Finite Group Databasebeta

← #27299669

Subgroups — #27299669

Order 13568.

944 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276400C211yesmaximal, characteristic, complemented
2#20276380C211yesmaximal, characteristic, complemented
3#20276369C212yesmaximal, complemented
4#20274690C212yesmaximal, complemented
5C53×#3499C211yesmaximal, characteristic
6#16810640—22no—
7#16810632—22no—
8#16810607C2212yes—
9#16810597C2212yes—
10#16810422—21no—
11#16810420—21no—
12#16810418—21no—
13#16810416—21no—
14#16810413—24no—
15C2×2−1+4×C53C411yescharacteristic, complemented
16C2×2−1+4×C53C2211yescharacteristic
17D4×Q8×C53—22no—
18C53×#547—22no—
19C53×#547—24no—
20C53×#536—21no—
21C53×#536—21no—
22C53×#536C411yescharacteristic, complemented
23C53×#536C2211yescharacteristic
24C53×#532—21no—
25C53×#532C2211yescharacteristic
26C2×C4⋊Q8×C53—21no—
27C2×C4⋊Q8×C53C411yescharacteristic, complemented
28C4×#41×C53C411yescharacteristic, complemented
29C2×#11719917—22no—
30#13133176—22no—
31#13133144—44no—
32#13133141—44no—
33#13133136—22no—
34#13133134—22no—
35#13133132—22no—
36#13133132—22no—
372−1+4×C53—42no—
382−1+4×C53—42no—
392−1+4×C53—42no—
402−1+4×C53—42no—
41C2×#41×C53—21no—
42C2×#41×C53—21no—
43C2×#41×C53—21no—
44C2×#41×C53—22no—
45C2×#41×C53—22no—
46C2×#41×C53C2×C411yescharacteristic
47C2×#41×C53C2×C411yescharacteristic
48C2×#41×C53D412yes—
49C2×#41×C53C2311yescharacteristic
50C22×Q8×C53—21no—
51C22×Q8×C53—21no—
52C22×Q8×C53—21no—
53C22×Q8×C53C2×C411yescharacteristic
54C22×Q8×C53C2×C411yescharacteristic
55C22×Q8×C53D412yes—
56C4⋊Q8×C53—24no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—22no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D411yescharacteristic
63#127×C53D411yescharacteristic
64#124×C53—21no—
65#124×C53—21no—
66#124×C53D411yescharacteristic
67#124×C53D411yescharacteristic
68#123×C53—24no—
69#123×C53—24no—
70#123×C53—22no—
71#123×C53D414yes—
72C22⋊Q8×C53—24no—
73C22⋊Q8×C53—24no—
74C22⋊Q8×C53—22no—
75C22⋊Q8×C53—22no—
76#121×C53—24no—
77#121×C53—22no—
78C4×Q8×C53—21no—
79C4×D4×C53—24no—
80C4×D4×C53—22no—
81C4×D4×C53—21no—
82C4×D4×C53—22no—
83#117×C53—21no—
84#117×C53C2×C411yescharacteristic
85C2×C4⋊C4×C53—24no—
86C2×C4⋊C4×C53—22no—
87C2×C42×C53—21no—
88C2×C42×C53C2×C411yescharacteristic
89#41×C53—22no—
90#41×C53—22no—
91#41×C53—24no—
92#41×C53—41no—
93#41×C53—41no—
94#41×C53—42no—
95#41×C53—44no—
96#41×C53—44no—
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—48no—
100#41×C53—48no—
101#41×C53—42no—
102#41×C53—42no—
103C2×Q8×C53—22no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—24no—
109C2×Q8×C53—41no—
110C2×Q8×C53—41no—
111C2×Q8×C53—42no—
112C2×Q8×C53—44no—
113C2×Q8×C53—44no—
114C2×Q8×C53—41no—
115C2×Q8×C53—41no—
116C2×Q8×C53—42no—
117C2×Q8×C53—42no—
118C2×Q8×C53C22⋊C412yes—
119C2×Q8×C53C22⋊C411yescharacteristic
120C2×Q8×C53C22⋊C411yescharacteristic
121C2×Q8×C53C2×D411yescharacteristic
122C2×D4×C53—21no—
123C2×D4×C53—22no—
124C2×D4×C53—21no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53C22⋊C411yescharacteristic
128C2×D4×C53C22⋊C411yescharacteristic
129C2×D4×C53C2×D411yescharacteristic
130C2×D4×C53C2×D412yes—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53C22⋊C412yes—
141C22×C4×C53C22×C411yescharacteristic
142C22×C4×C53C2×D412yes—
143C4⋊C4×C53—24no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—24no—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53C22⋊C412yes—
156C42×C53—22no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53C22⋊C411yescharacteristic
160C42×C53C22⋊C411yescharacteristic
161C22⋊Dic53—42no—
162C4⋊Dic53—42no—
163C4×Dic53—42no—
164#11719917—42no—
165#11719917—42no—
166C2×#88322—42no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53C2×C22⋊C411yescharacteristic
170C23×C53#12012yes—
171Q8×C53—21no—
172Q8×C53—22no—
173Q8×C53—22no—
174Q8×C53—41no—
175Q8×C53—44no—
176Q8×C53—42no—
177Q8×C53—42no—
178Q8×C53—44no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53#9912yes—
183D4×C53—24no—
184D4×C53—21no—
185D4×C53—42no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—44no—
189D4×C53—44no—
190D4×C53—42no—
191D4×C53—44no—
192D4×C53#9912yes—
193C2×C4×C53—24no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—21no—
205C2×C4×C53—44no—
206C2×C4×C53—44no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—42no—
211C2×C4×C53—44no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53#12012yes—
218C2×Dic53—82no—
219#88322—82no—
220#12717—531nomaximal, Sylow
221C22×C53—22no—
222C22×C53—21no—
223C22×C53—22no—
224C22×C53—41no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—41no—
229C22×C53#37911yescharacteristic
230C4×C53—22no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—22no—
234C4×C53—42no—
235C4×C53—41no—
236C4×C53—42no—
237C4×C53—41no—
238C4×C53—42no—
239C4×C53C2×#9911yescharacteristic
240C4×C53C2×#9911yescharacteristic
241Dic53—162no—
242#3499D5311yescharacteristic
243#2132—531no—
244#2126—531no—
245#2123—532no—
246#1796—532no—
247C2×C53—21no—
248C2×C53—42no—
249C2×C53—41no—
250C2×C53—42no—
251C2×C53—81no—
252C2×C53#178111yescharacteristic
253C2×2−1+4Dic5311yescharacteristic, complemented
254C2×2−1+4C2×D5311yescharacteristic
255D4×Q8—22no—
256#547—22no—
257#547—24no—
258#536—21no—
259#536—21no—
260#536Dic5311yescharacteristic, complemented
261#536C2×D5311yescharacteristic
262#532—21no—
263#532C2×D5311yescharacteristic
264C2×C4⋊Q8—21no—
265C2×C4⋊Q8Dic5311yescharacteristic, complemented
266C4×#41Dic5311yescharacteristic, complemented
267#421—1062no—
268#419—1062no—
269#413—532no—
270#410—532no—
271#356—1061no—
272#355—1061no—
273#354—1061no—
274#353—1061no—
275#352—1064no—
276C53#1271711yescharacteristic, complemented, Sylow
2772−1+4—42no—
2782−1+4—42no—
2792−1+4—42no—
2802−1+4—42no—
281C2×#41—21no—
282C2×#41—22no—
283C2×#41—21no—
284C2×#41—21no—
285C2×#41—22no—
286C2×#41C2×Dic5311yescharacteristic
287C2×#41C2×Dic5311yescharacteristic
288C2×#41C22⋊D5312yes—
289C2×#41C22×D5311yescharacteristic
290C22×Q8—21no—
291C22×Q8—21no—
292C22×Q8—21no—
293C22×Q8C2×Dic5311yescharacteristic
294C22×Q8C2×Dic5311yescharacteristic
295C22×Q8C22⋊D5312yes—
296C2×M16—1062no—
297C4⋊Q8—24no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—22no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C22⋊D5311yescharacteristic
304#127C22⋊D5311yescharacteristic
305#124—21no—
306#124—21no—
307#124C22⋊D5311yescharacteristic
308#124C22⋊D5311yescharacteristic
309#123—24no—
310#123—24no—
311#123—22no—
312#123C22⋊D5314yes—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317#121—24no—
318#121—22no—
319C4×Q8—21no—
320C4×D4—24no—
321C4×D4—21no—
322C4×D4—22no—
323C4×D4—22no—
324#117—21no—
325#117—1062no—
326#117C2×Dic5311yescharacteristic
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C42—21no—
330C2×C42C2×Dic5311yescharacteristic
331#104—2124no—
332#103—2124no—
333#101—1062no—
334#100—1062no—
335#99—1062no—
336#99—1062no—
337#41—22no—
338#41—22no—
339#41—24no—
340#41—41no—
341#41—41no—
342#41—42no—
343#41—44no—
344#41—44no—
345#41—41no—
346#41—41no—
347#41—48no—
348#41—48no—
349#41—42no—
350#41—42no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—42no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8C22⋊Dic5311yescharacteristic
367C2×Q8C22⋊Dic5312yes—
368C2×Q8C22⋊Dic5311yescharacteristic
369C2×Q8C2×C22⋊D5311yescharacteristic
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4C22⋊Dic5311yescharacteristic
376C2×D4C22⋊Dic5311yescharacteristic
377C2×D4C2×C22⋊D5311yescharacteristic
378C2×D4C2×C22⋊D5312yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4C22⋊Dic5312yes—
389C22×C4C22×Dic5311yescharacteristic
390C22×C4C2×C22⋊D5312yes—
391M16—2122no—
392M16—2122no—
393C2×C8—2122no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—2122no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—2122no—
408C22⋊C4C22⋊Dic5312yes—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—2122no—
413C42C22⋊Dic5311yescharacteristic
414C42C22⋊Dic5311yescharacteristic
415C23—21no—
416C23—21no—
417C23C2×C22⋊Dic5311yescharacteristic
418C23#1313318612yes—
419Q8—21no—
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—44no—
424Q8—41no—
425Q8—42no—
426Q8—42no—
427Q8—44no—
428Q8—41no—
429Q8—41no—
430Q8#1313313212yes—
431D4—24no—
432D4—21no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—44no—
437D4—44no—
438D4—42no—
439D4—44no—
440D4#1313313212yes—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—4242no—
464C2×C4C2×C22⋊Dic5311yescharacteristic
465C2×C4C2×C22⋊Dic5311yescharacteristic
466C2×C4#1313318612yes—
467C8—4242no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22#1681047911yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—22no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—41no—
485C4—42no—
486C4—8482no—
487C4C2×#1313313211yescharacteristic
488C4C2×#1313313211yescharacteristic
489C2—21no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—81no—
494C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3499C2
socleC2×C53#1781
cosocleC2×#41×C53C23
hypercenter#3499D53
hypocenterC53#12717
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20276400C2semidirect
2. #20276380C2semidirect
3. #20276369C2semidirect
4. #20274690C2semidirect
15. C2×2−1+4×C53C4semidirect
22. C53×#536C4semidirect
27. C2×C4⋊Q8×C53C4semidirect
28. C4×#41×C53C4semidirect
253. C2×2−1+4Dic53semidirect
260. #536Dic53semidirect
265. C2×C4⋊Q8Dic53semidirect
266. C4×#41Dic53semidirect
276. C53#12717semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202763802
#202763692
#202746902
C53×#34992
#1271753