Finite Group Databasebeta

← #27275461

Subgroups — #27275461

Order 13568.

1088 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D53C211yesmaximal, characteristic
2#20274692C211yesmaximal, characteristic
3C4×#13133129C211yesmaximal, characteristic
4C2×#16810355C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6C2×#16810353C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×Q8×D53C411yescharacteristic
9C2×Q8×Dic53C411yescharacteristic
10C2×C4×Dic106—21no—
11C2×C4×Dic106C2211yescharacteristic
12C2×#13133182—21no—
13C2×#13133182C411yescharacteristic
14C4×Q8×D53—22no—
15C4×Q8×D53—22no—
16C4×Q8×D53—22no—
17C4×Q8×D53—22no—
18C2×C4⋊C4×D53—21no—
19C2×C4⋊C4×D53C411yescharacteristic
20C2×C42×D53—21no—
21C2×C42×D53C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133129—21no—
24C2×#13133129—21no—
25C2×C4×#88322C2211yescharacteristic
26#16810377—21no—
27#16810377—21no—
28#16810355—22no—
29Q8⋊#88322—22no—
30#16810353—22no—
31C2×C4×Q8×C53C2211yescharacteristic
32C2×C4⋊C8×C53C2211yescharacteristic
33C2×C4×C8×C53C2211yescharacteristic
34C53×Q8⋊C8—22no—
35C22×Dic106—21no—
36C22×Dic106C2×C411yescharacteristic
37C2×Q8×D53—22no—
38C2×Q8×D53—22no—
39C2×Q8×D53—22no—
40C2×Q8×D53—22no—
41C2×Q8×D53—22no—
42C2×Q8×D53C814yescomplemented
43C22×C4×D53—21no—
44C22×C4×D53—21no—
45C22×C4×D53C2×C411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53D411yescharacteristic
48C22×#88322—21no—
49C22×#88322—21no—
50Q8×Dic53—22no—
51Q8×Dic53—22no—
52C4×Dic106—24no—
53C4×Dic106—24no—
54C4×Dic106—21no—
55C4×Dic106—21no—
56C4×Dic106D412yes—
57C4×Dic106D412yes—
58#13133182—24no—
59#13133182—24no—
60#13133182—22no—
61#13133182—22no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×#11719919—21no—
65C2×#11719919—21no—
66C2×#11719919—21no—
67C4⋊C4×D53—24no—
68C4⋊C4×D53—24no—
69C4⋊C4×D53—22no—
70C4⋊C4×D53—22no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73C2×C4×Dic53—21no—
74C2×C4×Dic53C2×C411yescharacteristic
75C2×C4×Dic53D411yescharacteristic
76C2×C4×Dic53D411yescharacteristic
77C42×D53—24no—
78C42×D53—24no—
79C42×D53—21no—
80C42×D53—21no—
81C42×D53—21no—
82C42×D53—21no—
83C4⋊#88322—21no—
84C4⋊#88322—21no—
85#13133129—24no—
86#13133129—42no—
87C4×#88322—21no—
88C4×#88322—21no—
89C22×Q8×C53C2×C411yescharacteristic
90C22×C8×C53—21no—
91C22×C8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53D412yes—
95C4×Q8×C53D412yes—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2×C411yescharacteristic
98C2×C42×C53—21no—
99C2×C42×C53C2311yescharacteristic
100C4⋊C8×C53—21no—
101C4⋊C8×C53—21no—
102C4×C8×C53—21no—
103C4×C8×C53—21no—
104C23×D53C22⋊C411yescharacteristic
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53C22⋊C412yes—
109C2×Q8×C53C2×C812yes—
110C2×Q8×C53M1612yes—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×Dic53—21no—
117C22×Dic53—21no—
118C22×Dic53C22⋊C411yescharacteristic
119C22×Dic53C22⋊C411yescharacteristic
120C22×Dic53C22⋊C411yescharacteristic
121Q8×D53—28no—
122Q8×D53—28no—
123Q8×D53—28no—
124Q8×D53—28no—
125C2×C4×D53—24no—
126C2×C4×D53—24no—
127C2×C4×D53—24no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C4×D53—24no—
137C2×C4×D53—22no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—41no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53C2×C812yes—
144C2×C4×D53M1612yes—
145C2×C4×D53SD1612yes—
146C2×C4×D53Q1612yes—
147C2×Dic106—24no—
148C2×Dic106—24no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—24no—
153C2×Dic106C22⋊C412yes—
154C2×Dic106C2×C812yes—
155C2×Dic106M1612yes—
156C2×C8×C53—22no—
157C2×C8×C53—41no—
158C2×C8×C53—41no—
159C2×C8×C53—41no—
160C2×C8×C53—41no—
161C2×C8×C53—41no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53C22⋊C412yes—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53C2×D411yescharacteristic
169C42×C53C2×D411yescharacteristic
170C42×C53C2×D411yescharacteristic
171C42×C53C2×D411yescharacteristic
172C4⋊Dic53—24no—
173C4⋊Dic53—21no—
174C4⋊Dic53C22⋊C412yes—
175#11719919—24no—
176#11719919—24no—
177#11719919—22no—
178#11719919—22no—
179C4×Dic53—24no—
180C4×Dic53—24no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C2×#88322—22no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C2×#88322—41no—
195C23×C53C2×C22⋊C411yescharacteristic
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53C22⋊C812yes—
201C22×D53#10312yes—
202C22×D53#10412yes—
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—22no—
206Q8×C53C22⋊C814yescomplemented
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53C22×C811yescharacteristic
222C2×C4×C53C2×M1611yescharacteristic
223C2×C4×C53C2×M1611yescharacteristic
224C2×C4×C53C2×M1611yescharacteristic
225C2×C4×C53C2×SD1611yescharacteristic
226C2×C4×C53C2×Q1611yescharacteristic
227C2×C4×C53#13711yescharacteristic
228C2×C4×C53#13711yescharacteristic
229C2×Dic53—21no—
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—22no—
233C2×Dic53—21no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53C22⋊C812yes—
243C2×Dic53#10312yes—
244C4×D53—216no—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—22no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—48no—
256Dic106—216no—
257Dic106—22no—
258Dic106—22no—
259Dic106—22no—
260Dic106C22⋊C814yescomplemented
261C8×C53—42no—
262C8×C53—81no—
263#88322—42no—
264#88322—81no—
265#8311—531nomaximal, Sylow
266C22×C53#37911yescharacteristic
267C22×C53C2×C22⋊C811yescharacteristic
268C22×C53#40711yescharacteristic
269C22×C53C2×#10311yescharacteristic
270C22×C53#41911yescharacteristic
271C22×C53C2×#10411yescharacteristic
272C22×C53#42111yescharacteristic
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53Q8⋊C814yescomplemented
280C4×C53—24no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—42no—
284C4×C53C2×C22⋊C811yescharacteristic
285C4×C53C2×C22⋊C811yescharacteristic
286C4×C53#40711yescharacteristic
287C4×C53#40711yescharacteristic
288C4×C53#44811yescharacteristic
289C4×C53#44811yescharacteristic
290C4×C53#45111yescharacteristic
291C4×C53#45111yescharacteristic
292Dic53—22no—
293Dic53—24no—
294Dic53—22no—
295Dic53—41no—
296Dic53—42no—
297C22×C4×Q8—531no—
298#1797—531no—
299C4×C22⋊C8—531no—
300C2×Q8⋊C8—531no—
301C2×Q8⋊C8—531no—
302C2×Q8⋊C8—531no—
303C2×Q8⋊C8D5311yescharacteristic, complemented
304C2×C53C2×Q8⋊C811yescharacteristic
305C2×C53#147911yescharacteristic
306C2×C53#148711yescharacteristic
307C2×C53#149111yescharacteristic
308C2×C53#177911yescharacteristic
309C2×C53#178711yescharacteristic
310C2×C53#178911yescharacteristic
311D53—24no—
312C23×Q8—531no—
313C2×C4×Q8—531no—
314C2×C4×Q8—531no—
315C2×C4×Q8—531no—
316C2×C4×Q8—1061no—
317C2×C4×Q8—1061no—
318C2×C4×Q8—1062no—
319C2×C4×Q8—1062no—
320C2×C4×Q8—1062no—
321C2×C4×Q8—1062no—
322C2×C4×Q8C2×D5311yescharacteristic
323C22×C4⋊C4—531no—
324C22×C4⋊C4—1061no—
325C22×C42—531no—
326C22×C42—1061no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8C2×D5311yescharacteristic
329C2×C22⋊C8—1061no—
330C2×C22⋊C8—1061no—
331C2×C4×C8—531no—
332C2×C4×C8C2×D5311yescharacteristic
333#336—1061no—
334#336—1061no—
335Q8⋊C8—22no—
336Q8⋊C8—1062no—
337Q8⋊C8—1062no—
338Q8⋊C8—1062no—
339C53#831111yescharacteristic, complemented, Sylow
340C22×Q8—534no—
341C22×Q8—531no—
342C22×Q8—1061no—
343C22×Q8—1062no—
344C22×Q8—1062no—
345C22×Q8—1062no—
346C22×Q8—1062no—
347C22×Q8—1062no—
348C22×Q8C4×D5311yescharacteristic
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—531no—
352C23×C4—1061no—
353C23×C4—1061no—
354C22×C8—21no—
355C22×C8—21no—
356C22×C8—1061no—
357C22×C8—1061no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—532no—
361C4×Q8—532no—
362C4×Q8—1064no—
363C4×Q8—1064no—
364C4×Q8—1062no—
365C4×Q8—1064no—
366C4×Q8—1064no—
367C4×Q8—1062no—
368C4×Q8—1062no—
369C4×Q8—1062no—
370C4×Q8—1061no—
371C4×Q8—1061no—
372C4×Q8D21212yes—
373C4×Q8C22⋊D5312yes—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1064no—
378C2×C4⋊C4—1064no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4C4×D5311yescharacteristic
385C2×C42—21no—
386C2×C42—531no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—1064no—
390C2×C42—1064no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C22⋊C8—1064no—
404C22⋊C8—2122no—
405C4×C8—21no—
406C4×C8—21no—
407C4×C8—1061no—
408C4×C8—1061no—
409C24—531no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—532no—
414C2×Q8—532no—
415C2×Q8—532no—
416C2×Q8—1064no—
417C2×Q8—1064no—
418C2×Q8—1064no—
419C2×Q8—1068no—
420C2×Q8—1068no—
421C2×Q8—1068no—
422C2×Q8—1068no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8C8×D5312yes—
427C2×Q8#1171991212yes—
428C2×Q8#1171992112yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—531no—
438C22×C4—1064no—
439C22×C4—1064no—
440C22×C4—1064no—
441C22×C4—1062no—
442C22×C4—1064no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—2121no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4C2×C4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C2×C8—22no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—1062no—
468C2×C8—2121no—
469C2×C8—2121no—
470C2×C8—2121no—
471C2×C8—2121no—
472C2×C8—2121no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—532no—
477C4⋊C4—1064no—
478C4⋊C4—1064no—
479C4⋊C4—1064no—
480C4⋊C4—1061no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4#1171992112yes—
484C42—22no—
485C42—22no—
486C42—1064no—
487C42—1064no—
488C42—1061no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1062no—
493C42—1061no—
494C42—1061no—
495C42—1062no—
496C42C2×D21211yescharacteristic
497C42D4×D5311yescharacteristic
498C42D4×D5311yescharacteristic
499C42C2×C22⋊D5311yescharacteristic
500C23—532no—
501C23—532no—
502C23—532no—
503C23—1061no—
504C23—1061no—
505C23—1061no—
506C23—1061no—
507C23C22⋊C4×D5311yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—534no—
512Q8—10616no—
513Q8—1062no—
514Q8—1062no—
515Q8—1062no—
516Q8#1313312914yescomplemented
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—532no—
526C2×C4—532no—
527C2×C4—10616no—
528C2×C4—1062no—
529C2×C4—1061no—
530C2×C4—1062no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1062no—
546C2×C4—1061no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2128no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4C22⋊C4×D5311yescharacteristic
553C2×C4C2×#1171992111yescharacteristic
554C2×C4#1313318511yescharacteristic
555C2×C4#1313318511yescharacteristic
556C2×C4#1313318711yescharacteristic
557C2×C4#1313318711yescharacteristic
558C2×C4C2×C8×D5311yescharacteristic
559C2×C4M16×D5311yescharacteristic
560C2×C4M16×D5311yescharacteristic
561C2×C4C2×#1171991211yescharacteristic
562C2×C4SD16×D5311yescharacteristic
563C2×C4Q16×D5311yescharacteristic
564C2×C4#1313324611yescharacteristic
565C2×C4#1313324911yescharacteristic
566C8—42no—
567C8—81no—
568C8—4241no—
569C8—2122no—
570C22—534no—
571C22—1062no—
572C22—1062no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22#1681047811yescharacteristic
578C22C22⋊C8×D5311yescharacteristic
579C22#1681058511yescharacteristic
580C22#103×D5311yescharacteristic
581C22#1681063011yescharacteristic
582C22#104×D5311yescharacteristic
583C22#1681063811yescharacteristic
584C4—24no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—1062no—
589C4—1064no—
590C4—1062no—
591C4—2121no—
592C4—2122no—
593C4C22⋊C8×D5311yescharacteristic
594C4C2×#1313312911yescharacteristic
595C4#1681058511yescharacteristic
596C4#1681058611yescharacteristic
597C4#1681076011yescharacteristic
598C4#1681076311yescharacteristic
599C4#1681077711yescharacteristic
600C4#1681078011yescharacteristic
601C2—1064no—
602C2Q8⋊C8×D5311yescharacteristic
603C2#2027310811yescharacteristic
604C2#2027314211yescharacteristic
605C2#2027316011yescharacteristic
606C2#2027462411yescharacteristic
607C2#2027464811yescharacteristic
608C2#2027465711yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#8311
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#16810355C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. C2×#16810353C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
42. C2×Q8×D53C8semidirect
206. Q8×C53C22⋊C8semidirect
260. Dic106C22⋊C8semidirect
279. C2×D53Q8⋊C8semidirect
303. C2×Q8⋊C8D53semidirect
339. C53#8311semidirect
516. Q8#13133129semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D532
#202746922
C4×#131331292
C2×#168103552
C2×Q8⋊#883222
C2×#168103532
C2×C53×Q8⋊C82
#831153