Finite Group Databasebeta

← #27367581

Subgroups — #27367581

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286022C211yesmaximal, characteristic
2#20285836C211yesmaximal, characteristic, complemented
3#20283461C211yesmaximal, characteristic, complemented
4#20274201C212yesmaximal, complemented
5#20274177C212yesmaximal, complemented
6#20274077C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3152C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16811026—21no—
15#16810971—21no—
16#16810936—22no—
17#16810903—21no—
18#16810900C2212yes—
19#16810900C2212yes—
20#16810900C2212yes—
21#16810857—22no—
22#16810818—21no—
23#16810815C2212yes—
24#16810702C2212yes—
25#16810696—22no—
26C8×D212C2212yes—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810591—21no—
31#16810376C2212yes—
32#16810374C2212yes—
33#16810369C2212yes—
34#16810369C2211yescharacteristic
35#16810369C2212yes—
36#16810369C2211yescharacteristic
37#16810367C2211yescharacteristic
38#16810367C2211yescharacteristic
39C2×#137×C53—21no—
40C2×#137×C53—21no—
41C53×#532C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#497—22no—
44C53×#494C2212yes—
45C53×#492C2212yes—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719922—22no—
49#13133225—44no—
50C2×#11719912—22no—
51C22×#88322—21no—
52C22×#88322—21no—
53C2×C8×D53—22no—
54#13133217D411yescharacteristic
55#13133214D411yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191D412yes—
59C4×D212—21no—
60C4×D212C2312yes—
61#13133150—22no—
62#13133150—22no—
63#13133150—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133138—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—22no—
77#13133129—22no—
78C4×#88322—21no—
79C2×#41×C53—21no—
80C22×Q8×C53—21no—
81#137×C53—44no—
82#137×C53—44no—
83C2×Q16×C53—22no—
84C2×Q16×C53—22no—
85C2×SD16×C53—22no—
86C2×SD16×C53—22no—
87C2×M16×C53D411yescharacteristic
88C2×M16×C53D411yescharacteristic
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53C2312yes—
93C4⋊Q8×C53C2311yescharacteristic
94#127×C53C2311yescharacteristic
95#124×C53—21no—
96#121×C53—21no—
97#121×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#97×C53D412yes—
104C4×C8×C53C2312yes—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—21no—
108C2×Q8×C53—22no—
109C2×Q8×C53—41no—
110C2×Q8×C53—42no—
111C2×Q8×C53—42no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—22no—
122#11719945—42no—
123#11719945—42no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212C2×D412yes—
127C2×C4×D53—22no—
128C2×Dic106—21no—
129Q16×C53—44no—
130Q16×C53—44no—
131SD16×C53—44no—
132SD16×C53—44no—
133M16×C53—22no—
134M16×C53D814yescomplemented
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149#11719922—82no—
150#11719921—22no—
151#11719921—22no—
152C4⋊Dic53—21no—
153C4⋊Dic53C2×D412yes—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—22no—
159C2×#88322—42no—
160C2×#88322—81no—
161#11719912—44no—
162C8×D53—44no—
163C23×C53—41no—
164C23×C53#12111yescharacteristic
165C22×D53—22no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—42no—
169Q8×C53—41no—
170Q8×C53—82no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—81no—
176D4×C53—82no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—82no—
183C2×C4×C53—82no—
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53C2×D812yes—
188C2×C4×C53AGL(1,ℤ/8)12yes—
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C22⋊D53—44no—
193C2×Dic53—22no—
194D212—22no—
195D212—22no—
196D212—42no—
197D212—81no—
198C4×D53—42no—
199C4×D53—42no—
200Dic106—42no—
201C8×C53—22no—
202C8×C53C2×D814yes—
203#88322—41no—
204#88322—41no—
205#88322—41no—
206#88322—42no—
207#88322—41no—
208#19231—531nomaximal, Sylow
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53#45912yes—
215C22×C53C2×#12111yescharacteristic
216C2×D53—42no—
217C2×D53—42no—
218C2×D53—42no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—81no—
223C4×C53—82no—
224C4×C53#45912yes—
225C4×C53C22×D811yescharacteristic
226C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C53#57711yescharacteristic
228C4×C53#57811yescharacteristic
229Dic53—42no—
230#3245—531no—
231#3226—531no—
232#3152D5311yescharacteristic, complemented
233#2935—531no—
234#1689—532no—
235#1685—532no—
236#1669—532no—
237#1665—532no—
238#1565—531no—
239#1540—531no—
240C2×#332—531no—
241C2×C53—22no—
242C2×C53—161no—
243C2×C53C2×#45911yescharacteristic
244C2×C53#306311yescharacteristic
245C2×C53#306911yescharacteristic
246D53—82no—
247C2×#137—21no—
248C2×#137—21no—
249C2×#131—1061no—
250#532—531no—
251#532C2×D5311yescharacteristic
252C2×C4⋊Q8C2×D5311yescharacteristic
253#500—1061no—
254#497—22no—
255#494C2×D5312yes—
256#492C2×D5312yes—
257#486—1061no—
258#480—1062no—
259#474—532no—
260#474—532no—
261#474—532no—
262#474—1061no—
263#466—1062no—
264#459—532no—
265#459—1061no—
266#436—532no—
267#435—1062no—
268C8×D4—532no—
269#431—531no—
270C2×C4⋊C8—531no—
271C2×C4⋊C8—531no—
272#408—1061no—
273C4×M16C2×D5311yescharacteristic
274#335—532no—
275#334—532no—
276#332—532no—
277#332—532no—
278#332—531no—
279#332—531no—
280#331—531no—
281#331—531no—
282C53#1923111yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1061no—
285C22×Q8—21no—
286#137—44no—
287#137—44no—
288C2×Q16—22no—
289C2×Q16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×D8—1062no—
293#131—2124no—
294C2×M16—1062no—
295C2×M16C22⋊D5311yescharacteristic
296C2×M16C22⋊D5311yescharacteristic
297C22×C8—1061no—
298C22×C8—1061no—
299C22×C8—1062no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—531no—
304C4⋊Q8C22×D5312yes—
305C4⋊Q8C22×D5311yescharacteristic
306#127—531no—
307#127C22×D5311yescharacteristic
308#124—21no—
309#124—1061no—
310#121—21no—
311#121—21no—
312#121—532no—
313#121—1061no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—532no—
317C4×D4—1061no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C42C22×D5311yescharacteristic
321#107—1062no—
322#107—1062no—
323#107—1062no—
324C4⋊C8—531no—
325C4⋊C8—531no—
326C4⋊C8—531no—
327C4⋊C8—531no—
328C4⋊C8—532no—
329C4⋊C8—1061no—
330C4⋊C8—1061no—
331#102—1062no—
332#102—1062no—
333#102—1062no—
334#102—1062no—
335C22⋊C8—1061no—
336C22⋊C8—1062no—
337C22⋊C8—1062no—
338#97C22⋊D5312yes—
339C4×C8—1061no—
340C4×C8C22×D5312yes—
341#41—42no—
342#41—42no—
343#41—2122no—
344#41—2122no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—1061no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C2×D4—532no—
356C2×D4—1062no—
357C2×D4—1061no—
358C2×D4—1061no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—1062no—
362C22×C4D4×D5311yescharacteristic
363C22×C4#1171994711yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365Q16—44no—
366Q16—44no—
367SD16—44no—
368SD16—44no—
369D8—4242no—
370M16—22no—
371M16—2124no—
372M16#1171992214yescomplemented
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1062no—
376C2×C8—1061no—
377C2×C8—1061no—
378C2×C8—4241no—
379C2×C8—2122no—
380C2×C8—2124no—
381C2×C8C2×C22⋊D5312yes—
382C2×C8C2×C22⋊D5312yes—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—532no—
390C4⋊C4—1061no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C42D4×D5311yescharacteristic
396C42#1171994711yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C23×D5311yescharacteristic
399C23—41no—
400C23—1062no—
401C23#1313318811yescharacteristic
402Q8—42no—
403Q8—41no—
404Q8—42no—
405Q8—41no—
406Q8—82no—
407Q8—2122no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—81no—
413D4—82no—
414D4—1062no—
415D4—1062no—
416D4—4241no—
417D4—2124no—
418D4—2122no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—1062no—
427C2×C4—2122no—
428C2×C4—2122no—
429C2×C4#1313318811yescharacteristic
430C2×C4#1313318811yescharacteristic
431C2×C4#1313318811yescharacteristic
432C2×C4C2×#1171992212yes—
433C2×C4#1313324312yes—
434C2×C4C2×D4×D5311yescharacteristic
435C2×C4C22×C22⋊D5311yescharacteristic
436C2×C4C2×#1171994711yescharacteristic
437C8—22no—
438C8—2121no—
439C8—2121no—
440C8—2121no—
441C8—2122no—
442C8—2121no—
443C8C2×#1171992214yes—
444C22—21no—
445C22—21no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—2122no—
450C22—2122no—
451C22—2122no—
452C22#1681081512yes—
453C22C2×#1313318811yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—82no—
459C4—2122no—
460C4#1681081512yes—
461C4C22×#1171992211yescharacteristic
462C4C2×#1313324311yescharacteristic
463C4#1681143211yescharacteristic
464C4#1681143911yescharacteristic
465C2—22no—
466C2—161no—
467C2—4242no—
468C2C2×#1681081511yescharacteristic
469C2#2028440011yescharacteristic
470C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#121
prime radical#20286022C2
hypercenter#3152D53
hypocenterC53#19231
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #20285836C2semidirect
3. #20283461C2semidirect
4. #20274201C2semidirect
5. #20274177C2semidirect
6. #20274077C2semidirect
7. #20274053C2semidirect
8. #20273467C2semidirect
9. #20273364C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3152C2semidirect
134. M16×C53D8semidirect
232. #3152D53semidirect
282. C53#19231semidirect
372. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202860222
#202858362
#202834612
#202742012
#202741772
#202740772
#202740532
#202734672
#202733642
C2×#168103692
C53×#31522
#1923153