Finite Group Databasebeta

← #27406576

Subgroups — #27406576

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810842C211yesmaximal, characteristic
2C2×#16810727C211yesmaximal, characteristic, complemented
3C2×#16810714C211yesmaximal, characteristic
4#20277790C211yesmaximal, characteristic
5#20277627C211yesmaximal, characteristic, complemented
6#20275476C211yesmaximal, characteristic, complemented
7#20275409C211yesmaximal, characteristic
8#20275388C211yesmaximal, characteristic, complemented
9#20275067C211yesmaximal, characteristic
10#20275053C211yesmaximal, characteristic, complemented
11#20274981C211yesmaximal, characteristic, complemented
12#20274974C211yesmaximal, characteristic
13#20274587C211yesmaximal, characteristic, complemented
14#20274546C211yesmaximal, characteristic, complemented
15C53×#1959C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×#13133182C2211yescharacteristic
21C2×#13133182C2211yescharacteristic
22C2×D4×Dic53C2211yescharacteristic
23#16810842C418yescomplemented
24#16810727—24no—
25#16810714—24no—
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133127C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35#16810530C2211yescharacteristic
36#16810526C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810518—21no—
40#16810518—21no—
41#16810516—21no—
42#16810516—21no—
43#16810498—21no—
44#16810492—21no—
45C4⋊C4×Dic53—21no—
46C4×C4⋊Dic53C2211yescharacteristic
47C4×C4⋊Dic53C2211yescharacteristic
48C22⋊C4×Dic53—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810377C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×#124×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C53×#384C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C2×#11719925—22no—
65C2×#11719925—22no—
66C2×#11719925C2×C414yescomplemented
67C2×#11719923—22no—
68C2×#11719923—22no—
69C2×#11719923C2×C414yes—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72Q8×Dic53—22no—
73Q8×Dic53—41no—
74C4×Dic106—22no—
75C4×Dic106C2×C414yescomplemented
76#13133182—22no—
77#13133182—22no—
78#13133182—41no—
79#13133182—41no—
80D4×Dic53—22no—
81D4×Dic53—41no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53D411yescharacteristic
84C2×C4⋊Dic53D411yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×C22⋊Dic53—21no—
94C2×C22⋊Dic53—21no—
95C2×C22⋊Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103#13133148—22no—
104#13133148—22no—
105C4⋊#88322C2×C414yescomplemented
106#13133142—22no—
107#13133142D414yes—
108#13133141C2×C414yescomplemented
109#13133138C2×C414yes—
110#13133127—22no—
111C8×Dic53—22no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124D411yescharacteristic
118#13133124D411yescharacteristic
119C22×Q8×C53C2311yescharacteristic
120C22×D4×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123#124×C53—22no—
124#124×C53C2×C414yes—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C22⋊C4×C53—21no—
130C2×C22⋊C4×C53—21no—
131C2×C42×C53C2311yescharacteristic
132#103×C53—22no—
133#103×C53—41no—
134#102×C53—22no—
135#102×C53—41no—
136#97×C53—22no—
137C24×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53C22×C412yes—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53C22×C412yes—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—81no—
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
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×C53C2×D412yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53—81no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—24no—
187C42×C53C22×C412yes—
188C42×C53#4112yes—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—81no—
192#11719925—28no—
193#11719923—28no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—22no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—41no—
200C4⋊Dic53—42no—
201C4⋊Dic53C22×C412yes—
202C4⋊Dic53C2×D412yes—
203#11719919—24no—
204#11719919—42no—
205#11719919—42no—
206#11719919—81no—
207#11719919—81no—
208C4×Dic53—24no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53—81no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322C22×C412yes—
222C2×#88322C22×C412yes—
223C23×C53—22no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C22×D411yescharacteristic
231Q8×C53—22no—
232Q8×C53—22no—
233Q8×C53—41no—
234Q8×C53—41no—
235D4×C53—22no—
236D4×C53—22no—
237D4×C53—41no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
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—41no—
251C2×C4×C53—41no—
252C2×C4×C53C4×D412yes—
253C2×C4×C53C4×D412yes—
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C532−1+411yescharacteristic
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—21no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—84no—
284C2×Dic53#13712yes—
285Dic106—22no—
286Dic106—22no—
287Dic106—22no—
288Dic106C4×D414yes—
289C8×C53—22no—
290C8×C53—41no—
291#88322—22no—
292#88322—22no—
293#23013—531nomaximal, Sylow
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53C2×C4×D411yescharacteristic
302C22×C53D4211yescharacteristic
303C22×C53#54811yescharacteristic
304C22×C53C2×#13711yescharacteristic
305C22×C53#57511yescharacteristic
306C22×C53#57711yescharacteristic
307C22×C53#57711yescharacteristic
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—42no—
311C4×C53—81no—
312C4×C53—81no—
313C4×C53C2×C4×D411yescharacteristic
314C4×C53#52011yescharacteristic
315C4×C53#54711yescharacteristic
316C4×C53#55311yescharacteristic
317Dic53—24no—
318Dic53—22no—
319Dic53—42no—
320Dic53—41no—
321Dic53—42no—
322Dic53—41no—
323Dic53—41no—
324Dic53—81no—
325C2×#464—531no—
326C2×#441—531no—
327C2×C4×SD16—531no—
328#2367—531no—
329Q8×C4⋊C4—531no—
330#1959D5311yescharacteristic
331#1948—531no—
332#1936—531no—
333#1933—531no—
334#1880—531no—
335#1878—531no—
336#1865—531no—
337#1864—531no—
338#1771—531no—
339#1763—531no—
340C2×C53—82no—
341C2×C53#236111yescharacteristic
342C2×C53C4×#13711yescharacteristic
343C2×C53#295211yescharacteristic
344C2×C53#326611yescharacteristic
345C2×C53#327111yescharacteristic
346C2×C53#336911yescharacteristic
347C2×C53#337111yescharacteristic
348C22×Q16—531no—
349C22×SD16—531no—
350C2×#124C2×D5311yescharacteristic
351C2×C4×Q8—531no—
352C2×C4×Q8—531no—
353C2×C4×Q8—531no—
354C2×C4×Q8—531no—
355C2×C4×D4—531no—
356#464—538no—
357#441—1064no—
358C4×SD16—1064no—
359C2×#106—531no—
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#103C2×D5311yescharacteristic
366C2×#103C2×D5311yescharacteristic
367C2×#102—531no—
368C2×#102C2×D5311yescharacteristic
369C2×#102C2×D5311yescharacteristic
370C2×#97—531no—
371C2×#97C2×D5311yescharacteristic
372C2×C4×C8—531no—
373#391—531no—
374#390—531no—
375#389—531no—
376#389—531no—
377#389—1061no—
378#389—1061no—
379#388—1061no—
380#388—1061no—
381#385—1061no—
382#384—1061no—
383#384C2×D5311yescharacteristic
384C4×C4⋊C4—531no—
385C4×C4⋊C4—531no—
386C4×C4⋊C4—1061no—
387C4×C22⋊C4—1061no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394C53#2301311yescharacteristic, complemented, Sylow
395C22×Q8—531no—
396C22×Q8C22×D5311yescharacteristic
397C22×D4C22×D5311yescharacteristic
398C23×C4—1061no—
399C2×Q16—534no—
400C2×Q16—1062no—
401C2×Q16—1062no—
402C2×SD16—534no—
403C2×SD16—1062no—
404C2×SD16—1062no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409#124—22no—
410#124C4×D5314yes—
411C4×Q8—534no—
412C4×Q8—1062no—
413C4×Q8—1062no—
414C4×Q8—1062no—
415C4×Q8—1062no—
416C4×Q8—2121no—
417C4×Q8—2121no—
418C4×Q8—2121no—
419C4×D4—1062no—
420C4×D4—2121no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449#106—1062no—
450#106—1062no—
451C4⋊C8—534no—
452#103—22no—
453#103—41no—
454#103—534no—
455#103—534no—
456#103—1062no—
457#102—22no—
458#102—41no—
459#102—534no—
460#97—22no—
461#97—1062no—
462C4×C8—1062no—
463#95—531no—
464#95—531no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—532no—
477C2×Q8—532no—
478C2×Q8—532no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—1061no—
482C2×Q8C2×C4×D5312yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D5312yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—4241no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4D4×D5311yescharacteristic
504C22×C4D4×D5311yescharacteristic
505C22×C4C23×D5311yescharacteristic
506Q16—1068no—
507SD16—1068no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—532no—
514C2×C8—532no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8D4×D5312yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—81no—
526C4⋊C4—532no—
527C4⋊C4—532no—
528C4⋊C4—1062no—
529C4⋊C4—1064no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—4241no—
535C4⋊C4—4241no—
536C4⋊C4—2121no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4—2122no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—4241no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C42—1064no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—4241no—
551C42—2122no—
552C42—2122no—
553C42—2122no—
554C42C2×C4×D5312yes—
555C42#1171994712yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D5311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—534no—
569Q8—1062no—
570Q8—1062no—
571Q8—1062no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—532no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1062no—
606C2×C4—4244no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4#1313317712yes—
615C2×C4#1313317712yes—
616C2×C4C22×C4×D5311yescharacteristic
617C2×C4C2×D4×D5311yescharacteristic
618C2×C4#41×D5311yescharacteristic
619C2×C4C2×#1171994711yescharacteristic
620C2×C4#1313326211yescharacteristic
621C2×C4#1313326411yescharacteristic
622C8—22no—
623C8—41no—
624C8—1062no—
625C8—1062no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#1313317711yescharacteristic
634C22#1681125811yescharacteristic
635C22#1681128111yescharacteristic
636C22#137×D5311yescharacteristic
637C22#1681141911yescharacteristic
638C22#1681143211yescharacteristic
639C22#1681143311yescharacteristic
640C4—22no—
641C4—22no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—1062no—
646C4—1064no—
647C4—4241no—
648C4—2121no—
649C4—2122no—
650C4—2121no—
651C4—2121no—
652C4—2122no—
653C4C4×D4×D5311yescharacteristic
654C4#1681110211yescharacteristic
655C4#1681127311yescharacteristic
656C4#1681132011yescharacteristic
657C2—82no—
658C2#2027773611yescharacteristic
659C2#2028353911yescharacteristic
660C2#2028359411yescharacteristic
661C2#2028622411yescharacteristic
662C2#2028629911yescharacteristic
663C2#2028745211yescharacteristic
664C2#2028748211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1959C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1959D53
hypocenterC53#23013
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16810727C2semidirect
5. #20277627C2semidirect
6. #20275476C2semidirect
8. #20275388C2semidirect
10. #20275053C2semidirect
11. #20274981C2semidirect
13. #20274587C2semidirect
14. #20274546C2semidirect
23. #16810842C4semidirect
66. C2×#11719925C2×C4semidirect
75. C4×Dic106C2×C4semidirect
105. C4⋊#88322C2×C4semidirect
108. #13133141C2×C4semidirect
394. C53#23013semidirect

Maximal subgroups

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

subgroupindex
C2×#168108422
C2×#168107272
C2×#168107142
#202777902
#202776272
#202754762
#202754092
#202753882
#202750672
#202750532
#202749812
#202749742
#202745872
#202745462
C53×#19592
#2301353