Finite Group Databasebeta

← #27441229

Subgroups — #27441229

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810909C211yesmaximal, characteristic, complemented
2C22×#13133142C211yesmaximal, characteristic
3#20278417C211yesmaximal, characteristic
4C22⋊C4×Dic106C211yesmaximal, characteristic
5#20276257C211yesmaximal, characteristic, complemented
6#20276095C211yesmaximal, characteristic, complemented
7#20276095C211yesmaximal, characteristic, complemented
8#20275857C211yesmaximal, characteristic, complemented
9#20274986C211yesmaximal, characteristic, complemented
10#20274986C211yesmaximal, characteristic, complemented
11#20274910C211yesmaximal, characteristic
12#20274740C211yesmaximal, characteristic
13#20274721C211yesmaximal, characteristic, complemented
14#20274651C211yesmaximal, characteristic
15C53×#1850C211yesmaximal, characteristic
16C23×Dic106C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133194—21no—
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810909—24no—
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133142—24no—
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133142C2211yescharacteristic
35C2×#13133142C2211yescharacteristic
36#16810555—21no—
37#16810544—21no—
38#16810529C2211yescharacteristic
39#16810529C2211yescharacteristic
40#16810527—21no—
41#16810527—21no—
42#16810521C2211yescharacteristic
43#16810521C2211yescharacteristic
44#16810498—21no—
45#16810493C2211yescharacteristic
46#16810483C2211yescharacteristic
47#16810483C2211yescharacteristic
48#16810481—21no—
49C22⋊C4×Dic53—21no—
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—22no—
62C22×Dic106—22no—
63C22×Dic106—21no—
64C22×Dic106—22no—
65C22×Dic106—21no—
66C22×Dic106—41no—
67C22×Dic106—41no—
68C22×Dic106Q814yescomplemented
69C22×Dic106C2311yescharacteristic
70C22×Dic106C2311yescharacteristic
71C23×Dic53—21no—
72C23×Dic53—21no—
73#13133217—22no—
74#13133217—41no—
75#13133194—44no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C4×Dic106—41no—
79C4×Dic106—41no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—22no—
85C2×C4⋊Dic53—41no—
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×C22⋊Dic53—21no—
97C2×C22⋊Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100#13133149—22no—
101#13133149—22no—
102#13133149—41no—
103#13133149—41no—
104#13133142—22no—
105#13133142—22no—
106#13133142—22no—
107#13133142—22no—
108#13133142—22no—
109#13133142—41no—
110#13133142—41no—
111#13133142D414yes—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120C23×C4×C53C2311yescharacteristic
121C22×C8×C53—22no—
122C22×C8×C53—22no—
123C22×C8×C53—41no—
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C22×C8×C53C2311yescharacteristic
128C2×C22⋊C4×C53D411yescharacteristic
129C2×C22⋊C4×C53D411yescharacteristic
130C2×C42×C53C2311yescharacteristic
131C2×C42×C53C2311yescharacteristic
132C4⋊C8×C53—22no—
133C4⋊C8×C53—41no—
134C22⋊C8×C53—22no—
135#95×C53D411yescharacteristic
136#95×C53D411yescharacteristic
137C24×C53C2×D411yescharacteristic
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×Q812yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—24no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—81no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—28no—
172C2×Dic106—24no—
173C2×Dic106—21no—
174C2×Dic106—44no—
175C2×Dic106—44no—
176C2×Dic106—44no—
177C2×Dic106—42no—
178C2×Dic106—42no—
179C2×Dic106—42no—
180C2×Dic106—42no—
181C2×Dic106—81no—
182C2×Dic106C2×D412yes—
183C2×Dic106C2×Q812yes—
184C2×Dic106C2×Q812yes—
185C2×Dic106#4112yes—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C2×C8×C53—21no—
189C2×C8×C53—21no—
190C2×C8×C53—21no—
191C2×C8×C53—21no—
192C2×C8×C53—21no—
193C2×C8×C53—21no—
194C2×C8×C53—21no—
195C2×C8×C53—21no—
196C2×C8×C53—21no—
197C2×C8×C53—44no—
198C2×C8×C53C2×D412yes—
199C22⋊C4×C53—22no—
200C22⋊C4×C53SD1614yes—
201C42×C53—21no—
202C42×C53—21no—
203C42×C53—21no—
204C42×C53—21no—
205C22⋊Dic53—42no—
206C22⋊Dic53—42no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211C4⋊Dic53—21no—
212C4⋊Dic53—21no—
213C4⋊Dic53—21no—
214C4⋊Dic53—21no—
215C4⋊Dic53—21no—
216C4⋊Dic53—21no—
217C4⋊Dic53—21no—
218C4⋊Dic53—42no—
219C4⋊Dic53—42no—
220C4⋊Dic53—42no—
221C4⋊Dic53—81no—
222C4⋊Dic53C2×D412yes—
223#11719919—42no—
224#11719919—42no—
225#11719919—42no—
226#11719919—42no—
227C4×Dic53—42no—
228C4×Dic53—42no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53C22⋊Q812yes—
234C23×C53C2×SD1612yes—
235C23×C53#13712yes—
236C23×C53C22×D411yescharacteristic
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×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53C22⋊Q812yes—
254C2×C4×C53C2×SD1612yes—
255C2×C4×C53C2×SD1612yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—48no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—84no—
288Dic106—22no—
289Dic106—22no—
290Dic106—22no—
291Dic106—22no—
292Dic106—22no—
293Dic106—48no—
294Dic106—41no—
295Dic106—41no—
296Dic106—84no—
297Dic106C22⋊Q814yescomplemented
298C8×C53—22no—
299C8×C53—41no—
300C8×C53—41no—
301C8×C53—41no—
302#25955—531nomaximal, Sylow
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—22no—
309C22×C53#47514yescomplemented
310C22×C53C2×C22⋊Q811yescharacteristic
311C22×C53D4211yescharacteristic
312C22×C53#54611yescharacteristic
313C22×C53C22×SD1611yescharacteristic
314C22×C53C2×#13711yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57811yescharacteristic
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—41no—
321C4×C53—41no—
322C4×C53C2×C22⋊Q811yescharacteristic
323C4×C53#54311yescharacteristic
324C4×C53#54611yescharacteristic
325C4×C53#55111yescharacteristic
326Dic53—24no—
327Dic53—42no—
328Dic53—42no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332Dic53—81no—
333Dic53—81no—
334C2×#475—531no—
335C22×#103—531no—
336#2440—531no—
337Q8×C22⋊C4—531no—
338#2096—531no—
339#2066—531no—
340#2066—531no—
341#2024—531no—
342#1865—531no—
343#1865—531no—
344#1853—531no—
345#1850D5311yescharacteristic
346#1807—531no—
347#1803—531no—
348#1787—531no—
349C2×C53—24no—
350C2×C53#243711yescharacteristic
351C2×C53C2×#47511yescharacteristic
352C2×C53#308211yescharacteristic
353C2×C53#327011yescharacteristic
354C2×C53D4×SD1611yescharacteristic
355C2×C53#329811yescharacteristic
356C2×C53#331611yescharacteristic
357C23×Q8—531no—
358C23×C8C2×D5311yescharacteristic
359C2×C4⋊Q8—531no—
360C2×C22⋊Q8—1061no—
361C2×C4×Q8—531no—
362C2×C4×Q8—531no—
363C22×C4⋊C4—531no—
364#475—1064no—
365C2×#106—531no—
366C2×#106—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×C4⋊C8C2×D5311yescharacteristic
370C2×C4⋊C8C2×D5311yescharacteristic
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#103—1064no—
380C2×C22⋊C8C2×D5311yescharacteristic
381#398—1061no—
382#395—1061no—
383#391—531no—
384#391—531no—
385#391—1061no—
386#391—1061no—
387#389—531no—
388#389—531no—
389#385—1061no—
390#384—531no—
391#380—531no—
392#380—531no—
393#380—1061no—
394C4×C22⋊C4—1061no—
395C4×C22⋊C4C2×D5311yescharacteristic
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#336C2×D5311yescharacteristic
401#336C2×D5311yescharacteristic
402C53#2595511yescharacteristic, complemented, Sylow
403C22×Q8—534no—
404C22×Q8—531no—
405C22×Q8—531no—
406C22×Q8—1062no—
407C22×Q8—1062no—
408C22×Q8—1061no—
409C22×Q8—1061no—
410C22×Q8—1062no—
411C22×Q8—2121no—
412C22×Q8—2121no—
413C23×C4—1061no—
414C23×C4—1061no—
415C23×C4C22×D5311yescharacteristic
416C22×C8—22no—
417C22×C8—22no—
418C22×C8—41no—
419C22×C8C22×D5311yescharacteristic
420C22×C8C22×D5311yescharacteristic
421C22×C8C22×D5311yescharacteristic
422C22×C8C22×D5311yescharacteristic
423C4⋊Q8—1062no—
424C4⋊Q8—2121no—
425C22⋊Q8—2124no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×Q8—2121no—
429C4×Q8—2121no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—2121no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4D21211yescharacteristic
449C2×C22⋊C4D21211yescharacteristic
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453C2×C42C22×D5311yescharacteristic
454#106—1062no—
455#106—1062no—
456#106—2121no—
457#106—2121no—
458C4⋊C8—22no—
459C4⋊C8—41no—
460#103—534no—
461#103—1062no—
462#103—1062no—
463#103—1062no—
464#103—1062no—
465#103—1062no—
466#103—2121no—
467#103—2121no—
468C22⋊C8—22no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95D21211yescharacteristic
478#95D21211yescharacteristic
479C24C2×D21211yescharacteristic
480C2×Q8—532no—
481C2×Q8—532no—
482C2×Q8—532no—
483C2×Q8—532no—
484C2×Q8—1068no—
485C2×Q8—1064no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8—1061no—
491C2×Q8—1061no—
492C2×Q8—1061no—
493C2×Q8—1061no—
494C2×Q8—4241no—
495C2×Q8—2122no—
496C2×Q8—2122no—
497C2×Q8—2122no—
498C2×Q8—2124no—
499C2×Q8—2124no—
500C2×Q8—2124no—
501C2×Q8—2122no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—1061no—
507C22×C4—1064no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—4241no—
516C22×C4—2122no—
517C22×C4—2122no—
518C22×C4—2122no—
519C22×C4—2122no—
520C22×C4C2×D21211yescharacteristic
521C22×C4C2×D21211yescharacteristic
522C22×C4C2×D21211yescharacteristic
523C22×C4C2×D21211yescharacteristic
524C22×C4C2×D21211yescharacteristic
525C22×C4#1171994712yes—
526C22×C4Q8×D5312yes—
527C22×C4C23×D5311yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—44no—
540C2×C8D4×D5312yes—
541C4⋊C4—532no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—4241no—
554C4⋊C4—2122no—
555C4⋊C4—2122no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4—2122no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C22⋊C4—22no—
562C22⋊C4—2122no—
563C22⋊C4—2122no—
564C22⋊C4#1171991314yes—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—2122no—
570C42—2122no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1313319712yes—
576C23C2×#1171991312yes—
577C23#1313325012yes—
578C23C22×D21211yescharacteristic
579Q8—534no—
580Q8—1062no—
581Q8—1062no—
582Q8—1062no—
583Q8—1062no—
584Q8—1062no—
585Q8—4244no—
586Q8—2128no—
587Q8—2121no—
588Q8—2121no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—4244no—
611C2×C4—2128no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4#1313319712yes—
632C2×C4C2×#1171991312yes—
633C2×C4C2×#1171991312yes—
634C2×C4C2×D4×D5311yescharacteristic
635C2×C4C2×Q8×D5311yescharacteristic
636C2×C4#41×D5311yescharacteristic
637C2×C4C2×#1171994711yescharacteristic
638C2×C4#1313326211yescharacteristic
639C2×C4#1313326311yescharacteristic
640C8—22no—
641C8—41no—
642C8—41no—
643C8—41no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1681090914yescomplemented
651C22C2×#1313319711yescharacteristic
652C22D4×D21211yescharacteristic
653C22#1681126811yescharacteristic
654C22C22×#1171991311yescharacteristic
655C22C2×#1313325011yescharacteristic
656C22#1681143511yescharacteristic
657C22#1681144011yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—1064no—
664C4—4241no—
665C4—4241no—
666C4—4241no—
667C4—4241no—
668C4—4241no—
669C4—2122no—
670C4—2122no—
671C4C22⋊Q8×D5311yescharacteristic
672C4#1681124811yescharacteristic
673C4#1681126011yescharacteristic
674C4#1681129611yescharacteristic
675C2—24no—
676C2#2027837911yescharacteristic
677C2C2×#1681090911yescharacteristic
678C2#2028455811yescharacteristic
679C2#2028629011yescharacteristic
680C2D4×#1171991311yescharacteristic
681C2#2028658111yescharacteristic
682C2#2028682811yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1850D53
hypocenterC53#25955
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×#16810909C2semidirect
5. #20276257C2semidirect
6. #20276095C2semidirect
7. #20276095C2semidirect
8. #20275857C2semidirect
9. #20274986C2semidirect
10. #20274986C2semidirect
13. #20274721C2semidirect
68. C22×Dic106Q8semidirect
297. Dic106C22⋊Q8semidirect
309. C22×C53#475semidirect
402. C53#25955semidirect
650. C22#16810909semidirect

Maximal subgroups

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

subgroupindex
C2×#168109092
C22×#131331422
#202784172
C22⋊C4×Dic1062
#202762572
#202760952
#202758572
#202749862
#202749102
#202747402
#202747212
#202746512
C53×#18502
#2595553