Finite Group Databasebeta

← #27368066

Subgroups — #27368066

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic, complemented
4#20274161C214yesmaximal, complemented
5#20274154C214yesmaximal, complemented
6#20273473C211yesmaximal, characteristic, complemented
7#20273387C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic, complemented
9C53×#3039C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133214C2211yescharacteristic
12#16810971—21no—
13#16810971—21no—
14#16810971C2214yes—
15#16810823—21no—
16#16810823C2212yes—
17#16810818—21no—
18#16810818C2212yes—
19Q8×#88322C2212yescomplemented
20D4×#88322—22no—
21D4×#88322C2212yescomplemented
22#16810676—21no—
23#16810676C2211yescharacteristic
24#16810647C2211yescharacteristic
25#16810627—21no—
26C2×#13133137—22no—
27C2×C4×#88322—21no—
28C2×C4×#88322C2211yescharacteristic
29#16810375C2214yescomplemented
30#16810369C2214yescomplemented
31#16810366C2214yes—
32C2×#137×C53—21no—
33C2×C4⋊Q8×C53C2211yescharacteristic
34C4×#41×C53C2211yescharacteristic
35C53×#462—21no—
36C53×#462C2212yescomplemented
37C53×#461—21no—
38C53×#461C2212yescomplemented
39C53×#423C2211yescharacteristic
40C53×#419—21no—
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44#13133242—44no—
45C2×#11719924—22no—
46C2×#11719922—22no—
47C2×#11719917—21no—
48C22×#88322—22no—
49C22×#88322D412yes—
50#13133214—21no—
51#13133214—21no—
52#13133214—21no—
53#13133214C2312yes—
54C4⋊#88322—21no—
55C4⋊#88322—22no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133137—22no—
59#13133137—22no—
60#13133137—24no—
61#13133137—24no—
62C22⋊#88322—22no—
63C22⋊#88322—21no—
64C4×#88322—21no—
65C4×#88322—22no—
66C4×#88322D411yescharacteristic
67C4×#88322D411yescharacteristic
68C4×#88322C2312yes—
69C2×#41×C53D411yescharacteristic
70C22×Q8×C53—21no—
71#137×C53—44no—
72C2×Q16×C53—22no—
73C2×SD16×C53—22no—
74C2×M16×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53C2312yes—
79C4×Q8×C53D411yescharacteristic
80C4×Q8×C53C2311yescharacteristic
81C4×D4×C53—21no—
82C4×D4×C53—21no—
83C4×D4×C53D411yescharacteristic
84C4×D4×C53C2311yescharacteristic
85#117×C53—21no—
86#117×C53D411yescharacteristic
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53C2311yescharacteristic
91C4⋊C8×C53—21no—
92C4⋊C8×C53C2312yes—
93#103×C53—22no—
94#103×C53—22no—
95C23×D53—41no—
96#41×C53—22no—
97#41×C53D814yes—
98C2×Q8×C53—22no—
99C2×Q8×C53—41no—
100C2×Q8×C53—42no—
101C2×Q8×C53—42no—
102C2×Q8×C53C2×D411yescharacteristic
103C2×D4×C53—21no—
104C2×D4×C53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—41no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—22no—
115C2×D212—44no—
116C2×D212—42no—
117C2×D212—42no—
118C2×D212—41no—
119C2×D212—41no—
120C2×D212—41no—
121C2×D212—82no—
122Q16×C53—44no—
123SD16×C53—44no—
124M16×C53—42no—
125C2×C8×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53C2×D411yescharacteristic
133C4⋊C4×C53C2×D411yescharacteristic
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C42×C53—21no—
138C42×C53—21no—
139C42×C53—21no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144#11719924—44no—
145#11719922—44no—
146#11719917—42no—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322—42no—
150C2×#88322—44no—
151C2×#88322C2×D412yes—
152C2×#88322C2×D412yes—
153C23×C53—21no—
154C23×C53#12111yescharacteristic
155C22×D53—42no—
156C22×D53—82no—
157C22×D53—82no—
158C22×D53—82no—
159Q8×C53—21no—
160Q8×C53—42no—
161Q8×C53—42no—
162Q8×C53—82no—
163Q8×C53C2×D812yes—
164D4×C53—24no—
165D4×C53—21no—
166D4×C53C2×D812yes—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—24no—
170C2×C4×C53—22no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—82no—
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53C2×D812yes—
181C2×C4×C53#13512yes—
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C2×#4111yescharacteristic
185D212—42no—
186D212—42no—
187D212—44no—
188D212—82no—
189D212—82no—
190D212—82no—
191D212—82no—
192C8×C53—42no—
193#88322—22no—
194#88322—22no—
195#88322—42no—
196#88322—42no—
197#19276—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#46612yes—
204C22×C53C2×#12111yescharacteristic
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—82no—
208C2×D53—164no—
209C4×C53—21no—
210C4×C53—22no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—82no—
215C4×C53#46612yes—
216C4×C53#47112yes—
217C4×C53C22×D811yescharacteristic
218C4×C53C2×#13511yescharacteristic
219C4×C53#57512yes—
220C2×#486—531no—
221#3039D5311yescharacteristic, complemented
222#3038—531no—
223C8×#41—531no—
224#1682—534no—
225#1681—534no—
226#1567—531no—
227#1546—531no—
228#1545—531no—
229C2×C53—22no—
230C2×C53—42no—
231C2×C53#304011yescharacteristic
232C2×C53C2×#46611yescharacteristic
233C2×C53C2×#47111yescharacteristic
234D53—162no—
235C2×#137—21no—
236C2×AGL(1,ℤ/8)—1061no—
237C2×C4⋊Q8C2×D5311yescharacteristic
238C2×#127—531no—
239C4×#41C2×D5311yescharacteristic
240#486—534no—
241#486—1061no—
242#486—1061no—
243#462—21no—
244#462C2×D5312yescomplemented
245#461—21no—
246#461C2×D5312yescomplemented
247#460—532no—
248#460—1061no—
249#459—532no—
250#459—1061no—
251C8×Q8—532no—
252C8×D4—532no—
253C8×D4—1062no—
254#431—531no—
255#431—1061no—
256#423—531no—
257#423C2×D5311yescharacteristic
258#419—21no—
259#418—1061no—
260C2×#102—1062no—
261C2×C4×C8—531no—
262C2×C4×C8—1061no—
263#335—534no—
264#332—534no—
265#331—534no—
266C53#1927611yescharacteristic, complemented, Sylow
267C2×#41C22⋊D5311yescharacteristic
268C22×Q8—21no—
269C22×D4—1061no—
270C22×D4—1061no—
271C22×D4—1061no—
272#137—44no—
273AGL(1,ℤ/8)—2124no—
274C2×Q16—22no—
275C2×SD16—22no—
276C2×SD16—1062no—
277C2×D8—1062no—
278C2×M16—21no—
279C2×M16—1061no—
280C22×C8—532no—
281C22×C8—1062no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8C22×D5312yes—
286#127—532no—
287#127—1061no—
288#127—1061no—
289#127—1061no—
290C4×Q8C22⋊D5311yescharacteristic
291C4×Q8C22×D5311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C22⋊D5311yescharacteristic
295C4×D4C22×D5311yescharacteristic
296#117—21no—
297#117C22⋊D5311yescharacteristic
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C42—21no—
301C2×C42C22×D5311yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—532no—
304C4⋊C8—532no—
305C4⋊C8—1062no—
306C4⋊C8—1061no—
307C4⋊C8C22×D5312yes—
308#103—22no—
309#103—22no—
310#102—1064no—
311#102—1064no—
312#102—1062no—
313#102—1062no—
314C22⋊C8—1062no—
315C22⋊C8—1061no—
316C4×C8—531no—
317C4×C8—531no—
318C4×C8—532no—
319C4×C8—1062no—
320C4×C8—1061no—
321C24—2121no—
322#41—22no—
323#41#1171992214yes—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C2×C22⋊D5311yescharacteristic
329C2×D4—21no—
330C2×D4—1062no—
331C2×D4—1062no—
332C2×D4—1062no—
333C2×D4—4242no—
334C2×D4—2124no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2122no—
339C2×D4—2122no—
340C2×D4C2×C22⋊D5311yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4D4×D5311yescharacteristic
346C22×C4#1171994711yescharacteristic
347C22×C4C2×C22⋊D5311yescharacteristic
348Q16—44no—
349SD16—44no—
350SD16—2124no—
351D8—2124no—
352M16—42no—
353M16—2122no—
354C2×C8—22no—
355C2×C8—532no—
356C2×C8—532no—
357C2×C8—1062no—
358C2×C8—1062no—
359C2×C8—2122no—
360C2×C8—2124no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C2×C22⋊D5311yescharacteristic
368C4⋊C4C2×C22⋊D5311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42D4×D5311yescharacteristic
376C42#1171994711yescharacteristic
377C42C2×C22⋊D5311yescharacteristic
378C42C23×D5311yescharacteristic
379C23—21no—
380C23—4242no—
381C23—4242no—
382C23—4242no—
383C23—2122no—
384C23#1313318911yescharacteristic
385Q8—21no—
386Q8—42no—
387Q8—42no—
388Q8—82no—
389Q8C2×#1171992212yes—
390D4—24no—
391D4—21no—
392D4—4242no—
393D4—4242no—
394D4—4242no—
395D4—4242no—
396D4—2122no—
397D4—2122no—
398D4—2124no—
399D4C2×#1171992212yes—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4#1313318911yescharacteristic
411C2×C4#1313318911yescharacteristic
412C2×C4#1313318911yescharacteristic
413C2×C4C2×#1171992212yes—
414C2×C4#1313323712yes—
415C2×C4C2×D4×D5311yescharacteristic
416C2×C4C22×C22⋊D5311yescharacteristic
417C2×C4C2×#1171994711yescharacteristic
418C8—42no—
419C8—1062no—
420C8—1062no—
421C8—2122no—
422C8—2122no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—8484no—
429C22—4242no—
430C22—4242no—
431C22—4242no—
432C22#1681085712yes—
433C22C2×#1313318911yescharacteristic
434C4—21no—
435C4—21no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4#1681085712yes—
441C4#1681088812yes—
442C4C22×#1171992211yescharacteristic
443C4C2×#1313323711yescharacteristic
444C4#1681142212yes—
445C2—22no—
446C2—42no—
447C2—8482no—
448C2#2028421111yescharacteristic
449C2C2×#1681085711yescharacteristic
450C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3039C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#3039D53
hypocenterC53#19276
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. C2×#16810971C2semidirect
3. #41×#88322C2semidirect
4. #20274161C2semidirect
5. #20274154C2semidirect
6. #20273473C2semidirect
7. #20273387C2semidirect
8. #20273384C2semidirect
9. C53×#3039C2semidirect
19. Q8×#88322C22semidirect
21. D4×#88322C22semidirect
29. #16810375C22semidirect
30. #16810369C22semidirect
36. C53×#462C22semidirect
38. C53×#461C22semidirect
221. #3039D53semidirect
244. #462C2×D53semidirect
246. #461C2×D53semidirect
266. C53#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202841952
#41×#883222
#202741612
#202741542
#202734732
#202733872
#202733842
C53×#30392
#1927653