Finite Group Databasebeta

← #27449737

Subgroups — #27449737

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810959C211yesmaximal, characteristic
2C2×C4×Dic212C211yesmaximal, characteristic, complemented
3#20278240C211yesmaximal, characteristic
4#20277718C211yesmaximal, characteristic
5#20276252C211yesmaximal, characteristic, complemented
6#20276038C211yesmaximal, characteristic, complemented
7#20275919C211yesmaximal, characteristic
8#20275725C211yesmaximal, characteristic
9#20275474C211yesmaximal, characteristic, complemented
10#20275448C211yesmaximal, characteristic, complemented
11#20275357C211yesmaximal, characteristic, complemented
12#20274972C211yesmaximal, characteristic, complemented
13#20274972C211yesmaximal, characteristic, complemented
14#20274905C211yesmaximal, characteristic
15C53×#1923C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810959—24no—
23C4×Dic212—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133142C2211yescharacteristic
35#16810559—21no—
36#16810557—21no—
37#16810529C2211yescharacteristic
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810520—21no—
42#16810516—21no—
43#16810510—21no—
44#16810506—21no—
45#16810495—21no—
46#16810493C2211yescharacteristic
47#16810483C2211yescharacteristic
48C4×C4⋊Dic53C2211yescharacteristic
49C22⋊C4×Dic53—21no—
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C2×C4×C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C2×Dic212—22no—
65C2×Dic212—22no—
66C2×Dic212—22no—
67#13133196—22no—
68#13133196—22no—
69#13133196—22no—
70#13133196—41no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C4×Dic106—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
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—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98#13133151—22no—
99#13133151—22no—
100#13133151—22no—
101#13133151—22no—
102#13133151—41no—
103#13133142—22no—
104#13133142—22no—
105#13133142—22no—
106#13133142—22no—
107#13133142—41no—
108#13133142D414yes—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117C23×C4×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C4⋊C8×C53—22no—
127C22⋊C8×C53—22no—
128C22⋊C8×C53—22no—
129C4×C8×C53—22no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
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—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106C2×D412yes—
167C2×Dic106#4112yes—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
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×C53C2×D412yes—
179C2×C8×C53#4112yes—
180C22⋊C4×C53—41no—
181C22⋊C4×C53Q1614yescomplemented
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C22⋊Dic53—24no—
187C22⋊Dic53—42no—
188C22⋊Dic53—81no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—42no—
200C4⋊Dic53—42no—
201C4⋊Dic53—81no—
202C4⋊Dic53C2×D412yes—
203C4⋊Dic53#4112yes—
204#11719919—24no—
205#11719919—42no—
206#11719919—42no—
207#11719919—42no—
208#11719919—81no—
209#11719919—81no—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53—81no—
213Dic212—28no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53C2×Q1612yes—
221C23×C53C22×D411yescharacteristic
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
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×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53C2×Q1612yes—
240C2×C4×C53C2×Q1612yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
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—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—84no—
272Dic106—22no—
273Dic106—22no—
274Dic106—22no—
275Dic106—22no—
276Dic106—41no—
277Dic106—41no—
278Dic106—41no—
279Dic106—41no—
280C8×C53—22no—
281C8×C53—22no—
282C8×C53—41no—
283C8×C53—41no—
284#26559—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53#52511yescharacteristic
293C22×C53D4211yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C22×Q1611yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57811yescharacteristic
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53#52511yescharacteristic
305C4×C53#53911yescharacteristic
306C4×C53#54711yescharacteristic
307C4×C53#55911yescharacteristic
308Dic53—42no—
309Dic53—42no—
310Dic53—42no—
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#484—531no—
317C2×C4×Q16—531no—
318#2423—531no—
319#2360—531no—
320#2095—531no—
321#2056—531no—
322#2036—531no—
323#2001—531no—
324#1947—531no—
325#1942—531no—
326#1928—531no—
327#1923D5311yescharacteristic
328#1863—531no—
329#1863—531no—
330#1852—531no—
331C2×C53—42no—
332C2×C53#245511yescharacteristic
333C2×C53#310411yescharacteristic
334C2×C53#312411yescharacteristic
335C2×C53#326811yescharacteristic
336C2×C53D4×Q1611yescharacteristic
337C2×C53#333811yescharacteristic
338C2×C53#335211yescharacteristic
339C22×Q16—531no—
340C2×C22⋊Q8—531no—
341C2×C22⋊Q8—531no—
342C2×C4×Q8—531no—
343C2×C4×Q8—531no—
344C22×C4⋊C4—531no—
345#484—1064no—
346C4×Q16—1064no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#107—531no—
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×C22⋊C8C2×D5311yescharacteristic
361C2×C4×C8C2×D5311yescharacteristic
362#399—1061no—
363#398—1061no—
364#391—531no—
365#391—531no—
366#389—531no—
367#389—531no—
368#389—1061no—
369#388—1061no—
370#387—1061no—
371#387—1061no—
372#385—1061no—
373#384—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377C4×C22⋊C4C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#336C2×D5311yescharacteristic
383#336C2×D5311yescharacteristic
384C53#2655911yescharacteristic, complemented, Sylow
385C22×Q8—531no—
386C22×Q8—531no—
387C23×C4—1061no—
388C23×C4C22×D5311yescharacteristic
389C2×Q16—1062no—
390C2×Q16—1062no—
391C2×Q16—1062no—
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396C22⋊Q8—1062no—
397C22⋊Q8—1062no—
398C22⋊Q8—1062no—
399C22⋊Q8—2121no—
400C4×Q8—1062no—
401C4×Q8—1062no—
402C4×Q8—2121no—
403C4×Q8—2121no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—2121no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4D21211yescharacteristic
425C2×C22⋊C4D21211yescharacteristic
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430C2×C42C22×D5311yescharacteristic
431#107—1062no—
432#107—1062no—
433#107—1062no—
434#107—1062no—
435#107—2121no—
436C4⋊C8—22no—
437#103—534no—
438#103—1062no—
439#103—1062no—
440#103—1062no—
441#103—1062no—
442#103—2121no—
443C22⋊C8—22no—
444C22⋊C8—22no—
445C4×C8—22no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95D21211yescharacteristic
455#95D21211yescharacteristic
456C24C2×D21211yescharacteristic
457C2×Q8—532no—
458C2×Q8—532no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—4241no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4C2×D21211yescharacteristic
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4C23×D5311yescharacteristic
492Q16—1068no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8D4×D5312yes—
504C2×C8#1171994712yes—
505C4⋊C4—532no—
506C4⋊C4—532no—
507C4⋊C4—1064no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—4241no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C22⋊C4—41no—
527C22⋊C4—1064no—
528C22⋊C4—4241no—
529C22⋊C4—2122no—
530C22⋊C4Dic21214yescomplemented
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—4241no—
536C42—2122no—
537C42—2122no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23C2×Dic21212yes—
545C23C22×D21211yescharacteristic
546Q8—1062no—
547Q8—1062no—
548Q8—1062no—
549Q8—1062no—
550Q8—2121no—
551Q8—2121no—
552Q8—2121no—
553Q8—2121no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—4244no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4C2×Dic21212yes—
597C2×C4C2×Dic21212yes—
598C2×C4C2×D4×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4#41×D5311yescharacteristic
601C2×C4C2×#1171994711yescharacteristic
602C2×C4#1313326311yescharacteristic
603C2×C4#1313326411yescharacteristic
604C8—22no—
605C8—22no—
606C8—41no—
607C8—41no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22#1681113511yescharacteristic
616C22D4×D21211yescharacteristic
617C22#1681127711yescharacteristic
618C22C22×Dic21211yescharacteristic
619C22#1681142311yescharacteristic
620C22#1681142811yescharacteristic
621C22#1681144011yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4—4241no—
631C4—2122no—
632C4—2122no—
633C4—2122no—
634C4—2122no—
635C4#1681113311yescharacteristic
636C4#1681121411yescharacteristic
637C4#1681127111yescharacteristic
638C4#1681135511yescharacteristic
639C2—42no—
640C2#2027859511yescharacteristic
641C2#2028475511yescharacteristic
642C2#2028497911yescharacteristic
643C2#2028626011yescharacteristic
644C2D4×Dic21211yescharacteristic
645C2#2028710811yescharacteristic
646C2#2028728911yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1923D53
hypocenterC53#26559
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic212C2semidirect
5. #20276252C2semidirect
6. #20276038C2semidirect
9. #20275474C2semidirect
10. #20275448C2semidirect
11. #20275357C2semidirect
12. #20274972C2semidirect
13. #20274972C2semidirect
181. C22⋊C4×C53Q16semidirect
384. C53#26559semidirect
530. C22⋊C4Dic212semidirect

Maximal subgroups

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

subgroupindex
C2×#168109592
C2×C4×Dic2122
#202782402
#202777182
#202762522
#202760382
#202759192
#202757252
#202754742
#202754482
#202753572
#202749722
#202749052
C53×#19232
#2655953