Finite Group Databasebeta

← #27280360

Subgroups — #27280360

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#20284711C211yesmaximal, characteristic
2C42×D212C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273139C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C53×#1962C211yesmaximal, characteristic, complemented
8C2×C4×D212—21no—
9C2×C4×D212C2211yescharacteristic
10C2×C42×D53—21no—
11#16810934—21no—
12#16810900C412yescomplemented
13#16810815C412yescomplemented
14#16810756—21no—
15C4×#11719922—22no—
16#16810676C2211yescharacteristic
17#16810676C2211yescharacteristic
18C4×C4⋊Dic53—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20C4×#11719921—22no—
21C4×#11719921—21no—
22#16810357—22no—
23#16810350—22no—
24C53×#532C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C2×#104×C53—21no—
27C2×#104×C53—21no—
28C43×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—22no—
31C22×C4×D53—21no—
32C2×#11719922—22no—
33C4×D212—28no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212C2×C412yes—
37C4×D212D412yes—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—21no—
44C42×D53—42no—
45C42×D53—42no—
46#13133150—22no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yescomplemented
49#13133138—22no—
50#13133137—22no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—22no—
56#13133124D412yes—
57C2×#41×C53—21no—
58C2×M16×C53—21no—
59C4⋊Q8×C53C2×C411yescharacteristic, complemented
60#127×C53C2×C411yescharacteristic, complemented
61#124×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53D411yescharacteristic
69C2×C42×C53D411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#104×C53—42no—
74#104×C53—42no—
75C23×D53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—21no—
92C22×Dic53—22no—
93C2×D212—21no—
94C2×D212—24no—
95C2×D212C22⋊C412yes—
96C2×C4×D53—28no—
97C2×C4×D53—22no—
98C2×C4×D53—22no—
99C2×C4×D53—44no—
100C2×C4×D53—44no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105M16×C53—41no—
106M16×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C42×C53—24no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—42no—
117C42×C53D812yescomplemented
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53#4111yescharacteristic
122#11719922—44no—
123#11719921—28no—
124#11719921—22no—
125#11719921—22no—
126C4⋊Dic53—24no—
127C4⋊Dic53—21no—
128C4⋊Dic53C22⋊C412yes—
129C4×Dic53—44no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132C2×#88322—22no—
133C2×#88322—41no—
134C23×C53—41no—
135C23×C53#12011yescharacteristic
136C22×D53—24no—
137C22×D53—22no—
138C22×D53—42no—
139C22×D53—41no—
140C22×D53—41no—
141Q8×C53—41no—
142Q8×C53—41no—
143D4×C53—42no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53#13511yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×Dic53—24no—
171C2×Dic53—22no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176D212—24no—
177D212—22no—
178D212#10414yescomplemented
179C4×D53—416no—
180C4×D53—42no—
181C4×D53—42no—
182C4×D53—42no—
183C4×D53—42no—
184C8×C53—81no—
185#88322—42no—
186#88322—81no—
187#9233—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53#38611yescharacteristic
194C22×C53#44711yescharacteristic
195C22×C53#44911yescharacteristic
196C2×D53—24no—
197C2×D53—44no—
198C2×D53—44no—
199C2×D53—42no—
200C2×D53—42no—
201C4×C53—24no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—42no—
206C4×C53—81no—
207C4×C53C2×#10412yes—
208C4×C53C4×D811yescharacteristic
209C4×C53#44211yescharacteristic
210C4×C53#45911yescharacteristic
211C4×C53#46311yescharacteristic
212Dic53—44no—
213Dic53—42no—
214#3100—531no—
215C42×D4—531no—
216#1962D5311yescharacteristic, complemented
217#1502—531no—
218#1486—531no—
219#1303—531no—
220#1276—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—161no—
224C2×C53#187911yescharacteristic
225C2×C53#189611yescharacteristic
226C2×C53#190611yescharacteristic
227D53—44no—
228#532C2×D5311yescharacteristic
229C2×C4×D4—531no—
230C2×C4×D4—1061no—
231C22×C42—1061no—
232#480—1061no—
233#474—532no—
234#459—532no—
235#447—1061no—
236C4×D8—1062no—
237#431—531no—
238#431—531no—
239#423C2×D5311yescharacteristic
240C2×#104—21no—
241C2×#104—21no—
242C4×C4⋊C4—531no—
243C4×C4⋊C4—1061no—
244C4×C22⋊C4—1062no—
245C4×C22⋊C4—1061no—
246C43C2×D5311yescharacteristic
247#327—1062no—
248D4⋊C8—1062no—
249C53#923311yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C22×D4—531no—
252C23×C4—1062no—
253C23×C4—1061no—
254C2×D8—1062no—
255C2×M16—21no—
256C4⋊Q8C4×D5311yescharacteristic, complemented
257#127C4×D5311yescharacteristic, complemented
258#124—21no—
259#121—21no—
260#121—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—532no—
264C4×D4—532no—
265C4×D4—1068no—
266C4×D4—1061no—
267C4×D4—1061no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—1061no—
270C2×C22⋊C4—1062no—
271C2×C22⋊C4—1061no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—1062no—
275C2×C42—1061no—
276C2×C42—2122no—
277C2×C42—2122no—
278C2×C42C22⋊D5311yescharacteristic
279C2×C42C22⋊D5311yescharacteristic
280C2×C42C22×D5311yescharacteristic
281#107—1062no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—532no—
285C4⋊C8—1061no—
286#104—42no—
287#104—42no—
288#102—1062no—
289#102—1062no—
290C22⋊C8—1061no—
291C22⋊C8—1061no—
292C4×C8—1061no—
293C4×C8—1061no—
294#95—532no—
295#95—1062no—
296C24—1061no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301C2×Q8—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—532no—
307C2×D4—1064no—
308C2×D4—1061no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—1068no—
313C22×C4—1062no—
314C22×C4—1061no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—2124no—
318C22×C4—2124no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4D4×D5311yescharacteristic
324C22×C4D4×D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326D8—2124no—
327M16—41no—
328M16—41no—
329C2×C8—41no—
330C2×C8—1062no—
331C2×C8—2121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—532no—
335C4⋊C4—1064no—
336C4⋊C4—1061no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1068no—
340C22⋊C4—1062no—
341C22⋊C4—1062no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—42no—
347C42—2124no—
348C42—2121no—
349C42—2121no—
350C42#1171992212yescomplemented
351C42C2×C4×D5311yescharacteristic
352C42#1171994511yescharacteristic
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C23—41no—
356C23—1064no—
357C23—1062no—
358C23—2122no—
359C23—2121no—
360C23—2121no—
361C23#1313318511yescharacteristic
362Q8—41no—
363Q8—41no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—534no—
371D4—1064no—
372D4—1062no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—1064no—
385C2×C4—1062no—
386C2×C4—21216no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4C22⋊C4×D5311yescharacteristic
396C2×C4#1313317711yescharacteristic
397C2×C4C4×C22⋊D5311yescharacteristic
398C2×C4#1313318811yescharacteristic
399C2×C4#1313319011yescharacteristic
400C2×C4#1313319911yescharacteristic
401C2×C4C2×#1171992211yescharacteristic
402C2×C4#1313323711yescharacteristic
403C2×C4#1313324211yescharacteristic
404C2×C4#1313324311yescharacteristic
405C8—81no—
406C8—4241no—
407C8—2122no—
408C22—21no—
409C22—21no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—1064no—
414C22—2124no—
415C22—2124no—
416C22—2122no—
417C22—2122no—
418C22#1681049911yescharacteristic
419C22#1681075611yescharacteristic
420C22#1681076611yescharacteristic
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—81no—
427C4—2124no—
428C4—2122no—
429C4#104×D5312yes—
430C4C4×#1171992211yescharacteristic
431C4#1681073311yescharacteristic
432C4#1681081511yescharacteristic
433C4#1681083511yescharacteristic
434C2—21no—
435C2—21no—
436C2—161no—
437C2—2124no—
438C2#2027506611yescharacteristic
439C2#2027515511yescharacteristic
440C2#2027521911yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1962C2
socleC22×C53#386
prime radical#20284711C2
cosocleC2×C42×C53C23
hypercenter#1962D53
hypocenterC53#9233
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. C42×D212C2semidirect
3. #20273210C2semidirect
4. #20273139C2semidirect
5. #20272616C2semidirect
6. #20272548C2semidirect
7. C53×#1962C2semidirect
12. #16810900C4semidirect
13. #16810815C4semidirect
48. C4⋊#88322C2×C4semidirect
59. C4⋊Q8×C53C2×C4semidirect
60. #127×C53C2×C4semidirect
117. C42×C53D8semidirect
178. D212#104semidirect
216. #1962D53semidirect
249. C53#9233semidirect
256. C4⋊Q8C4×D53semidirect
257. #127C4×D53semidirect
350. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847112
C42×D2122
#202732102
#202731392
#202726162
#202725482
C53×#19622
#923353