Finite Group Databasebeta

← #27354150

Subgroups — #27354150

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284764C211yesmaximal, characteristic
2#20283860C211yesmaximal, characteristic, complemented
3#20283738C211yesmaximal, characteristic, complemented
4#20273578C211yesmaximal, characteristic, complemented
5#20273571C211yesmaximal, characteristic, complemented
6#20273550C211yesmaximal, characteristic, complemented
7#20273543C211yesmaximal, characteristic, complemented
8#20273538C211yesmaximal, characteristic, complemented
9#20273526C211yesmaximal, characteristic, complemented
10#20273514C211yesmaximal, characteristic, complemented
11#20273509C211yesmaximal, characteristic, complemented
12#20273139C211yesmaximal, characteristic, complemented
13#20273121C211yesmaximal, characteristic, complemented
14#20273015C211yesmaximal, characteristic, complemented
15C53×#1574C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810959C411yescharacteristic
19#16810942C411yescharacteristic
20#16810924C411yescharacteristic
21#16810924C411yescharacteristic
22#16810839C411yescharacteristic
23#16810839C411yescharacteristic
24#16810776C411yescharacteristic
25#16810759C411yescharacteristic
26#16810750C2211yescharacteristic, complemented
27#16810747C2211yescharacteristic, complemented
28C4×#11719913C2211yescharacteristic
29C4×#11719913C2211yescharacteristic
30C4×D424C2211yescharacteristic
31C4×D424C2211yescharacteristic
32#16810700C2211yescharacteristic, complemented
33#16810699—21no—
34#16810692—21no—
35C8×D212—21no—
36C8×D212—21no—
37#16810688C2211yescharacteristic
38#16810688C2211yescharacteristic
39#16810688C2211yescharacteristic
40#16810688C2211yescharacteristic
41#16810688C2211yescharacteristic, complemented
42#16810683—21no—
43#16810676C2211yescharacteristic
44#16810676C2211yescharacteristic
45#16810648—21no—
46C22⋊C8×D53—21no—
47C22⋊C8×D53—21no—
48#16810353C2211yescharacteristic, complemented
49#16810353C2211yescharacteristic, complemented
50#16810350C2211yescharacteristic
51#16810350C2211yescharacteristic
52#16810350C2211yescharacteristic
53#16810350C2211yescharacteristic
54#16810350C2211yescharacteristic, complemented
55#16810350C2211yescharacteristic, complemented
56#16810345C2211yescharacteristic, complemented
57#16810345C2211yescharacteristic, complemented
58#16810344C2211yescharacteristic, complemented
59C8×#88322C2211yescharacteristic, complemented
60C53×#431C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#423C2211yescharacteristic
63C53×#335C2211yescharacteristic, complemented
64C53×#335C2211yescharacteristic, complemented
65C53×#334C2211yescharacteristic, complemented
66C53×#334C2211yescharacteristic, complemented
67C22×D212C2×C411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719913C2×C411yescharacteristic
70C2×#11719913C2×C411yescharacteristic
71C2×D424C2×C411yescharacteristic
72C2×D424C2×C411yescharacteristic
73C2×#11719912—21no—
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76C2×C8×D53—21no—
77C2×C8×D53—21no—
78C2×C8×D53—21no—
79#13133211—21no—
80#13133209C2×C411yescharacteristic
81#13133205C2×C411yescharacteristic
82#13133200—21no—
83#13133196C2×C411yescharacteristic
84#13133191C2×C411yescharacteristic
85C4×Dic106C2311yescharacteristic
86C4×D212—21no—
87C4×D212—21no—
88C4×D212C2×C411yescharacteristic
89C4×D212C2×C411yescharacteristic
90C4×D212C2311yescharacteristic
91C4×D212C2311yescharacteristic
92C4×D212C2311yescharacteristic
93C4×C22⋊D53—21no—
94#13133174—21no—
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×#11719921—21no—
97#13133151C2×C411yescharacteristic
98#13133151C2×C411yescharacteristic
99#13133149C2×C411yescharacteristic
100#13133149C2×C411yescharacteristic
101C4⋊#88322—21no—
102C4⋊#88322D411yescharacteristic
103C4⋊#88322D411yescharacteristic
104#13133146—21no—
105#13133146—21no—
106#13133142C2×C411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133139C2×C411yescharacteristic
110#13133139C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133139C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114C22⋊#88322—21no—
115C22⋊#88322D411yescharacteristic
116C22⋊#88322D411yescharacteristic
117#13133129—21no—
118#13133129—21no—
119#13133129—21no—
120#13133129—21no—
121#13133129—21no—
122#13133129—21no—
123#13133127—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C4×#88322C2311yescharacteristic
128C8×Dic53—21no—
129C2×M16×C53—21no—
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2×C411yescharacteristic
133C4⋊C8×C53C2×C411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138C22⋊C8×C53—21no—
139C22⋊C8×C53C2×C411yescharacteristic
140C22⋊C8×C53C2×C411yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C23×D53—21no—
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4111yescharacteristic
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C2×D212—21no—
151C2×D212—22no—
152C2×D212—22no—
153C2×D212C22×C411yescharacteristic
154C2×D212C22×C411yescharacteristic
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×Dic106C22×C411yescharacteristic
164M16×C53—81no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C22×C411yescharacteristic
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C22×C411yescharacteristic
173C42×C53C22×C411yescharacteristic
174C42×C53C2×D411yescharacteristic
175C42×C53#4111yescharacteristic
176C42×C53C2411yescharacteristic
177C22⋊Dic53—21no—
178#11719921—21no—
179#11719921—21no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53C22×C411yescharacteristic
188C4⋊Dic53C22×C411yescharacteristic
189C4⋊Dic53C22×C411yescharacteristic
190C4⋊Dic53C22×C411yescharacteristic
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195C4×Dic53—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C2×D411yescharacteristic
199C2×#88322C2×D411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202D424—22no—
203D424—22no—
204#11719913—22no—
205#11719913—22no—
206#11719912—81no—
207C8×D53—81no—
208C8×D53—81no—
209C8×D53—81no—
210C8×D53—81no—
211C8×D53—81no—
212C23×C53C4×D411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—24no—
216C22×D53—21no—
217C22×D53—81no—
218C22×D53—81no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—81no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—81no—
237D212—24no—
238D212—21no—
239D212—21no—
240D212—21no—
241D212—21no—
242D212#13112yes—
243D212#13112yes—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—41no—
248C4×D53—41no—
249C4×D53—41no—
250C4×D53—81no—
251C4×D53—81no—
252Dic106—21no—
253Dic106—21no—
254C8×C53—21no—
255C8×C53—21no—
256C8×C53—21no—
257C8×C53—21no—
258C8×C53—41no—
259C8×C53—41no—
260C8×C53—41no—
261C8×C53—41no—
262#88322—21no—
263#88322—21no—
264#88322—21no—
265#88322—21no—
266#88322—41no—
267#88322—41no—
268#17918—531nomaximal, Sylow
269C22×C53—21no—
270C22×C53—21no—
271C22×C53—21no—
272C22×C53C2×C4×D411yescharacteristic
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—41no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—84no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53C2×#13111yescharacteristic
288C4×C53C2×#13111yescharacteristic
289C4×C53#57611yescharacteristic
290C4×C53#57711yescharacteristic
291Dic53—41no—
292Dic53—41no—
293Dic53—41no—
294Dic53—41no—
295Dic53—41no—
296#3105—531no—
297#2993—531no—
298#2972—531no—
299#1586—531no—
300#1585—531no—
301#1582—531no—
302#1581—531no—
303#1580—531no—
304SD16⋊C8—531no—
305C8×SD16—531no—
306C8×D8—531no—
307#1574D5311yescharacteristic, complemented
308#1486—531no—
309#1482—531no—
310#1454—531no—
311C2×C53—41no—
312C2×C53#292911yescharacteristic
313C2×C53#294611yescharacteristic
314C2×C53#295711yescharacteristic
315D53—42no—
316D53—42no—
317D53—81no—
318#529—531no—
319C2×C4×D4—531no—
320#484—531no—
321#481—531no—
322#478—531no—
323#478—531no—
324#463—531no—
325#463—531no—
326#450—531no—
327#447—531no—
328#446—531no—
329C8×Q8—531no—
330C4×SD16—531no—
331C4×SD16—531no—
332C4×D8—531no—
333C4×D8—531no—
334#436—531no—
335#436—1061no—
336#435—1061no—
337C8×D4—531no—
338C8×D4—531no—
339C8×D4—531no—
340C8×D4—531no—
341C8×D4—531no—
342C8×D4—1061no—
343C8×D4—1061no—
344#433—1061no—
345#431—531no—
346#431—531no—
347#431C2×D5311yescharacteristic
348#431C2×D5311yescharacteristic
349#424—1061no—
350#423C2×D5311yescharacteristic
351C2×C22⋊C8—1061no—
352C2×C22⋊C8—1061no—
353#335C2×D5311yescharacteristic, complemented
354#335C2×D5311yescharacteristic, complemented
355#334C2×D5311yescharacteristic, complemented
356#334C2×D5311yescharacteristic, complemented
357Q8⋊C8—531no—
358Q8⋊C8—531no—
359D4⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362D4⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8—531no—
365#322—531no—
366#322—531no—
367#322—531no—
368C82—531no—
369C53#1791811yescharacteristic, complemented, Sylow
370C22×D4—531no—
371C23×C4—1061no—
372C2×SD16—531no—
373C2×SD16—531no—
374C2×D8—531no—
375C2×D8—531no—
376C2×M16—21no—
377C2×M16—1061no—
378C22×C8—1061no—
379C22×C8—1061no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383#126—1061no—
384#125—531no—
385#124—531no—
386#123—1061no—
387C22⋊Q8—531no—
388#121—531no—
389C4×Q8—531no—
390C4×D4—531no—
391C4×D4—531no—
392C4×D4—531no—
393C4×D4—531no—
394C4×D4—531no—
395C4×D4—1061no—
396C4×D4—1061no—
397C4×D4—1061no—
398#117—1061no—
399C2×C4⋊C4—531no—
400C2×C22⋊C4—1061no—
401C2×C42C22×D5311yescharacteristic
402#107—531no—
403#107—531no—
404#106—531no—
405#106—531no—
406C4⋊C8—21no—
407C4⋊C8—531no—
408C4⋊C8—531no—
409C4⋊C8—1061no—
410C4⋊C8—1061no—
411C4⋊C8—1061no—
412C4⋊C8C4×D5311yescharacteristic
413C4⋊C8C4×D5311yescharacteristic
414C4⋊C8C22×D5311yescharacteristic
415C4⋊C8C22×D5311yescharacteristic
416C4⋊C8C22×D5311yescharacteristic
417C4⋊C8C22×D5311yescharacteristic
418#103—531no—
419#103—531no—
420#102—531no—
421#102—531no—
422#102—531no—
423#102—531no—
424#102—531no—
425#102—531no—
426C22⋊C8—21no—
427C22⋊C8—531no—
428C22⋊C8—531no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8C4×D5311yescharacteristic
437C22⋊C8C4×D5311yescharacteristic
438#97—1061no—
439C4×C8—21no—
440C4×C8—531no—
441C4×C8—531no—
442C4×C8—531no—
443C4×C8—531no—
444C4×C8—1061no—
445C4×C8C22×D5311yescharacteristic
446C4×C8C22×D5311yescharacteristic
447C24—1061no—
448C2×Q8—531no—
449C2×D4—531no—
450C2×D4—531no—
451C2×D4—531no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—1061no—
455C2×D4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—4241no—
463C22×C4—4241no—
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4#1171994911yescharacteristic
467SD16—1062no—
468SD16—1062no—
469D8—1062no—
470D8—1062no—
471M16—81no—
472M16—4241no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—531no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—531no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—4241no—
484C2×C8—4241no—
485C2×C8—4241no—
486C2×C8—4241no—
487C2×C8—4241no—
488C2×C8C2×C4×D5311yescharacteristic
489C2×C8C2×C4×D5311yescharacteristic
490C2×C8C2×C4×D5311yescharacteristic
491C2×C8C2×C4×D5311yescharacteristic
492C4⋊C4—531no—
493C4⋊C4—531no—
494C4⋊C4—531no—
495C4⋊C4—531no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C22⋊C4—1061no—
502C22⋊C4—1061no—
503C22⋊C4—1061no—
504C22⋊C4—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C42—1061no—
511C42C2×C4×D5311yescharacteristic
512C42D4×D5311yescharacteristic
513C42#1171994911yescharacteristic
514C42C23×D5311yescharacteristic
515C23—1064no—
516C23—1061no—
517C23—1061no—
518C23—1061no—
519C23—4241no—
520C23—4241no—
521C23#1313317911yescharacteristic
522Q8—1061no—
523Q8—1061no—
524D4—532no—
525D4—532no—
526D4—1064no—
527D4—1061no—
528D4—1061no—
529D4—1061no—
530D4—1061no—
531D4—4241no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—4241no—
543C2×C4—4241no—
544C2×C4—4241no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4#1313317911yescharacteristic
552C2×C4#1313317911yescharacteristic
553C2×C4#1313317911yescharacteristic
554C2×C4C22×C4×D5311yescharacteristic
555C2×C4C2×D4×D5311yescharacteristic
556C2×C4C2×#1171994911yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—1061no—
566C8—1061no—
567C8—1061no—
568C8—1061no—
569C8—2121no—
570C8—2121no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—1062no—
575C22—1062no—
576C22—4244no—
577C22—2121no—
578C22—2121no—
579C22—2121no—
580C22—2121no—
581C22—2121no—
582C22—2121no—
583C22—2121no—
584C22C2×#1313317911yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—2121no—
590C4—2121no—
591C4—2121no—
592C4—2121no—
593C4—2121no—
594C4#131×D5311yescharacteristic
595C4#131×D5311yescharacteristic
596C4#1681142711yescharacteristic
597C4#1681143211yescharacteristic
598C2—41no—
599C2—4241no—
600C2—2122no—
601C2—2122no—
602C2#2028340811yescharacteristic
603C2#2028354511yescharacteristic
604C2#2028363111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1574C2
socleC22×C53C2×C4×D4
prime radical#16810759C4
cosocleC42×C53C24
hypercenter#1574D53
hypocenterC53#17918
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
2. #20283860C2semidirect
3. #20283738C2semidirect
4. #20273578C2semidirect
5. #20273571C2semidirect
6. #20273550C2semidirect
7. #20273543C2semidirect
8. #20273538C2semidirect
9. #20273526C2semidirect
10. #20273514C2semidirect
11. #20273509C2semidirect
12. #20273139C2semidirect
13. #20273121C2semidirect
14. #20273015C2semidirect
15. C53×#1574C2semidirect
26. #16810750C22semidirect
27. #16810747C22semidirect
32. #16810700C22semidirect
41. #16810688C22semidirect
48. #16810353C22semidirect
49. #16810353C22semidirect
54. #16810350C22semidirect
55. #16810350C22semidirect
56. #16810345C22semidirect
57. #16810345C22semidirect
58. #16810344C22semidirect
59. C8×#88322C22semidirect
63. C53×#335C22semidirect
64. C53×#335C22semidirect
65. C53×#334C22semidirect
66. C53×#334C22semidirect
307. #1574D53semidirect
353. #335C2×D53semidirect
354. #335C2×D53semidirect
355. #334C2×D53semidirect
356. #334C2×D53semidirect
369. C53#17918semidirect

Maximal subgroups

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

subgroupindex
#202847642
#202838602
#202837382
#202735782
#202735712
#202735502
#202735432
#202735382
#202735262
#202735142
#202735092
#202731392
#202731212
#202730152
C53×#15742
#1791853