Finite Group Databasebeta

← #27412816

Subgroups — #27412816

Order 13568.

960 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810672C211yesmaximal, characteristic, complemented
2C2×#16810671C211yesmaximal, characteristic, complemented
3C2×#16810605C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5C2×#16810605C211yesmaximal, characteristic
6#20276115C214yesmaximal, complemented
7#20276112C214yesmaximal, complemented
8C2×#16810381C211yesmaximal, characteristic, complemented
9C2×C53×#430C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16810672—22no—
15#16810672C2214yes—
16#16810671—22no—
17#16810671C2214yes—
18C2×#13133153—22no—
19C2×#13133152—22no—
20#16810605—22no—
21#16810605—22no—
22#16810605—22no—
23#16810605—22no—
24#16810605—22no—
25#16810605—22no—
26C2×#13133135C2212yes—
27C2×#13133135C2212yes—
28C2×#13133135C2212yes—
29C2×#13133133C2212yes—
30C2×#13133133C2212yes—
31C2×#13133133C2212yes—
32#16810381C2214yes—
33C22×M16×C53C2211yescharacteristic
34C22×M16×C53C2211yescharacteristic
35C22×M16×C53C2211yescharacteristic
36C53×#430—22no—
37C53×#430C2214yes—
38C2×#108×C53—22no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945—21no—
46C2×#11719945—21no—
47C2×#11719945—21no—
48C2×#11719945—21no—
49C2×#11719945—21no—
50C2×#11719945—21no—
51C22×Dic106C2311yescharacteristic
52C22×D212C2311yescharacteristic
53C22×C22⋊D53Q812yes—
54C22×C4×D53Q812yes—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917—21no—
62C2×#11719917—21no—
63C2×#11719917—21no—
64C2×#11719917—21no—
65C2×#11719917—21no—
66C2×#11719917—21no—
67C2×#11719917—21no—
68C2×#11719917D412yes—
69C2×#11719917D412yes—
70C2×#11719917C2312yes—
71C2×#11719917C2312yes—
72C2×#11719917C2312yes—
73C22×#88322—21no—
74C22×#88322—21no—
75C22×#88322—21no—
76#13133153—28no—
77#13133152—28no—
78#13133135—24no—
79#13133135—24no—
80#13133135—24no—
81#13133133—24no—
82#13133133—24no—
83#13133133—24no—
84C23×C4×C53C2311yescharacteristic
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53—21no—
92C2×M16×C53—21no—
93C2×M16×C53—21no—
94C2×M16×C53—21no—
95C2×M16×C53—21no—
96C2×M16×C53D412yes—
97C2×M16×C53D412yes—
98C2×M16×C53D412yes—
99C2×M16×C53D412yes—
100C2×M16×C53C2312yes—
101C2×M16×C53C2312yes—
102C2×M16×C53C2312yes—
103C22×C8×C53—21no—
104C22×C8×C53—21no—
105C22×C8×C53—21no—
106#108×C53—28no—
107C24×C53C2×Q811yescharacteristic
108C23×D53C2×Q812yes—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×Q811yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C22×Dic53C2×Q812yes—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141#11719945—41no—
142#11719945—41no—
143#11719945—41no—
144#11719945—41no—
145#11719945—41no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164M16×C53—22no—
165M16×C53—22no—
166M16×C53—22no—
167M16×C53—22no—
168M16×C53—22no—
169M16×C53—22no—
170M16×C53—22no—
171M16×C53—22no—
172M16×C53—41no—
173M16×C53—41no—
174M16×C53C2×D414yes—
175M16×C53C2×D414yes—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—22no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—22no—
182C2×C8×C53—22no—
183C2×C8×C53—41no—
184C2×C8×C53—41no—
185#11719917—22no—
186#11719917—22no—
187#11719917—22no—
188#11719917—22no—
189#11719917—22no—
190#11719917—22no—
191#11719917—22no—
192#11719917—41no—
193#11719917—41no—
194#11719917—41no—
195#11719917—41no—
196#11719917C2×D414yes—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—41no—
203C2×#88322—41no—
204C2×#88322—41no—
205C2×#88322—41no—
206C23×C53—81no—
207C23×C53C22⋊Q811yescharacteristic
208C23×C53C22⋊Q811yescharacteristic
209C23×C53C22⋊Q811yescharacteristic
210C23×C53C22⋊Q811yescharacteristic
211C23×C53C22⋊Q811yescharacteristic
212C23×C53C22⋊Q811yescharacteristic
213C23×C53C22×Q811yescharacteristic
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—82no—
218C2×C4×C53#12111yescharacteristic
219C2×C4×C53#12111yescharacteristic
220C2×C4×C53#12111yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53C22⋊Q811yescharacteristic
231C2×C4×C53C22⋊Q811yescharacteristic
232C2×C4×C53C22⋊Q811yescharacteristic
233C2×C4×C53C22⋊Q811yescharacteristic
234C2×C4×C53C22⋊Q811yescharacteristic
235C2×C4×C53C22⋊Q811yescharacteristic
236C2×C4×C53#12611yescharacteristic
237C2×C4×C53#12611yescharacteristic
238C2×C4×C53#12611yescharacteristic
239C2×C4×C53#12611yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C22⋊D53—42no—
247C22⋊D53—42no—
248C22⋊D53—42no—
249C22⋊D53—42no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253D212—41no—
254D212—41no—
255D212—41no—
256D212—41no—
257C4×D53—42no—
258C4×D53—42no—
259C4×D53—42no—
260C4×D53—42no—
261Dic106—41no—
262Dic106—41no—
263Dic106—41no—
264Dic106—41no—
265C8×C53—24no—
266C8×C53—24no—
267C8×C53—42no—
268#88322—24no—
269#88322—42no—
270#88322—42no—
271#23622—531nomaximal, Sylow
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53#39611yescharacteristic
280C22×C53#39611yescharacteristic
281C22×C53#39611yescharacteristic
282C22×C53#39611yescharacteristic
283C22×C53C2×C22⋊Q811yescharacteristic
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53C2×C22⋊Q811yescharacteristic
286C2×D53—42no—
287C2×D53—82no—
288C2×D53—82no—
289C2×D53—82no—
290C4×C53#39611yescharacteristic
291C4×C53#39611yescharacteristic
292C4×C53#39611yescharacteristic
293C4×C53#39611yescharacteristic
294C4×C53C2×#12111yescharacteristic
295C4×C53C2×#12111yescharacteristic
296C4×C53C2×#12111yescharacteristic
297C4×C53C2×#12611yescharacteristic
298Dic53—42no—
299C2×#430—531no—
300C2×#430—531no—
301C2×#430D5311yescharacteristic, complemented
302C2×#413—531no—
303C2×#413—531no—
304C2×#413—531no—
305#2070—534no—
306#2069—534no—
307C2×#338—531no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53C2×#39611yescharacteristic
316D53—82no—
317C22×#41—531no—
318C22×M16—531no—
319C22×M16—531no—
320C22×M16—531no—
321C22×M16C2×D5311yescharacteristic
322C22×M16C2×D5311yescharacteristic
323C22×M16C2×D5311yescharacteristic
324#430—22no—
325#430—534no—
326#430—534no—
327#430—1062no—
328#430—1062no—
329#430C2×D5314yes—
330C2×#108—22no—
331C2×#108—1062no—
332C2×#108—1062no—
333#413—1062no—
334#413—1062no—
335#413—1062no—
336#413—1062no—
337#413—1062no—
338#413—1062no—
339C2×#101—532no—
340C2×#101—532no—
341C2×#101—532no—
342C2×#100—532no—
343C2×#100—532no—
344C2×#100—532no—
345#338—534no—
346C53#2362211yescharacteristic, complemented, Sylow
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C2×#41—1061no—
353C2×#41—1061no—
354C2×#41—1061no—
355C2×#41—1061no—
356C2×#41—1061no—
357C2×#41—1061no—
358C2×#41—1061no—
359C22×Q8—531no—
360C22×D4—532no—
361C22×D4—531no—
362C23×C4—532no—
363C23×C4C22×D5311yescharacteristic
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—532no—
376C2×M16—532no—
377C2×M16—532no—
378C2×M16—532no—
379C2×M16—532no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16—1061no—
392C2×M16—1061no—
393C2×M16D21212yes—
394C2×M16D21212yes—
395C2×M16C22⋊D5312yes—
396C2×M16C22⋊D5312yes—
397C2×M16C22×D5312yes—
398C2×M16C22×D5312yes—
399C2×M16C22×D5312yes—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—21no—
403C22×C8—1061no—
404C22×C8—1061no—
405C22×C8—1061no—
406#108—28no—
407#108—1068no—
408#108—1068no—
409#101—1064no—
410#101—1064no—
411#101—1064no—
412#100—1064no—
413#100—1064no—
414#100—1064no—
415C24—532no—
416C24Q8×D5311yescharacteristic
417#41—2121no—
418#41—2121no—
419#41—2121no—
420#41—2121no—
421#41—2121no—
422#41—2121no—
423#41—2121no—
424#41—2121no—
425#41—2121no—
426#41—2121no—
427#41—2121no—
428#41—2121no—
429#41—2121no—
430#41—2121no—
431#41—2121no—
432#41—2121no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—1062no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—1062no—
443C2×D4—1062no—
444C2×D4—1062no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C22×C4—532no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4C2×D21211yescharacteristic
459C22×C4C2×D21211yescharacteristic
460C22×C4#1171994511yescharacteristic
461C22×C4#1171994511yescharacteristic
462C22×C4D4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4#1171994711yescharacteristic
465C22×C4#1171994711yescharacteristic
466C22×C4Q8×D5311yescharacteristic
467C22×C4#1171994911yescharacteristic
468C22×C4#1171994911yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C23×D5311yescharacteristic
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—534no—
483M16—1062no—
484M16—1062no—
485M16—1062no—
486M16—1062no—
487M16—1062no—
488M16—1062no—
489M16—1062no—
490M16—2121no—
491M16—2121no—
492M16—2121no—
493M16—2121no—
494M16C2×D21214yes—
495M16C2×C22⋊D5314yes—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C2×C8—1062no—
509C2×C8—1062no—
510C2×C8—2121no—
511C2×C8—2121no—
512C2×C8—2121no—
513C2×C8—2121no—
514C23—81no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—4242no—
519C23#1313319311yescharacteristic
520C23#1313319311yescharacteristic
521C23#1313319511yescharacteristic
522C23#1313319511yescharacteristic
523C23#1313319711yescharacteristic
524C23#1313319711yescharacteristic
525C23C2×Q8×D5311yescharacteristic
526Q8—2121no—
527Q8—2121no—
528Q8—2121no—
529Q8—2121no—
530D4—2122no—
531D4—2122no—
532D4—2122no—
533D4—2122no—
534D4—2121no—
535D4—2121no—
536D4—2121no—
537D4—2121no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4#1313318911yescharacteristic
546C2×C4#1313318911yescharacteristic
547C2×C4#1313318911yescharacteristic
548C2×C4#1313318911yescharacteristic
549C2×C4#1313319111yescharacteristic
550C2×C4#1313319111yescharacteristic
551C2×C4#1313319111yescharacteristic
552C2×C4#1313319111yescharacteristic
553C2×C4#1313319211yescharacteristic
554C2×C4#1313319211yescharacteristic
555C2×C4#1313319211yescharacteristic
556C2×C4#1313319211yescharacteristic
557C2×C4#1313319311yescharacteristic
558C2×C4#1313319311yescharacteristic
559C2×C4#1313319511yescharacteristic
560C2×C4#1313319511yescharacteristic
561C2×C4#1313319711yescharacteristic
562C2×C4#1313319711yescharacteristic
563C2×C4#1313321211yescharacteristic
564C2×C4#1313321211yescharacteristic
565C2×C4#1313321211yescharacteristic
566C2×C4#1313321211yescharacteristic
567C2×C4C2×D4×D5311yescharacteristic
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4C22×D21211yescharacteristic
570C2×C4C2×#1171994711yescharacteristic
571C2×C4C2×#1171994511yescharacteristic
572C2×C4C2×#1171994911yescharacteristic
573C8—24no—
574C8—24no—
575C8—42no—
576C8—1064no—
577C8—2122no—
578C8—2122no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—4242no—
587C22—4242no—
588C22—4242no—
589C22—2122no—
590C22#1681054511yescharacteristic
591C22#1681054511yescharacteristic
592C22#1681054511yescharacteristic
593C22#1681054511yescharacteristic
594C22C2×#1313319311yescharacteristic
595C22C2×#1313319511yescharacteristic
596C22C2×#1313319711yescharacteristic
597C4—2122no—
598C4#1681054511yescharacteristic
599C4#1681054511yescharacteristic
600C4#1681054511yescharacteristic
601C4#1681054511yescharacteristic
602C4C2×#1313318911yescharacteristic
603C4C2×#1313319111yescharacteristic
604C4C2×#1313319211yescharacteristic
605C4C2×#1313321211yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—4242no—
614C2C2×#1681054511yescharacteristic
subgroupG/H
centerC4#16810545
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#430C2
socleC2×C53C2×#396
prime radicalC22×#11719945C22
hypercenterC2×#430D53
hypocenterC53#23622
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
1. C2×#16810672C2semidirect
2. C2×#16810671C2semidirect
6. #20276115C2semidirect
7. #20276112C2semidirect
8. C2×#16810381C2semidirect
9. C2×C53×#430C2semidirect
301. C2×#430D53semidirect
346. C53#23622semidirect

Maximal subgroups

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

subgroupindex
C2×#168106722
C2×#168106712
C2×#168106052
#202761152
#202761122
C2×#168103812
C2×C53×#4302
#2362253