Finite Group Databasebeta

← #27367587

Subgroups — #27367587

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#20285833C211yesmaximal, characteristic, complemented
3#20285174C211yesmaximal, characteristic, complemented
4#20274204C212yesmaximal, complemented
5#20274180C212yesmaximal, complemented
6#20274080C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273365C211yesmaximal, characteristic, complemented
10C2×#16810370C211yesmaximal, characteristic, complemented
11C53×#2935C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133250—21no—
14#16811173C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16#16811026—21no—
17#16811017—22no—
18#16811003C2212yes—
19#16810996C2212yes—
20#16810969—21no—
21#16810934—22no—
22#16810903C2212yes—
23#16810903C2212yes—
24#16810903C2212yes—
25#16810900—21no—
26#16810857—22no—
27#16810818C2212yes—
28#16810815—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810376C2212yes—
32#16810374C2212yes—
33#16810370C2212yes—
34#16810370C2211yescharacteristic
35#16810370C2212yes—
36#16810370C2211yescharacteristic
37#16810366C2211yescharacteristic
38#16810366C2211yescharacteristic
39C2×#131×C53—21no—
40C53×#532C2211yescharacteristic
41C53×#436C2212yes—
42C53×#435—22no—
43C8×D4×C53C2212yes—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#408—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×Dic106—21no—
49#13133250—44no—
50#13133250—44no—
51C2×Dic212—22no—
52C2×Dic212—22no—
53C2×#11719913—22no—
54C2×#11719913—22no—
55C2×#11719922—22no—
56C22×#88322—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217—21no—
60#13133217C2312yes—
61#13133217C2311yescharacteristic
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133150—22no—
71#13133150—22no—
72#13133150—22no—
73C4⋊#88322—21no—
74C4⋊#88322C2312yes—
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2311yescharacteristic
77#13133138—22no—
78#13133138—22no—
79#13133138—22no—
80#13133137—22no—
81C22⋊#88322—21no—
82C4×#88322—21no—
83C2×#41×C53—21no—
84#131×C53—44no—
85C2×M16×C53—22no—
86C2×M16×C53D411yescharacteristic
87C2×M16×C53D411yescharacteristic
88C22×C8×C53—22no—
89C22×C8×C53—21no—
90C4⋊Q8×C53D411yescharacteristic
91#127×C53D411yescharacteristic
92#124×C53—21no—
93#121×C53—21no—
94#121×C53D412yes—
95C4×D4×C53—21no—
96C4×D4×C53C2312yes—
97C2×C42×C53C2311yescharacteristic
98C4⋊C8×C53—21no—
99C4⋊C8×C53C2311yescharacteristic
100C4⋊C8×C53C2311yescharacteristic
101C22⋊C8×C53—22no—
102C22⋊C8×C53—22no—
103#97×C53D412yes—
104C4×C8×C53C2312yes—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—21no—
108C2×D4×C53—22no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53C2×D412yes—
112C22×C4×C53—22no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C2×C22⋊D53—41no—
117C22×Dic53—41no—
118#11719945—42no—
119#11719945—42no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212—21no—
123C2×C4×D53—41no—
124C2×Dic106—21no—
125C2×Dic106—22no—
126C2×Dic106—42no—
127C2×Dic106—42no—
128C2×Dic106—41no—
129M16×C53—22no—
130M16×C53—44no—
131M16×C53D814yescomplemented
132C2×C8×C53—21no—
133C2×C8×C53—21no—
134C2×C8×C53—42no—
135C2×C8×C53—44no—
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C2×D412yes—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53#4111yescharacteristic
145C42×C53C2411yescharacteristic
146#11719922—82no—
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—22no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—21no—
153C4⋊Dic53—41no—
154C4⋊Dic53—41no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322—22no—
158C2×#88322—81no—
159Dic212—44no—
160Dic212—44no—
161#11719913—44no—
162#11719913—44no—
163C23×C53—22no—
164C23×C53#12111yescharacteristic
165C22×D53—41no—
166Q8×C53—42no—
167D4×C53—22no—
168D4×C53—22no—
169D4×C53—44no—
170D4×C53—42no—
171D4×C53—81no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—22no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53C2×D812yes—
182C2×C4×C53AGL(1,ℤ/8)12yes—
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C2×#4111yescharacteristic
186C22⋊D53—82no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—82no—
190D212—41no—
191D212—41no—
192D212—41no—
193D212—41no—
194D212—81no—
195C4×D53—82no—
196Dic106—42no—
197Dic106—41no—
198Dic106—42no—
199Dic106—41no—
200Dic106—82no—
201C8×C53—22no—
202C8×C53—41no—
203C8×C53—41no—
204C8×C53C2×D814yes—
205#88322—41no—
206#88322—42no—
207#88322—41no—
208#19231—531nomaximal, Sylow
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53#45912yes—
215C22×C53C2×#12111yescharacteristic
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—81no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—42no—
223C4×C53#45912yes—
224C4×C53C22×D811yescharacteristic
225C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C53#57711yescharacteristic
227C4×C53#57811yescharacteristic
228Dic53—82no—
229Dic53—81no—
230#3245—531no—
231#3226—531no—
232#3152—531no—
233#2935D5311yescharacteristic, complemented
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—82no—
243C2×C53C2×#45911yescharacteristic
244C2×C53#306311yescharacteristic
245C2×C53#306911yescharacteristic
246D53—161no—
247C2×#137—1061no—
248C2×#137—1061no—
249C2×#131—21no—
250#532—531no—
251#532C2×D5311yescharacteristic
252C2×C4⋊Q8—531no—
253#500—1061no—
254#497—1062no—
255#494—532no—
256#492—532no—
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#436C2×D5312yes—
267#435—22no—
268C8×D4C2×D5312yes—
269#431—531no—
270C2×C4⋊C8—531no—
271C2×C4⋊C8C2×D5311yescharacteristic
272#408—21no—
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—1061no—
286#137—2124no—
287#137—2124no—
288C2×Q16—1062no—
289C2×Q16—1062no—
290C2×SD16—1062no—
291C2×SD16—1062no—
292C2×D8—1062no—
293#131—44no—
294C2×M16—22no—
295C2×M16C22⋊D5311yescharacteristic
296C2×M16C22⋊D5311yescharacteristic
297C22×C8—21no—
298C22×C8—22no—
299C22×C8—1061no—
300C4⋊Q8—532no—
301C4⋊Q8—531no—
302C4⋊Q8—1061no—
303C4⋊Q8—1061no—
304C4⋊Q8—1061no—
305C4⋊Q8D21211yescharacteristic
306#127—531no—
307#127D21211yescharacteristic
308#124—21no—
309#124—1061no—
310#121—21no—
311#121—1061no—
312#121—1061no—
313#121D21212yes—
314C4×D4—21no—
315C4×D4—1061no—
316C4×D4—1061no—
317C4×D4C22×D5312yes—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4—1061no—
320C2×C42C22×D5311yescharacteristic
321#107—1062no—
322#107—1062no—
323#107—1062no—
324C4⋊C8—21no—
325C4⋊C8—531no—
326C4⋊C8—531no—
327C4⋊C8—532no—
328C4⋊C8—1061no—
329C4⋊C8C22×D5311yescharacteristic
330C4⋊C8C22×D5311yescharacteristic
331#102—1062no—
332#102—1062no—
333#102—1062no—
334#102—1062no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8—1061no—
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—1061no—
347C2×Q8—1062no—
348C2×Q8—2122no—
349C2×Q8—2122no—
350C2×Q8—2121no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—1061no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—2121no—
358C2×D4C2×D21212yes—
359C22×C4—22no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4C2×D21211yescharacteristic
363C22×C4#1171994511yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365Q16—2124no—
366Q16—2124no—
367SD16—2124no—
368SD16—2124no—
369D8—4242no—
370M16—22no—
371M16—44no—
372M16#1171992214yescomplemented
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—42no—
376C2×C8—44no—
377C2×C8—1061no—
378C2×C8—1061no—
379C2×C8—1062no—
380C2×C8—4241no—
381C2×C8C2×C22⋊D5312yes—
382C2×C8C2×C22⋊D5312yes—
383C4⋊C4—21no—
384C4⋊C4—1062no—
385C4⋊C4—1061no—
386C4⋊C4—1062no—
387C4⋊C4—1061no—
388C4⋊C4—2121no—
389C4⋊C4—2121no—
390C4⋊C4C2×D21212yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42C2×D21211yescharacteristic
396C42#1171994511yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C23×D5311yescharacteristic
399C23—22no—
400C23—2121no—
401C23#1313319111yescharacteristic
402Q8—42no—
403Q8—4242no—
404Q8—2122no—
405Q8—2121no—
406Q8—2122no—
407Q8—2121no—
408D4—22no—
409D4—22no—
410D4—44no—
411D4—42no—
412D4—81no—
413D4—4241no—
414D4—4242no—
415D4—2121no—
416D4—2121no—
417D4—2121no—
418D4—2121no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—4242no—
426C2×C4—4242no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4#1313319111yescharacteristic
430C2×C4#1313319111yescharacteristic
431C2×C4#1313319111yescharacteristic
432C2×C4C2×#1171992212yes—
433C2×C4#1313324212yes—
434C2×C4C22×C22⋊D5311yescharacteristic
435C2×C4C22×D21211yescharacteristic
436C2×C4C2×#1171994511yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8—2121no—
441C8—2122no—
442C8—2121no—
443C8C2×#1171992214yes—
444C22—21no—
445C22—21no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22#1681081812yes—
453C22C2×#1313319111yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—4241no—
459C4—4242no—
460C4#1681081812yes—
461C4C22×#1171992211yescharacteristic
462C4C2×#1313324211yescharacteristic
463C4#1681143511yescharacteristic
464C4#1681144011yescharacteristic
465C2—22no—
466C2—82no—
467C2—8481no—
468C2C2×#1681081811yescharacteristic
469C2#2028440611yescharacteristic
470C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2935C2
socleC22×C53C2×#121
prime radical#20286022C2
hypercenter#2935D53
hypocenterC53#19231
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #20285833C2semidirect
3. #20285174C2semidirect
4. #20274204C2semidirect
5. #20274180C2semidirect
6. #20274080C2semidirect
7. #20274056C2semidirect
8. #20273467C2semidirect
9. #20273365C2semidirect
10. C2×#16810370C2semidirect
11. C53×#2935C2semidirect
131. M16×C53D8semidirect
233. #2935D53semidirect
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
#202858332
#202851742
#202742042
#202741802
#202740802
#202740562
#202734672
#202733652
C2×#168103702
C53×#29352
#1923153