Finite Group Databasebeta

← #27280357

Subgroups — #27280357

Order 13568.

724 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#20284715C211yesmaximal, characteristic
2#20275543C211yesmaximal, characteristic, complemented
3#20273209C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C42×D4×C53C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810936—21no—
10#16810903C412yescomplemented
11#16810818C412yescomplemented
12#16810757—21no—
13C4×#11719922—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17C2×#13133145—21no—
18C2×#13133145—21no—
19#16810356—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C2211yescharacteristic
23C22×C42×C53—21no—
24C4×C4⋊C4×C53—21no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C4×C22⋊C4×C53—21no—
27C4×C22⋊C4×C53—22no—
28C43×C53C2211yescharacteristic
29C2×#11719945—21no—
30C2×#11719922—22no—
31C2×#11719917—21no—
32#13133217C2×C411yescharacteristic, complemented
33#13133214C2×C411yescharacteristic, complemented
34#13133205—21no—
35#13133191—21no—
36#13133191—21no—
37C4×D212—21no—
38C4×D212—21no—
39#13133150—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yescomplemented
44#13133145—42no—
45#13133145—42no—
46#13133138—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—22no—
54C23×C4×C53—21no—
55C4×D4×C53—28no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53C2×C412yes—
59C4×D4×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—22no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—22no—
65C2×C42×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53—42no—
69C2×C42×C53—42no—
70C2×C42×C53D411yescharacteristic
71C2×C42×C53D411yescharacteristic
72C2×C42×C53C2311yescharacteristic
73#95×C53—22no—
74#95×C53D412yes—
75C24×C53—21no—
76C2×D4×C53—24no—
77C2×D4×C53—21no—
78C2×D4×C53C22⋊C412yes—
79C22×C4×C53—21no—
80C22×C4×C53—28no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—44no—
87C22×C4×C53—44no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C2×C22⋊D53—41no—
96#11719945—41no—
97#11719945—41no—
98#11719945—41no—
99#11719945—41no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×C4×D53—41no—
104C2×Dic106—21no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53C22⋊C412yes—
108C22⋊C4×C53—28no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C42×C53—24no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53—42no—
116C42×C53—44no—
117C42×C53—41no—
118C42×C53—41no—
119C42×C53D812yescomplemented
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53#4111yescharacteristic
124#11719922—44no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129#11719917—41no—
130#11719917—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53—24no—
135C23×C53—22no—
136C23×C53—42no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53#12011yescharacteristic
140C22×D53—41no—
141D4×C53—24no—
142D4×C53—22no—
143D4×C53#10414yescomplemented
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—24no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—416no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C22⋊D53—81no—
174C22⋊D53—81no—
175C2×Dic53—41no—
176D212—42no—
177D212—41no—
178D212—41no—
179D212—81no—
180C4×D53—81no—
181C4×D53—81no—
182Dic106—41no—
183Dic106—41no—
184#88322—42no—
185#88322—81no—
186#88322—81no—
187#9233—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—24no—
191C22×C53—44no—
192C22×C53—44no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53#38611yescharacteristic
196C22×C53#44711yescharacteristic
197C22×C53#44911yescharacteristic
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—24no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—42no—
206C4×C53—44no—
207C4×C53—42no—
208C4×C53C2×#10412yes—
209C4×C53C4×D811yescharacteristic
210C4×C53#44211yescharacteristic
211C4×C53#45911yescharacteristic
212C4×C53#46311yescharacteristic
213Dic53—81no—
214#3100—531no—
215C42×D4D5311yescharacteristic, complemented
216#1962—531no—
217#1502—531no—
218#1486—531no—
219#1303—531no—
220#1276—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—44no—
224C2×C53#187911yescharacteristic
225C2×C53#189611yescharacteristic
226C2×C53#190611yescharacteristic
227D53—161no—
228#532—531no—
229C2×C4×D4—21no—
230C2×C4×D4C2×D5311yescharacteristic
231C22×C42—21no—
232#480—1061no—
233#474—532no—
234#459—532no—
235#447—1061no—
236C4×D8—1062no—
237#431—531no—
238#431—531no—
239#423—531no—
240C2×#104—1061no—
241C2×#104—1061no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2×D5311yescharacteristic
244C4×C22⋊C4—21no—
245C4×C22⋊C4—22no—
246C43C2×D5311yescharacteristic
247#327—1062no—
248D4⋊C8—1062no—
249C53#923311yescharacteristic, complemented, Sylow
250C2×#41—1061no—
251C22×D4C22⋊D5311yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C2×D8—1062no—
255C2×M16—1061no—
256C4⋊Q8—531no—
257#127—531no—
258#124—1061no—
259#121—1061no—
260#121—1061no—
261C4×D4—28no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—1061no—
265C4×D4—1061no—
266C4×D4C4×D5312yes—
267C4×D4D21212yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊D5311yescharacteristic
279C2×C42C22⋊D5311yescharacteristic
280C2×C42C22×D5311yescharacteristic
281#107—1062no—
282C4⋊C8—532no—
283C4⋊C8—1061no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286#104—2122no—
287#104—2122no—
288#102—1062no—
289#102—1062no—
290C22⋊C8—1061no—
291C22⋊C8—1061no—
292C4×C8—1061no—
293C4×C8—1061no—
294#95—22no—
295#95C22⋊D5312yes—
296C24—21no—
297#41—2121no—
298#41—2121no—
299#41—2121no—
300#41—2121no—
301C2×Q8—1061no—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—2121no—
308C2×D4#1171992112yes—
309C22×C4—21no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—2121no—
323C22×C4C2×C22⋊D5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326D8—2124no—
327M16—2121no—
328M16—2121no—
329C2×C8—1062no—
330C2×C8—2121no—
331C2×C8—2121no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4#1171992112yes—
337C22⋊C4—28no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—2121no—
341C22⋊C4—2121no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—44no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42#1171992212yescomplemented
351C42C2×C4×D5311yescharacteristic
352C42C2×D21211yescharacteristic
353C42#1171994511yescharacteristic
354C42#1171994511yescharacteristic
355C23—24no—
356C23—22no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—2121no—
361C23#1313318611yescharacteristic
362Q8—2121no—
363Q8—2121no—
364D4—24no—
365D4—22no—
366D4—4241no—
367D4—4241no—
368D4—4241no—
369D4—2122no—
370D4—2121no—
371D4—2121no—
372D4#1313314514yescomplemented
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—2121no—
395C2×C4C2×#1171992111yescharacteristic
396C2×C4C4×C22⋊D5311yescharacteristic
397C2×C4C4×C22⋊D5311yescharacteristic
398C2×C4#1313319111yescharacteristic
399C2×C4#1313319111yescharacteristic
400C2×C4#1313320011yescharacteristic
401C2×C4C2×#1171992211yescharacteristic
402C2×C4#1313323711yescharacteristic
403C2×C4#1313324211yescharacteristic
404C2×C4#1313324311yescharacteristic
405C8—4241no—
406C8—4241no—
407C8—2122no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—4241no—
416C22—4241no—
417C22—4241no—
418C22#1681050211yescharacteristic
419C22#1681075711yescharacteristic
420C22#1681076811yescharacteristic
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—44no—
426C4—42no—
427C4—42no—
428C4—4241no—
429C4C2×#1313314512yes—
430C4C4×#1171992211yescharacteristic
431C4#1681073311yescharacteristic
432C4#1681081811yescharacteristic
433C4#1681083811yescharacteristic
434C2—21no—
435C2—21no—
436C2—44no—
437C2—8481no—
438C2#2027506311yescharacteristic
439C2#2027515811yescharacteristic
440C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC42×D4×C53C2
socleC22×C53#386
prime radical#20284715C2
cosocleC2×C42×C53C23
hypercenterC42×D4D53
hypocenterC53#9233
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #20275543C2semidirect
3. #20273209C2semidirect
4. #20273140C2semidirect
5. #20272618C2semidirect
6. #20272548C2semidirect
7. C42×D4×C53C2semidirect
10. #16810903C4semidirect
11. #16810818C4semidirect
32. #13133217C2×C4semidirect
33. #13133214C2×C4semidirect
43. C4⋊#88322C2×C4semidirect
119. C42×C53D8semidirect
143. D4×C53#104semidirect
215. C42×D4D53semidirect
249. C53#9233semidirect
350. C42#11719922semidirect
372. D4#13133145semidirect

Maximal subgroups

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

subgroupindex
#202847152
#202755432
#202732092
#202731402
#202726182
#202725482
C42×D4×C532
#923353