Finite Group Databasebeta

← #27366955

Subgroups — #27366955

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285172C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274207C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274083C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic
10#20273387C211yesmaximal, characteristic, complemented
11C2×C53×#475C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13C2×#13133248—21no—
14#16811173C2211yescharacteristic
15#16811015—22no—
16#16811002C2212yes—
17#16810997C2212yes—
18#16810941—21no—
19#16810908C2211yescharacteristic
20#16810908C2211yescharacteristic
21#16810823C2211yescharacteristic
22#16810823C2211yescharacteristic
23#16810762—21no—
24#16810754—22no—
25#16810754C2212yes—
26Q8×#88322C2212yes—
27C4×#11719924—21no—
28C4×#11719924—21no—
29#16810676C2211yescharacteristic
30#16810647—21no—
31#16810647C2211yescharacteristic
32C4×#11719917—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yes—
35#16810373C2212yes—
36#16810369C2212yes—
37#16810369C2212yes—
38#16810368C2212yes—
39#16810366C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#475—21no—
43C53×#475—21no—
44C53×#475C2212yes—
45C53×#475C2212yes—
46C2×#106×C53—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#103×C53—21no—
49C2×#11719945—21no—
50#13133248—44no—
51#13133248—44no—
52C2×#11719925—22no—
53C2×#11719925—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719924—21no—
57C2×#11719924—21no—
58C2×#11719917—21no—
59C2×#11719917D411yescharacteristic
60C2×#11719917D411yescharacteristic
61#13133217C2311yescharacteristic
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133148—21no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76#13133141—21no—
77#13133141—21no—
78#13133137—21no—
79#13133137—21no—
80C22⋊#88322—21no—
81#13133128—22no—
82#13133128D412yes—
83C4×#88322—21no—
84C4×#88322—22no—
85C4×#88322C2312yes—
86C22×Q8×C53—21no—
87C22×Q8×C53D411yescharacteristic
88C22×C8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53C2312yes—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53D411yescharacteristic
96C4×Q8×C53D411yescharacteristic
97C4×Q8×C53C2311yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105#106×C53—22no—
106#106×C53—22no—
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#103×C53—22no—
110#103×C53—22no—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—41no—
115C2×Q8×C53—41no—
116C2×Q8×C53—42no—
117C2×Q8×C53—42no—
118C2×Q8×C53C2×D411yescharacteristic
119C2×Q8×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—41no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×C22⋊D53—41no—
127#11719945—42no—
128#11719945—42no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×C4×D53—41no—
133C2×Dic106—21no—
134C2×C8×C53—22no—
135C2×C8×C53—42no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53C2×D411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53#4111yescharacteristic
154C42×C53C2411yescharacteristic
155#11719925—44no—
156#11719925—44no—
157#11719924—44no—
158#11719924—42no—
159#11719924—42no—
160#11719924—44no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165#11719917—22no—
166#11719917—42no—
167#11719917SD1614yescomplemented
168C2×#88322—22no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322C2×D412yes—
172C2×#88322C2×D412yes—
173C23×C53#12111yescharacteristic
174C22×D53—41no—
175Q8×C53—21no—
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—42no—
179Q8×C53—42no—
180Q8×C53—42no—
181Q8×C53—82no—
182Q8×C53#13712yescomplemented
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—81no—
193C2×C4×C53—82no—
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53C2×SD1612yes—
198C2×C4×C53AGL(1,ℤ/8)12yes—
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C2×#4111yescharacteristic
202C22⋊D53—82no—
203C2×Dic53—41no—
204D212—41no—
205D212—41no—
206D212—41no—
207D212—41no—
208D212—81no—
209C4×D53—82no—
210Dic106—41no—
211Dic106—41no—
212C8×C53—42no—
213#88322—22no—
214#88322—42no—
215#88322—41no—
216#88322—41no—
217#88322C2×SD1614yes—
218#19171—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53#46012yes—
222C22×C53C2×#12111yescharacteristic
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—81no—
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—82no—
234C4×C53#46012yes—
235C4×C53C22×SD1611yescharacteristic
236C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C53C2×#13711yescharacteristic
238C4×C53#57511yescharacteristic
239Dic53—81no—
240#3152—531no—
241#3103—531no—
242C2×#475D5311yescharacteristic, complemented
243Q8×M16—531no—
244#1690—532no—
245#1686—532no—
246#1670—532no—
247#1666—532no—
248#1565—531no—
249#1547—531no—
250#1546—531no—
251C2×C53—22no—
252C2×C53C2×#46011yescharacteristic
253C2×C53#305611yescharacteristic
254C2×C53#306911yescharacteristic
255D53—161no—
256C2×#137—1061no—
257C2×#137—1061no—
258#532—531no—
259C2×C4⋊Q8C2×D5311yescharacteristic
260C2×C4×Q8C2×D5311yescharacteristic
261#497—1062no—
262#494—532no—
263#492—532no—
264#481—1061no—
265#475—21no—
266#475—21no—
267#475—531no—
268#475—531no—
269#475C2×D5312yes—
270#475C2×D5312yes—
271#460—531no—
272#460—531no—
273#448—1061no—
274#446—532no—
275#446—1062no—
276C8×Q8—532no—
277C4×SD16—1061no—
278C4×SD16—1061no—
279#431—531no—
280C2×#106—21no—
281#423—531no—
282#423—1061no—
283C2×C4⋊C8C2×D5311yescharacteristic
284C2×#103—21no—
285C4×M16—531no—
286C4×M16—1061no—
287#335—532no—
288#334—532no—
289#332—532no—
290#332—532no—
291#332—532no—
292#331—532no—
293C53#1917111yescharacteristic, complemented, Sylow
294C2×#41—1061no—
295C22×Q8—21no—
296C22×Q8C22⋊D5311yescharacteristic
297#137—2124no—
298#137—2124no—
299C2×Q16—1062no—
300C2×Q16—1062no—
301C2×SD16—1062no—
302C2×SD16—1062no—
303C2×SD16—1061no—
304C2×SD16—1061no—
305C2×M16—531no—
306C2×M16—531no—
307C2×M16—1061no—
308C22×C8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—531no—
313C4⋊Q8C22×D5312yes—
314#127—531no—
315#124—1061no—
316#121—1061no—
317#121—1061no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D5311yescharacteristic
321C4×Q8C22⋊D5311yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×Q8C22×D5311yescharacteristic
324C4×D4—1061no—
325C4×D4—1061no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C22⋊D5311yescharacteristic
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#106—22no—
333#106—22no—
334#106—1061no—
335#106—1061no—
336C4⋊C8—21no—
337C4⋊C8—532no—
338C4⋊C8—531no—
339C4⋊C8—531no—
340C4⋊C8—1062no—
341C4⋊C8—1062no—
342C4⋊C8—1061no—
343C4⋊C8C22×D5312yes—
344#103—22no—
345#103—22no—
346#103—1061no—
347#103—1061no—
348#102—1061no—
349#102—1061no—
350C22⋊C8—1061no—
351#97—532no—
352#97—1062no—
353C4×C8—532no—
354C4×C8—1062no—
355C4×C8—1061no—
356#41—2122no—
357#41—2122no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—1061no—
366C2×Q8C2×C22⋊D5311yescharacteristic
367C2×Q8C2×C22⋊D5311yescharacteristic
368C2×D4—1061no—
369C2×D4—1061no—
370C2×D4—1061no—
371C2×D4—2121no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—2121no—
376C22×C4D4×D5311yescharacteristic
377C22×C4#1171994711yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—2124no—
380Q16—2124no—
381SD16—2124no—
382SD16—2124no—
383SD16—2122no—
384SD16—2122no—
385M16—534no—
386M16—1062no—
387M16—2122no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—532no—
391C2×C8—532no—
392C2×C8—1062no—
393C2×C8—1061no—
394C2×C8—1061no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4C2×C22⋊D5311yescharacteristic
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42D4×D5311yescharacteristic
415C42#1171994711yescharacteristic
416C42C2×C22⋊D5311yescharacteristic
417C42C23×D5311yescharacteristic
418C23—2121no—
419C23#1313318911yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—82no—
427Q8—2121no—
428Q8—2121no—
429Q8#1313324812yescomplemented
430D4—4241no—
431D4—4242no—
432D4—2121no—
433D4—2121no—
434D4—2121no—
435D4—2121no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—4242no—
448C2×C4—2121no—
449C2×C4#1313318911yescharacteristic
450C2×C4#1313318911yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4SD16×D5312yes—
453C2×C4#1313324412yes—
454C2×C4C2×D4×D5311yescharacteristic
455C2×C4C22×C22⋊D5311yescharacteristic
456C2×C4C2×#1171994711yescharacteristic
457C8—42no—
458C8—534no—
459C8—1062no—
460C8—2122no—
461C8—2121no—
462C8—2121no—
463C22—21no—
464C22—21no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22#1681082112yes—
469C22C2×#1313318911yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—4241no—
479C4#1681082112yes—
480C4C2×SD16×D5311yescharacteristic
481C4C2×#1313324411yescharacteristic
482C4C2×#1313324811yescharacteristic
483C4#1681142111yescharacteristic
484C2—22no—
485C2—8481no—
486C2C2×#1681082111yescharacteristic
487C2#2028434111yescharacteristic
488C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#475C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#475D53
hypocenterC53#19171
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. Q8×#11719917C2semidirect
4. #20274207C2semidirect
5. #20274189C2semidirect
6. #20274083C2semidirect
7. #20274065C2semidirect
8. #20273465C2semidirect
10. #20273387C2semidirect
11. C2×C53×#475C2semidirect
167. #11719917SD16semidirect
182. Q8×C53#137semidirect
242. C2×#475D53semidirect
293. C53#19171semidirect
429. Q8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202847462
Q8×#117199172
#202742072
#202741892
#202740832
#202740652
#202734652
#202733902
#202733872
C2×C53×#4752
#1917153