Finite Group Databasebeta

← #27299709

Subgroups — #27299709

Order 13568.

846 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#20276361C212yesmaximal, complemented
2C2×#16810410C212yesmaximal, complemented
3#20274826C211yesmaximal, characteristic, complemented
4#20274823C211yesmaximal, characteristic, complemented
5C53×#3490C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8#16810413—22no—
9#16810410—22no—
10#16810410—22no—
11#16810410—22no—
12#16810393—22no—
13#16810389—22no—
14C53×#551—22no—
15C53×#551—22no—
16C53×#546—22no—
17C53×#546—22no—
18C53×#543C411yescharacteristic, complemented
19C53×#538—22no—
20C53×#534C2211yescharacteristic
21C53×#525—22no—
22C2×#120×C53C411yescharacteristic, complemented
23C2×#120×C53C2211yescharacteristic
24C53×#518C411yescharacteristic, complemented
25C53×#518C2211yescharacteristic
26C22×C22⋊C4×C53C411yescharacteristic, complemented
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29#13133134—22no—
30#13133134—22no—
31#13133132—22no—
32#13133132—22no—
33C25×C53C2×C411yescharacteristic
34C2×#41×C53—22no—
35C22×D4×C53—21no—
36C22×D4×C53—21no—
37C22×D4×C53C2×C411yescharacteristic
38C22×D4×C53C2311yescharacteristic
39C23×C4×C53—21no—
40#126×C53—22no—
41#126×C53—22no—
42#124×C53—22no—
43#124×C53—22no—
44#123×C53—22no—
45#123×C53—22no—
46#123×C53—22no—
47#123×C53—22no—
48#123×C53—21no—
49#123×C53D412yes—
50C22⋊Q8×C53—22no—
51C22⋊Q8×C53—22no—
52C22⋊Q8×C53—22no—
53C22⋊Q8×C53—22no—
54#121×C53—22no—
55#121×C53—22no—
56#121×C53—22no—
57#121×C53—22no—
58#120×C53—22no—
59#120×C53—21no—
60#120×C53—21no—
61#120×C53—21no—
62#120×C53—22no—
63#120×C53D412yes—
64#120×C53D412yes—
65#120×C53D412yes—
66C4×D4×C53—22no—
67C4×D4×C53—22no—
68C4×D4×C53—22no—
69C4×D4×C53—22no—
70C4×D4×C53—22no—
71C4×D4×C53—22no—
72#117×C53—22no—
73#117×C53—21no—
74#117×C53D412yes—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53—21no—
81C2×C22⋊C4×C53C2×C411yescharacteristic
82C2×C22⋊C4×C53C2×C411yescharacteristic
83C2×C22⋊C4×C53C2×C411yescharacteristic
84C2×C22⋊C4×C53C2×C411yescharacteristic
85C2×C22⋊C4×C53D412yes—
86C24×C53—21no—
87C24×C53—21no—
88C24×C53—42no—
89C24×C53—82no—
90C24×C53C22⋊C411yescharacteristic
91C24×C53C22⋊C411yescharacteristic
92C24×C53C22⋊C411yescharacteristic
93C24×C53C22×C411yescharacteristic
94C24×C53C2×D411yescharacteristic
95#41×C53—84no—
96C2×Q8×C53—22no—
97C2×D4×C53—22no—
98C2×D4×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—22no—
103C2×D4×C53—22no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53—82no—
107C2×D4×C53—82no—
108C2×D4×C53C2×D412yes—
109C2×D4×C53C2×D412yes—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—82no—
119C22×C4×C53C22⋊C411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×Dic53—42no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—22no—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—22no—
133C22⋊C4×C53—21no—
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53C22⋊C412yes—
143C22⋊C4×C53C22⋊C412yes—
144C22⋊C4×C53C22⋊C412yes—
145C22⋊C4×C53C22⋊C412yes—
146C42×C53—22no—
147C42×C53—22no—
148C22⋊Dic53—42no—
149C22⋊Dic53—42no—
150#11719917—42no—
151#11719917—42no—
152C2×#88322—42no—
153C23×C53—21no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—42no—
159C23×C53—41no—
160C23×C53—42no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—42no—
166C23×C53—81no—
167C23×C53—81no—
168C23×C53—81no—
169C23×C53—81no—
170C23×C53—81no—
171C23×C53—81no—
172C23×C53—84no—
173C23×C53—84no—
174C23×C53C2×C22⋊C411yescharacteristic
175C23×C53C2×C22⋊C411yescharacteristic
176C23×C53C2×C22⋊C411yescharacteristic
177C23×C53#12012yes—
178Q8×C53—82no—
179D4×C53—42no—
180D4×C53—42no—
181D4×C53—42no—
182D4×C53—44no—
183D4×C53—44no—
184D4×C53—42no—
185D4×C53—82no—
186D4×C53—82no—
187D4×C53—82no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—44no—
195C2×C4×C53—41no—
196C2×C4×C53—81no—
197C2×C4×C53—82no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53—81no—
201C2×C4×C53—82no—
202C2×C4×C53—82no—
203C2×C4×C53#12012yes—
204C2×Dic53—82no—
205C2×Dic53—162no—
206#88322—82no—
207#12725—531nomaximal, Sylow
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—42no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—42no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—82no—
226C22×C53—82no—
227C22×C53—84no—
228C22×C53—84no—
229C22×C53—82no—
230C22×C53—161no—
231C22×C53#37911yescharacteristic
232C22×C53#41012yes—
233C4×C53—22no—
234C4×C53—42no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—42no—
238C4×C53—41no—
239Dic53—162no—
240#3490D5311yescharacteristic
241#2121—532no—
242C2×AΣL(1,𝔽16)—532no—
243#1829—531no—
244#1828—531no—
245C2×C53—21no—
246C2×C53—22no—
247C2×C53—42no—
248C2×C53—42no—
249C2×C53—41no—
250C2×C53—41no—
251C2×C53—81no—
252C2×C53—81no—
253C2×C53#178211yescharacteristic
254#551—22no—
255#551—22no—
256#546—22no—
257#546—22no—
258#543Dic5311yescharacteristic, complemented
259#538—22no—
260#534C2×D5311yescharacteristic
261#525—22no—
262C2×#120Dic5311yescharacteristic, complemented
263C2×#120C2×D5311yescharacteristic
264#518Dic5311yescharacteristic, complemented
265#518C2×D5311yescharacteristic
266C22×C22⋊C4Dic5311yescharacteristic, complemented
267C2×#100—532no—
268C2×#99—532no—
269#352—1062no—
270AΣL(1,𝔽16)—1062no—
271AΣL(1,𝔽16)—1062no—
272AΣL(1,𝔽16)—1062no—
273#344—1062no—
274#342—1062no—
275C53#1272511yescharacteristic, complemented, Sylow
276C25C2×Dic5311yescharacteristic
277C2×#41—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2×Dic5311yescharacteristic
281C22×D4C22×D5311yescharacteristic
282C23×C4—21no—
283C2×M16—1062no—
284#126—22no—
285#126—22no—
286#124—22no—
287#124—22no—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292#123—21no—
293#123C22⋊D5312yes—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301#121—22no—
302#120—21no—
303#120—21no—
304#120—21no—
305#120—22no—
306#120—22no—
307#120C22⋊D5312yes—
308#120C22⋊D5312yes—
309#120C22⋊D5312yes—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316#117—22no—
317#117—21no—
318#117C22⋊D5312yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—1062no—
326C2×C22⋊C4C2×Dic5311yescharacteristic
327C2×C22⋊C4C2×Dic5311yescharacteristic
328C2×C22⋊C4C2×Dic5311yescharacteristic
329C2×C22⋊C4C2×Dic5311yescharacteristic
330C2×C22⋊C4C22⋊D5312yes—
331#100—1062no—
332#100—1062no—
333#99—1062no—
334#99—1062no—
335C24—21no—
336C24—21no—
337C24—42no—
338C24—82no—
339C24C22⋊Dic5311yescharacteristic
340C24C22⋊Dic5311yescharacteristic
341C24C22⋊Dic5311yescharacteristic
342C24C22×Dic5311yescharacteristic
343C24C2×C22⋊D5311yescharacteristic
344#41—84no—
345C2×Q8—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4C2×C22⋊D5312yes—
358C2×D4C2×C22⋊D5312yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—82no—
368C22×C4—2122no—
369C22×C4C22⋊Dic5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371M16—2122no—
372M16—2122no—
373C2×C8—2122no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—2122no—
395C22⋊C4—2122no—
396C22⋊C4C22⋊Dic5312yes—
397C22⋊C4C22⋊Dic5312yes—
398C22⋊C4C22⋊Dic5312yes—
399C22⋊C4C22⋊Dic5312yes—
400C42—22no—
401C42—22no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—84no—
422C23—84no—
423C23C2×C22⋊Dic5311yescharacteristic
424C23C2×C22⋊Dic5311yescharacteristic
425C23C2×C22⋊Dic5311yescharacteristic
426C23#1313318612yes—
427Q8—82no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—44no—
432D4—44no—
433D4—42no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—8482no—
453C2×C4—4242no—
454C2×C4#1313318612yes—
455C8—4242no—
456C22—22no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—82no—
474C22—82no—
475C22—84no—
476C22—84no—
477C22—82no—
478C22—161no—
479C22#1681047911yescharacteristic
480C22#1681059712yes—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—41no—
487C4—8482no—
488C2—21no—
489C2—22no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2—81no—
495C2—81no—
496C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC53×#3490C2
socleC2×C53#1782
cosocleC22×D4×C53C23
hypercenter#3490D53
hypocenterC53#12725
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3490

Complements

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

NH ≅ G/Ntype
1. #20276361C2semidirect
2. C2×#16810410C2semidirect
3. #20274826C2semidirect
4. #20274823C2semidirect
18. C53×#543C4semidirect
22. C2×#120×C53C4semidirect
24. C53×#518C4semidirect
26. C22×C22⋊C4×C53C4semidirect
258. #543Dic53semidirect
262. C2×#120Dic53semidirect
264. #518Dic53semidirect
266. C22×C22⋊C4Dic53semidirect
275. C53#12725semidirect

Maximal subgroups

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

subgroupindex
#202763612
C2×#168104102
#202748262
#202748232
C53×#34902
#1272553