22no—
463#117—28no—
464#117—22no—
465#117—22no—
466C2×C4⋊C4—21no—
467C2×C4⋊C4—21no—
468C2×C4⋊C4—21no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—22no—
473C2×C4⋊C4—22no—
474C2×C4⋊C4—531no—
475C2×C4⋊C4—531no—
476C2×C4⋊C4—531no—
477C2×C4⋊C4—1061no—
478C2×C4⋊C4—1061no—
479C2×C4⋊C4—1061no—
480C2×C4⋊C4C22⋊D5311yescharacteristic
481C2×C4⋊C4C22×D5311yescharacteristic
482C2×C4⋊C4C22×D5311yescharacteristic
483C2×C4⋊C4C22×D5311yescharacteristic
484C2×C22⋊C4—21no—
485C2×C22⋊C4—21no—
486C2×C22⋊C4—21no—
487C2×C22⋊C4—1061no—
488C2×C22⋊C4—1061no—
489C2×C22⋊C4—1061no—
490C2×C22⋊C4—1061no—
491C2×C22⋊C4C22⋊D5311yescharacteristic
492C2×C22⋊C4C22⋊D5311yescharacteristic
493C2×C42—21no—
494C2×C42—21no—
495C2×C42—1061no—
496C2×C42C22×D5311yescharacteristic
497#107—1062no—
498#107—1062no—
499#106—1062no—
500C4⋊C8—1062no—
501#103—1062no—
502#103—1062no—
503#103—1062no—
504#103—1062no—
505#103—2121no—
506#103—2121no—
507C22⋊C8—1062no—
508C22⋊C8—1062no—
509C22⋊C8—2121no—
510C22⋊C8—2121no—
511C4×C8—1062no—
512#95—1061no—
513#95—1061no—
514#95—1061no—
515#95—1061no—
516C24C2×C22⋊D5311yescharacteristic
517C24C2×C22⋊D5311yescharacteristic
518C2×Q8—21no—
519C2×Q8—21no—
520C2×Q8—21no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—532no—
524C2×Q8—1061no—
525C2×Q8—1061no—
526C2×Q8—1061no—
527C2×Q8—1061no—
528C2×Q8—1061no—
529C2×Q8C2×C22⋊D5312yes—
530C2×D4—22no—
531C2×D4—22no—
532C2×D4—41no—
533C22×C4—24no—
534C22×C4—21no—
535C22×C4—21no—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—21no—
546C22×C4—21no—
547C22×C4—21no—
548C22×C4—21no—
549C22×C4—21no—
550C22×C4—42no—
551C22×C4—42no—
552C22×C4—1061no—
553C22×C4—1061no—
554C22×C4—1061no—
555C22×C4—1061no—
556C22×C4C2×C22⋊D5311yescharacteristic
557C22×C4C2×C22⋊D5311yescharacteristic
558C22×C4C2×C22⋊D5311yescharacteristic
559C22×C4C2×C22⋊D5311yescharacteristic
560C22×C4C23×D5311yescharacteristic
561Q16—1068no—
562C2×C8—532no—
563C2×C8—1061no—
564C2×C8—1061no—
565C2×C8—1061no—
566C2×C8—1061no—
567C2×C8—1061no—
568C2×C8—1061no—
569C2×C8—1061no—
570C2×C8—1061no—
571C2×C8—1061no—
572C2×C8—1061no—
573C2×C8—1061no—
574C4⋊C4—24no—
575C4⋊C4—24no—
576C4⋊C4—24no—
577C4⋊C4—24no—
578C4⋊C4—24no—
579C4⋊C4—24no—
580C4⋊C4—22no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—1061no—
585C4⋊C4—1061no—
586C4⋊C4—1061no—
587C4⋊C4—1061no—
588C4⋊C4—1061no—
589C4⋊C4—1061no—
590C4⋊C4—2122no—
591C4⋊C4—2122no—
592C4⋊C4—2122no—
593C4⋊C4C2×C22⋊D5312yes—
594C4⋊C4C2×C22⋊D5312yes—
595C4⋊C4C2×C22⋊D5312yes—
596C22⋊C4—24no—
597C22⋊C4—24no—
598C22⋊C4—24no—
599C22⋊C4—22no—
600C22⋊C4—22no—
601C22⋊C4—2122no—
602C22⋊C4—2122no—
603C22⋊C4—2122no—
604C22⋊C4—2122no—
605C42—24no—
606C42—24no—
607C42—2122no—
608C42C2×C22⋊D5312yes—
609C42C2×C22⋊D5312yes—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—21no—
620C23—21no—
621C23—21no—
622C23—21no—
623C23#1313323712yes—
624C23C22×C22⋊D5311yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629Q8—1062no—
630Q8—1062no—
631Q8—2121no—
632Q8—2121no—
633D4—44no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—48no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—1062no—
666C2×C4—2121no—
667C2×C4—2121no—
668C2×C4—2121no—
669C2×C4—2121no—
670C2×C4—2121no—
671C2×C4—2121no—
672C2×C4—2121no—
673C2×C4—2121no—
674C2×C4—2121no—
675C2×C4—2121no—
676C2×C4—2121no—
677C2×C4#1313318612yes—
678C2×C4#1313318612yes—
679C2×C4C22×C22⋊D5311yescharacteristic
680C2×C4C22×C22⋊D5311yescharacteristic
681C2×C4#41×D5311yescharacteristic
682C2×C4#41×D5311yescharacteristic
683C2×C4#1313326311yescharacteristic
684C2×C4#1313326411yescharacteristic
685C8—1062no—
686C8—2121no—
687C8—2121no—
688C8—2121no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22C2×#1313318611yescharacteristic
704C22#1681126111yescharacteristic
705C22#1681127211yescharacteristic
706C22C2×#1313323711yescharacteristic
707C22#1681142111yescharacteristic
708C22#1681143811yescharacteristic
709C22#1681143811yescharacteristic
710C4—22no—
711C4—22no—
712C4—42no—
713C4—42no—
714C4—41no—
715C4—41no—
716C4—42no—
717C4—42no—
718C4—4241no—
719C4—4241no—
720C4—4241no—
721C4—2122no—
722C4#1681113011yescharacteristic
723C4#1681120111yescharacteristic
724C4#1681128011yescharacteristic
725C4#1681129211yescharacteristic
726C2—42no—
727C2—81no—
728C2#2027805411yescharacteristic
729C2#2028393311yescharacteristic
730C2#2028401011yescharacteristic
731C2#2028659611yescharacteristic
732C2#2028697711yescharacteristic
733C2#2028707711yescharacteristic
734C2#2028728411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#552C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#552D53
hypocenterC53#24127
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16810927C2semidirect
2. C2×C4×#11719925C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
6. #20275723C2semidirect
7. #20275723C2semidirect
8. #20275681C2semidirect
10. #20275027C2semidirect
11. #20275022C2semidirect
12. #20274914C2semidirect
13. #20274903C2semidirect
14. #20274730C2semidirect
430. C53#24127semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C2×C4×#117199252
#202796242
#202759152
#202759082
#202757232
#202756812
#202755102
#202750272
#202750222
#202749142
#202749032
#202747302
C2×C53×#5522
#2412753