Finite Group Databasebeta

← #27367023

Subgroups — #27367023

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285109C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3C2×#16810833C211yesmaximal, characteristic, complemented
4#20274283C212yesmaximal, complemented
5#20274269C212yesmaximal, complemented
6#20274159C212yesmaximal, complemented
7#20274145C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273401C211yesmaximal, characteristic, complemented
10#20273393C211yesmaximal, characteristic
11Q8×M16×C53C211yesmaximal, characteristic, complemented
12C22×#11719925—21no—
13#16811173C2211yescharacteristic
14C2×#13133217C2211yescharacteristic
15#16810983—22no—
16#16810974C2212yes—
17#16810970C2212yes—
18#16810941—21no—
19#16810908C2211yescharacteristic
20#16810908C2211yescharacteristic
21#16810833—21no—
22#16810833—21no—
23#16810833C2212yes—
24#16810833C2212yes—
25#16810823C2211yescharacteristic
26#16810823C2211yescharacteristic
27#16810762—21no—
28C4×#11719924—21no—
29C4×#11719924—21no—
30#16810676C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32#16810625—21no—
33#16810625—21no—
34C2×#13133140—21no—
35#16810376C2212yes—
36#16810374C2212yes—
37#16810370C2212yes—
38#16810368C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yes—
44C8×Q8×C53C2212yes—
45C53×#423—21no—
46C53×#423C2211yescharacteristic
47C4×M16×C53—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719925—22no—
52C2×#11719925—22no—
53C2×#11719925—44no—
54C2×#11719924—21no—
55C2×#11719924—21no—
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#13133148—21no—
71#13133148—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2312yes—
76#13133142—22no—
77#13133142—22no—
78#13133141—21no—
79#13133141—21no—
80#13133140—22no—
81#13133140—22no—
82#13133139—22no—
83#13133139—22no—
84#13133137—21no—
85#13133137—21no—
86C22⋊#88322—21no—
87C4×#88322—21no—
88C22×Q8×C53D411yescharacteristic
89C2×M16×C53—21no—
90C2×M16×C53D411yescharacteristic
91C2×M16×C53D411yescharacteristic
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53D411yescharacteristic
95C4×Q8×C53D411yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C4×Q8×C53C2311yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102C4⋊C8×C53—22no—
103C4⋊C8×C53—22no—
104C4⋊C8×C53—21no—
105C4⋊C8×C53C2312yes—
106#97×C53—22no—
107#97×C53D412yes—
108C4×C8×C53—22no—
109C4×C8×C53C2312yes—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—41no—
113C2×Q8×C53C2×D411yescharacteristic
114C2×Q8×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C2×C22⋊D53—41no—
121C22×Dic53—41no—
122#11719945—42no—
123#11719945—42no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×Dic106—22no—
129C2×Dic106—21no—
130C2×Dic106—42no—
131C2×Dic106—42no—
132C2×Dic106—41no—
133M16×C53—22no—
134M16×C53—22no—
135M16×C53—42no—
136C2×C8×C53—22no—
137C2×C8×C53C2×D412yes—
138C2×C8×C53C2×D412yes—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C2×D411yescharacteristic
145C4⋊C4×C53C2×D411yescharacteristic
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719925—44no—
155#11719925—44no—
156#11719924—42no—
157#11719924—42no—
158#11719921—41no—
159#11719921—41no—
160C4⋊Dic53—22no—
161C4⋊Dic53—21no—
162C4⋊Dic53—22no—
163C4⋊Dic53—21no—
164C4⋊Dic53—41no—
165C4⋊Dic53—41no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—22no—
169C2×#88322—42no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172Q8×C53—21no—
173Q8×C53—21no—
174Q8×C53—21no—
175Q8×C53—42no—
176Q8×C53AGL(1,ℤ/8)12yes—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—81no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#13512yes—
190C2×C4×C53#13712yes—
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C22⋊D53—82no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—82no—
198D212—41no—
199D212—41no—
200D212—41no—
201D212—41no—
202D212—81no—
203C4×D53—82no—
204Dic106—41no—
205Dic106—42no—
206Dic106—41no—
207Dic106—42no—
208Dic106—82no—
209C8×C53—22no—
210C8×C53—22no—
211C8×C53—42no—
212#88322—42no—
213#88322—41no—
214#88322—41no—
215#19177—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53#46412yes—
219C22×C53C2×#12111yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—22no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—41no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53#46412yes—
231C4×C53C2×#13511yescharacteristic
232C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C53C2×#13711yescharacteristic
234C4×C53#57511yescharacteristic
235Dic53—81no—
236Dic53—82no—
237#3141—531no—
238#3103—531no—
239C2×#462—531no—
240Q8×M16D5311yescharacteristic, complemented
241#1702—532no—
242#1700—532no—
243#1682—532no—
244#1680—532no—
245#1565—531no—
246#1549—531no—
247#1547—531no—
248C2×C53—22no—
249C2×C53C2×#46411yescharacteristic
250C2×C53#305511yescharacteristic
251C2×C53#306911yescharacteristic
252D53—161no—
253C22×Q16—1061no—
254#532—531no—
255C2×C4⋊Q8—531no—
256C2×C4×Q8C2×D5311yescharacteristic
257#489—1062no—
258#487—532no—
259#486—532no—
260#481—1061no—
261#475—531no—
262#475—531no—
263#462—532no—
264#462—532no—
265#462—1061no—
266#462—1061no—
267#460—531no—
268#460—531no—
269#448—1061no—
270#446—22no—
271#446C2×D5312yes—
272C8×Q8C2×D5312yes—
273C4×SD16—1061no—
274C4×SD16—1061no—
275#431—531no—
276#423—21no—
277#423C2×D5311yescharacteristic
278C2×C4⋊C8—531no—
279#417—1061no—
280#417—1061no—
281C2×#103—1061no—
282C4×M16—21no—
283C4×M16C2×D5311yescharacteristic
284#335—532no—
285#334—532no—
286#332—532no—
287#332—532no—
288#332—532no—
289#331—532no—
290C53#1917711yescharacteristic, complemented, Sylow
291C2×#41—1061no—
292C22×Q8—1061no—
293C22×Q8D21211yescharacteristic
294C2×Q16—1062no—
295C2×Q16—1062no—
296C2×Q16—2124no—
297C2×SD16—1061no—
298C2×SD16—1061no—
299C2×M16—21no—
300C2×M16C22⋊D5311yescharacteristic
301C2×M16C22⋊D5311yescharacteristic
302C22×C8—1061no—
303C4⋊Q8—532no—
304C4⋊Q8—531no—
305C4⋊Q8—1061no—
306C4⋊Q8—1061no—
307C4⋊Q8—1061no—
308#127—531no—
309#124—1061no—
310#121—1061no—
311#121—1061no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8D21211yescharacteristic
315C4×Q8D21211yescharacteristic
316C4×Q8C22×D5311yescharacteristic
317C4×Q8C22×D5311yescharacteristic
318C4×D4—1061no—
319C4×D4—1061no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4D21211yescharacteristic
324C2×C42—21no—
325C2×C42C22×D5311yescharacteristic
326#106—1061no—
327#106—1061no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—21no—
331C4⋊C8—532no—
332C4⋊C8—531no—
333C4⋊C8—531no—
334C4⋊C8—1061no—
335C4⋊C8C22×D5312yes—
336#103—1061no—
337#103—1061no—
338#103—1062no—
339#103—1062no—
340#103—1062no—
341#103—1062no—
342#102—1062no—
343#102—1062no—
344#102—1061no—
345#102—1061no—
346C22⋊C8—1061no—
347#97—22no—
348#97C22⋊D5312yes—
349C4×C8—22no—
350C4×C8—1061no—
351C4×C8C22×D5312yes—
352#41—2122no—
353#41—2122no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×Q8—1061no—
358C2×Q8—1062no—
359C2×Q8—2122no—
360C2×Q8—2122no—
361C2×Q8—2121no—
362C2×Q8C2×D21211yescharacteristic
363C2×Q8C2×D21211yescharacteristic
364C2×D4—1061no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—2121no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—2121no—
371C22×C4—2121no—
372C22×C4C2×D21211yescharacteristic
373C22×C4#1171994511yescharacteristic
374C22×C4C2×C22⋊D5311yescharacteristic
375Q16—2124no—
376Q16—2124no—
377SD16—2122no—
378SD16—2122no—
379M16—22no—
380M16—22no—
381M16—42no—
382C2×C8—22no—
383C2×C8—1061no—
384C2×C8—1061no—
385C2×C8—1062no—
386C2×C8—2122no—
387C2×C8C2×C22⋊D5312yes—
388C2×C8C2×C22⋊D5312yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—1062no—
395C4⋊C4—1061no—
396C4⋊C4—1062no—
397C4⋊C4—1061no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4C2×D21211yescharacteristic
401C4⋊C4C2×D21211yescharacteristic
402C22⋊C4—2121no—
403C22⋊C4—2121no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42C2×D21211yescharacteristic
409C42#1171994511yescharacteristic
410C42C2×C22⋊D5311yescharacteristic
411C42C23×D5311yescharacteristic
412C23—2121no—
413C23#1313319111yescharacteristic
414Q8—21no—
415Q8—21no—
416Q8—21no—
417Q8—42no—
418Q8—4242no—
419Q8—2121no—
420Q8—2122no—
421Q8—2121no—
422Q8—2122no—
423Q8#1313324512yes—
424D4—4241no—
425D4—4242no—
426D4—2121no—
427D4—2121no—
428D4—2121no—
429D4—2121no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—4242no—
440C2×C4—4242no—
441C2×C4—2121no—
442C2×C4—2122no—
443C2×C4#1313319111yescharacteristic
444C2×C4#1313319111yescharacteristic
445C2×C4#1313319111yescharacteristic
446C2×C4#1313323712yes—
447C2×C4#1313324812yes—
448C2×C4C22×C22⋊D5311yescharacteristic
449C2×C4C22×D21211yescharacteristic
450C2×C4C2×#1171994511yescharacteristic
451C8—22no—
452C8—22no—
453C8—42no—
454C8—2122no—
455C8—2121no—
456C8—2121no—
457C22—21no—
458C22—21no—
459C22—4241no—
460C22—4241no—
461C22—4241no—
462C22#1681084512yes—
463C22C2×#1313319111yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—4242no—
472C4—4241no—
473C4#1681084512yes—
474C4C2×#1313323711yescharacteristic
475C4C2×#1313324511yescharacteristic
476C4C2×#1313324811yescharacteristic
477C4#1681142311yescharacteristic
478C2—22no—
479C2—8481no—
480C2C2×#1681084511yescharacteristic
481C2#2028433411yescharacteristic
482C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterQ8×M16D53
hypocenterC53#19177
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810833C2semidirect
4. #20274283C2semidirect
5. #20274269C2semidirect
6. #20274159C2semidirect
7. #20274145C2semidirect
8. #20273467C2semidirect
9. #20273401C2semidirect
11. Q8×M16×C53C2semidirect
240. Q8×M16D53semidirect
290. C53#19177semidirect

Maximal subgroups

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

subgroupindex
#202851092
#202847462
C2×#168108332
#202742832
#202742692
#202741592
#202741452
#202734672
#202734012
#202733932
Q8×M16×C532
#1917753