Finite Group Databasebeta

← #27442949

Subgroups — #27442949

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811032C211yesmaximal, characteristic, complemented
2C2×#16810665C211yesmaximal, characteristic
3#20278428C211yesmaximal, characteristic
4#20278417C211yesmaximal, characteristic
5#20276193C211yesmaximal, characteristic, complemented
6#20276188C211yesmaximal, characteristic, complemented
7#20276054C211yesmaximal, characteristic, complemented
8#20276049C211yesmaximal, characteristic, complemented
9#20275453C211yesmaximal, characteristic, complemented
10#20275448C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274910C211yesmaximal, characteristic
13#20274872C211yesmaximal, characteristic, complemented
14#20274748C211yesmaximal, characteristic
15C53×#1753C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133194—21no—
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16811032—24no—
22#16810665—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133149C2211yescharacteristic
31C2×#13133149C2211yescharacteristic
32C2×#13133149C2211yescharacteristic
33#16810559—21no—
34#16810555—21no—
35#16810546—21no—
36#16810544—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810520—21no—
42#16810506—21no—
43#16810498—21no—
44#16810493C2211yescharacteristic
45#16810493C2211yescharacteristic
46#16810483C2211yescharacteristic
47#16810483C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C2×C4×C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106—21no—
62C23×Dic53—21no—
63#13133217—22no—
64#13133217—41no—
65#13133209—22no—
66#13133209—41no—
67#13133194—44no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×C22⋊Dic53—21no—
93C2×C22⋊Dic53—21no—
94C2×C22⋊Dic53—21no—
95C2×C4×Dic53—21no—
96#13133151—22no—
97#13133151—22no—
98#13133151—22no—
99#13133151D414yes—
100#13133149—22no—
101#13133149—22no—
102#13133149—22no—
103#13133149—22no—
104#13133149—41no—
105#13133149—41no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C23×C4×C53C2311yescharacteristic
113C2×M16×C53—22no—
114C2×M16×C53—22no—
115C2×M16×C53Q814yescomplemented
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C22⋊C8×C53Q814yescomplemented
125#97×C53—22no—
126C4×C8×C53—22no—
127#95×C53D411yescharacteristic
128#95×C53D411yescharacteristic
129C24×C53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×Q812yes—
140C22×C4×C53C2×Q812yes—
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106—81no—
156M16×C53—28no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C2×D412yes—
164C2×C8×C53C2×Q812yes—
165C2×C8×C53C2×Q812yes—
166C2×C8×C53C2×Q812yes—
167C2×C8×C53C2×Q812yes—
168C2×C8×C53#4112yes—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—81no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—42no—
196C4⋊Dic53—81no—
197C4⋊Dic53—81no—
198C4⋊Dic53C2×D412yes—
199C4⋊Dic53C2×D412yes—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203#11719919—42no—
204#11719919—42no—
205#11719919—81no—
206C4×Dic53—42no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C4⋊Q812yes—
214C23×C53C22×D411yescharacteristic
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53C4⋊Q812yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53#13712yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C22×Q811yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
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—84no—
265Dic106—84no—
266C8×C53—22no—
267C8×C53—22no—
268C8×C53—22no—
269C8×C53C22⋊Q814yescomplemented
270#26078—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×C4⋊Q811yescharacteristic
279C22×C53D4211yescharacteristic
280C22×C53#54711yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53C2×C22⋊Q811yescharacteristic
291C4×C53#54411yescharacteristic
292C4×C53D4×Q811yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—42no—
295Dic53—42no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302C2×#501—531no—
303C2×#428—531no—
304#2441—531no—
305#2440—531no—
306#2084—531no—
307#2083—531no—
308#2059—531no—
309#2058—531no—
310#1943—531no—
311#1942—531no—
312#1854—531no—
313#1853—531no—
314#1844—531no—
315#1809—531no—
316#1753D5311yescharacteristic
317C2×C53—42no—
318C2×C53#244011yescharacteristic
319C2×C53#316311yescharacteristic
320C2×C53#316511yescharacteristic
321C2×C53#327211yescharacteristic
322C2×C53#327811yescharacteristic
323C2×C53#334411yescharacteristic
324C2×C53#334511yescharacteristic
325C22×M16C2×D5311yescharacteristic
326C2×C4⋊Q8—531no—
327C2×#125—531no—
328C2×C22⋊Q8—1061no—
329C2×#117—531no—
330C22×C4⋊C4—531no—
331#501—1064no—
332#428—1064no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×C22⋊C8C2×D5311yescharacteristic
344C2×#97C2×D5311yescharacteristic
345C2×C4×C8C2×D5311yescharacteristic
346#399—1061no—
347#398—1061no—
348#396—1061no—
349#395—1061no—
350#389—531no—
351#389—531no—
352#389—531no—
353#389—531no—
354#389—1061no—
355#387—1061no—
356#385—1061no—
357#384—531no—
358#384—531no—
359#380—531no—
360#380—531no—
361C4×C22⋊C4C2×D5311yescharacteristic
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336C2×D5311yescharacteristic
369#336C2×D5311yescharacteristic
370C53#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1061no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic10614yescomplemented
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C4⋊Q8—1062no—
382C4⋊Q8—2121no—
383#125—1062no—
384#125—2121no—
385C22⋊Q8—2124no—
386#117—1062no—
387#117—2121no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—2121no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4D21211yescharacteristic
414C2×C22⋊C4D21211yescharacteristic
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417C2×C42C22×D5311yescharacteristic
418#107—534no—
419#107—1062no—
420#107—1062no—
421#107—1062no—
422#106—1062no—
423#106—1062no—
424#106—1062no—
425#106—1062no—
426#106—2121no—
427#106—2121no—
428C22⋊C8Dic10614yescomplemented
429#97—22no—
430C4×C8—22no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95D21211yescharacteristic
438#95D21211yescharacteristic
439C24C2×D21211yescharacteristic
440C2×Q8—4241no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4C2×Dic10612yes—
459C22×C4C2×Dic10612yes—
460C22×C4C2×D21211yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21211yescharacteristic
465C22×C4C23×D5311yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×Dic10612yes—
474C2×C8C2×Dic10612yes—
475C2×C8C2×Dic10612yes—
476C2×C8C2×Dic10612yes—
477C2×C8D4×D5312yes—
478C2×C8#1171994712yes—
479C4⋊C4—532no—
480C4⋊C4—532no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—4241no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2122no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1313321712yes—
524C23C22×D21211yescharacteristic
525Q8—4244no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—4244no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4#1313321712yes—
568C2×C4#1313324512yes—
569C2×C4#1313325012yes—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C22×Dic10611yescharacteristic
572C2×C4C22×Dic10611yescharacteristic
573C2×C4C2×#1171994711yescharacteristic
574C2×C4#1313326311yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1313319414yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313321711yescharacteristic
588C22D4×D21211yescharacteristic
589C22#1681127711yescharacteristic
590C22C2×#1313324511yescharacteristic
591C22C2×#1313325011yescharacteristic
592C22#1681143511yescharacteristic
593C22#1681143511yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—4241no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—2122no—
606C4—2122no—
607C4C2×#1313319411yescharacteristic
608C4#1681125111yescharacteristic
609C4D4×Dic10611yescharacteristic
610C4#1681133011yescharacteristic
611C2—42no—
612C2#2027841711yescharacteristic
613C2#2028524611yescharacteristic
614C2#2028526111yescharacteristic
615C2#2028631711yescharacteristic
616C2#2028638311yescharacteristic
617C2#2028717211yescharacteristic
618C2#2028718711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1753D53
hypocenterC53#26078
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16811032C2semidirect
5. #20276193C2semidirect
6. #20276188C2semidirect
7. #20276054C2semidirect
8. #20276049C2semidirect
9. #20275453C2semidirect
10. #20275448C2semidirect
13. #20274872C2semidirect
115. C2×M16×C53Q8semidirect
124. C22⋊C8×C53Q8semidirect
269. C8×C53C22⋊Q8semidirect
370. C53#26078semidirect
376. C2×M16Dic106semidirect
428. C22⋊C8Dic106semidirect
579. C8#13133194semidirect

Maximal subgroups

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

subgroupindex
C2×#168110322
C2×#168106652
#202784282
#202784172
#202761932
#202761882
#202760542
#202760492
#202754532
#202754482
#202749162
#202749102
#202748722
#202747482
C53×#17532
#2607853