Finite Group Databasebeta

← #27367592

Subgroups — #27367592

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285640C211yesmaximal, characteristic
3#20283461C211yesmaximal, characteristic
4#20274225C212yesmaximal, complemented
5#20274211C212yesmaximal, complemented
6#20274101C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16810997—21no—
15#16810973—21no—
16#16810910C2212yes—
17#16810908—21no—
18#16810854—22no—
19#16810825C2212yes—
20#16810825C2212yes—
21#16810825C2212yes—
22#16810823—21no—
23#16810762—22no—
24#16810702C2212yes—
25#16810696—22no—
26C8×D212C2212yes—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810591—21no—
31#16810376C2212yes—
32#16810374C2212yes—
33#16810372C2211yescharacteristic
34#16810372C2211yescharacteristic
35#16810369C2212yes—
36#16810369C2211yescharacteristic
37#16810369C2212yes—
38#16810369C2211yescharacteristic
39C53×#533C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#501—22no—
42C53×#500C2212yes—
43C53×#498C2212yes—
44C53×#428—21no—
45C53×#428—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719924—22no—
49C2×#11719924—22no—
50C2×#11719924—22no—
51#13133225—44no—
52C2×#11719912—22no—
53C22×#88322—21no—
54C22×#88322—21no—
55C2×C8×D53—22no—
56#13133217D411yescharacteristic
57#13133214D411yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191D412yes—
61C4×D212—21no—
62C4×D212C2312yes—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133141—22no—
72#13133141—22no—
73#13133141—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—22no—
77#13133129—22no—
78C4×#88322—21no—
79C22×Q8×C53—21no—
80C2×M16×C53D411yescharacteristic
81C2×M16×C53D411yescharacteristic
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53—21no—
85C4⋊Q8×C53C2311yescharacteristic
86C4⋊Q8×C53C2312yes—
87C4⋊Q8×C53C2311yescharacteristic
88#125×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93#117×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C42×C53C2311yescharacteristic
97#107×C53—22no—
98#107×C53—22no—
99#106×C53—22no—
100#106×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C2×Q8×C53—21no—
104C2×Q8×C53—22no—
105C2×Q8×C53—21no—
106C2×Q8×C53—41no—
107C2×Q8×C53—81no—
108C2×Q8×C53—81no—
109C22×C4×C53—41no—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C2×C22⋊D53—22no—
114#11719945—42no—
115#11719945—42no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212C2×D412yes—
119C2×C4×D53—22no—
120C2×Dic106—21no—
121M16×C53—22no—
122M16×C53SD1614yescomplemented
123C2×C8×C53C2×D412yes—
124C2×C8×C53C2×D412yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C42×C53—41no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141#11719924—82no—
142#11719924—82no—
143#11719924—82no—
144#11719921—22no—
145#11719921—22no—
146C4⋊Dic53—21no—
147C4⋊Dic53C2×D412yes—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—21no—
151C2×#88322—21no—
152C2×#88322—22no—
153C2×#88322—42no—
154C2×#88322—81no—
155#11719912—44no—
156C8×D53—44no—
157C23×C53#12111yescharacteristic
158C22×D53—22no—
159Q8×C53—42no—
160Q8×C53—41no—
161Q8×C53—41no—
162Q8×C53—42no—
163Q8×C53—82no—
164Q8×C53—81no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—82no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53C2×SD1612yes—
176C2×C4×C53#13712yes—
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C22⋊D53—44no—
181C2×Dic53—22no—
182D212—22no—
183D212—22no—
184D212—42no—
185D212—81no—
186C4×D53—42no—
187C4×D53—42no—
188Dic106—42no—
189C8×C53—22no—
190C8×C53C2×SD1614yes—
191#88322—41no—
192#88322—41no—
193#88322—42no—
194#88322—41no—
195#88322—41no—
196#19232—531nomaximal, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53#46112yes—
200C22×C53C2×#12111yescharacteristic
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—42no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—82no—
208C4×C53—81no—
209C4×C53—81no—
210C4×C53#46112yes—
211C4×C53C22×SD1611yescharacteristic
212C4×C53C2×#13711yescharacteristic
213C4×C53#57611yescharacteristic
214C4×C53#57711yescharacteristic
215Dic53—42no—
216#3246—531no—
217#3206—531no—
218#3165D5311yescharacteristic, complemented
219#2935—531no—
220#1693—532no—
221#1691—532no—
222#1673—532no—
223#1671—532no—
224#1565—531no—
225#1542—531no—
226C2×#332—531no—
227C2×C53—22no—
228C2×C53C2×#46111yescharacteristic
229C2×C53#306211yescharacteristic
230C2×C53#306911yescharacteristic
231D53—82no—
232C2×#131—1061no—
233#533C2×D5311yescharacteristic
234#532—531no—
235C2×C4⋊Q8C2×D5311yescharacteristic
236#501—22no—
237#500C2×D5312yes—
238#498C2×D5312yes—
239#492—1061no—
240#487—1061no—
241#476—532no—
242#475—1061no—
243#465—1062no—
244#461—532no—
245#461—532no—
246#461—532no—
247#460—1061no—
248#448—1062no—
249#436—532no—
250#435—1062no—
251C8×D4—532no—
252#431—531no—
253#428—21no—
254#428—21no—
255C2×C4⋊C8—531no—
256C2×C4⋊C8—531no—
257#408—1061no—
258C4×M16C2×D5311yescharacteristic
259#335—532no—
260#334—532no—
261#333—531no—
262#333—531no—
263#332—532no—
264#332—532no—
265#332—531no—
266#332—531no—
267C53#1923211yescharacteristic, complemented, Sylow
268C2×#41—1061no—
269C22×Q8—21no—
270C2×SD16—1062no—
271C2×SD16—1062no—
272C2×SD16—1062no—
273#131—2124no—
274C2×M16—1062no—
275C2×M16C22⋊D5311yescharacteristic
276C2×M16C22⋊D5311yescharacteristic
277C22×C8—1061no—
278C22×C8—1061no—
279C22×C8—1062no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—531no—
284C4⋊Q8C22×D5312yes—
285C4⋊Q8C22×D5311yescharacteristic
286C4⋊Q8C22×D5311yescharacteristic
287#127—531no—
288#125—21no—
289#124—1061no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292#121—532no—
293#121—1061no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×D4—532no—
297C4×D4—1061no—
298#117—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C42C22×D5311yescharacteristic
302#107—22no—
303#107—22no—
304#106—22no—
305#106—22no—
306#106—1062no—
307C4⋊C8—531no—
308C4⋊C8—531no—
309C4⋊C8—532no—
310C4⋊C8—531no—
311C4⋊C8—531no—
312C4⋊C8—1061no—
313C4⋊C8—1061no—
314#103—1062no—
315#103—1062no—
316#103—1062no—
317#102—1062no—
318C22⋊C8—1061no—
319C22⋊C8—1062no—
320C22⋊C8—1062no—
321#97C22⋊D5312yes—
322C4×C8—1061no—
323C4×C8C22×D5312yes—
324#41—2122no—
325#41—2122no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—41no—
330C2×Q8—81no—
331C2×Q8—81no—
332C2×Q8—1061no—
333C2×D4—532no—
334C2×D4—1062no—
335C2×D4—1061no—
336C2×D4—1061no—
337C22×C4—41no—
338C22×C4—1062no—
339C22×C4D4×D5311yescharacteristic
340C22×C4#1171994711yescharacteristic
341C22×C4C2×C22⋊D5311yescharacteristic
342SD16—4242no—
343SD16—4242no—
344SD16—4242no—
345M16—22no—
346M16—2124no—
347M16#1171992414yescomplemented
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1062no—
351C2×C8—1061no—
352C2×C8—1061no—
353C2×C8—4241no—
354C2×C8—2122no—
355C2×C8—2124no—
356C2×C8C2×C22⋊D5312yes—
357C2×C8C2×C22⋊D5312yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—532no—
369C4⋊C4—1061no—
370C22⋊C4—41no—
371C22⋊C4—1062no—
372C22⋊C4—1062no—
373C42—41no—
374C42D4×D5311yescharacteristic
375C42#1171994711yescharacteristic
376C42C2×C22⋊D5311yescharacteristic
377C42C23×D5311yescharacteristic
378C23—1062no—
379C23#1313318811yescharacteristic
380Q8—41no—
381Q8—42no—
382Q8—42no—
383Q8—41no—
384Q8—81no—
385Q8—82no—
386Q8—2122no—
387D4—1062no—
388D4—1062no—
389D4—4241no—
390D4—2124no—
391D4—2122no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—1062no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4#1313318811yescharacteristic
403C2×C4#1313318811yescharacteristic
404C2×C4#1313318811yescharacteristic
405C2×C4C2×#1171992412yes—
406C2×C4#1313324812yes—
407C2×C4C2×D4×D5311yescharacteristic
408C2×C4C22×C22⋊D5311yescharacteristic
409C2×C4C2×#1171994711yescharacteristic
410C8—22no—
411C8—2121no—
412C8—2121no—
413C8—2122no—
414C8—2121no—
415C8—2121no—
416C8C2×#1171992414yes—
417C22—21no—
418C22—21no—
419C22—2122no—
420C22—2122no—
421C22—2122no—
422C22#1681082512yes—
423C22C2×#1313318811yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—82no—
429C4—81no—
430C4—2122no—
431C4#1681082512yes—
432C4C22×#1171992411yescharacteristic
433C4C2×#1313324811yescharacteristic
434C4#1681142711yescharacteristic
435C4#1681143311yescharacteristic
436C2—22no—
437C2—4242no—
438C2C2×#1681082511yescharacteristic
439C2#2028438911yescharacteristic
440C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3165D53
hypocenterC53#19232
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20274225C2semidirect
5. #20274211C2semidirect
6. #20274101C2semidirect
7. #20274087C2semidirect
8. #20273467C2semidirect
9. #20273372C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3165C2semidirect
122. M16×C53SD16semidirect
218. #3165D53semidirect
267. C53#19232semidirect
347. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202856402
#202834612
#202742252
#202742112
#202741012
#202740872
#202734672
#202733722
C2×#168103692
C53×#31652
#1923253