Finite Group Databasebeta

← #27406393

Subgroups — #27406393

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810846C211yesmaximal, characteristic
2C2×C4×Dic212C211yesmaximal, characteristic, complemented
3C2×C4×#11719913C211yesmaximal, characteristic
4#20277784C211yesmaximal, characteristic
5#20277635C211yesmaximal, characteristic, complemented
6#20275526C211yesmaximal, characteristic
7#20275485C211yesmaximal, characteristic, complemented
8#20275398C211yesmaximal, characteristic
9#20275391C211yesmaximal, characteristic, complemented
10#20275069C211yesmaximal, characteristic
11#20275055C211yesmaximal, characteristic, complemented
12#20274986C211yesmaximal, characteristic, complemented
13#20274965C211yesmaximal, characteristic
14C4×#13133142C211yesmaximal, characteristic, complemented
15C8×C4⋊C4×C53C211yesmaximal, characteristic, complemented
16C22×Dic212C2211yescharacteristic
17C22×#11719913C2211yescharacteristic
18C2×#13133205C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C2×C4×Dic106C2211yescharacteristic
22C2×C4×Dic106C2211yescharacteristic
23C2×C4×D212C2211yescharacteristic
24#16810846C418yescomplemented
25C4×Dic212—24no—
26C4×#11719913—24no—
27C2×#13133151C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133139C2211yescharacteristic
35C2×#13133139C2211yescharacteristic
36C2×#13133139C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810525C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810518—21no—
42#16810518—21no—
43#16810513—21no—
44#16810512—21no—
45#16810494—21no—
46#16810489C2211yescharacteristic
47#16810487—21no—
48C4×C4⋊Dic53C2211yescharacteristic
49#16810474—21no—
50#16810470—21no—
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54#16810386C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C2×C4×C8×C53C2211yescharacteristic
58C2×C4×C8×C53C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C53×#336C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C22×Dic106C2311yescharacteristic
64C22×D212C2311yescharacteristic
65C22×C4×D53—21no—
66C2×Dic212—22no—
67C2×Dic212—22no—
68C2×Dic212C2×C414yescomplemented
69C2×#11719913—22no—
70C2×#11719913—22no—
71C2×#11719913C2×C414yes—
72#13133205—22no—
73#13133205C2×C414yes—
74C4×Dic106—22no—
75C4×Dic106—22no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C4×Dic106—41no—
79C4×Dic106—41no—
80C4×Dic106—41no—
81C4×Dic106C2×C414yescomplemented
82C4×D212—22no—
83C4×D212—41no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105#13133151—22no—
106#13133149—22no—
107#13133142—22no—
108#13133142—22no—
109#13133142—41no—
110#13133142C2×C414yescomplemented
111#13133142D414yes—
112#13133139—22no—
113#13133139—41no—
114#13133139C2×C414yes—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121#13133124—21no—
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C2×C4⋊C4×C53D411yescharacteristic
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C42×C53C2311yescharacteristic
129C2×C42×C53C2311yescharacteristic
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53C2×C414yescomplemented
132C4×C8×C53—22no—
133C4×C8×C53—22no—
134C4×C8×C53—22no—
135#95×C53D411yescharacteristic
136#95×C53D411yescharacteristic
137C23×D53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212C22×C412yes—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106C22×C412yes—
170C2×Dic106C22×C412yes—
171C2×Dic106C2×D412yes—
172C2×Dic106#4112yes—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53C22×C412yes—
183C2×C8×C53C22×C412yes—
184C2×C8×C53C2×D412yes—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53C22×C412yes—
192C42×C53#4112yes—
193#11719921—24no—
194#11719921—24no—
195#11719921—42no—
196#11719921—42no—
197#11719921—81no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C412yes—
207C4⋊Dic53C2×D412yes—
208#11719919—24no—
209#11719919—42no—
210#11719919—42no—
211#11719919—42no—
212#11719919—42no—
213#11719919—42no—
214C4×Dic53—24no—
215C4×Dic53—42no—
216C4×Dic53—42no—
217C4×Dic53—42no—
218C4×Dic53—81no—
219Dic212—28no—
220#11719913—28no—
221C23×C53C22×D411yescharacteristic
222C22×D53—22no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
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×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53C4×D412yes—
244C2×C4×C53C4×D412yes—
245C2×C4×C53#13512yes—
246C2×C4×C53#13512yes—
247C2×C4×C53C23×C411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
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—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274D212—22no—
275D212—22no—
276D212—41no—
277D212—41no—
278C4×D53—84no—
279Dic106—22no—
280Dic106—22no—
281Dic106—22no—
282Dic106—22no—
283Dic106—22no—
284Dic106—41no—
285Dic106—41no—
286Dic106C4×D414yes—
287C8×C53—22no—
288C8×C53—22no—
289C8×C53—22no—
290C8×C53—41no—
291#23000—531nomaximal, Sylow
292C22×C53C2×C4×D411yescharacteristic
293C22×C53D4211yescharacteristic
294C22×C53#54811yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53C2×#13511yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57811yescharacteristic
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53C2×C4×D411yescharacteristic
313C4×C53#52011yescharacteristic
314C4×C53#54711yescharacteristic
315C4×C53#55311yescharacteristic
316Dic53—24no—
317Dic53—42no—
318Dic53—42no—
319Dic53—42no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323C2×#464—531no—
324C2×C4×Q16—531no—
325C2×C4×SD16—531no—
326#2367—531no—
327#2351—531no—
328#1958—531no—
329#1949—531no—
330#1934—531no—
331#1933—531no—
332#1880—531no—
333#1878—531no—
334#1865—531no—
335#1862—531no—
336C8×C4⋊C4D5311yescharacteristic, complemented
337C4×#103—531no—
338C2×C53#236111yescharacteristic
339C2×C53C4×#13511yescharacteristic
340C2×C53#294711yescharacteristic
341C2×C53#328211yescharacteristic
342C2×C53#328511yescharacteristic
343C2×C53#336411yescharacteristic
344C2×C53#336511yescharacteristic
345D53—82no—
346C22×Q16—531no—
347C22×SD16—531no—
348C2×#124—531no—
349C2×C4×Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×Q8—531no—
352C2×C4×Q8—531no—
353C2×C4×D4—531no—
354#464—538no—
355C4×Q16—1064no—
356C4×SD16—1064no—
357C2×#107—531no—
358C2×#106—531no—
359C2×C4⋊C8C2×D5311yescharacteristic
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×#102—531no—
368C2×C4×C8C2×D5311yescharacteristic
369C2×C4×C8C2×D5311yescharacteristic
370C2×C4×C8C2×D5311yescharacteristic
371#391—531no—
372#390—531no—
373#389—531no—
374#389—531no—
375#389—1061no—
376#389—1061no—
377#388—1061no—
378#388—1061no—
379#385—1061no—
380#382—531no—
381#382—1061no—
382C4×C4⋊C4—531no—
383C4×C4⋊C4—1061no—
384C4×C4⋊C4C2×D5311yescharacteristic
385C4×C22⋊C4—1061no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#336C2×D5311yescharacteristic
391#336C2×D5311yescharacteristic
392C53#2300011yescharacteristic, complemented, Sylow
393C22×Q8—531no—
394C22×Q8—531no—
395C22×D4—531no—
396C23×C4—1061no—
397C2×Q16—534no—
398C2×Q16—1062no—
399C2×Q16—1062no—
400C2×SD16—534no—
401C2×SD16—1062no—
402C2×SD16—1062no—
403C22×C8C22×D5311yescharacteristic
404C22×C8C22×D5311yescharacteristic
405C22×C8C22×D5311yescharacteristic
406C22×C8C22×D5311yescharacteristic
407#124—534no—
408#124—1062no—
409C4×Q8—534no—
410C4×Q8—1062no—
411C4×Q8—1062no—
412C4×Q8—1062no—
413C4×Q8—1062no—
414C4×Q8—2121no—
415C4×Q8—2121no—
416C4×Q8—2121no—
417C4×D4—1062no—
418C4×D4—2121no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4D21211yescharacteristic
431C2×C4⋊C4D21211yescharacteristic
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—1061no—
438C2×C42—1061no—
439C2×C42—1061no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42C22×D5311yescharacteristic
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5311yescharacteristic
445#107—1062no—
446#106—1062no—
447C4⋊C8C4×D5314yescomplemented
448#103—534no—
449#103—534no—
450#103—1062no—
451#103—1062no—
452#103—2121no—
453#102—534no—
454#102—1062no—
455#102—2121no—
456C4×C8—22no—
457C4×C8—22no—
458C4×C8—22no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95D21211yescharacteristic
467#95D21211yescharacteristic
468C24—1061no—
469C2×Q8—532no—
470C2×Q8—532no—
471C2×Q8—532no—
472C2×Q8—532no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×D4—532no—
482C2×D4—1061no—
483C2×D4—1061no—
484C2×D4—1061no—
485C2×D4—1061no—
486C2×D4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—4241no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4C2×D21211yescharacteristic
498C22×C4C2×D21211yescharacteristic
499C22×C4C2×D21211yescharacteristic
500C22×C4C2×D21211yescharacteristic
501C22×C4C2×D21211yescharacteristic
502C22×C4C2×D21211yescharacteristic
503C22×C4C23×D5311yescharacteristic
504Q16—1068no—
505SD16—1068no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8C2×C4×D5312yes—
516C2×C8C2×C4×D5312yes—
517C2×C8D4×D5312yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—532no—
521C4⋊C4—532no—
522C4⋊C4—1064no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C22⋊C4—1064no—
537C22⋊C4—1064no—
538C22⋊C4—4241no—
539C22⋊C4—2122no—
540C22⋊C4—2122no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1064no—
546C42—4241no—
547C42—2122no—
548C42—2122no—
549C42—2122no—
550C42C2×C4×D5312yes—
551C42#1171994512yes—
552C23—1062no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23C22×D21211yescharacteristic
560Q8—534no—
561Q8—1062no—
562Q8—1062no—
563Q8—1062no—
564Q8—1062no—
565Q8—1062no—
566Q8—2121no—
567Q8—2121no—
568D4—1062no—
569D4—1062no—
570D4—2121no—
571D4—2121no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—4244no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4C4×D21212yes—
609C2×C4C4×D21212yes—
610C2×C4#1313323912yes—
611C2×C4#1313323912yes—
612C2×C4C22×C4×D5311yescharacteristic
613C2×C4C2×D4×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4C2×#1171994511yescharacteristic
616C2×C4#1313326311yescharacteristic
617C2×C4#1313326511yescharacteristic
618C8—22no—
619C8—22no—
620C8—22no—
621C8—41no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22C2×C4×D21211yescharacteristic
630C22D4×D21211yescharacteristic
631C22#1681128511yescharacteristic
632C22C2×#1313323911yescharacteristic
633C22C2×#1313323911yescharacteristic
634C22#1681143511yescharacteristic
635C22#1681144011yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—1064no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—2122no—
647C4—2122no—
648C4—2122no—
649C4C4×D4×D5311yescharacteristic
650C4#1681110511yescharacteristic
651C4#1681127411yescharacteristic
652C4#1681131711yescharacteristic
653C2—4242no—
654C2#2027773311yescharacteristic
655C2C4×#1313323911yescharacteristic
656C2#2028356011yescharacteristic
657C2#2028642711yescharacteristic
658C2#2028645811yescharacteristic
659C2#2028740211yescharacteristic
660C2#2028741311yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC8×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC8×C4⋊C4D53
hypocenterC53#23000
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic212C2semidirect
5. #20277635C2semidirect
7. #20275485C2semidirect
9. #20275391C2semidirect
11. #20275055C2semidirect
12. #20274986C2semidirect
14. C4×#13133142C2semidirect
15. C8×C4⋊C4×C53C2semidirect
24. #16810846C4semidirect
68. C2×Dic212C2×C4semidirect
81. C4×Dic106C2×C4semidirect
110. #13133142C2×C4semidirect
131. C4⋊C8×C53C2×C4semidirect
336. C8×C4⋊C4D53semidirect
392. C53#23000semidirect
447. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108462
C2×C4×Dic2122
C2×C4×#117199132
#202777842
#202776352
#202755262
#202754852
#202753982
#202753912
#202750692
#202750552
#202749862
#202749652
C4×#131331422
C8×C4⋊C4×C532
#2300053