Finite Group Databasebeta

← #27367156

Subgroups — #27367156

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2C2×#16810997C211yesmaximal, characteristic, complemented
3#20283427C211yesmaximal, characteristic, complemented
4#20274182C214yesmaximal, complemented
5#20274058C214yesmaximal, complemented
6#20273461C211yesmaximal, characteristic, complemented
7#20273387C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C53×#3226C211yesmaximal, characteristic, complemented
10C22×#11719924—21no—
11C22×#11719924—21no—
12C2×#13133224—21no—
13C2×#13133214C2211yescharacteristic
14#16810999—21no—
15#16810997—21no—
16#16810997—21no—
17#16810997C2212yes—
18#16810997C2212yes—
19#16810971—21no—
20#16810818C2214yes—
21#16810756—22no—
22#16810701—21no—
23#16810701C2212yescomplemented
24D4×#88322—21no—
25D4×#88322C2212yescomplemented
26#16810676C2211yescharacteristic
27#16810647C2211yescharacteristic
28#16810590—21no—
29C2×C4×#88322C2211yescharacteristic
30#16810375C2212yescomplemented
31#16810373C2212yescomplemented
32#16810369C2214yescomplemented
33#16810366C2214yes—
34C53×#532C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#500—21no—
37C53×#486—21no—
38C53×#480—22no—
39C53×#474C2214yescomplemented
40C53×#431C2211yescharacteristic
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C2×#11719924—22no—
45C2×#11719924—22no—
46C2×#11719924—22no—
47C2×#11719924—22no—
48C2×#11719924—44no—
49C2×#11719924—44no—
50C2×#11719922—24no—
51#13133224—44no—
52C2×#11719917—22no—
53C2×#11719917—21no—
54C22×#88322—22no—
55C22×#88322D411yescharacteristic
56C22×#88322D411yescharacteristic
57#13133214—21no—
58#13133214—21no—
59#13133214—21no—
60#13133214C2312yes—
61C4⋊#88322—21no—
62C4⋊#88322C2312yes—
63C4⋊#88322C2312yes—
64#13133137—24no—
65C22⋊#88322—22no—
66C22⋊#88322—22no—
67C22⋊#88322—21no—
68C4×#88322—21no—
69C4×#88322D411yescharacteristic
70C4×#88322D411yescharacteristic
71C4×#88322C2311yescharacteristic
72C4×#88322C2311yescharacteristic
73C2×#41×C53—21no—
74C22×Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53D411yescharacteristic
79C4⋊Q8×C53C2312yes—
80#127×C53D411yescharacteristic
81#124×C53—21no—
82#121×C53—21no—
83#121×C53D412yes—
84C4×D4×C53—21no—
85C4×D4×C53C2312yes—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C42×C53C2311yescharacteristic
89#107×C53—24no—
90C4⋊C8×C53C2312yes—
91#102×C53—24no—
92C22⋊C8×C53—21no—
93C4×C8×C53—21no—
94C23×D53—41no—
95#41×C53—42no—
96#41×C53—42no—
97C2×Q8×C53—21no—
98C2×Q8×C53—22no—
99C2×Q8×C53—41no—
100C2×Q8×C53—42no—
101C2×Q8×C53—42no—
102C2×D4×C53—22no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53C2×D412yes—
106C22×C4×C53—22no—
107C22×C4×C53—41no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—42no—
115C2×D212—42no—
116C2×D212—41no—
117C2×D212—41no—
118C2×D212—41no—
119C2×D212—82no—
120C2×D212—82no—
121C2×C8×C53—22no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53C2×D412yes—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—22no—
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53#4111yescharacteristic
133C42×C53C2411yescharacteristic
134#11719924—44no—
135#11719924—44no—
136#11719924—44no—
137#11719924—44no—
138#11719922—84no—
139#11719917—42no—
140#11719917—44no—
141C2×#88322—22no—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—44no—
145C2×#88322D814yescomplemented
146C2×#88322C2×D411yescharacteristic
147C2×#88322C2×D411yescharacteristic
148C2×#88322C2×D411yescharacteristic
149C2×#88322C2×D411yescharacteristic
150C23×C53—22no—
151C23×C53#12111yescharacteristic
152C22×D53—42no—
153C22×D53—82no—
154C22×D53—82no—
155C22×D53—82no—
156Q8×C53—42no—
157Q8×C53—42no—
158Q8×C53—42no—
159Q8×C53—82no—
160D4×C53—24no—
161D4×C53—44no—
162D4×C53—42no—
163D4×C53—42no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—82no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C2×D812yes—
175C2×C4×C53AGL(1,ℤ/8)12yes—
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179D212—42no—
180D212—42no—
181D212—44no—
182D212—82no—
183D212—82no—
184D212—82no—
185D212—82no—
186C8×C53—42no—
187#88322—21no—
188#88322—21no—
189#88322—42no—
190#88322—42no—
191#88322C2×D812yes—
192#88322C2×D812yes—
193#19186—531nomaximal, Sylow
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—42no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53#45912yes—
200C22×C53C2×#12111yescharacteristic
201C2×D53—82no—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—164no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—42no—
208C4×C53—82no—
209C4×C53#45912yes—
210C4×C53#47112yes—
211C4×C53C22×D811yescharacteristic
212C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
213C4×C53#57712yes—
214#3226D5311yescharacteristic, complemented
215#3205—531no—
216C2×#492—531no—
217#2931—531no—
218#1685—534no—
219#1665—534no—
220#1564—531no—
221#1546—531no—
222#1545—531no—
223C2×C53—22no—
224C2×C53—82no—
225C2×C53C2×#45911yescharacteristic
226C2×C53#305711yescharacteristic
227C2×C53C2×#47111yescharacteristic
228D53—162no—
229C22×SD16—1061no—
230C22×SD16—1061no—
231C2×#131—1061no—
232#532C2×D5311yescharacteristic
233C2×C4⋊Q8C2×D5311yescharacteristic
234C2×#127—531no—
235#500—21no—
236#493—1061no—
237#492—532no—
238#492—532no—
239#492—1061no—
240#492—1061no—
241#486—21no—
242#486—1061no—
243#480—22no—
244#474C2×D5314yescomplemented
245#459—534no—
246#447—1062no—
247#436—532no—
248#436—1061no—
249C8×D4—532no—
250C8×D4—1061no—
251#431—531no—
252#431C2×D5311yescharacteristic
253#423—531no—
254#408—1061no—
255C2×C4×C8—531no—
256#335—532no—
257#334—532no—
258#332—534no—
259#331—534no—
260C53#1918611yescharacteristic, complemented, Sylow
261C2×#41—21no—
262C22×Q8—21no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4—1061no—
266C2×SD16—1062no—
267C2×SD16—1062no—
268C2×SD16—1062no—
269C2×SD16—1062no—
270C2×SD16—2124no—
271C2×SD16—2124no—
272C2×D8—1064no—
273#131—2124no—
274C2×M16—1062no—
275C2×M16—1061no—
276C22×C8—531no—
277C22×C8—531no—
278C22×C8—1062no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C22⋊D5311yescharacteristic
283C4⋊Q8C22×D5312yes—
284#127—532no—
285#127—1061no—
286#127—1061no—
287#127—1061no—
288#127C22⋊D5311yescharacteristic
289#124—21no—
290#121—21no—
291#121C22⋊D5312yes—
292C4×D4—21no—
293C4×D4C22×D5312yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C42C22×D5311yescharacteristic
297#107—24no—
298C4⋊C8—532no—
299C4⋊C8—532no—
300C4⋊C8—1061no—
301C4⋊C8C22×D5312yes—
302#102—24no—
303#102—1064no—
304C22⋊C8—21no—
305C22⋊C8—1062no—
306C22⋊C8—1062no—
307C22⋊C8—1061no—
308C4×C8—21no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—531no—
312C4×C8—531no—
313C4×C8—1061no—
314C24—2121no—
315#41—42no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—1062no—
328C2×D4—4242no—
329C2×D4—4242no—
330C2×D4—2121no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2122no—
334C2×D4—2122no—
335C2×D4C2×C22⋊D5312yes—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4D4×D5311yescharacteristic
339C22×C4#1171994711yescharacteristic
340C22×C4C2×C22⋊D5311yescharacteristic
341SD16—2124no—
342SD16—2124no—
343SD16—2124no—
344SD16—2124no—
345D8—4244no—
346M16—2124no—
347M16—2122no—
348C2×C8—22no—
349C2×C8—534no—
350C2×C8—531no—
351C2×C8—531no—
352C2×C8—531no—
353C2×C8—531no—
354C2×C8—1062no—
355C2×C8—1062no—
356C2×C8—1062no—
357C2×C8—2124no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4C2×C22⋊D5312yes—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C42D4×D5311yescharacteristic
367C42#1171994711yescharacteristic
368C42C2×C22⋊D5311yescharacteristic
369C42C23×D5311yescharacteristic
370C23—22no—
371C23—4242no—
372C23—4242no—
373C23—4242no—
374C23—2122no—
375C23#1313318911yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—82no—
380D4—24no—
381D4—44no—
382D4—42no—
383D4—42no—
384D4—4242no—
385D4—4242no—
386D4—4242no—
387D4—4242no—
388D4—2122no—
389D4—2122no—
390D4—2124no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4#1313318911yescharacteristic
399C2×C4#1313318911yescharacteristic
400C2×C4#1313318911yescharacteristic
401C2×C4D8×D5312yes—
402C2×C4#1313324112yes—
403C2×C4C2×D4×D5311yescharacteristic
404C2×C4C22×C22⋊D5311yescharacteristic
405C2×C4C2×#1171994711yescharacteristic
406C8—42no—
407C8—532no—
408C8—532no—
409C8—1061no—
410C8—1061no—
411C8—2122no—
412C8—2122no—
413C22—21no—
414C22—21no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—8484no—
419C22—4242no—
420C22—4242no—
421C22—4242no—
422C22#1681081612yes—
423C22C2×#1313318911yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—82no—
428C4#1681081612yes—
429C4#1681088812yes—
430C4C2×D8×D5311yescharacteristic
431C4C2×#1313324111yescharacteristic
432C4#1681143112yes—
433C2—22no—
434C2—82no—
435C2—8482no—
436C2C2×#1681081611yescharacteristic
437C2#2028435611yescharacteristic
438C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3226C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#3226D53
hypocenterC53#19186
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
2. C2×#16810997C2semidirect
3. #20283427C2semidirect
4. #20274182C2semidirect
5. #20274058C2semidirect
6. #20273461C2semidirect
7. #20273387C2semidirect
8. #20273382C2semidirect
9. C53×#3226C2semidirect
23. #16810701C22semidirect
25. D4×#88322C22semidirect
30. #16810375C22semidirect
31. #16810373C22semidirect
32. #16810369C22semidirect
39. C53×#474C22semidirect
145. C2×#88322D8semidirect
214. #3226D53semidirect
244. #474C2×D53semidirect
260. C53#19186semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168109972
#202834272
#202741822
#202740582
#202734612
#202733872
#202733822
C53×#32262
#1918653