Finite Group Databasebeta

← #27299086

Subgroups — #27299086

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810607C211yesmaximal, characteristic
2#20283160C211yesmaximal, characteristic
3#20272843C214yesmaximal, complemented
4C2×C53×#410C211yesmaximal, characteristic
5C22×#11719917—21no—
6C22×#11719917C411yescharacteristic, complemented
7C22×#11719917C411yescharacteristic, complemented
8#16810607—21no—
9#16810607—21no—
10#16810607C412yescomplemented
11#16810607C2212yes—
12C2×#13133136C412yescomplemented
13C2×#13133134—21no—
14#16810590—21no—
15#16810590—21no—
16#16810590C412yescomplemented
17#16810590C2212yes—
18#16810587—21no—
19#16810587—21no—
20C22×#41×C53C2211yescharacteristic
21C2×#117×C53—21no—
22C53×#410—21no—
23C53×#410—21no—
24C53×#410C412yes—
25C53×#410C2212yes—
26C2×#99×C53—21no—
27C2×#99×C53C412yes—
28C2×#11719917—21no—
29C2×#11719917—21no—
30C2×#11719917—21no—
31C2×#11719917—22no—
32C2×#11719917—22no—
33C2×#11719917—22no—
34C2×#11719917—41no—
35C2×#11719917—41no—
36C2×#11719917—41no—
37C2×#11719917—41no—
38C2×#11719917—41no—
39C2×#11719917—41no—
40C2×#11719917—41no—
41C2×#11719917C2×C412yescomplemented
42C22×#88322—21no—
43C22×#88322—41no—
44C22×#88322C2×C412yescomplemented
45#13133136—22no—
46#13133136C2×C414yescomplemented
47#13133134—22no—
48#13133134—22no—
49C22⋊#88322—22no—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C22⋊#88322—22no—
53C2×#41×C53—41no—
54C2×#41×C53—41no—
55C2×#41×C53—41no—
56C2×#41×C53—41no—
57C2×#41×C53C2×C411yescharacteristic
58C2×#41×C53C2×C411yescharacteristic
59C2×#41×C53C2×C411yescharacteristic
60C2×#41×C53D411yescharacteristic
61C2×#41×C53D411yescharacteristic
62C2×#41×C53D411yescharacteristic
63C2×#41×C53Q811yescharacteristic
64C2×#41×C53C2311yescharacteristic
65C22×Q8×C53C2×C411yescharacteristic
66C22×D4×C53C2×C411yescharacteristic
67C22×D4×C53D411yescharacteristic
68C22×D4×C53Q811yescharacteristic
69C23×C4×C53C2×C411yescharacteristic
70C23×C4×C53D411yescharacteristic
71C23×C4×C53D411yescharacteristic
72#117×C53—22no—
73#117×C53—22no—
74#117×C53—41no—
75#117×C53—41no—
76C2×C4⋊C4×C53—41no—
77C2×C22⋊C4×C53—22no—
78C2×C42×C53—41no—
79#99×C53—22no—
80#99×C53—22no—
81#99×C53—22no—
82#99×C53C2×C414yes—
83C24×C53C22⋊C411yescharacteristic
84C24×C53C4⋊C411yescharacteristic
85C24×C53C4⋊C411yescharacteristic
86#41×C53—41no—
87#41×C53—41no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—41no—
93#41×C53—41no—
94#41×C53—41no—
95#41×C53—41no—
96#41×C53—41no—
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—41no—
100#41×C53—41no—
101#41×C53—41no—
102C2×Q8×C53—41no—
103C2×Q8×C53—41no—
104C2×Q8×C53C4211yescharacteristic
105C2×Q8×C53C22⋊C411yescharacteristic
106C2×Q8×C53C4⋊C411yescharacteristic
107C2×Q8×C53C22×C411yescharacteristic
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—41no—
113C2×D4×C53—41no—
114C2×D4×C53C4211yescharacteristic
115C2×D4×C53C22⋊C411yescharacteristic
116C2×D4×C53C22⋊C411yescharacteristic
117C2×D4×C53C22⋊C411yescharacteristic
118C2×D4×C53C4⋊C411yescharacteristic
119C2×D4×C53C4⋊C411yescharacteristic
120C2×D4×C53C4⋊C411yescharacteristic
121C2×D4×C53C4⋊C411yescharacteristic
122C2×D4×C53C4⋊C411yescharacteristic
123C2×D4×C53C22×C411yescharacteristic
124C2×D4×C53C2×D411yescharacteristic
125C2×D4×C53C2×Q811yescharacteristic
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—42no—
139C22×C4×C53C4211yescharacteristic
140C22×C4×C53C4211yescharacteristic
141C22×C4×C53C22⋊C411yescharacteristic
142C22×C4×C53C22⋊C411yescharacteristic
143C22×C4×C53C22⋊C411yescharacteristic
144C22×C4×C53C22⋊C411yescharacteristic
145C22×C4×C53C22⋊C411yescharacteristic
146C22×C4×C53C22⋊C411yescharacteristic
147C22×C4×C53C22⋊C411yescharacteristic
148C22×C4×C53C4⋊C411yescharacteristic
149C22×C4×C53C4⋊C411yescharacteristic
150C22×C4×C53C4⋊C411yescharacteristic
151C22×C4×C53C4⋊C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C4⋊C4×C53—42no—
156C4⋊C4×C53—42no—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—42no—
159C42×C53—42no—
160C42×C53—42no—
161#11719917—24no—
162#11719917—42no—
163#11719917—42no—
164#11719917—42no—
165#11719917—42no—
166#11719917—41no—
167#11719917—41no—
168#11719917—41no—
169#11719917—41no—
170#11719917—42no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—42no—
174C2×#88322—42no—
175C2×#88322—42no—
176C2×#88322—42no—
177C2×#88322—42no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—81no—
185C23×C53—81no—
186C23×C53—81no—
187C23×C53#9511yescharacteristic
188C23×C53#9511yescharacteristic
189C23×C53#9511yescharacteristic
190C23×C53#9511yescharacteristic
191C23×C53C2×C22⋊C411yescharacteristic
192C23×C53C2×C4⋊C411yescharacteristic
193C23×C53C2×C4⋊C411yescharacteristic
194Q8×C53—41no—
195Q8×C53—41no—
196Q8×C53—41no—
197Q8×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—41no—
205D4×C53—41no—
206D4×C53—41no—
207D4×C53—41no—
208D4×C53—41no—
209D4×C53—41no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
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—21no—
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—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—44no—
241C2×C4×C53—84no—
242C2×C4×C53#9511yescharacteristic
243C2×C4×C53#9511yescharacteristic
244C2×C4×C53#9511yescharacteristic
245C2×C4×C53#9511yescharacteristic
246C2×C4×C53#9911yescharacteristic
247C2×C4×C53#9911yescharacteristic
248C2×C4×C53#9911yescharacteristic
249C2×C4×C53#9911yescharacteristic
250C2×C4×C53C2×C4211yescharacteristic
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C4⋊C411yescharacteristic
254#88322—44no—
255#88322—42no—
256#88322—42no—
257#12589—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53#34211yescharacteristic
278C22×C53#34211yescharacteristic
279C22×C53C2×#9511yescharacteristic
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—84no—
287C4×C53#34211yescharacteristic
288C4×C53#34211yescharacteristic
289C4×C53C2×#9911yescharacteristic
290C4×C53C2×#9911yescharacteristic
291C2×#413—531no—
292C2×#410D5311yescharacteristic
293#2879—531no—
294#1395—534no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—41no—
299C2×C53—41no—
300C2×C53—41no—
301C2×C53—41no—
302C2×C53—41no—
303C2×C53—41no—
304C2×C53C2×#34211yescharacteristic
305C22×#41C2×D5311yescharacteristic
306C22×M16—531no—
307C22×M16—531no—
308C22×M16—1061no—
309C2×#117—21no—
310#413—532no—
311#413—532no—
312#413—1061no—
313#413—1061no—
314C2×#101—532no—
315C2×#100—1061no—
316#410—21no—
317#410—21no—
318#410Dic5312yes—
319#410C2×D5312yes—
320C2×#99—21no—
321C2×#99Dic5312yes—
322#408—532no—
323#408—532no—
324#408—1061no—
325#408—1061no—
326#407—1061no—
327#407—1061no—
328C53#1258911yescharacteristic, complemented, Sylow
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic10611yescharacteristic
334C2×#41C4×D5311yescharacteristic
335C2×#41C4×D5311yescharacteristic
336C2×#41D21211yescharacteristic
337C2×#41C2×Dic5311yescharacteristic
338C2×#41C22⋊D5311yescharacteristic
339C2×#41C22⋊D5311yescharacteristic
340C2×#41C22×D5311yescharacteristic
341C22×Q8C4×D5311yescharacteristic
342C22×D4Dic10611yescharacteristic
343C22×D4C2×Dic5311yescharacteristic
344C22×D4C22⋊D5311yescharacteristic
345C23×C4C4×D5311yescharacteristic
346C23×C4D21211yescharacteristic
347C23×C4C22⋊D5311yescharacteristic
348C2×M16—532no—
349C2×M16—1061no—
350C2×M16—1061no—
351C2×M16—1061no—
352C2×M16—1062no—
353C2×M16—1062no—
354C2×M16—1062no—
355C2×M16—2121no—
356C2×M16—2121no—
357C2×M16—2121no—
358C2×M16—2121no—
359C2×M16—2121no—
360C2×M16—2121no—
361C2×M16—2121no—
362C22×C8—532no—
363C22×C8—1061no—
364C22×C8—2121no—
365#117—22no—
366#117—22no—
367#117—41no—
368#117—41no—
369C2×C4⋊C4—41no—
370C2×C22⋊C4—22no—
371C2×C42—41no—
372#101—534no—
373#101—1062no—
374#100—1062no—
375#100—1062no—
376#99—22no—
377#99—22no—
378#99—22no—
379#99C2×Dic5314yes—
380C22⋊C8—1062no—
381C22⋊C8—1062no—
382C22⋊C8—1062no—
383C22⋊C8—1062no—
384C24#1171991911yescharacteristic
385C24C4⋊Dic5311yescharacteristic
386C24C22⋊Dic5311yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic5311yescharacteristic
406C2×Q8#1171991911yescharacteristic
407C2×Q8#1171992111yescharacteristic
408C2×Q8C2×C4×D5311yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic5311yescharacteristic
416C2×D4#1171991911yescharacteristic
417C2×D4#1171991911yescharacteristic
418C2×D4#1171991911yescharacteristic
419C2×D4C4⋊Dic5311yescharacteristic
420C2×D4C4⋊Dic5311yescharacteristic
421C2×D4#1171992111yescharacteristic
422C2×D4C22⋊Dic5311yescharacteristic
423C2×D4C22⋊Dic5311yescharacteristic
424C2×D4C2×Dic10611yescharacteristic
425C2×D4C22×Dic5311yescharacteristic
426C2×D4C2×C22⋊D5311yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4C4×Dic5311yescharacteristic
441C22×C4C4×Dic5311yescharacteristic
442C22×C4#1171991911yescharacteristic
443C22×C4#1171991911yescharacteristic
444C22×C4#1171991911yescharacteristic
445C22×C4C4⋊Dic5311yescharacteristic
446C22×C4#1171992111yescharacteristic
447C22×C4#1171992111yescharacteristic
448C22×C4#1171992111yescharacteristic
449C22×C4#1171992111yescharacteristic
450C22×C4#1171992111yescharacteristic
451C22×C4#1171992111yescharacteristic
452C22×C4C22⋊Dic5311yescharacteristic
453C22×C4C2×C4×D5311yescharacteristic
454C22×C4C2×D21211yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456M16—1064no—
457M16—2122no—
458M16—2122no—
459M16—2122no—
460M16—2122no—
461M16—2121no—
462M16—2121no—
463M16—2121no—
464M16—2121no—
465M16—2122no—
466C2×C8—1062no—
467C2×C8—1062no—
468C2×C8—2122no—
469C2×C8—2122no—
470C2×C8—2122no—
471C2×C8—2122no—
472C2×C8—2122no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C42—42no—
478C42—42no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23#1313312411yescharacteristic
489C23#1313312411yescharacteristic
490C23#1313312411yescharacteristic
491C23#1313312411yescharacteristic
492C23C2×C22⋊Dic5311yescharacteristic
493C23C2×#1171991911yescharacteristic
494C23C2×C4⋊Dic5311yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—84no—
543C2×C4#1313312411yescharacteristic
544C2×C4#1313312411yescharacteristic
545C2×C4#1313312411yescharacteristic
546C2×C4#1313312411yescharacteristic
547C2×C4#1313313111yescharacteristic
548C2×C4#1313313111yescharacteristic
549C2×C4#1313313111yescharacteristic
550C2×C4#1313313111yescharacteristic
551C2×C4C2×C4×Dic5311yescharacteristic
552C2×C4C2×#1171992111yescharacteristic
553C2×C4C2×#1171992111yescharacteristic
554C2×C4C2×#1171991911yescharacteristic
555C8—2124no—
556C8—2122no—
557C8—2122no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1681038911yescharacteristic
578C22#1681038911yescharacteristic
579C22C2×#1313312411yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—84no—
587C4#1681038911yescharacteristic
588C4#1681038911yescharacteristic
589C4C2×#1313313111yescharacteristic
590C4C2×#1313313111yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2—41no—
600C2C2×#1681038911yescharacteristic
subgroupG/H
centerC4C2×#13133131
commutatorC2×C4×C53C2×C42
FrattiniC2×#41C22×D53
FittingC2×C53×#410C2
socleC2×C53C2×#342
prime radicalC22×#41×C53C22
cosocleC2×#41×C53C23
hypercenterC2×#410D53
hypocenterC53#12589
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #20272843C2semidirect
6. C22×#11719917C4semidirect
7. C22×#11719917C4semidirect
10. #16810607C4semidirect
12. C2×#13133136C4semidirect
16. #16810590C4semidirect
41. C2×#11719917C2×C4semidirect
44. C22×#88322C2×C4semidirect
46. #13133136C2×C4semidirect
328. C53#12589semidirect

Maximal subgroups

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

subgroupindex
C2×#168106072
#202831602
#202728432
C2×C53×#4102
#1258953