Finite Group Databasebeta

← #27280982

Subgroups — #27280982

Order 13568.

558 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#20283832C211yesmaximal, characteristic, complemented
2#20275643C211yesmaximal, characteristic, complemented
3#20274735C211yesmaximal, characteristic, complemented
4#20273383C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#3038C211yesmaximal, characteristic
8#16810754—21no—
9#16810701—21no—
10#16810695—22no—
11#16810680—21no—
12#16810676C2211yescharacteristic
13#16810647—21no—
14#16810647C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133144—21no—
17C2×#13133138—22no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810382—21no—
20#16810367—21no—
21#16810367—21no—
22C2×AGL(1,ℤ/8)×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C4×#41×C53C2211yescharacteristic
25C53×#460—21no—
26C53×#460C412yescomplemented
27C53×#459—21no—
28C53×#459C412yescomplemented
29C53×#423C2211yescharacteristic
30C53×#418—21no—
31C2×#11719917—22no—
32C2×#11719917—21no—
33C22×#88322D412yes—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D411yescharacteristic
42C4⋊#88322D411yescharacteristic
43#13133144—42no—
44#13133138—44no—
45C22⋊#88322—22no—
46C22⋊#88322—21no—
47#13133128—22no—
48C4×#88322—21no—
49#13133124—21no—
50C2×#41×C53D411yescharacteristic
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54AGL(1,ℤ/8)×C53—42no—
55AGL(1,ℤ/8)×C53—42no—
56C2×SD16×C53—22no—
57C2×D8×C53—22no—
58C2×M16×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53C2×C412yes—
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53D411yescharacteristic
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69#117×C53—21no—
70#117×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73C4⋊C8×C53—21no—
74C4⋊C8×C53C2×C412yes—
75#102×C53—22no—
76#102×C53—22no—
77C24×C53—41no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53—21no—
81#41×C53—21no—
82C2×Q8×C53C22⋊C411yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—82no—
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—41no—
105SD16×C53—44no—
106D8×C53—44no—
107M16×C53—41no—
108M16×C53—41no—
109C2×C8×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128#11719917—41no—
129#11719917—41no—
130#11719917—82no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—22no—
134C2×#88322—42no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53—42no—
138C23×C53—82no—
139C23×C53—82no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53#12111yescharacteristic
143Q8×C53—21no—
144Q8×C53—21no—
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—44no—
148D4×C53—41no—
149D4×C53—41no—
150D4×C53—42no—
151D4×C53—42no—
152D4×C53—82no—
153D4×C53—82no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—81no—
157D4×C53—81no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—81no—
169C2×C4×C53C2×C22⋊C411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C53#13711yescharacteristic
179C2×Dic53—42no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C8×C53—42no—
183#88322—42no—
184#88322—41no—
185#88322—41no—
186#88322—81no—
187#9322—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—82no—
196C22×C53—162no—
197C22×C53—162no—
198C22×C53#38611yescharacteristic
199C22×C53#46811yescharacteristic
200C22×C53#46911yescharacteristic
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53#41711yescharacteristic
208C4×C53#41811yescharacteristic
209C4×C53#45311yescharacteristic
210C4×C53#45411yescharacteristic
211C4×C53#46811yescharacteristic
212C4×C53#46911yescharacteristic
213Dic53—82no—
214#3038D5311yescharacteristic
215#2989—531no—
216#1982—531no—
217#1806—531no—
218#1545—531no—
219#1303—531no—
220#1292—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—81no—
224C2×C53—162no—
225C2×C53#188311yescharacteristic
226C2×C53#189511yescharacteristic
227C2×C53#190311yescharacteristic
228C2×AGL(1,ℤ/8)—21no—
229C2×#127C2×D5311yescharacteristic
230C4×#41C2×D5311yescharacteristic
231#460—21no—
232#460Dic5312yescomplemented
233#459—21no—
234#459Dic5312yescomplemented
235#446—1061no—
236#436—1061no—
237#435—1062no—
238#432—1061no—
239#431—531no—
240#423—531no—
241#423—1061no—
242#423C2×D5311yescharacteristic
243C2×C4⋊C8—531no—
244C2×#104—1061no—
245#418—21no—
246C2×#102—1062no—
247C4×C4⋊C4—531no—
248#339—1061no—
249#331—1061no—
250#331—1061no—
251C53#932211yescharacteristic, complemented, Sylow
252C2×#41C22⋊D5311yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256AGL(1,ℤ/8)—42no—
257AGL(1,ℤ/8)—42no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×M16—21no—
261C2×M16—1062no—
262C2×M16—1061no—
263C22×C8—532no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic5312yes—
268C4×Q8C2×Dic5311yescharacteristic
269C4×Q8C22⋊D5311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic5311yescharacteristic
273C4×D4C22⋊D5311yescharacteristic
274#117—21no—
275#117C22⋊D5311yescharacteristic
276C2×C4⋊C4—1061no—
277C2×C42—21no—
278C2×C42—1061no—
279C2×C42C22×D5311yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—531no—
282C4⋊C8—531no—
283C4⋊C8—1062no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8C2×Dic5312yes—
289#104—2122no—
290#102—22no—
291#102—22no—
292#102—2124no—
293C22⋊C8—1062no—
294C22⋊C8—1061no—
295#97—1062no—
296C4×C8—1061no—
297#95—1061no—
298C24—41no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—21no—
303C2×Q8C22⋊Dic5311yescharacteristic
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—82no—
318C2×D4C22⋊Dic5311yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—2121no—
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—44no—
327D8—44no—
328M16—41no—
329M16—41no—
330M16—4242no—
331M16—2121no—
332M16—2121no—
333C2×C8—22no—
334C2×C8—1061no—
335C2×C8—1061no—
336C2×C8—1062no—
337C2×C8—2121no—
338C2×C8—2122no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1062no—
342C4⋊C4—2121no—
343C4⋊C4C22⋊Dic5311yescharacteristic
344C4⋊C4C22⋊Dic5311yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—2121no—
352C42—2121no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C22×Dic5311yescharacteristic
356C42C2×C22⋊D5311yescharacteristic
357C23—21no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23#1313318911yescharacteristic
364Q8—21no—
365Q8—21no—
366D4—21no—
367D4—21no—
368D4—44no—
369D4—41no—
370D4—41no—
371D4—42no—
372D4—42no—
373D4—82no—
374D4—82no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—2122no—
393C2×C4C2×C22⋊Dic5311yescharacteristic
394C2×C4D4×Dic5311yescharacteristic
395C2×C4D4×Dic5311yescharacteristic
396C2×C4#1313318511yescharacteristic
397C2×C4#1313318911yescharacteristic
398C2×C4#1313319811yescharacteristic
399C2×C4#1313324211yescharacteristic
400C2×C4#1313324311yescharacteristic
401C2×C4#1313324311yescharacteristic
402C2×C4#1313324711yescharacteristic
403C8—42no—
404C8—4241no—
405C8—2122no—
406C8—2121no—
407C8—2121no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—162no—
417C22—162no—
418C22#1681050311yescharacteristic
419C22#1681086711yescharacteristic
420C22#1681087411yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4242no—
428C4#1681062411yescharacteristic
429C4#1681062811yescharacteristic
430C4#1681078911yescharacteristic
431C4#1681079411yescharacteristic
432C4#1681086711yescharacteristic
433C4#1681087411yescharacteristic
434C2—21no—
435C2—21no—
436C2—81no—
437C2—162no—
438C2#2027509311yescharacteristic
439C2#2027515411yescharacteristic
440C2#2027519711yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3038C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3038D53
hypocenterC53#9322
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20283832C2semidirect
2. #20275643C2semidirect
3. #20274735C2semidirect
4. #20273383C2semidirect
5. #20272616C2semidirect
6. #20272589C2semidirect
26. C53×#460C4semidirect
28. C53×#459C4semidirect
232. #460Dic53semidirect
234. #459Dic53semidirect
251. C53#9322semidirect

Maximal subgroups

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

subgroupindex
#202838322
#202756432
#202747352
#202733832
#202726162
#202725892
C53×#30382
#932253