Finite Group Databasebeta

← #27299837

Subgroups — #27299837

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276464C211yesmaximal, characteristic, complemented
2C2×#16810439C211yesmaximal, characteristic, complemented
3#20276451C211yesmaximal, characteristic, complemented
4#20276435C211yesmaximal, characteristic, complemented
5#20276343C211yesmaximal, characteristic, complemented
6#20274769C211yesmaximal, characteristic, complemented
7C2×C53×#578C211yesmaximal, characteristic
8C2×#13133157C2211yescharacteristic
9C2×#13133157C2211yescharacteristic
10C2×#13133153C2211yescharacteristic
11#16810661C2211yescharacteristic
12#16810640—21no—
13#16810439—22no—
14#16810439—21no—
15#16810436—21no—
16#16810433—22no—
17#16810427—22no—
18#16810407—22no—
19#16810395—21no—
20C2×2−1+4×C53—21no—
21C53×#578—22no—
22C53×#578—22no—
23C53×#578—22no—
24C53×#578C414yescomplemented
25C2×#137×C53—22no—
26C2×#137×C53—21no—
27C2×#135×C53—21no—
28C2×#135×C53C2211yescharacteristic
29C22×Q16×C53—21no—
30C22×Q16×C53C2211yescharacteristic
31C2×#131×C53C2211yescharacteristic
32C2×#11719917—21no—
33#13133176—21no—
34#13133157—21no—
35#13133157—21no—
36#13133157—21no—
37#13133157—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133151—21no—
43#13133149—21no—
44#13133144—42no—
452−1+4×C53—24no—
462−1+4×C53—24no—
472−1+4×C53—42no—
482−1+4×C53—42no—
49C2×#41×C53—22no—
50C2×#41×C53—21no—
51C2×#41×C53—42no—
52C2×#41×C53—41no—
53C2×#41×C53D411yescharacteristic
54C22×Q8×C53—22no—
55C22×Q8×C53—21no—
56C22×Q8×C53—41no—
57#137×C53—28no—
58#137×C53—24no—
59#137×C53—28no—
60#137×C53—24no—
61#135×C53—24no—
62#135×C53—24no—
63#135×C53—21no—
64#135×C53—21no—
65#135×C53C2×C412yes—
66#135×C53D412yes—
67C2×Q16×C53—22no—
68C2×Q16×C53—24no—
69C2×Q16×C53—24no—
70C2×Q16×C53—21no—
71C2×Q16×C53—21no—
72C2×Q16×C53—21no—
73C2×Q16×C53C2×C412yes—
74C2×Q16×C53D411yescharacteristic
75C2×Q16×C53D412yes—
76C2×Q16×C53D412yes—
77C2×SD16×C53—22no—
78C2×SD16×C53—21no—
79C2×D8×C53D411yescharacteristic
80#131×C53—21no—
81#131×C53—21no—
82#131×C53C2×C412yes—
83#131×C53D412yes—
84C2×M16×C53—21no—
85C2×M16×C53D411yescharacteristic
86C22×C8×C53—21no—
87C22×C8×C53C2311yescharacteristic
88#41×C53—21no—
89#41×C53—21no—
90#41×C53—28no—
91#41×C53—22no—
92#41×C53—22no—
93#41×C53—44no—
94#41×C53—48no—
95#41×C53—48no—
96#41×C53—44no—
97#41×C53—44no—
98#41×C53—41no—
99#41×C53—41no—
100#41×C53C22⋊C412yes—
101#41×C53D812yes—
102C2×Q8×C53—28no—
103C2×Q8×C53—22no—
104C2×Q8×C53—22no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—42no—
108C2×Q8×C53—44no—
109C2×Q8×C53—44no—
110C2×Q8×C53—44no—
111C2×Q8×C53—41no—
112C2×Q8×C53—41no—
113C2×Q8×C53—41no—
114C2×Q8×C53SD1611yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—42no—
118C2×D4×C53—41no—
119C2×D4×C53SD1611yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126Q16×C53—28no—
127Q16×C53—22no—
128Q16×C53—21no—
129Q16×C53—22no—
130Q16×C53—22no—
131Q16×C53C22⋊C412yes—
132Q16×C53C22⋊C414yes—
133SD16×C53—28no—
134SD16×C53—22no—
135SD16×C53—22no—
136D8×C53—21no—
137D8×C53C22⋊C412yes—
138M16×C53—21no—
139M16×C53—22no—
140M16×C53—22no—
141M16×C53C22⋊C412yes—
142C2×C8×C53—21no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C2×C8×C53C22×C411yescharacteristic
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C22⋊Dic53—41no—
152C4⋊Dic53—41no—
153C4×Dic53—41no—
154#11719917—41no—
155#11719917—41no—
156C2×#88322—41no—
157#11719908—41no—
158#11719908—41no—
159C23×C53—21no—
160C23×C53—41no—
161C23×C53AGL(1,ℤ/8)11yescharacteristic
162Q8×C53—21no—
163Q8×C53—22no—
164Q8×C53—24no—
165Q8×C53—48no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—42no—
169Q8×C53—42no—
170Q8×C53#10212yes—
171D4×C53—22no—
172D4×C53—22no—
173D4×C53—22no—
174D4×C53—21no—
175D4×C53—48no—
176D4×C53—42no—
177D4×C53—42no—
178D4×C53—41no—
179D4×C53#10212yes—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—48no—
184C2×C4×C53—48no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53C2×C22⋊C411yescharacteristic
192C2×C4×C53#12011yescharacteristic
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×SD1611yescharacteristic
196C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
197C2×Dic53—81no—
198C8×C53—22no—
199C8×C53C2×C22⋊C411yescharacteristic
200C8×C53C2×C22⋊C411yescharacteristic
201C8×C53#12011yescharacteristic
202C8×C53#12011yescharacteristic
203#88322—81no—
204#12751—531nomaximal, Sylow
205C22×C53—22no—
206C22×C53—41no—
207C22×C53—42no—
208C22×C53—42no—
209C22×C53—82no—
210C22×C53#41811yescharacteristic
211C22×C53#44711yescharacteristic
212C22×C53#45011yescharacteristic
213C4×C53—22no—
214C4×C53—44no—
215C4×C53—42no—
216C4×C53#37911yescharacteristic
217C4×C53C2×#10211yescharacteristic
218C4×C53#44711yescharacteristic
219C4×C53#45011yescharacteristic
220Dic53—161no—
221C2×#578D5311yescharacteristic
222#2148—531no—
223C2×#362—531no—
224#2145—531no—
225#2141—531no—
226#2117—531no—
227#1814—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—42no—
232C2×C53—82no—
233C2×C53#178611yescharacteristic
234C2×2−1+4—21no—
235#578—22no—
236#578—22no—
237#578—22no—
238#578Dic5314yescomplemented
239C2×#137—22no—
240C2×#137—21no—
241C2×#135—21no—
242C2×#135C2×D5311yescharacteristic
243C22×Q16—21no—
244C22×Q16C2×D5311yescharacteristic
245C2×#131C2×D5311yescharacteristic
246C2×M32—531no—
247C2×M32—531no—
248C2×#108—531no—
249#427—531no—
250#421—1061no—
251#362—1062no—
252#362—1061no—
253#361—1061no—
254#360—1062no—
255#358—1062no—
256#350—1062no—
257#344—1061no—
258C53#1275111yescharacteristic, complemented, Sylow
2592−1+4—24no—
2602−1+4—24no—
2612−1+4—42no—
2622−1+4—42no—
263C2×#41—22no—
264C2×#41—21no—
265C2×#41—42no—
266C2×#41—41no—
267C2×#41C22⋊D5311yescharacteristic
268C22×Q8—22no—
269C22×Q8—21no—
270C22×Q8—41no—
271#137—24no—
272#137—28no—
273#137—28no—
274#137—24no—
275#135—24no—
276#135—24no—
277#135—21no—
278#135—21no—
279#135C2×Dic5312yes—
280#135C22⋊D5312yes—
281C2×Q16—22no—
282C2×Q16—24no—
283C2×Q16—24no—
284C2×Q16—21no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16C2×Dic5312yes—
288C2×Q16C22⋊D5311yescharacteristic
289C2×Q16C22⋊D5312yes—
290C2×Q16C22⋊D5312yes—
291C2×SD16—22no—
292C2×SD16—21no—
293C2×D8C22⋊D5311yescharacteristic
294#131—21no—
295#131—21no—
296#131C2×Dic5312yes—
297#131C22⋊D5312yes—
298C2×M16—21no—
299C2×M16—1061no—
300C2×M16C22⋊D5311yescharacteristic
301C22×C8—21no—
302C22×C8C22×D5311yescharacteristic
303#117—1061no—
304M32—1061no—
305M32—1061no—
306M32—1061no—
307M32—1061no—
308C2×C16—1061no—
309C2×C16—1061no—
310#108—1061no—
311#108—1061no—
312#107—1061no—
313#106—1061no—
314#104—2122no—
315#41—21no—
316#41—21no—
317#41—28no—
318#41—22no—
319#41—22no—
320#41—44no—
321#41—48no—
322#41—48no—
323#41—44no—
324#41—44no—
325#41—41no—
326#41—41no—
327#41#1171992212yes—
328#41C22⋊Dic5312yes—
329C2×Q8—28no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8#1171992311yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4#1171992311yescharacteristic
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×C22⋊D5311yescharacteristic
353Q16—28no—
354Q16—22no—
355Q16—21no—
356Q16—22no—
357Q16—22no—
358Q16C22⋊Dic5312yes—
359Q16C22⋊Dic5314yes—
360SD16—28no—
361SD16—22no—
362SD16—22no—
363D8—21no—
364D8C22⋊Dic5312yes—
365M16—21no—
366M16—22no—
367M16—22no—
368M16—2121no—
369M16—2121no—
370M16C22⋊Dic5312yes—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—2121no—
375C2×C8C22×Dic5311yescharacteristic
376C2×C8C2×C22⋊D5311yescharacteristic
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C2×C22⋊D5311yescharacteristic
379C2×C8C2×C22⋊D5311yescharacteristic
380C2×C8C2×C22⋊D5311yescharacteristic
381C4⋊C4—2121no—
382C22⋊C4—2121no—
383C42—2121no—
384C16—2121no—
385C16—2121no—
386C23—21no—
387C23—41no—
388C23#1313324311yescharacteristic
389Q8—21no—
390Q8—24no—
391Q8—22no—
392Q8—48no—
393Q8—41no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8#1313313812yes—
398D4—22no—
399D4—22no—
400D4—21no—
401D4—22no—
402D4—48no—
403D4—42no—
404D4—42no—
405D4—41no—
406D4#1313313812yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—4241no—
419C2×C4C2×C22⋊Dic5311yescharacteristic
420C2×C4#1313318611yescharacteristic
421C2×C4#1313318611yescharacteristic
422C2×C4C2×#1171992211yescharacteristic
423C2×C4C2×#1171992311yescharacteristic
424C2×C4#1313324311yescharacteristic
425C8—22no—
426C8—4241no—
427C8C2×C22⋊Dic5311yescharacteristic
428C8C2×C22⋊Dic5311yescharacteristic
429C8#1313318611yescharacteristic
430C8#1313318611yescharacteristic
431C22—22no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22#1681062811yescharacteristic
437C22#1681075711yescharacteristic
438C22#1681077411yescharacteristic
439C4—22no—
440C4—44no—
441C4—42no—
442C4—8481no—
443C4#1681047911yescharacteristic
444C4C2×#1313313811yescharacteristic
445C4#1681075711yescharacteristic
446C4#1681077411yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2#2027464611yescharacteristic
subgroupG/H
centerC2#20274646
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#578C2
socleC2×C53#1786
cosocleC22×C8×C53C23
hypercenterC2×#578D53
hypocenterC53#12751
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. #20276464C2semidirect
2. C2×#16810439C2semidirect
3. #20276451C2semidirect
4. #20276435C2semidirect
5. #20276343C2semidirect
6. #20274769C2semidirect
24. C53×#578C4semidirect
238. #578Dic53semidirect
258. C53#12751semidirect

Maximal subgroups

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

subgroupindex
#202764642
C2×#168104392
#202764512
#202764352
#202763432
#202747692
C2×C53×#5782
#1275153