Finite Group Databasebeta

← #27294548

Subgroups — #27294548

Order 13568.

778 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16810390C211yesmaximal, characteristic, complemented
2#20273042C211yesmaximal, characteristic, complemented
3#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2404C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7#16810587—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C22×#41×C53C411yescharacteristic, complemented
13C2×#120×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#393C411yescharacteristic, complemented
16C53×#393C2211yescharacteristic
17C53×#392C411yescharacteristic, complemented
18C53×#392C2211yescharacteristic
19C53×#386—21no—
20C53×#386—21no—
21C53×#386—21no—
22C53×#381C411yescharacteristic, complemented
23C23×Dic53—41no—
24C2×#11719917—41no—
25C2×C22⋊Dic53—24no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—41no—
28C2×C22⋊Dic53—42no—
29C22⋊#88322—22no—
30C22⋊#88322—41no—
31C25×C53D411yescharacteristic
32C2×#41×C53—41no—
33C2×#41×C53—41no—
34C2×#41×C53—44no—
35C22×Q8×C53C2×C411yescharacteristic, complemented
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53—21no—
39C22×D4×C53C2×C411yescharacteristic, complemented
40C23×C4×C53—21no—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43#120×C53—44no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—41no—
50C2×C22⋊C4×C53—41no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic, complemented
59#95×C53C2×C411yescharacteristic, complemented
60C24×C53—21no—
61C24×C53—21no—
62C24×C53—22no—
63C24×C53—21no—
64C24×C53—21no—
65C24×C53—21no—
66C24×C53—42no—
67C24×C53—81no—
68C24×C53C22⋊C411yescharacteristic
69C24×C53C22⋊C411yescharacteristic
70C24×C53C2×D411yescharacteristic
71#41×C53—44no—
72#41×C53—44no—
73#41×C53—44no—
74#41×C53—44no—
75C2×Q8×C53—42no—
76C2×Q8×C53—41no—
77C2×D4×C53—24no—
78C2×D4×C53—41no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—81no—
89C2×D4×C53—81no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—81no—
102C22×C4×C53C22⋊C411yescharacteristic
103C22×C4×C53C22⋊C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×Dic53—44no—
107C22×Dic53—42no—
108C22×Dic53—82no—
109C22×Dic53—82no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—42no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—81no—
118C22⋊C4×C53—81no—
119C22⋊Dic53—24no—
120C22⋊Dic53—44no—
121C22⋊Dic53—42no—
122#11719917—82no—
123C2×#88322—42no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—22no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53—84no—
152C23×C53—82no—
153C23×C53—82no—
154C23×C53#9911yescharacteristic
155C23×C53#9911yescharacteristic
156C23×C53#10412yes—
157C23×C53C2×C22⋊C411yescharacteristic
158C23×C53AGL(1,ℤ/8)11yescharacteristic
159C23×C53#13711yescharacteristic
160Q8×C53—42no—
161Q8×C53—42no—
162D4×C53—44no—
163D4×C53—42no—
164D4×C53—42no—
165D4×C53—44no—
166D4×C53—84no—
167D4×C53—82no—
168D4×C53—82no—
169C2×C4×C53—22no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—48no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—81no—
191C2×Dic53—44no—
192C2×Dic53—88no—
193C2×Dic53—82no—
194C2×Dic53—82no—
195#88322—82no—
196#11753—531nomaximal, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—22no—
206C22×C53—21no—
207C22×C53—42no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—44no—
213C22×C53—44no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—42no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—82no—
226C22×C53—82no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53AΣL(1,𝔽16)12yes—
230C22×C53C2×#9911yescharacteristic
231C22×C53#41711yescharacteristic
232C22×C53C2×#10411yescharacteristic
233C4×C53—41no—
234C4×C53—42no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53—41no—
238C4×C53—81no—
239C4×C53—81no—
240Dic53—84no—
241#2404D5311yescharacteristic
242C2×#342—531no—
243#1462—531no—
244#1351—532no—
245#1343—532no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—22no—
249C2×C53—41no—
250C2×C53—41no—
251C2×C53—42no—
252C2×C53—41no—
253C2×C53—82no—
254C2×C53#150411yescharacteristic
255C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
256C2×C53#212711yescharacteristic
257C22×#41Dic5311yescharacteristic, complemented
258C2×#120—21no—
259C22×C22⋊C4—1061no—
260C22×C22⋊C4C2×D5311yescharacteristic
261#407—1061no—
262#393Dic5311yescharacteristic, complemented
263#393C2×D5311yescharacteristic
264#392Dic5311yescharacteristic, complemented
265#392C2×D5311yescharacteristic
266#386—21no—
267#386—21no—
268#386—21no—
269#381Dic5311yescharacteristic, complemented
270#342—532no—
271#342—1061no—
272#323—532no—
273#323—1061no—
274C53#1175311yescharacteristic, complemented, Sylow
275C25C22⋊D5311yescharacteristic
276C2×#41—41no—
277C2×#41—41no—
278C2×#41—44no—
279C22×Q8C2×Dic5311yescharacteristic, complemented
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C2×Dic5311yescharacteristic, complemented
284C23×C4—21no—
285C23×C4—2121no—
286C23×C4C2×Dic5311yescharacteristic
287C23×C4C22⋊D5311yescharacteristic
288C2×M16—2121no—
289#120—44no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—1064no—
298C2×C22⋊C4—1062no—
299C2×C22⋊C4—2121no—
300C2×C22⋊C4—2122no—
301C2×C22⋊C4C2×Dic5311yescharacteristic
302C2×C22⋊C4C22⋊D5311yescharacteristic
303C2×C22⋊C4C22⋊D5311yescharacteristic
304C2×C22⋊C4C22×D5311yescharacteristic
305C22⋊C8—1062no—
306C22⋊C8—2121no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95C2×Dic5311yescharacteristic, complemented
311#95C2×Dic5311yescharacteristic, complemented
312C24—21no—
313C24—21no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—42no—
319C24—81no—
320C24C22⋊Dic5311yescharacteristic
321C24C22⋊Dic5311yescharacteristic
322C24C2×C22⋊D5311yescharacteristic
323#41—44no—
324#41—44no—
325#41—44no—
326#41—44no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×D4—24no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—81no—
341C2×D4—81no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—81no—
354C22×C4—4242no—
355C22×C4—4242no—
356C22×C4—2124no—
357C22×C4—2122no—
358C22×C4C22⋊Dic5311yescharacteristic
359C22×C4C22⋊Dic5311yescharacteristic
360C22×C4C22×Dic5311yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362M16—4242no—
363C2×C8—2122no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—81no—
372C22⋊C4—81no—
373C22⋊C4—1064no—
374C22⋊C4—2124no—
375C22⋊C4—2122no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—41no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23—84no—
404C23—82no—
405C23—82no—
406C23#1313313211yescharacteristic
407C23#1313313211yescharacteristic
408C23#1313314412yes—
409C23C2×C22⋊Dic5311yescharacteristic
410C23#1313324211yescharacteristic
411C23#1313324711yescharacteristic
412Q8—42no—
413Q8—42no—
414D4—44no—
415D4—42no—
416D4—42no—
417D4—44no—
418D4—84no—
419D4—82no—
420D4—82no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—48no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—4248no—
444C2×C4—4242no—
445C2×C4—4242no—
446C2×C4—2124no—
447C8—4242no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22#1681041012yes—
481C22C2×#1313313211yescharacteristic
482C22#1681062411yescharacteristic
483C22C2×#1313314411yescharacteristic
484C4—41no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4—4244no—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—42no—
498C2—41no—
499C2—82no—
500C2#2027321911yescharacteristic
501C2C2×#1681041011yescharacteristic
502C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2404C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2404D53
hypocenterC53#11753
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#16810390C2semidirect
2. #20273042C2semidirect
3. #20272736C2semidirect
4. #20272712C2semidirect
12. C22×#41×C53C4semidirect
15. C53×#393C4semidirect
17. C53×#392C4semidirect
22. C53×#381C4semidirect
35. C22×Q8×C53C2×C4semidirect
39. C22×D4×C53C2×C4semidirect
58. #95×C53C2×C4semidirect
59. #95×C53C2×C4semidirect
257. C22×#41Dic53semidirect
262. #393Dic53semidirect
264. #392Dic53semidirect
269. #381Dic53semidirect
274. C53#11753semidirect
279. C22×Q8C2×Dic53semidirect
283. C22×D4C2×Dic53semidirect
310. #95C2×Dic53semidirect
311. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168103902
#202730422
#202727362
#202727122
C53×#24042
#1175353