Finite Group Databasebeta

← #27389758

Subgroups — #27389758

Order 13568.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic53C211yesmaximal, characteristic, complemented
2C2×#16810724C212yesmaximal, complemented
3#121×Dic53C211yesmaximal, characteristic
4#20275072C212yesmaximal, complemented
5#20275033C211yesmaximal, characteristic
6#20274971C212yesmaximal, complemented
7#20274905C211yesmaximal, characteristic, complemented
8#20274650C211yesmaximal, characteristic, complemented
9#20274547C212yesmaximal, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C2×C53×#448C211yesmaximal, characteristic
12C2×Q8×Dic53—28no—
13C2×Q8×Dic53—21no—
14C2×Q8×Dic53—21no—
15C2×Q8×Dic53C2212yes—
16C2×Q8×Dic53C2212yes—
17C2×D4×Dic53—21no—
18C2×D4×Dic53C2211yescharacteristic
19C22×C4⋊Dic53—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4×Dic53—21no—
22C22×C4×Dic53C2211yescharacteristic
23#16810724—28no—
24C2×#13133151C2212yes—
25C2×#13133141C2212yes—
26C2×#13133141C2212yes—
27C2×#13133138C2212yes—
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133127C2212yes—
30#16810526C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810516—21no—
33#16810503—21no—
34#16810483C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C22⋊C4×Dic53—21no—
37#16810386C2212yes—
38#16810385C2212yes—
39#16810377C2212yes—
40C23×Q8×C53C2211yescharacteristic
41C22×SD16×C53C2212yes—
42C2×#121×C53C2211yescharacteristic
43C53×#448C418yescomplemented
44C2×#103×C53C2212yes—
45C2×C22⋊C8×C53C2211yescharacteristic
46C23×Dic53—21no—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C22×#88322C2312yes—
51Q8×Dic53—28no—
52Q8×Dic53—28no—
53Q8×Dic53—24no—
54Q8×Dic53—24no—
55Q8×Dic53—24no—
56Q8×Dic53D418yescomplemented
57D4×Dic53—24no—
58D4×Dic53—44no—
59C2×C4⋊Dic53—28no—
60C2×C4⋊Dic53—24no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2312yes—
68C2×C22⋊Dic53—21no—
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C22⋊Dic53—21no—
72C2×C4×Dic53—28no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53D411yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81#13133151—24no—
82#13133141—24no—
83#13133141—24no—
84#13133138—24no—
85C22⋊#88322—22no—
86#13133127—24no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90C22×Q8×C53—24no—
91C22×Q8×C53—24no—
92C22×Q8×C53—21no—
93C22×Q8×C53—22no—
94C22×Q8×C53C2×C414yescomplemented
95C22×Q8×C53C2312yes—
96C22×D4×C53—21no—
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C2×SD16×C53—24no—
101C2×SD16×C53—24no—
102C2×SD16×C53C2×C418yescomplemented
103C22×C8×C53C2312yes—
104#121×C53—22no—
105#121×C53C2×C414yes—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108#103×C53C2×C418yescomplemented
109C22⋊C8×C53C2×C414yescomplemented
110C24×C53—21no—
111C24×C53C2×D411yescharacteristic
112C2×Q8×C53—216no—
113C2×Q8×C53—24no—
114C2×Q8×C53—24no—
115C2×Q8×C53—216no—
116C2×Q8×C53—24no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53C22×C414yes—
121C2×Q8×C53C2×D414yes—
122C2×Q8×C53#4114yes—
123C2×D4×C53—24no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53C22×C412yes—
130C22×C4×C53—24no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—24no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—41no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160SD16×C53—216no—
161C2×C8×C53—22no—
162C2×C8×C53—22no—
163C2×C8×C53C22×C414yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C412yes—
166C22⋊C4×C53—24no—
167C22⋊Dic53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—42no—
171C4⋊Dic53—24no—
172C4⋊Dic53—28no—
173C4⋊Dic53—22no—
174C4⋊Dic53—28no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178C4⋊Dic53C2×D414yes—
179C4×Dic53—28no—
180C4×Dic53—22no—
181C4×Dic53—28no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—42no—
185C4×Dic53C2×D414yes—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53C4×D412yes—
201C23×C53AGL(1,ℤ/8)12yes—
202C23×C53#13712yes—
203C23×C53C22×D411yescharacteristic
204Q8×C53—216no—
205Q8×C53—24no—
206Q8×C53—24no—
207Q8×C53—24no—
208Q8×C53C4×D418yescomplemented
209D4×C53—24no—
210D4×C53—48no—
211D4×C53—42no—
212C2×C4×C53—216no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—21no—
216C2×C4×C53—24no—
217C2×C4×C53—21no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C4×D412yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D412yes—
226C2×C4×C53C2×#4112yes—
227C2×C4×C53C2×#4111yescharacteristic
228C2×Dic53—28no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—24no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—44no—
240C2×Dic53—48no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—84no—
250C2×Dic53#12012yes—
251C2×Dic53#12012yes—
252C2×Dic53#13712yes—
253C8×C53—24no—
254#88322—42no—
255#21454—531nomaximal, Sylow
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53#44014yescomplemented
270C22×C53C2×C4×D411yescharacteristic
271C22×C53C2×#12011yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C53C2×#13711yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C4×C53—28no—
278C4×C53—22no—
279C4×C53—42no—
280C4×C53C2×C4×D412yes—
281C4×C53#52512yes—
282Dic53—22no—
283Dic53—22no—
284Dic53—22no—
285Dic53—41no—
286Dic53—44no—
287Dic53—81no—
288Dic53—81no—
289C22×C4×Q8—531no—
290C2×#448D5311yescharacteristic
291C2×#440—532no—
292C4×#121—531no—
293#1880—532no—
294#1874—531no—
295#1863—532no—
296#1852—531no—
297#1787—531no—
298#1763—532no—
299#1749—531no—
300C2×C53—24no—
301C2×C53—82no—
302C2×C53C4×#12011yescharacteristic
303C2×C53C2×#44011yescharacteristic
304C2×C53C4×#13711yescharacteristic
305C2×C53#300411yescharacteristic
306C2×C53#300611yescharacteristic
307C2×C53#311211yescharacteristic
308C2×C53#311711yescharacteristic
309C23×Q8C2×D5311yescharacteristic
310C22×SD16C2×D5312yes—
311C2×#121C2×D5311yescharacteristic
312C2×C4×Q8—532no—
313C2×C4×Q8—532no—
314C2×C4×Q8—1061no—
315C2×C4×Q8—1061no—
316C2×C4×Q8—1068no—
317C2×C4×D4—531no—
318C2×C4×D4—1061no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—1061no—
321C22×C42—531no—
322C22×C42—1061no—
323#448Dic5318yescomplemented
324#440—1068no—
325C2×#107—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#103C2×D5312yes—
329C2×#102—532no—
330C2×C22⋊C8—531no—
331C2×C22⋊C8C2×D5311yescharacteristic
332C2×#97—532no—
333#390—531no—
334#389—531no—
335#388—1061no—
336#386—1061no—
337#380—531no—
338C4×C4⋊C4—531no—
339C4×C22⋊C4—1061no—
340#340—532no—
341#340—532no—
342#336—532no—
343C53#2145411yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C2×Dic5314yescomplemented
349C22×Q8C22×D5312yes—
350C22×D4—21no—
351C22×D4C22×D5311yescharacteristic
352C23×C4—21no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—1061no—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C2×SD16—24no—
359C2×SD16—24no—
360C2×SD16C2×Dic5318yescomplemented
361C22×C8—532no—
362C22×C8C22×D5312yes—
363#121—22no—
364#121C2×Dic5314yes—
365C4×Q8—538no—
366C4×Q8—1064no—
367C4×Q8—1064no—
368C4×Q8—1068no—
369C4×Q8—1068no—
370C4×Q8—1064no—
371C4×D4—1064no—
372C4×D4—2124no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1068no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—1064no—
393C2×C42—1062no—
394C2×C42—1062no—
395C2×C42—1068no—
396C2×C42—1061no—
397#107—1064no—
398#103—1064no—
399#103—1064no—
400#103C2×Dic5318yescomplemented
401#102—1064no—
402C22⋊C8—1062no—
403C22⋊C8C2×Dic5314yescomplemented
404#97—1064no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408C24—21no—
409C24D4×D5311yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8D4×D5314yes—
419C2×Q8#1171994714yes—
420C2×Q8C22×Dic5314yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Dic5312yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—2122no—
449C22×C4—2121no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4D4×D5311yescharacteristic
455C22×C4#1171994712yes—
456C22×C4C22×Dic5312yes—
457C22×C4C23×D5311yescharacteristic
458SD16—216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8C22×Dic5314yes—
465C4⋊C4—21no—
466C4⋊C4—534no—
467C4⋊C4—1068no—
468C4⋊C4—1068no—
469C4⋊C4—1064no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—2122no—
474C4⋊C4C22×Dic5312yes—
475C22⋊C4—24no—
476C22⋊C4—2122no—
477C22⋊C4—2122no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C42—534no—
481C42—1062no—
482C42—1062no—
483C42—1068no—
484C42—1068no—
485C42—1062no—
486C42—2122no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23D4×Dic5312yes—
499C23#1313324412yes—
500C23#1313324612yes—
501C23C2×D4×D5311yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8D4×Dic5318yescomplemented
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—1068no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1064no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—4244no—
536C2×C4—2124no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2128no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4D4×Dic5312yes—
547C2×C4C23×Dic5311yescharacteristic
548C2×C4C2×D4×D5312yes—
549C2×C4C2×#1171994712yes—
550C2×C4C2×#1171994711yescharacteristic
551C8—24no—
552C8—2122no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22#1681072414yescomplemented
567C22C2×D4×Dic5311yescharacteristic
568C22#120×D5311yescharacteristic
569C22#1681112911yescharacteristic
570C22C2×#1313324411yescharacteristic
571C22#137×D5311yescharacteristic
572C22C2×#1313324611yescharacteristic
573C22#1681142211yescharacteristic
574C4—28no—
575C4—22no—
576C4—42no—
577C4—1062no—
578C4—1062no—
579C4—1062no—
580C4—4241no—
581C4—4241no—
582C4—2121no—
583C4—2124no—
584C4C2×D4×Dic5312yes—
585C4#1681112912yes—
586C2—24no—
587C2—82no—
588C2#120×Dic5311yescharacteristic
589C2C2×#1681072411yescharacteristic
590C2#137×Dic5311yescharacteristic
591C2#2028394611yescharacteristic
592C2#2028396611yescharacteristic
593C2#2028483211yescharacteristic
594C2#2028488111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#448C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#448D53
hypocenterC53#21454
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic53C2semidirect
2. C2×#16810724C2semidirect
4. #20275072C2semidirect
6. #20274971C2semidirect
7. #20274905C2semidirect
8. #20274650C2semidirect
9. #20274547C2semidirect
10. #20274479C2semidirect
43. C53×#448C4semidirect
56. Q8×Dic53D4semidirect
94. C22×Q8×C53C2×C4semidirect
102. C2×SD16×C53C2×C4semidirect
108. #103×C53C2×C4semidirect
109. C22⋊C8×C53C2×C4semidirect
208. Q8×C53C4×D4semidirect
269. C22×C53#440semidirect
323. #448Dic53semidirect
343. C53#21454semidirect
348. C22×Q8C2×Dic53semidirect
360. C2×SD16C2×Dic53semidirect
400. #103C2×Dic53semidirect
403. C22⋊C8C2×Dic53semidirect
506. Q8D4×Dic53semidirect
566. C22#16810724semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic532
C2×#168107242
#121×Dic532
#202750722
#202750332
#202749712
#202749052
#202746502
#202745472
#202744792
C2×C53×#4482
#2145453