Finite Group Databasebeta

← #27361429

Subgroups — #27361429

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C211yesmaximal, characteristic, complemented
2C2×#16810975C211yesmaximal, characteristic, complemented
3#20284530C211yesmaximal, characteristic, complemented
4#20273880C214yesmaximal, complemented
5#20273866C214yesmaximal, complemented
6#20273387C211yesmaximal, characteristic, complemented
7#20273166C211yesmaximal, characteristic, complemented
8#20273097C211yesmaximal, characteristic, complemented
9C53×#3499C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16811019—21no—
12#16810975—22no—
13#16810975C2214yes—
14#16810969—21no—
15#16810939—22no—
16#16810918C2212yes—
17#16810913C2212yes—
18#16810913C2214yes—
19#16810908—21no—
20#16810903—21no—
21#16810676C2211yescharacteristic
22#16810666—21no—
23#16810647C2211yescharacteristic
24#16810624—21no—
25C2×#13133140—22no—
26C2×C4×#88322C2211yescharacteristic
27#16810369C2214yes—
28Q8⋊#88322C2212yes—
29D4⋊#88322C2214yes—
30D4⋊#88322C2212yes—
31C2×2−1+4×C53—21no—
32D4×Q8×C53—21no—
33D4×Q8×C53C2212yes—
34C53×#547—22no—
35C53×#547—21no—
36C53×#547—22no—
37C53×#547C2212yes—
38C53×#536—22no—
39C53×#536—21no—
40C53×#532—21no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53—21no—
43C2×C4⋊Q8×C53C2211yescharacteristic
44C4×#41×C53C2211yescharacteristic
45C22×Dic106—21no—
46C2×#11719917—21no—
47C22×#88322—21no—
48#13133217—21no—
49#13133217—21no—
50#13133217—21no—
51#13133217C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54#13133150—22no—
55#13133148—24no—
56#13133148—22no—
57C4⋊#88322—21no—
58C4⋊#88322C2312yes—
59C4⋊#88322C2312yes—
60#13133141—22no—
61#13133140—28no—
62#13133138—22no—
63#13133138—24no—
64C22⋊#88322—21no—
65C4×#88322—21no—
66C4×#88322—21no—
67C4×#88322C2312yes—
682−1+4×C53—44no—
692−1+4×C53—44no—
70C2×#41×C53—22no—
71C2×#41×C53—24no—
72C2×#41×C53—22no—
73C2×#41×C53—21no—
74C2×#41×C53D411yescharacteristic
75C22×Q8×C53—22no—
76C22×Q8×C53—22no—
77C22×Q8×C53—21no—
78C4⋊Q8×C53—22no—
79C4⋊Q8×C53—22no—
80C4⋊Q8×C53—22no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53D411yescharacteristic
84C4⋊Q8×C53D412yes—
85C4⋊Q8×C53C2312yes—
86#127×C53D411yescharacteristic
87#124×C53—22no—
88#124×C53—21no—
89#123×C53—24no—
90#123×C53—24no—
91#123×C53—24no—
92C22⋊Q8×C53—24no—
93C22⋊Q8×C53—24no—
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53—22no—
96#121×C53—22no—
97#121×C53—22no—
98#121×C53—21no—
99#121×C53D412yes—
100C4×Q8×C53D411yescharacteristic
101C4×Q8×C53C2311yescharacteristic
102C4×D4×C53—22no—
103C4×D4×C53—21no—
104C4×D4×C53—22no—
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53D411yescharacteristic
108C4×D4×C53C2311yescharacteristic
109C4×D4×C53C2312yes—
110#117×C53—21no—
111#117×C53D411yescharacteristic
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53D412yes—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#41×C53—22no—
119#41×C53—48no—
120#41×C53—416no—
121#41×C53—44no—
122#41×C53—44no—
123#41×C53—42no—
124#41×C53—42no—
125#41×C53SD1614yes—
126C2×Q8×C53—24no—
127C2×Q8×C53—22no—
128C2×Q8×C53—21no—
129C2×Q8×C53—22no—
130C2×Q8×C53—48no—
131C2×Q8×C53—44no—
132C2×Q8×C53—44no—
133C2×Q8×C53—42no—
134C2×Q8×C53—42no—
135C2×Q8×C53C2×D411yescharacteristic
136C2×D4×C53—21no—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53C2×D412yes—
142C2×D4×C53C2×D411yescharacteristic
143C22×C4×C53—21no—
144C22×C4×C53—24no—
145C22×C4×C53—24no—
146C22×C4×C53—22no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—22no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×Dic53—41no—
154C2×Dic106—22no—
155C2×Dic106—44no—
156C2×Dic106—41no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C2×D412yes—
166C4⋊C4×C53C2×D414yes—
167C4⋊C4×C53C2×D411yescharacteristic
168C4⋊C4×C53C2×D411yescharacteristic
169C22⋊C4×C53—24no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—22no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53C2×D411yescharacteristic
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53C2411yescharacteristic
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—41no—
186C4⋊Dic53—41no—
187#11719917—42no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—42no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53#12011yescharacteristic
195Q8×C53—21no—
196Q8×C53—48no—
197Q8×C53—42no—
198Q8×C53—42no—
199Q8×C53—42no—
200Q8×C53—44no—
201Q8×C53C2×SD1612yes—
202D4×C53—24no—
203D4×C53—21no—
204D4×C53—24no—
205D4×C53—48no—
206D4×C53—44no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53C2×SD1612yes—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—24no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—48no—
219C2×C4×C53—48no—
220C2×C4×C53—44no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53C2×SD1612yes—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×Dic53—42no—
234C2×Dic53—82no—
235Dic106—44no—
236Dic106—82no—
237#88322—42no—
238#88322—42no—
239#88322—42no—
240#18641—531nomaximal, Sylow
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—22no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53#45012yes—
250C22×C53C2×#12011yescharacteristic
251C4×C53—22no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—44no—
255C4×C53—42no—
256C4×C53—41no—
257C4×C53—41no—
258C4×C53#45012yes—
259C4×C53#45412yes—
260C4×C53C22×SD1611yescharacteristic
261C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C53#57812yes—
263Dic53—82no—
264#3499D5311yescharacteristic
265#3227—531no—
266C2×#487—531no—
267#3079—531no—
268#1637—534no—
269#1635—534no—
270#1546—531no—
271#1492—531no—
272#1476—531no—
273C2×C53—22no—
274C2×C53—42no—
275C2×C53—82no—
276C2×C53C2×#45011yescharacteristic
277C2×C53#301111yescharacteristic
278C2×C53C2×#45411yescharacteristic
279C2×2−1+4—21no—
280D4×Q8—21no—
281D4×Q8C2×D5312yes—
282#547—22no—
283#547—21no—
284#547—22no—
285#547C2×D5312yes—
286#536—22no—
287#536—21no—
288#532—21no—
289#532C2×D5311yescharacteristic
290C2×C4⋊Q8—21no—
291C2×C4⋊Q8—531no—
292C2×C4⋊Q8C2×D5311yescharacteristic
293C4×#41C2×D5311yescharacteristic
294#498—1061no—
295#487—534no—
296#487—1062no—
297#486—1061no—
298#481—1062no—
299#477—532no—
300#476—532no—
301#476—534no—
302#475—1061no—
303#474—1061no—
304#431—531no—
305#428—1061no—
306#423—531no—
307#417—1061no—
308C2×#103—1062no—
309C2×C4×C8—531no—
310#332—534no—
311Q8⋊C8—532no—
312D4⋊C8—532no—
313D4⋊C8—534no—
314C53#1864111yescharacteristic, complemented, Sylow
3152−1+4—44no—
3162−1+4—44no—
317C2×#41—22no—
318C2×#41—24no—
319C2×#41—22no—
320C2×#41—21no—
321C2×#41C22⋊D5311yescharacteristic
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—21no—
325C22×Q8—1061no—
326C2×M16—1061no—
327C22×C8—1061no—
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—532no—
334C4⋊Q8—1061no—
335C4⋊Q8—1061no—
336C4⋊Q8—1061no—
337C4⋊Q8C22⋊D5311yescharacteristic
338C4⋊Q8C22⋊D5312yes—
339C4⋊Q8C22×D5312yes—
340#127C22⋊D5311yescharacteristic
341#124—22no—
342#124—21no—
343#123—24no—
344#123—24no—
345#123—24no—
346C22⋊Q8—24no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—22no—
350#121—22no—
351#121—22no—
352#121—21no—
353#121C22⋊D5312yes—
354C4×Q8C22⋊D5311yescharacteristic
355C4×Q8C22×D5311yescharacteristic
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4C22⋊D5311yescharacteristic
362C4×D4C22×D5311yescharacteristic
363C4×D4C22×D5312yes—
364#117—21no—
365#117C22⋊D5311yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4C22⋊D5312yes—
372C2×C42—21no—
373C2×C42C22×D5311yescharacteristic
374#107—1062no—
375#106—1062no—
376#106—1064no—
377C4⋊C8—532no—
378C4⋊C8—532no—
379C4⋊C8—1061no—
380#103—1062no—
381#103—1068no—
382#102—1062no—
383#102—1064no—
384C22⋊C8—1061no—
385C4×C8—532no—
386C4×C8—1061no—
387C4×C8—1061no—
388#41—22no—
389#41—48no—
390#41—416no—
391#41—44no—
392#41—44no—
393#41—42no—
394#41—42no—
395#41#1171992314yes—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—1062no—
406C2×Q8—2124no—
407C2×Q8—2121no—
408C2×Q8C2×C22⋊D5311yescharacteristic
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C22⋊D5312yes—
415C2×D4C2×C22⋊D5311yescharacteristic
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—2121no—
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C2×C22⋊D5311yescharacteristic
427M16—2122no—
428C2×C8—1062no—
429C2×C8—1062no—
430C2×C8—1062no—
431C2×C8—2122no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—1062no—
441C4⋊C4—1062no—
442C4⋊C4—2121no—
443C4⋊C4—2121no—
444C4⋊C4C2×C22⋊D5314yes—
445C4⋊C4C2×C22⋊D5312yes—
446C4⋊C4C2×C22⋊D5311yescharacteristic
447C4⋊C4C2×C22⋊D5311yescharacteristic
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42C2×C22⋊D5311yescharacteristic
459C42C2×C22⋊D5311yescharacteristic
460C42C2×C22⋊D5311yescharacteristic
461C42C23×D5311yescharacteristic
462C23—21no—
463C23—22no—
464C23#1313318611yescharacteristic
465Q8—21no—
466Q8—48no—
467Q8—42no—
468Q8—44no—
469Q8—42no—
470Q8—42no—
471Q8—4242no—
472Q8—2124no—
473Q8C2×#1171992312yes—
474D4—24no—
475D4—21no—
476D4—24no—
477D4—48no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4C2×#1171992312yes—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—4242no—
498C2×C4—2122no—
499C2×C4#1313318611yescharacteristic
500C2×C4#1313318611yescharacteristic
501C2×C4#1313318611yescharacteristic
502C2×C4C2×#1171992312yes—
503C2×C4#1313324312yes—
504C2×C4C22×C22⋊D5311yescharacteristic
505C2×C4C22×C22⋊D5311yescharacteristic
506C2×C4C22×C22⋊D5311yescharacteristic
507C8—2122no—
508C8—2122no—
509C8—2122no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22#1681077412yes—
519C22C2×#1313318611yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—41no—
524C4—41no—
525C4—44no—
526C4—42no—
527C4—4242no—
528C4#1681077412yes—
529C4#1681079612yes—
530C4C22×#1171992311yescharacteristic
531C4C2×#1313324311yescharacteristic
532C4#1681143812yes—
533C2—22no—
534C2—42no—
535C2—82no—
536C2C2×#1681077411yescharacteristic
537C2#2028401011yescharacteristic
538C2C2×#1681079611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3499C2
socleC22×C53C2×#120
hypercenter#3499D53
hypocenterC53#18641
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. C2×#16810975C2semidirect
3. #20284530C2semidirect
4. #20273880C2semidirect
5. #20273866C2semidirect
6. #20273387C2semidirect
7. #20273166C2semidirect
8. #20273097C2semidirect
314. C53#18641semidirect

Maximal subgroups

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

subgroupindex
#202858402
C2×#168109752
#202845302
#202738802
#202738662
#202733872
#202731662
#202730972
C53×#34992
#1864153