Finite Group Databasebeta

← #27387094

Subgroups — #27387094

Order 13568.

934 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic53C211yesmaximal, characteristic
2#20277177C211yesmaximal, characteristic
3C4×Q8×Dic53C211yesmaximal, characteristic, complemented
4#20275338C211yesmaximal, characteristic
5#20275332C211yesmaximal, characteristic
6#20275301C211yesmaximal, characteristic, complemented
7#20275299C211yesmaximal, characteristic, complemented
8#20275072C211yesmaximal, characteristic
9#20274969C211yesmaximal, characteristic, complemented
10#20274967C211yesmaximal, characteristic, complemented
11#20274607C211yesmaximal, characteristic, complemented
12#103×Dic53C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic
14#20274459C211yesmaximal, characteristic, complemented
15C53×#1880C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133173—21no—
20SD16×Dic53C418yescomplemented
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810517—21no—
34#16810511—21no—
35C4⋊C4×Dic53—21no—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53—21no—
41C4×C4⋊Dic53C2211yescharacteristic
42C22⋊C4×Dic53—21no—
43C42×Dic53—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—22no—
64Q8×Dic53—22no—
65Q8×Dic53—22no—
66Q8×Dic53C2×C414yes—
67D4×Dic53—22no—
68D4×Dic53C2×C414yes—
69#13133173—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133150—22no—
94#13133150—22no—
95#13133149C2×C414yes—
96#13133141—22no—
97#13133141C2×C414yes—
98#13133138—22no—
99#13133138C2×C414yes—
100C8×Dic53—22no—
101C8×Dic53C2×C414yes—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C2×SD16×C53—22no—
111C2×SD16×C53—22no—
112C2×SD16×C53C2×C414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4×Q8×C53—22no—
116C4×Q8×C53—22no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C42×C53—21no—
123C2×C42×C53C2311yescharacteristic
124#103×C53—22no—
125#102×C53—22no—
126C24×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C22×C412yes—
131C2×Q8×C53#4112yes—
132C2×Q8×C53#4112yes—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53C22×C412yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—24no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156SD16×C53—28no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53#4112yes—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C42×C53—24no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—24no—
174C22⋊Dic53—42no—
175C22⋊Dic53—42no—
176C4⋊Dic53—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53#4112yes—
186#11719919—42no—
187#11719919—42no—
188C4×Dic53—24no—
189C4×Dic53—22no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—24no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—22no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53C22×C412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53#11714yescomplemented
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—41no—
223D4×C53—41no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53#13512yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—48no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C4×D412yes—
266C2×Dic53C4×D412yes—
267C2×Dic53#13512yes—
268C2×Dic53AGL(1,ℤ/8)12yes—
269C2×Dic53#13712yes—
270C8×C53—22no—
271C8×C53—41no—
272#88322—22no—
273#88322—41no—
274#21120—531nomaximal, Sylow
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53C2×C4×D411yescharacteristic
283C22×C53#52511yescharacteristic
284C22×C53#52511yescharacteristic
285C22×C53C2×#13511yescharacteristic
286C22×C53C2×#13511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C4×C53—22no—
290C4×C53—24no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—81no—
294C4×C53C2×#11711yescharacteristic
295C4×C53C4×#4111yescharacteristic
296C4×C53#52911yescharacteristic
297C4×C53#52911yescharacteristic
298Dic53—22no—
299Dic53—22no—
300Dic53—41no—
301Dic53—42no—
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53#44014yes—
306C2×C4×SD16—531no—
307#2293—531no—
308C42×Q8—531no—
309#1925—531no—
310#1925—531no—
311#1920—531no—
312#1920—531no—
313#1880—531no—
314#1880D5311yescharacteristic
315#1863—531no—
316#1863—531no—
317C4×#107—531no—
318C4×#103—531no—
319C4×#102—531no—
320#1744—531no—
321C2×C53—82no—
322C2×C53#229011yescharacteristic
323C2×C53C4×#13511yescharacteristic
324C2×C53C2×#44011yescharacteristic
325C2×C53#310611yescharacteristic
326C2×C53#310611yescharacteristic
327C2×C53#310811yescharacteristic
328C2×C53#310811yescharacteristic
329C22×SD16C2×D5311yescharacteristic
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4—531no—
334C2×#117—1061no—
335C4×SD16—538no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#106—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103C2×D5311yescharacteristic
342C2×#102—531no—
343C2×#102—531no—
344C2×#102C2×D5311yescharacteristic
345C2×C4×C8—531no—
346C2×C4×C8—531no—
347#390—531no—
348#390C2×D5311yescharacteristic
349#389—531no—
350#389—531no—
351#388—1061no—
352#388—1061no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C4⋊C4—1061no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—1061no—
361C43—531no—
362C43—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371#336—531no—
372#336C2×D5311yescharacteristic
373C53#2112011yescharacteristic, complemented, Sylow
374C22×Q8C22×D5311yescharacteristic
375C22×D4C22×D5311yescharacteristic
376C23×C4—1061no—
377C2×SD16—22no—
378C2×SD16—22no—
379C2×SD16C4×D5314yes—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—534no—
387C4×Q8—1062no—
388C4×Q8—1062no—
389C4×Q8—1062no—
390C4×D4—534no—
391C4×D4—1062no—
392#117—2124no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C42—21no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423#107—1062no—
424#107—1062no—
425#106—534no—
426#103—22no—
427#103—534no—
428#103—1062no—
429#102—22no—
430#102—534no—
431#102—1062no—
432C4×C8—534no—
433C4×C8—1062no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D5312yes—
445C2×Q8#1171994712yes—
446C2×Q8#1171994712yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—1064no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470SD16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8C2×C4×D5312yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1064no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4#1171994712yes—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—1064no—
503C22⋊C4—2122no—
504C22⋊C4—2122no—
505C42—24no—
506C42—21no—
507C42—21no—
508C42—532no—
509C42—532no—
510C42—1062no—
511C42—1062no—
512C42—1062no—
513C42—1064no—
514C42—1064no—
515C42—1062no—
516C42—1061no—
517C42—1061no—
518C42—2122no—
519C42—2122no—
520C42—2122no—
521C42—2122no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D5311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1313317314yescomplemented
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—2128no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4#1313323812yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C8—22no—
585C8—41no—
586C8—1062no—
587C8—2121no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D5311yescharacteristic
596C22#1681112911yescharacteristic
597C22#1681113311yescharacteristic
598C22#135×D5311yescharacteristic
599C22C2×#1313323811yescharacteristic
600C22AGL(1,ℤ/8)×D5311yescharacteristic
601C22#137×D5311yescharacteristic
602C4—22no—
603C4—24no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—534no—
608C4—1062no—
609C4—1062no—
610C4—4241no—
611C4—2122no—
612C4—2122no—
613C4—2121no—
614C4—2122no—
615C4C2×#1313317311yescharacteristic
616C4C4×#1171994711yescharacteristic
617C4#1681115011yescharacteristic
618C4#1681115411yescharacteristic
619C2—82no—
620C2#2027715511yescharacteristic
621C2C4×#1313323811yescharacteristic
622C2#440×D5311yescharacteristic
623C2#2028476811yescharacteristic
624C2#2028477411yescharacteristic
625C2#2028479011yescharacteristic
626C2#2028479411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1880C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1880D53
hypocenterC53#21120
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. C4×Q8×Dic53C2semidirect
6. #20275301C2semidirect
7. #20275299C2semidirect
9. #20274969C2semidirect
10. #20274967C2semidirect
11. #20274607C2semidirect
12. #103×Dic53C2semidirect
14. #20274459C2semidirect
20. SD16×Dic53C4semidirect
219. Q8×C53#117semidirect
373. C53#21120semidirect
533. Q8#13133173semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic532
#202771772
C4×Q8×Dic532
#202753382
#202753322
#202753012
#202752992
#202750722
#202749692
#202749672
#202746072
#103×Dic532
#102×Dic532
#202744592
C53×#18802
#2112053