Finite Group Databasebeta

← #27446651

Subgroups — #27446651

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810919C211yesmaximal, characteristic, complemented
2C22×#13133151C211yesmaximal, characteristic
3#20278417C211yesmaximal, characteristic
4#20277718C211yesmaximal, characteristic
5#20276252C211yesmaximal, characteristic, complemented
6#20276208C211yesmaximal, characteristic, complemented
7#20276109C211yesmaximal, characteristic, complemented
8#20276044C211yesmaximal, characteristic, complemented
9#20275474C211yesmaximal, characteristic, complemented
10#20275308C211yesmaximal, characteristic, complemented
11#20275138C211yesmaximal, characteristic
12#20275020C211yesmaximal, characteristic
13#20274972C211yesmaximal, characteristic, complemented
14#20274905C211yesmaximal, characteristic
15C53×#1850C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133194—21no—
19C2×C4×Dic106C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810919—24no—
23C2×#13133151—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133151C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34#16810555—21no—
35#16810544—21no—
36#16810529C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810520—21no—
41#16810510—21no—
42#16810498—21no—
43#16810495—21no—
44#16810493C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4×C4⋊Dic53C2211yescharacteristic
47C22⋊C4×Dic53—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C23×Dic53—21no—
65#13133217—22no—
66#13133217—41no—
67#13133196—22no—
68#13133196Q814yescomplemented
69#13133194—44no—
70C4×Dic106—22no—
71C4×Dic106—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53—41no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×C22⋊Dic53—21no—
94C2×C22⋊Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97#13133151—22no—
98#13133151—22no—
99#13133151—22no—
100#13133151—22no—
101#13133151—41no—
102#13133151D414yes—
103#13133142—22no—
104#13133142—22no—
105#13133142—22no—
106#13133142—41no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114C23×C4×C53C2311yescharacteristic
115C22×C8×C53—22no—
116C22×C8×C53—22no—
117C22×C8×C53—22no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C4⋊C8×C53—22no—
127C4⋊C8×C53—22no—
128C22⋊C8×C53—22no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53C2×D411yescharacteristic
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—81no—
157C22×Dic53—81no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—42no—
164C2×Dic106—42no—
165C2×Dic106—81no—
166C2×Dic106C2×Q812yes—
167C2×C8×C53—28no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53C2×D412yes—
179C2×C8×C53#4112yes—
180C22⋊C4×C53—22no—
181C22⋊C4×C53Q1614yes—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C22⋊Dic53—24no—
187C22⋊Dic53—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—42no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—81no—
204C4⋊Dic53C2×D412yes—
205C4⋊Dic53C2×D412yes—
206C4⋊Dic53C2×Q812yes—
207#11719919—24no—
208#11719919—42no—
209#11719919—42no—
210#11719919—42no—
211#11719919—81no—
212C4×Dic53—42no—
213C4×Dic53—81no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C22⋊Q812yes—
219C23×C53C2×Q1612yes—
220C23×C53AGL(1,ℤ/8)12yes—
221C23×C53C22×D411yescharacteristic
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53C22⋊Q812yes—
239C2×C4×C53C2×Q1612yes—
240C2×C4×C53C2×Q1612yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—84no—
272C2×Dic53—84no—
273Dic106—22no—
274Dic106—22no—
275Dic106—41no—
276Dic106—41no—
277Dic106—84no—
278C8×C53—22no—
279C8×C53—22no—
280C8×C53—41no—
281C8×C53—41no—
282#26342—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53#47714yescomplemented
290C22×C53C2×C22⋊Q811yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C22×Q1611yescharacteristic
294C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C53#57611yescharacteristic
296C22×C53#57811yescharacteristic
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53#52511yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53D4×Q811yescharacteristic
305C4×C53#55411yescharacteristic
306Dic53—42no—
307Dic53—42no—
308Dic53—42no—
309Dic53—42no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314C2×#477—531no—
315C22×#107—531no—
316#2440—531no—
317#2360—531no—
318#2095—531no—
319#2087—531no—
320#2068—531no—
321#2057—531no—
322#1947—531no—
323#1921—531no—
324#1892—531no—
325#1872—531no—
326#1863—531no—
327#1852—531no—
328#1850D5311yescharacteristic
329C2×C53—24no—
330C2×C53#244911yescharacteristic
331C2×C53C2×#47711yescharacteristic
332C2×C53#308311yescharacteristic
333C2×C53#326411yescharacteristic
334C2×C53D4×Q1611yescharacteristic
335C2×C53#333811yescharacteristic
336C2×C53#334311yescharacteristic
337C23×C8C2×D5311yescharacteristic
338C2×C4⋊Q8—531no—
339C2×C22⋊Q8—531no—
340C2×C22⋊Q8—1061no—
341C2×C4×Q8—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—531no—
344#477—1064no—
345C2×#107—531no—
346C2×#107—531no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#107—1064no—
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×C4⋊C8C2×D5311yescharacteristic
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×C22⋊C8C2×D5311yescharacteristic
359#398—1061no—
360#395—1061no—
361#391—531no—
362#389—531no—
363#389—531no—
364#389—531no—
365#389—1061no—
366#387—1061no—
367#385—1061no—
368#385—1061no—
369#384—531no—
370#380—531no—
371C4×C4⋊C4—531no—
372C4×C22⋊C4—1061no—
373C4×C22⋊C4C2×D5311yescharacteristic
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#336C2×D5311yescharacteristic
381#336C2×D5311yescharacteristic
382C53#2634211yescharacteristic, complemented, Sylow
383C22×Q8—531no—
384C22×Q8—1061no—
385C23×C4—1061no—
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—22no—
391C22×C8C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C4⋊Q8—1062no—
396C4⋊Q8—2121no—
397C22⋊Q8—534no—
398C22⋊Q8—1062no—
399C22⋊Q8—2124no—
400C4×Q8—1062no—
401C4×Q8—2121no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—2121no—
422C2×C4⋊C4—2121no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4D21211yescharacteristic
426C2×C22⋊C4D21211yescharacteristic
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430C2×C42C22×D5311yescharacteristic
431#107—534no—
432#107—1062no—
433#107—1062no—
434#107—1062no—
435#107—1062no—
436#107—2121no—
437C4⋊C8—22no—
438C4⋊C8—22no—
439#103—1062no—
440#103—1062no—
441#103—1062no—
442#103—2121no—
443C22⋊C8—22no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95D21211yescharacteristic
452#95D21211yescharacteristic
453C24C2×D21211yescharacteristic
454C2×Q8—532no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—4241no—
461C2×Q8—2122no—
462C2×Q8—2122no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—4241no—
476C22×C4—4241no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4C2×D21211yescharacteristic
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4C2×D21211yescharacteristic
485C22×C4C2×D21211yescharacteristic
486C22×C4#1171994712yes—
487C22×C4Q8×D5312yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—28no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8D4×D5312yes—
501C2×C8#1171994712yes—
502C4⋊C4—532no—
503C4⋊C4—532no—
504C4⋊C4—532no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C22⋊C4—22no—
527C22⋊C4—1064no—
528C22⋊C4—2122no—
529C22⋊C4Dic21214yes—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—4241no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1313319712yes—
541C23C2×Dic21212yes—
542C23#1313324512yes—
543C23C22×D21211yescharacteristic
544Q8—1062no—
545Q8—1062no—
546Q8—4244no—
547Q8—2121no—
548Q8—2121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—4244no—
570C2×C4—4244no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4#1313319712yes—
592C2×C4C2×Dic21212yes—
593C2×C4C2×Dic21212yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4C2×Q8×D5311yescharacteristic
596C2×C4C2×#1171994711yescharacteristic
597C2×C4C2×#1171994711yescharacteristic
598C2×C4#1313326311yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—22no—
601C8—22no—
602C8—41no—
603C8—41no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1681091914yescomplemented
611C22C2×#1313319711yescharacteristic
612C22D4×D21211yescharacteristic
613C22#1681127711yescharacteristic
614C22C22×Dic21211yescharacteristic
615C22C2×#1313324511yescharacteristic
616C22#1681142811yescharacteristic
617C22#1681144011yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4—4241no—
627C4—2122no—
628C4—2122no—
629C4—2122no—
630C4—2122no—
631C4#1681112911yescharacteristic
632C4#1681121711yescharacteristic
633C4#1681128611yescharacteristic
634C4#1681133111yescharacteristic
635C2—24no—
636C2#2027852211yescharacteristic
637C2C2×#1681091911yescharacteristic
638C2#2028456811yescharacteristic
639C2#2028620011yescharacteristic
640C2D4×Dic21211yescharacteristic
641C2#2028710811yescharacteristic
642C2#2028715711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1850D53
hypocenterC53#26342
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#16810919C2semidirect
5. #20276252C2semidirect
6. #20276208C2semidirect
7. #20276109C2semidirect
8. #20276044C2semidirect
9. #20275474C2semidirect
10. #20275308C2semidirect
13. #20274972C2semidirect
68. #13133196Q8semidirect
289. C22×C53#477semidirect
382. C53#26342semidirect
610. C22#16810919semidirect

Maximal subgroups

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

subgroupindex
C2×#168109192
C22×#131331512
#202784172
#202777182
#202762522
#202762082
#202761092
#202760442
#202754742
#202753082
#202751382
#202750202
#202749722
#202749052
C53×#18502
#2634253