Finite Group Databasebeta

← #27387017

Subgroups — #27387017

Order 13568.

1054 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic53C211yesmaximal, characteristic
2#20277177C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic, complemented
4#20275338C211yesmaximal, characteristic, complemented
5#20275332C211yesmaximal, characteristic, complemented
6#20275287C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275065C211yesmaximal, characteristic
9#20274962C211yesmaximal, characteristic, complemented
10#20274960C211yesmaximal, characteristic, complemented
11#20274607C211yesmaximal, characteristic, complemented
12#102×Dic53C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic, complemented
14#20274459C211yesmaximal, characteristic, complemented
15C53×#1879C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133173—21no—
20C22×C4×Dic53—21no—
21D8×Dic53C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810517—21no—
35#16810511—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C22⋊C4×Dic53—21no—
43C4×C22⋊Dic53—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×D8×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65D4×Dic53—22no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yes—
70D4×Dic53C2×C414yescomplemented
71#13133173—44no—
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×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—44no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133151C2×C414yescomplemented
95#13133150—22no—
96#13133150—22no—
97#13133138—22no—
98#13133138—22no—
99#13133138C2×C414yes—
100#13133138C2×C414yescomplemented
101C8×Dic53—22no—
102C8×Dic53C2×C414yescomplemented
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53C2311yescharacteristic
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C2×D8×C53—22no—
113C2×D8×C53—22no—
114C2×D8×C53C2×C414yes—
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C4×D4×C53—22no—
118C4×D4×C53—22no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C42×C53C2311yescharacteristic
125#102×C53—22no—
126#102×C53—22no—
127C24×C53—21no—
128C24×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53C22×C412yes—
138C2×D4×C53C22×C412yes—
139C2×D4×C53#4112yes—
140C2×D4×C53#4112yes—
141C22×C4×C53—24no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—24no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—24no—
154C22×Dic53—24no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166D8×C53—28no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53C22×C412yes—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53#4112yes—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—42no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—24no—
183C22⋊Dic53—24no—
184C22⋊Dic53—24no—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53C22×C412yes—
193C4⋊Dic53C22×C412yes—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—42no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—22no—
200C4×Dic53—41no—
201C4×Dic53—41no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C4×Dic53D814yescomplemented
207C4×Dic53C22×C412yes—
208C4×Dic53#4112yes—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C22×C412yes—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53C22×D411yescharacteristic
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—22no—
233D4×C53—22no—
234D4×C53—22no—
235D4×C53—41no—
236D4×C53—41no—
237D4×C53#11714yescomplemented
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—48no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53C2×D812yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—48no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—48no—
275C2×Dic53—48no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53C4×D412yes—
280C2×Dic53C4×D412yes—
281C2×Dic53C2×D812yes—
282C2×Dic53C2×D812yes—
283C2×Dic53#13512yes—
284C8×C53—22no—
285C8×C53—41no—
286#88322—22no—
287#88322—41no—
288#21113—531nomaximal, Sylow
289C22×C53—24no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53C2×C4×D411yescharacteristic
304C22×C53#52511yescharacteristic
305C22×C53#52511yescharacteristic
306C22×C53C22×D811yescharacteristic
307C22×C53C2×#13511yescharacteristic
308C22×C53#57511yescharacteristic
309C22×C53#57511yescharacteristic
310C4×C53—22no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—81no—
314C4×C53C2×#11711yescharacteristic
315C4×C53C4×#4111yescharacteristic
316C4×C53#52911yescharacteristic
317C4×C53#52911yescharacteristic
318Dic53—22no—
319Dic53—22no—
320Dic53—42no—
321Dic53—41no—
322Dic53—42no—
323Dic53—42no—
324Dic53—81no—
325Dic53C4×D814yes—
326C2×C4×D8—531no—
327#2293—531no—
328C42×D4—531no—
329#1925—531no—
330#1925—531no—
331#1918—531no—
332#1918—531no—
333#1879—531no—
334#1879D5311yescharacteristic
335#1862—531no—
336#1862—531no—
337C4×#107—531no—
338C4×#102—531no—
339C4×#102—531no—
340#1744—531no—
341C2×C53—44no—
342C2×C53—82no—
343C2×C53#229011yescharacteristic
344C2×C53C2×C4×D811yescharacteristic
345C2×C53#294311yescharacteristic
346C2×C53#310011yescharacteristic
347C2×C53#310011yescharacteristic
348C2×C53#310211yescharacteristic
349C2×C53#310211yescharacteristic
350C22×D8C2×D5311yescharacteristic
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4C2×D5311yescharacteristic
355C2×#117—1061no—
356C22×C42—1061no—
357C4×D8—538no—
358C2×#107—531no—
359C2×#107—531no—
360C2×#107—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102C2×D5311yescharacteristic
366C2×#102C2×D5311yescharacteristic
367C2×C4×C8—531no—
368C2×C4×C8—531no—
369#390—531no—
370#390C2×D5311yescharacteristic
371#389—531no—
372#389—531no—
373#388—1061no—
374#388—1061no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—531no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4—531no—
379C4×C22⋊C4—1061no—
380C4×C22⋊C4—1061no—
381C4×C22⋊C4—1061no—
382C4×C22⋊C4—1061no—
383C43—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340C2×D5311yescharacteristic
390#336—531no—
391#336—531no—
392#336—531no—
393#336C2×D5311yescharacteristic
394C53#2111311yescharacteristic, complemented, Sylow
395C22×D4C22×D5311yescharacteristic
396C22×D4C22×D5311yescharacteristic
397C23×C4—21no—
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4—1061no—
401C2×D8—22no—
402C2×D8—22no—
403C2×D8C4×D5314yes—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—534no—
411C4×D4—534no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—1062no—
415C4×D4—1062no—
416#117—2124no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1061no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42—2124no—
444C2×C42C22×D5311yescharacteristic
445#107—534no—
446#107—1062no—
447#107—1062no—
448#102—22no—
449#102—22no—
450#102—534no—
451#102—534no—
452#102—1062no—
453#102—1062no—
454C4×C8—534no—
455C4×C8—1062no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C24—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D5312yes—
473C2×D4C2×C4×D5312yes—
474C2×D4#1171994712yes—
475C2×D4#1171994712yes—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4D4×D5311yescharacteristic
499C22×C4D4×D5311yescharacteristic
500C22×C4C23×D5311yescharacteristic
501D8—28no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—532no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8C2×C4×D5312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—532no—
518C4⋊C4—532no—
519C4⋊C4—532no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4#1171994712yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—1064no—
532C22⋊C4—1064no—
533C22⋊C4—1064no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C42—21no—
537C42—21no—
538C42—534no—
539C42—532no—
540C42—532no—
541C42—1062no—
542C42—1061no—
543C42—1061no—
544C42—2121no—
545C42—2121no—
546C42—2122no—
547C42—2122no—
548C42—2122no—
549C42—2122no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D4×D5311yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4#1313317314yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—532no—
586C2×C4—532no—
587C2×C4—532no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1061no—
597C2×C4—1062no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—2128no—
601C2×C4—2128no—
602C2×C4—2128no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4D8×D5312yes—
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4C2×#1171994711yescharacteristic
617C2×C4C2×#1171994711yescharacteristic
618C2×C4C2×#1171994711yescharacteristic
619C8—22no—
620C8—41no—
621C8—1062no—
622C8—2121no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C4×D4×D5311yescharacteristic
638C22#1681112911yescharacteristic
639C22#1681113311yescharacteristic
640C22C2×D8×D5311yescharacteristic
641C22#135×D5311yescharacteristic
642C22#1681141911yescharacteristic
643C22#1681141911yescharacteristic
644C4—22no—
645C4—42no—
646C4—41no—
647C4—81no—
648C4—534no—
649C4—1062no—
650C4—1062no—
651C4—4241no—
652C4—2122no—
653C4—2121no—
654C4—2122no—
655C4—2122no—
656C4C2×#1313317311yescharacteristic
657C4C4×#1171994711yescharacteristic
658C4#1681115011yescharacteristic
659C4#1681115411yescharacteristic
660C2—44no—
661C2—82no—
662C2#2027715511yescharacteristic
663C2C4×D8×D5311yescharacteristic
664C2#2028351211yescharacteristic
665C2#2028471011yescharacteristic
666C2#2028471411yescharacteristic
667C2#2028472611yescharacteristic
668C2#2028473011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1879C2
socleC23×C53C22×D4
prime radicalC22×D8×C53C22
cosocleC22×C4×C53C24
hypercenter#1879D53
hypocenterC53#21113
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #20277177C2semidirect
3. C4×D4×Dic53C2semidirect
4. #20275338C2semidirect
5. #20275332C2semidirect
6. #20275287C2semidirect
7. #20275285C2semidirect
9. #20274962C2semidirect
10. #20274960C2semidirect
11. #20274607C2semidirect
12. #102×Dic53C2semidirect
13. #102×Dic53C2semidirect
14. #20274459C2semidirect
21. D8×Dic53C4semidirect
70. D4×Dic53C2×C4semidirect
94. #13133151C2×C4semidirect
100. #13133138C2×C4semidirect
102. C8×Dic53C2×C4semidirect
206. C4×Dic53D8semidirect
237. D4×C53#117semidirect
394. C53#21113semidirect
572. D4#13133173semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic532
#202771772
C4×D4×Dic532
#202753382
#202753322
#202752872
#202752852
#202750652
#202749622
#202749602
#202746072
#102×Dic532
#202744592
C53×#18792
#2111353