Finite Group Databasebeta

← #27299903

Subgroups — #27299903

Order 13568.

936 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810418C211yesmaximal, characteristic, complemented
2C2×#16810413C211yesmaximal, characteristic, complemented
3C2×#16810410C211yesmaximal, characteristic
4#20275129C211yesmaximal, characteristic, complemented
5#20275124C211yesmaximal, characteristic
6#20275104C211yesmaximal, characteristic, complemented
7C2×C53×#551C211yesmaximal, characteristic
8#16811098C2211yescharacteristic
9#16810632—21no—
10C2×#13133134C2211yescharacteristic
11C2×#13133132C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13#16810486—21no—
14#16810479—21no—
15#16810418—22no—
16#16810413—22no—
17#16810410—22no—
18#16810389—21no—
19#16810389—21no—
20#16810383—21no—
21C53×#551—22no—
22C53×#551—22no—
23C53×#551—22no—
24C53×#551C414yescomplemented
25C2×#126×C53—21no—
26C2×#124×C53C2211yescharacteristic
27C2×#123×C53—21no—
28C2×#123×C53C2211yescharacteristic
29C2×C22⋊Q8×C53—21no—
30C2×#120×C53C2211yescharacteristic
31C2×C4×D4×C53—21no—
32C2×#117×C53—21no—
33C22×C22⋊C4×C53—21no—
34C2×#11719917—21no—
35D4×Dic53—42no—
36#13133176—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—42no—
42#13133141—42no—
43#13133134—21no—
44#13133134—21no—
45#13133132—21no—
46#13133132—21no—
47#13133132—21no—
48#13133132—21no—
49#13133124—42no—
50C25×C53D411yescharacteristic
51C22×Q8×C53D411yescharacteristic
52C22×D4×C53—21no—
53C22×D4×C53C2311yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53D411yescharacteristic
57#126×C53—24no—
58#126×C53—24no—
59#124×C53—21no—
60#124×C53—21no—
61#124×C53C2×C412yes—
62#124×C53D412yes—
63#123×C53—24no—
64#123×C53—24no—
65#123×C53—21no—
66#123×C53—21no—
67#123×C53C2×C412yes—
68#123×C53D412yes—
69C22⋊Q8×C53—24no—
70C22⋊Q8×C53—24no—
71#120×C53—21no—
72#120×C53—21no—
73#120×C53C2×C412yes—
74#120×C53D412yes—
75C4×D4×C53—24no—
76C4×D4×C53—24no—
77#117×C53—24no—
78#117×C53—24no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C22⋊C4×C53—21no—
84C2×C22⋊C4×C53—21no—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53—21no—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53D411yescharacteristic
92C2×C22⋊C4×C53D411yescharacteristic
93C2×C42×C53—21no—
94C2×C42×C53D411yescharacteristic
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—44no—
98C24×C53—41no—
99C24×C53—41no—
100C24×C53C22⋊C412yes—
101C24×C53C2×D411yescharacteristic
102C24×C53C2×D411yescharacteristic
103C2×Q8×C53—21no—
104C2×Q8×C53—41no—
105C2×Q8×C53—41no—
106C2×Q8×C53C22⋊C412yescomplemented
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—44no—
112C2×D4×C53C22×C411yescharacteristic
113C2×D4×C53C2×D411yescharacteristic
114C2×D4×C53C2×D411yescharacteristic
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—24no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—24no—
123C22×C4×C53—21no—
124C22×C4×C53—44no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—81no—
128C22×C4×C53—81no—
129C22×C4×C53C22⋊C412yescomplemented
130C22×C4×C53C2×D411yescharacteristic
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—41no—
134C22×Dic53—41no—
135C4⋊C4×C53—24no—
136C4⋊C4×C53—24no—
137C4⋊C4×C53—24no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—24no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—22no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53C22⋊C412yes—
153C22⋊C4×C53C22⋊C412yescomplemented
154C42×C53—24no—
155C42×C53—21no—
156C42×C53C22⋊C412yescomplemented
157C22⋊Dic53—41no—
158C22⋊Dic53—41no—
159C22⋊Dic53—41no—
160C22⋊Dic53—41no—
161C22⋊Dic53—41no—
162C22⋊Dic53—41no—
163C22⋊Dic53—41no—
164C22⋊Dic53—84no—
165C4⋊Dic53—22no—
166C4×Dic53—41no—
167#11719917—41no—
168#11719917—41no—
169C2×#88322—41no—
170C23×C53—24no—
171C23×C53—22no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—44no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—44no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—44no—
185C23×C53—81no—
186C23×C53—84no—
187C23×C53—82no—
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53C2×C22⋊C411yescharacteristic
190C23×C53#12011yescharacteristic
191C23×C53#12011yescharacteristic
192C23×C53#12011yescharacteristic
193Q8×C53—42no—
194Q8×C53—81no—
195D4×C53—41no—
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—44no—
200D4×C53—44no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—48no—
211C2×C4×C53—42no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—81no—
218C2×C4×C53—84no—
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×Dic53—42no—
222C2×Dic53—82no—
223C2×Dic53—82no—
224C2×Dic53—81no—
225C2×Dic53—81no—
226C2×Dic53—82no—
227C2×Dic53—82no—
228#88322—81no—
229#12761—531nomaximal, Sylow
230C22×C53—22no—
231C22×C53—24no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53—42no—
241C22×C53—44no—
242C22×C53—44no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—44no—
246C22×C53—42no—
247C22×C53—81no—
248C22×C53—81no—
249C22×C53—84no—
250C22×C53—82no—
251C22×C53#37911yescharacteristic
252C22×C53B(4,𝔽2)11yescharacteristic
253C22×C53#45811yescharacteristic
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—42no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—42no—
260C4×C53—42no—
261Dic53—82no—
262Dic53—161no—
263Dic53—161no—
264C2×#551D5311yescharacteristic
265C2×#354—531no—
266C2×#352—531no—
267C2×AΣL(1,𝔽16)—531no—
268#1890—531no—
269#1889—531no—
270#1885—531no—
271C2×C53—21no—
272C2×C53—21no—
273C2×C53—41no—
274C2×C53—41no—
275C2×C53—44no—
276C2×C53—41no—
277C2×C53—41no—
278C2×C53#179011yescharacteristic
279C2×C53#219712yes—
280#551—22no—
281#551—22no—
282#551—22no—
283#551Dic5314yescomplemented
284C2×#126—21no—
285C2×#124C2×D5311yescharacteristic
286C2×#123—21no—
287C2×#123C2×D5311yescharacteristic
288C2×C22⋊Q8—21no—
289C2×#120C2×D5311yescharacteristic
290#518—531no—
291C2×C4×D4—21no—
292C2×#117—21no—
293C22×C22⋊C4—21no—
294#419—1061no—
295C2×#100—531no—
296C2×#99—531no—
297C2×#99—531no—
298#381—1061no—
299#379—1061no—
300#354—1062no—
301#352—1062no—
302AΣL(1,𝔽16)—1062no—
303#342—1061no—
304#342—1061no—
305#339—1061no—
306C53#1276111yescharacteristic, complemented, Sylow
307C25C22⋊D5311yescharacteristic
308C22×Q8C22⋊D5311yescharacteristic
309C22×D4—21no—
310C22×D4C22×D5311yescharacteristic
311C23×C4—21no—
312C23×C4—21no—
313C23×C4C22⋊D5311yescharacteristic
314C2×M16—1061no—
315#126—24no—
316#126—24no—
317#124—21no—
318#124—21no—
319#124C2×Dic5312yes—
320#124C22⋊D5312yes—
321#123—24no—
322#123—21no—
323#123—21no—
324#123—24no—
325#123C2×Dic5312yes—
326#123C22⋊D5312yes—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329#120—21no—
330#120—21no—
331#120C2×Dic5312yes—
332#120C22⋊D5312yes—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—2122no—
336#117—24no—
337#117—24no—
338#117—1061no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1061no—
352C2×C22⋊C4—1061no—
353C2×C22⋊C4—1061no—
354C2×C22⋊C4—1061no—
355C2×C22⋊C4—2122no—
356C2×C22⋊C4C22⋊D5311yescharacteristic
357C2×C22⋊C4C22⋊D5311yescharacteristic
358C2×C42—21no—
359C2×C42C22⋊D5311yescharacteristic
360#103—2122no—
361#100—1061no—
362#100—1061no—
363#99—1061no—
364#99—1061no—
365#99—1061no—
366#99—1061no—
367#95—2122no—
368C24—21no—
369C24—21no—
370C24—41no—
371C24—44no—
372C24—41no—
373C24C22⋊Dic5312yes—
374C24C2×C22⋊D5311yescharacteristic
375C24C2×C22⋊D5311yescharacteristic
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8C22⋊Dic5312yescomplemented
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—44no—
385C2×D4C22×Dic5311yescharacteristic
386C2×D4C2×C22⋊D5311yescharacteristic
387C2×D4C2×C22⋊D5311yescharacteristic
388C2×D4C2×C22⋊D5311yescharacteristic
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—44no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—1062no—
403C22×C4—1062no—
404C22×C4—2121no—
405C22×C4—2121no—
406C22×C4C22⋊Dic5312yescomplemented
407C22×C4C2×C22⋊D5311yescharacteristic
408M16—2121no—
409M16—2121no—
410C2×C8—2121no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—1062no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—4244no—
430C22⋊C4—2121no—
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4C22⋊Dic5312yes—
438C22⋊C4C22⋊Dic5312yescomplemented
439C42—24no—
440C42—21no—
441C42—2121no—
442C42C22⋊Dic5312yescomplemented
443C23—24no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—41no—
456C23—41no—
457C23—44no—
458C23—81no—
459C23—84no—
460C23—82no—
461C23C2×C22⋊Dic5311yescharacteristic
462C23C2×C22⋊Dic5311yescharacteristic
463C23#1313318611yescharacteristic
464C23#1313318611yescharacteristic
465C23#1313318611yescharacteristic
466Q8—42no—
467Q8—81no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—44no—
473D4—44no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—81no—
491C2×C4—84no—
492C2×C4—4241no—
493C2×C4—4241no—
494C2×C4—4242no—
495C2×C4—4242no—
496C2×C4—4242no—
497C2×C4—4242no—
498C2×C4—2122no—
499C2×C4C2×C22⋊Dic5311yescharacteristic
500C2×C4#1313318611yescharacteristic
501C8—4241no—
502C22—22no—
503C22—24no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—84no—
522C22—82no—
523C22#1681047911yescharacteristic
524C22#1681081011yescharacteristic
525C22#1681081311yescharacteristic
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—8481no—
534C4—8481no—
535C4—4242no—
536C2—21no—
537C2—21no—
538C2—41no—
539C2—41no—
540C2—44no—
541C2—41no—
542C2—41no—
543C2#2027466311yescharacteristic
544C2#2027670512yes—
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#551C2
socleC22×C53B(4,𝔽2)
prime radicalC2×#120×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#551D53
hypocenterC53#12761
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#551

Complements

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

NH ≅ G/Ntype
1. C2×#16810418C2semidirect
2. C2×#16810413C2semidirect
4. #20275129C2semidirect
6. #20275104C2semidirect
24. C53×#551C4semidirect
106. C2×Q8×C53C22⋊C4semidirect
129. C22×C4×C53C22⋊C4semidirect
153. C22⋊C4×C53C22⋊C4semidirect
156. C42×C53C22⋊C4semidirect
283. #551Dic53semidirect
306. C53#12761semidirect
379. C2×Q8C22⋊Dic53semidirect
406. C22×C4C22⋊Dic53semidirect
438. C22⋊C4C22⋊Dic53semidirect
442. C42C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
C2×#168104132
C2×#168104102
#202751292
#202751242
#202751042
C2×C53×#5512
#1276153