Finite Group Databasebeta

← #27434286

Subgroups — #27434286

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810931C211yesmaximal, characteristic, complemented
2#20279881C211yesmaximal, characteristic
3#20279624C211yesmaximal, characteristic
4#20276148C211yesmaximal, characteristic, complemented
5#20276143C211yesmaximal, characteristic, complemented
6#20275919C211yesmaximal, characteristic, complemented
7#20275912C211yesmaximal, characteristic, complemented
8#20275680C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275148C211yesmaximal, characteristic, complemented
11#20275148C211yesmaximal, characteristic, complemented
12#20274748C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic, complemented
14C4×#13133142C211yesmaximal, characteristic, complemented
15C2×C53×#436C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810931—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29#16810561—21no—
30#16810554—21no—
31#16810537—21no—
32#16810537—21no—
33#16810517—21no—
34#16810517—21no—
35#16810511—21no—
36#16810511—21no—
37#16810489C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C53×#436—24no—
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62#13133209—22no—
63#13133209—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67#13133196—41no—
68C4×Dic106—22no—
69C4×Dic106—22no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92#13133151—22no—
93#13133151—41no—
94#13133149—22no—
95#13133149—41no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53D411yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C2×M16×C53—22no—
113C2×M16×C53—22no—
114C2×M16×C53—22no—
115C2×M16×C53—22no—
116C2×M16×C53—41no—
117C2×M16×C53—41no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C4×D4×C53—22no—
123C4×D4×C53—22no—
124C2×C4⋊C4×C53D411yescharacteristic
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C42×C53C2311yescharacteristic
128C4⋊C8×C53D414yes—
129C22⋊C8×C53—22no—
130C22⋊C8×C53—22no—
131C4×C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—22no—
135C2×D4×C53—41no—
136C2×D4×C53—41no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106#4112yes—
168M16×C53—44no—
169M16×C53—44no—
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×C53—21no—
179C2×C8×C53C2×D412yes—
180C2×C8×C53C2×D412yes—
181C2×C8×C53#4112yes—
182C4⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C42×C53C2×D412yes—
186C42×C53#4112yes—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—81no—
194C22⋊Dic53—81no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
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⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209#11719919—42no—
210#11719919—42no—
211#11719919—42no—
212#11719919—42no—
213#11719919—42no—
214C4×Dic53—42no—
215C4×Dic53—81no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53C22×D411yescharacteristic
231D4×C53—44no—
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—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53#12112yes—
251C2×C4×C53#12112yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
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—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282Dic106—22no—
283Dic106—22no—
284Dic106—41no—
285Dic106—41no—
286C8×C53—22no—
287C8×C53—22no—
288C8×C53—22no—
289C8×C53—41no—
290#25345—531nomaximal, Sylow
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53C2×#12111yescharacteristic
306C22×C53#54611yescharacteristic
307C22×C53#54711yescharacteristic
308C22×C53#57511yescharacteristic
309C22×C53#57511yescharacteristic
310C22×C53#57711yescharacteristic
311C22×C53#57811yescharacteristic
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53#52911yescharacteristic
317C4×C53#54211yescharacteristic
318C4×C53#54611yescharacteristic
319C4×C53#54811yescharacteristic
320Dic53—42no—
321Dic53—81no—
322Dic53—81no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328C2×#479—531no—
329C2×#436D5311yescharacteristic
330#2562—531no—
331#2536—531no—
332#2077—531no—
333#2076—531no—
334#2036—531no—
335#2035—531no—
336#1991—531no—
337#1949—531no—
338#1894—531no—
339#1894—531no—
340#1809—531no—
341#1809—531no—
342C4×#103—531no—
343C2×C53—81no—
344C2×C53—81no—
345C2×C53#242711yescharacteristic
346C2×C53#305411yescharacteristic
347C2×C53#306311yescharacteristic
348C2×C53#332611yescharacteristic
349C2×C53#332711yescharacteristic
350C2×C53#335111yescharacteristic
351C2×C53#335211yescharacteristic
352C22×M16C2×D5311yescharacteristic
353C22×M16C2×D5311yescharacteristic
354C2×#125—531no—
355C2×C22⋊Q8—531no—
356C2×C22⋊Q8—531no—
357C2×C4×Q8—531no—
358C2×C4×D4C2×D5311yescharacteristic
359#479—1064no—
360#436—24no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#106—531no—
364C2×#106—531no—
365C2×C4⋊C8C2×D5311yescharacteristic
366C2×#103—531no—
367C2×#103—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×C22⋊C8C2×D5311yescharacteristic
371C2×C22⋊C8C2×D5311yescharacteristic
372C2×C4×C8C2×D5311yescharacteristic
373#400—1061no—
374#398—1061no—
375#394—1061no—
376#394—1061no—
377#388—1061no—
378#388—1061no—
379#388—1061no—
380#388—1061no—
381#382—531no—
382#380—531no—
383#380—531no—
384#380—531no—
385#380—531no—
386#380—531no—
387#380—531no—
388C4×C4⋊C4—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397C53#2534511yescharacteristic, complemented, Sylow
398C22×Q8—531no—
399C22×D4D21211yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—41no—
407C2×M16—41no—
408C22×C8C22×D5311yescharacteristic
409C22×C8C22×D5311yescharacteristic
410C22×C8C22×D5311yescharacteristic
411C22×C8C22×D5311yescharacteristic
412#125—1062no—
413#125—1062no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—2121no—
417C22⋊Q8—2121no—
418C4×Q8—1062no—
419C4×Q8—1062no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4D21211yescharacteristic
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4D21211yescharacteristic
444C2×C22⋊C4D21211yescharacteristic
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42C22×D5311yescharacteristic
448#107—1062no—
449#107—2121no—
450#106—1062no—
451#106—2121no—
452C4⋊C8C22⋊D5314yes—
453#103—1062no—
454#103—1062no—
455#103—1062no—
456#103—1062no—
457C22⋊C8—22no—
458C22⋊C8—22no—
459C4×C8—22no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469C24C2×D21211yescharacteristic
470C24C2×D21211yescharacteristic
471C2×Q8—532no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×D4—22no—
478C2×D4—41no—
479C2×D4—41no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4C2×D21211yescharacteristic
501C22×C4C2×D21211yescharacteristic
502C22×C4C2×D21211yescharacteristic
503C22×C4C2×D21211yescharacteristic
504C22×C4C23×D5311yescharacteristic
505M16—44no—
506M16—44no—
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×C8—21no—
516C2×C8#1171994712yes—
517C2×C8C2×C22⋊D5312yes—
518C2×C8C2×C22⋊D5312yes—
519C4⋊C4—22no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
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—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—4241no—
542C22⋊C4—4241no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C22⋊C4—2122no—
546C22⋊C4—2122no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C42—4241no—
550C42—2122no—
551C42#1171994512yes—
552C42C2×C22⋊D5312yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22×D21211yescharacteristic
568Q8—1062no—
569Q8—1062no—
570Q8—2121no—
571Q8—2121no—
572D4—44no—
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—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—1062no—
592C2×C4—2121no—
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×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4#1313319112yes—
616C2×C4#1313319112yes—
617C2×C4C22×C22⋊D5311yescharacteristic
618C2×C4#41×D5311yescharacteristic
619C2×C4C2×#1171994711yescharacteristic
620C2×C4C2×#1171994511yescharacteristic
621C2×C4#1313326211yescharacteristic
622C2×C4#1313326411yescharacteristic
623C8—22no—
624C8—22no—
625C8—22no—
626C8—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1313319111yescharacteristic
642C22#1681126811yescharacteristic
643C22#1681127711yescharacteristic
644C22#1681142311yescharacteristic
645C22#1681142311yescharacteristic
646C22#1681143511yescharacteristic
647C22#1681144011yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—4241no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4—2122no—
660C4#1681115511yescharacteristic
661C4#1681123811yescharacteristic
662C4#1681126711yescharacteristic
663C4#1681128011yescharacteristic
664C2—81no—
665C2—81no—
666C2#2027828611yescharacteristic
667C2#2028432411yescharacteristic
668C2#2028440611yescharacteristic
669C2#2028697011yescharacteristic
670C2#2028698511yescharacteristic
671C2#2028727711yescharacteristic
672C2#2028728911yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#436C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#436D53
hypocenterC53#25345
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×#16810931C2semidirect
4. #20276148C2semidirect
5. #20276143C2semidirect
6. #20275919C2semidirect
7. #20275912C2semidirect
8. #20275680C2semidirect
9. #20275485C2semidirect
10. #20275148C2semidirect
11. #20275148C2semidirect
12. #20274748C2semidirect
13. #20274748C2semidirect
14. C4×#13133142C2semidirect
397. C53#25345semidirect

Maximal subgroups

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

subgroupindex
C2×#168109312
#202798812
#202796242
#202761482
#202761432
#202759192
#202759122
#202756802
#202754852
#202751482
#202747482
C4×#131331422
C2×C53×#4362
#2534553