Finite Group Databasebeta

← #27278937

Subgroups — #27278937

Order 13568.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284723C211yesmaximal, characteristic
2#20275659C211yesmaximal, characteristic, complemented
3C4×#13133138C211yesmaximal, characteristic
4#20273169C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7D4×M16×C53C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810953—21no—
10#16810913C411yescharacteristic, complemented
11#16810913C411yescharacteristic, complemented
12#16810828C411yescharacteristic, complemented
13#16810828C411yescharacteristic, complemented
14#16810774—21no—
15C4×#11719923—21no—
16C4×#11719923—21no—
17#16810676C2211yescharacteristic
18C2×#13133138—21no—
19C2×C4×#88322C2211yescharacteristic
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810384—21no—
23#16810384—21no—
24#16810353—22no—
25C22×M16×C53—21no—
26C2×C4×D4×C53C2211yescharacteristic
27C53×#436—21no—
28C53×#435—22no—
29C8×D4×C53—21no—
30C53×#423C2211yescharacteristic
31C53×#407—21no—
32C4×M16×C53C2211yescharacteristic
33C2×#11719923—21no—
34C2×#11719923—21no—
35C22×#88322—21no—
36#13133217C2×C411yescharacteristic
37#13133217C2×C411yescharacteristic
38#13133209—21no—
39#13133196—21no—
40#13133196—21no—
41C4×Dic106—21no—
42C4×Dic106—21no—
43#13133176—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133148—21no—
47#13133148—21no—
48C4⋊#88322C2×C411yescharacteristic
49C4⋊#88322C2×C411yescharacteristic
50#13133140—21no—
51#13133140—21no—
52#13133138—24no—
53#13133138—21no—
54#13133138—21no—
55C22⋊#88322—21no—
56C4×#88322—21no—
57C4×#88322—21no—
58C4×#88322—21no—
59#13133124—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62C2×M16×C53—22no—
63C2×M16×C53—21no—
64C2×M16×C53—42no—
65C2×M16×C53—42no—
66C2×M16×C53D411yescharacteristic
67C2×M16×C53D411yescharacteristic
68C22×C8×C53—22no—
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53C2×C411yescharacteristic
72C4×D4×C53C2×C411yescharacteristic
73C4×D4×C53D411yescharacteristic
74C4×D4×C53D411yescharacteristic
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C22⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53D412yes—
80C22⋊C8×C53—22no—
81C22⋊C8×C53—22no—
82#97×C53—21no—
83C4×C8×C53—21no—
84C24×C53—21no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53C22⋊C411yescharacteristic
89C2×D4×C53C22⋊C411yescharacteristic
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×Dic53—41no—
103C2×Dic106—21no—
104C2×Dic106—21no—
105M16×C53—21no—
106M16×C53—21no—
107M16×C53—21no—
108M16×C53—44no—
109M16×C53—41no—
110M16×C53—41no—
111M16×C53SD1612yescomplemented
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—22no—
115C2×C8×C53—44no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53C22⋊C411yescharacteristic
118C4⋊C4×C53C22⋊C411yescharacteristic
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C42×C53C22×C411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C22⋊Dic53—41no—
128#11719923—42no—
129#11719923—42no—
130C4⋊Dic53—22no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4⋊Dic53—41no—
136#11719919—41no—
137#11719919—41no—
138C4×Dic53—41no—
139C4×Dic53—41no—
140C4×Dic53—41no—
141C2×#88322—22no—
142C2×#88322—21no—
143C2×#88322—21no—
144C2×#88322—41no—
145C2×#88322—41no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—42no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—81no—
152C23×C53#12111yescharacteristic
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—21no—
156D4×C53—42no—
157D4×C53#10412yes—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53C2×C22⋊C411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×C4×C53C4×D411yescharacteristic
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12311yescharacteristic
179C2×C4×C53C2×SD1611yescharacteristic
180C2×C4×C53#13511yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C53#13711yescharacteristic
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188Dic106—42no—
189Dic106—41no—
190Dic106—41no—
191C8×C53—22no—
192C8×C53—42no—
193C8×C53—41no—
194#88322—42no—
195#88322—41no—
196#88322—41no—
197#8989—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—82no—
207C22×C53—82no—
208C22×C53#38611yescharacteristic
209C22×C53#46111yescharacteristic
210C22×C53#46311yescharacteristic
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—41no—
215C4×C53—41no—
216C4×C53C2×#10411yescharacteristic
217C4×C53#42111yescharacteristic
218C4×C53C4×SD1611yescharacteristic
219C4×C53#44011yescharacteristic
220C4×C53#45011yescharacteristic
221C4×C53#45211yescharacteristic
222Dic53—82no—
223Dic53—81no—
224Dic53—81no—
225#3101—531no—
226D4×M16D5311yescharacteristic
227#1986—531no—
228C4×#102—531no—
229#1493—531no—
230#1303—531no—
231#1278—531no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—42no—
235C2×C53—81no—
236C2×C53#187111yescharacteristic
237C2×C53#187711yescharacteristic
238C2×C53#188111yescharacteristic
239C22×M16—21no—
240#533—531no—
241C2×C4×D4C2×D5311yescharacteristic
242#483—1061no—
243#476—531no—
244#476—531no—
245#461—531no—
246#461—531no—
247#450—1061no—
248C4×SD16—1061no—
249C4×SD16—1061no—
250#436—21no—
251#435—22no—
252C8×D4—21no—
253#431—531no—
254#423C2×D5311yescharacteristic
255C2×#102—1061no—
256#407—21no—
257C4×M16C2×D5311yescharacteristic
258C2×C4×C8—531no—
259C4×C4⋊C4—531no—
260#340—1061no—
261#339—1061no—
262#339—1061no—
263Q8⋊C8—1062no—
264C53#898911yescharacteristic, complemented, Sylow
265C22×D4D21211yescharacteristic
266C23×C4—21no—
267C2×SD16—1061no—
268C2×SD16—1061no—
269C2×M16—22no—
270C2×M16—21no—
271C2×M16—42no—
272C2×M16—42no—
273C2×M16C22⋊D5311yescharacteristic
274C2×M16C22⋊D5311yescharacteristic
275C22×C8—22no—
276C22×C8—1061no—
277C4⋊Q8—531no—
278C4⋊Q8—531no—
279#125—1061no—
280C22⋊Q8—1061no—
281C22⋊Q8—1061no—
282C4×Q8—1061no—
283C4×Q8—1061no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C4×D5311yescharacteristic
287C4×D4C4×D5311yescharacteristic
288C4×D4C22⋊D5311yescharacteristic
289C4×D4C22⋊D5311yescharacteristic
290#117—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4D21211yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—1061no—
295C2×C42C22×D5311yescharacteristic
296#106—1061no—
297#106—1061no—
298C4⋊C8—21no—
299C4⋊C8—531no—
300C4⋊C8—531no—
301C4⋊C8C22⋊D5312yes—
302#103—1061no—
303#103—1061no—
304#102—1064no—
305#102—1061no—
306#102—1061no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—1061no—
310#97—21no—
311C4×C8—21no—
312C4×C8—1061no—
313C4×C8—1061no—
314C4×C8—1061no—
315#95—1061no—
316C24—21no—
317C2×Q8—1061no—
318C2×Q8—1061no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4#1171992111yescharacteristic
323C2×D4#1171992111yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—2121no—
334C22×C4C2×D21211yescharacteristic
335C22×C4#1171994511yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—2122no—
338SD16—2122no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16—44no—
343M16—41no—
344M16—41no—
345M16#1171992312yescomplemented
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—44no—
350C2×C8—1062no—
351C2×C8—1061no—
352C2×C8—1061no—
353C2×C8—2121no—
354C2×C8—2121no—
355C4⋊C4—21no—
356C4⋊C4—1062no—
357C4⋊C4—1061no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C4⋊C4—1061no—
361C4⋊C4—2121no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C4⋊C4#1171992111yescharacteristic
365C4⋊C4#1171992111yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—2121no—
371C42—2121no—
372C42—2121no—
373C42—2121no—
374C42C2×C4×D5311yescharacteristic
375C42#1171994511yescharacteristic
376C42C2×C22⋊D5311yescharacteristic
377C42C2×C22⋊D5311yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1313319111yescharacteristic
385Q8—2122no—
386Q8—2121no—
387Q8—2121no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1313314312yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2122no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C2×#1171992111yescharacteristic
414C2×C4C4×C22⋊D5311yescharacteristic
415C2×C4C4×C22⋊D5311yescharacteristic
416C2×C4#1313318611yescharacteristic
417C2×C4#1313319111yescharacteristic
418C2×C4#1313320011yescharacteristic
419C2×C4C2×#1171992311yescharacteristic
420C2×C4#1313323711yescharacteristic
421C2×C4#1313324311yescharacteristic
422C2×C4#1313324711yescharacteristic
423C8—22no—
424C8—42no—
425C8—41no—
426C8—2122no—
427C8—2121no—
428C8—2121no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1681050211yescharacteristic
440C22#1681082811yescharacteristic
441C22#1681083811yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—4242no—
448C4—4241no—
449C4—4241no—
450C4C2×#1313314311yescharacteristic
451C4#1681063911yescharacteristic
452C4C4×#1171992311yescharacteristic
453C4#1681072311yescharacteristic
454C4#1681077411yescharacteristic
455C4#1681078511yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2027501711yescharacteristic
461C2#2027504911yescharacteristic
462C2#2027507711yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingD4×M16×C53C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterD4×M16D53
hypocenterC53#8989
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #20275659C2semidirect
4. #20273169C2semidirect
5. #20272616C2semidirect
6. #20272553C2semidirect
10. #16810913C4semidirect
11. #16810913C4semidirect
12. #16810828C4semidirect
13. #16810828C4semidirect
111. M16×C53SD16semidirect
264. C53#8989semidirect
345. M16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847232
#202756592
C4×#131331382
#202731692
#202726162
#202725532
D4×M16×C532
#898953