Finite Group Databasebeta

← #27352683

Subgroups — #27352683

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D53C211yesmaximal, characteristic
2#20283347C211yesmaximal, characteristic
3#20283343C211yesmaximal, characteristic
4#20273430C214yesmaximal, complemented
5#20273398C214yesmaximal, complemented
6#20273388C211yesmaximal, characteristic, complemented
7#20273387C211yesmaximal, characteristic, complemented
8#20273386C211yesmaximal, characteristic, complemented
9C53×#1546C211yesmaximal, characteristic, complemented
10C22×Q8×D53—21no—
11C2×#13133217C2211yescharacteristic
12C2×#13133216—21no—
13C2×#13133216C412yes—
14C2×#13133215—21no—
15C4⋊Q8×D53—21no—
16C4⋊Q8×D53—21no—
17C4⋊Q8×D53—21no—
18C4⋊Q8×D53—21no—
19C4⋊Q8×D53C412yes—
20C4⋊Q8×D53C412yes—
21C4⋊Q8×D53C2212yes—
22C4⋊Q8×D53C2212yes—
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53C412yes—
25C2×C42×D53C2211yescharacteristic
26#16810679—21no—
27#16810679C2212yes—
28#16810679C2212yes—
29#16810677—21no—
30#16810676C2211yescharacteristic
31#16810675—21no—
32#16810675C2212yes—
33#16810674—21no—
34#16810647C2211yescharacteristic
35#16810645C2212yes—
36#16810586—21no—
37#16810585—21no—
38M16×Dic53—21no—
39C4×#11719912—21no—
40#16810370C2214yescomplemented
41#16810369C2214yescomplemented
42#16810368C2214yescomplemented
43C2×C4⋊Q8×C53C2211yescharacteristic
44C53×#431C2211yescharacteristic
45C53×#423C2211yescharacteristic
46C53×#332C2214yescomplemented
47C22×Dic106—21no—
48C22×Dic106—22no—
49C2×Q8×D53—22no—
50C2×Q8×D53—22no—
51C2×Q8×D53—22no—
52C2×Q8×D53—22no—
53C2×Q8×D53—44no—
54C2×Q8×D53—44no—
55C22×C4×D53—22no—
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D411yescharacteristic
59C2×#11719917—21no—
60C2×#11719912—22no—
61#13133217—21no—
62#13133217—21no—
63#13133217C2×C412yes—
64#13133217C2312yes—
65#13133216—22no—
66#13133216—22no—
67#13133216—22no—
68#13133216—22no—
69#13133216—22no—
70#13133216—22no—
71#13133216C2×C414yes—
72#13133216C2×C414yes—
73#13133215—22no—
74#13133215—22no—
75#13133215—22no—
76#13133215—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2×C412yes—
79C2×#11719919—22no—
80C2×#11719919—22no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—22no—
83C4⋊C4×D53—22no—
84C4⋊C4×D53—22no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—22no—
87C4⋊C4×D53C2×C414yes—
88C4⋊C4×D53C2×C414yes—
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C42×D53C2×C411yescharacteristic
93C42×D53C2×C411yescharacteristic
94C42×D53D411yescharacteristic
95C42×D53D411yescharacteristic
96C42×D53D411yescharacteristic
97C42×D53D411yescharacteristic
98C42×D53C2311yescharacteristic
99C42×D53C2311yescharacteristic
100C4⋊#88322—21no—
101C4⋊#88322C2312yes—
102C4⋊#88322C2312yes—
103#13133146—22no—
104C22⋊#88322—21no—
105#13133129—22no—
106#13133129—22no—
107#13133129—22no—
108#13133127—22no—
109#13133127—22no—
110C4×#88322—21no—
111C8×Dic53—22no—
112C22×Q8×C53—21no—
113C2×M16×C53—21no—
114C4⋊Q8×C53—21no—
115C4⋊Q8×C53—21no—
116C4⋊Q8×C53C2×C412yes—
117C4⋊Q8×C53C2312yes—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53C2×C412yes—
120C2×C42×C53C2311yescharacteristic
121C4⋊C8×C53—21no—
122C4⋊C8×C53C2312yes—
123C4⋊C8×C53C2312yes—
124C22⋊C8×C53—21no—
125C4×C8×C53—21no—
126C23×D53C22⋊C411yescharacteristic
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—42no—
130C2×Q8×C53—42no—
131C22×C4×C53—22no—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×Dic53—22no—
136C22×Dic53C22⋊C411yescharacteristic
137C22×Dic53C22⋊C411yescharacteristic
138C22×Dic53C22⋊C411yescharacteristic
139Q8×D53—44no—
140Q8×D53—44no—
141Q8×D53—44no—
142Q8×D53—44no—
143Q8×D53—48no—
144Q8×D53—48no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—24no—
150C2×C4×D53—24no—
151C2×C4×D53—48no—
152C2×C4×D53C22⋊C411yescharacteristic
153C2×C4×D53C22⋊C411yescharacteristic
154C2×C4×D53C22⋊C411yescharacteristic
155C2×C4×D53C22⋊C411yescharacteristic
156C2×C4×D53C22⋊C411yescharacteristic
157C2×C4×D53C22⋊C411yescharacteristic
158C2×C4×D53C22×C411yescharacteristic
159C2×C4×D53C22×C411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53C2×D411yescharacteristic
162C2×C4×D53C2×D411yescharacteristic
163C2×C4×D53C2×D411yescharacteristic
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106—24no—
167C2×Dic106—24no—
168C2×Dic106—48no—
169C2×Dic106—42no—
170C2×Dic106—42no—
171M16×C53—42no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C22×C414yes—
178C42×C53C22×C411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53C2×D411yescharacteristic
181C42×C53C2411yescharacteristic
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53C22×C414yes—
186#11719919—24no—
187#11719919—24no—
188#11719919—24no—
189#11719919—24no—
190C4×Dic53C22⋊C411yescharacteristic
191C4×Dic53C22⋊C411yescharacteristic
192C4×Dic53C22⋊C411yescharacteristic
193C4×Dic53C22⋊C411yescharacteristic
194C4×Dic53C22⋊C411yescharacteristic
195C4×Dic53C22⋊C411yescharacteristic
196C4×Dic53C22×C411yescharacteristic
197C4×Dic53C22×C411yescharacteristic
198C4×Dic53C2×D411yescharacteristic
199C4×Dic53C2×D411yescharacteristic
200C4×Dic53C2×D411yescharacteristic
201C4×Dic53C2×D411yescharacteristic
202#11719917—42no—
203C2×#88322—22no—
204C2×#88322—22no—
205#11719912—44no—
206C23×C53C2×C22⋊C411yescharacteristic
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53C2×C22⋊C411yescharacteristic
211C22×D53C2×C22⋊C411yescharacteristic
212Q8×C53—42no—
213Q8×C53—42no—
214Q8×C53—44no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—44no—
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53AGL(1,ℤ/8)12yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—24no—
231C2×Dic53—44no—
232C2×Dic53C2×C22⋊C411yescharacteristic
233C2×Dic53C2×C22⋊C411yescharacteristic
234C2×Dic53C2×C22⋊C411yescharacteristic
235C2×Dic53C2×C22⋊C411yescharacteristic
236C2×Dic53C2×C22⋊C411yescharacteristic
237C2×Dic53C2×C22⋊C411yescharacteristic
238C4×D53—22no—
239C4×D53—22no—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—21no—
243C4×D53—21no—
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—48no—
251C4×D53—48no—
252C4×D53#10114yes—
253C4×D53#13714yes—
254Dic106—48no—
255Dic106—48no—
256Dic106—42no—
257Dic106—42no—
258Dic106—44no—
259C8×C53—42no—
260C8×C53—42no—
261#88322—42no—
262#88322—42no—
263#17742—531nomaximal, Sylow
264C22×C53—21no—
265C22×C53—21no—
266C22×C53#41812yes—
267C22×C53C22×C22⋊C411yescharacteristic
268C2×D53—28no—
269C2×D53—21no—
270C2×D53—21no—
271C2×D53—21no—
272C2×D53—21no—
273C2×D53C2×#10112yes—
274C2×D53#41912yes—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—44no—
278C4×C53C2×#10112yes—
279C4×C53#41812yes—
280C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C4×C53C2×#13712yes—
283Dic53—21no—
284Dic53—21no—
285Dic53—21no—
286Dic53—21no—
287Dic53—44no—
288Dic53C2×#10112yes—
289Dic53#41912yes—
290C22×C4⋊Q8—531no—
291#2920—531no—
292#2920—531no—
293#1556—534no—
294#1549—534no—
295#1546—531no—
296#1546—531no—
297#1546—531no—
298#1546D5311yescharacteristic, complemented
299C2×C53—22no—
300C2×C53C22×#10111yescharacteristic
301C2×C53C2×#41811yescharacteristic
302C2×C53C2×#41911yescharacteristic
303D53—22no—
304D53—22no—
305C23×Q8—1061no—
306C2×C4⋊Q8—532no—
307C2×C4⋊Q8—532no—
308C2×C4⋊Q8—532no—
309C2×C4⋊Q8—532no—
310C2×C4⋊Q8—532no—
311C2×C4⋊Q8—531no—
312C2×C4⋊Q8—1061no—
313C2×C4⋊Q8—1061no—
314C2×C4⋊Q8—1061no—
315C2×C4⋊Q8—1061no—
316C2×C4⋊Q8—1061no—
317C2×C4⋊Q8—1061no—
318C2×C4⋊Q8C2×D5311yescharacteristic
319C22×C4⋊C4—532no—
320C22×C4⋊C4—1061no—
321C22×C42—531no—
322#432—532no—
323#432—532no—
324#432—1061no—
325#432—1061no—
326#431—531no—
327#431—532no—
328#431—1061no—
329#431—1061no—
330#431C2×D5311yescharacteristic
331#423—531no—
332#423—532no—
333#423C2×D5311yescharacteristic
334#407—1061no—
335#407—1061no—
336C4×M16—1061no—
337C4×M16—1061no—
338#332—534no—
339#332—534no—
340#332—534no—
341#332C2×D5314yescomplemented
342C53#1774211yescharacteristic, complemented, Sylow
343C22×Q8—21no—
344C22×Q8—1062no—
345C22×Q8—1062no—
346C22×Q8—1062no—
347C22×Q8—1062no—
348C22×Q8—1062no—
349C22×Q8—1061no—
350C22×Q8—2124no—
351C22×Q8—2124no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—1062no—
356C2×M16—21no—
357C2×M16—1062no—
358C2×M16—1061no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—534no—
362C4⋊Q8—534no—
363C4⋊Q8—532no—
364C4⋊Q8—532no—
365C4⋊Q8—1062no—
366C4⋊Q8—1062no—
367C4⋊Q8—1062no—
368C4⋊Q8—1062no—
369C4⋊Q8—1062no—
370C4⋊Q8—1062no—
371C4⋊Q8—1062no—
372C4⋊Q8—1062no—
373C4⋊Q8—1062no—
374C4⋊Q8—1062no—
375C4⋊Q8—1061no—
376C4⋊Q8—1061no—
377C4⋊Q8C4×D5312yes—
378C4⋊Q8C22×D5312yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C4×D5312yes—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42C22×D5311yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—532no—
407C4⋊C8—532no—
408C4⋊C8—1062no—
409C4⋊C8—1061no—
410C4⋊C8C22×D5312yes—
411C4⋊C8C22×D5312yes—
412C22⋊C8—21no—
413C22⋊C8—1062no—
414C22⋊C8—1062no—
415C22⋊C8—1062no—
416C22⋊C8—1061no—
417#97—1062no—
418#97—1062no—
419C4×C8—21no—
420C4×C8—1062no—
421C4×C8—1061no—
422C24—531no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—1064no—
428C2×Q8—1064no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8—2128no—
432C2×Q8—2128no—
433C2×Q8—2128no—
434C2×Q8—2124no—
435C2×Q8—2124no—
436C2×Q8—2124no—
437C2×Q8—2124no—
438C2×Q8—2122no—
439C2×Q8—2122no—
440C22×C4—22no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1064no—
457C22×C4—1064no—
458C22×C4—1064no—
459C22×C4—1064no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—2128no—
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467M16—42no—
468M16—2124no—
469M16—2122no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—1062no—
473C2×C8—1062no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—534no—
478C4⋊C4—1064no—
479C4⋊C4—1064no—
480C4⋊C4—1064no—
481C4⋊C4—1064no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—1062no—
485C4⋊C4C2×C4×D5314yes—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—531no—
490C42—531no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42C2×C4×D5311yescharacteristic
499C42D4×D5311yescharacteristic
500C42D4×D5311yescharacteristic
501C42C23×D5311yescharacteristic
502C23—531no—
503C23—531no—
504C23—1062no—
505C23—1062no—
506C23—1062no—
507C23C22⋊C4×D5311yescharacteristic
508Q8—44no—
509Q8—42no—
510Q8—42no—
511Q8—2128no—
512Q8—2128no—
513Q8—2124no—
514Q8—2122no—
515Q8—2122no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—534no—
523C2×C4—534no—
524C2×C4—531no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1062no—
543C2×C4—1064no—
544C2×C4—2128no—
545C2×C4—2128no—
546C2×C4—2124no—
547C2×C4C22⋊C4×D5311yescharacteristic
548C2×C4C22⋊C4×D5311yescharacteristic
549C2×C4C22⋊C4×D5311yescharacteristic
550C2×C4#1313324112yes—
551C2×C4#1313324412yes—
552C2×C4C22×C4×D5311yescharacteristic
553C2×C4C2×D4×D5311yescharacteristic
554C2×C4C2×D4×D5311yescharacteristic
555C8—42no—
556C8—42no—
557C8—2122no—
558C8—2122no—
559C22—21no—
560C22—21no—
561C22—532no—
562C22—532no—
563C22—1068no—
564C22—1061no—
565C22—1061no—
566C22—1061no—
567C22—1061no—
568C22#1681062612yes—
569C22C2×C22⋊C4×D5311yescharacteristic
570C4—21no—
571C4—21no—
572C4—44no—
573C4—532no—
574C4—532no—
575C4—1061no—
576C4—1061no—
577C4—1061no—
578C4—1061no—
579C4—2124no—
580C4#101×D5312yes—
581C4#1681062612yes—
582C4C2×#1313324111yescharacteristic
583C4C2×#1313324411yescharacteristic
584C4#137×D5312yes—
585C2—22no—
586C2—1062no—
587C2—1062no—
588C2C2×#101×D5311yescharacteristic
589C2C2×#1681062611yescharacteristic
590C2#419×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1546C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenter#1546D53
hypocenterC53#17742
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #20273430C2semidirect
5. #20273398C2semidirect
6. #20273388C2semidirect
7. #20273387C2semidirect
8. #20273386C2semidirect
9. C53×#1546C2semidirect
40. #16810370C22semidirect
41. #16810369C22semidirect
42. #16810368C22semidirect
46. C53×#332C22semidirect
298. #1546D53semidirect
341. #332C2×D53semidirect
342. C53#17742semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D532
#202833472
#202833432
#202734302
#202733982
#202733882
#202733872
#202733862
C53×#15462
#1774253