Finite Group Databasebeta

← #27367671

Subgroups — #27367671

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285172C211yesmaximal, characteristic
3#20283664C211yesmaximal, characteristic, complemented
4#20274207C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274083C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273362C211yesmaximal, characteristic
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3229C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13C2×#13133248—21no—
14#16811173C2211yescharacteristic
15#16811015—22no—
16#16811002C2212yes—
17#16810997C2212yes—
18#16810910—21no—
19#16810908C2212yes—
20#16810854—22no—
21#16810825—21no—
22#16810823C2212yes—
23#16810754—22no—
24#16810754C2212yes—
25Q8×#88322C2212yes—
26#16810650—21no—
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810579—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yes—
32#16810373C2212yes—
33#16810369C2212yes—
34#16810369C2212yes—
35#16810368C2211yescharacteristic
36#16810368C2211yescharacteristic
37#16810366C2211yescharacteristic
38#16810366C2211yescharacteristic
39C53×#533C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#498—21no—
42C53×#487—21no—
43C53×#483—22no—
44C53×#475C2212yes—
45C53×#475C2212yes—
46C53×#431C2211yescharacteristic
47C2×#11719945—21no—
48#13133248—44no—
49#13133248—44no—
50C2×#11719925—22no—
51C2×#11719925—22no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719917D411yescharacteristic
56C2×#11719917D411yescharacteristic
57C22×#88322—21no—
58C22×#88322—21no—
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133148—22no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322—22no—
71C4⋊#88322C2311yescharacteristic
72C4⋊#88322C2312yes—
73C4⋊#88322C2311yescharacteristic
74#13133141—22no—
75#13133137—22no—
76#13133128—22no—
77#13133128D412yes—
78C4×#88322—22no—
79C4×#88322C2312yes—
80C22×Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53D411yescharacteristic
85C4⋊Q8×C53D411yescharacteristic
86C4⋊Q8×C53C2312yes—
87#125×C53—21no—
88C22⋊Q8×C53—21no—
89C22⋊Q8×C53D412yes—
90C4×Q8×C53—21no—
91C4×Q8×C53C2312yes—
92#117×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C42×C53C2311yescharacteristic
96#106×C53—22no—
97#106×C53—22no—
98C4⋊C8×C53C2311yescharacteristic
99C4⋊C8×C53C2311yescharacteristic
100#103×C53—22no—
101#103×C53—22no—
102C22⋊C8×C53—21no—
103C4×C8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—22no—
106C2×Q8×C53—41no—
107C2×Q8×C53—42no—
108C2×Q8×C53—42no—
109C2×Q8×C53C2×D412yes—
110C22×C4×C53—41no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C2×C22⋊D53—41no—
115#11719945—42no—
116#11719945—42no—
117C2×D212—21no—
118C2×D212—21no—
119C2×D212—21no—
120C2×C4×D53—41no—
121C2×Dic106—21no—
122C2×C8×C53—21no—
123C2×C8×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53C2×D412yes—
134C22⋊C4×C53—22no—
135C42×C53—22no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140#11719925—44no—
141#11719925—44no—
142#11719924—44no—
143#11719924—44no—
144#11719924—82no—
145#11719921—41no—
146#11719921—41no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149#11719917—22no—
150#11719917SD1614yescomplemented
151C2×#88322—21no—
152C2×#88322—21no—
153C2×#88322—22no—
154C2×#88322—42no—
155C2×#88322—81no—
156C2×#88322C2×D412yes—
157C2×#88322C2×D412yes—
158C23×C53#12111yescharacteristic
159C22×D53—41no—
160Q8×C53—22no—
161Q8×C53—22no—
162Q8×C53—42no—
163Q8×C53—42no—
164Q8×C53—81no—
165Q8×C53—82no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—82no—
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53C2×SD1612yes—
177C2×C4×C53AGL(1,ℤ/8)12yes—
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C22×D411yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C22⋊D53—82no—
182C2×Dic53—41no—
183D212—41no—
184D212—41no—
185D212—41no—
186D212—41no—
187D212—81no—
188C4×D53—82no—
189Dic106—41no—
190Dic106—41no—
191C8×C53—41no—
192C8×C53—41no—
193#88322—22no—
194#88322—41no—
195#88322—41no—
196#88322—42no—
197#88322C2×SD1614yes—
198#19239—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53#46012yes—
202C22×C53C2×#12111yescharacteristic
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—42no—
210C4×C53—42no—
211C4×C53—82no—
212C4×C53#46012yes—
213C4×C53C22×SD1611yescharacteristic
214C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
215C4×C53#57711yescharacteristic
216C4×C53#57811yescharacteristic
217Dic53—81no—
218#3246—531no—
219#3229D5311yescharacteristic, complemented
220#3152—531no—
221#2962—531no—
222#1690—532no—
223#1686—532no—
224#1670—532no—
225#1666—532no—
226#1565—531no—
227#1540—531no—
228C2×#332—531no—
229C2×C53—22no—
230C2×C53C2×#46011yescharacteristic
231C2×C53#306311yescharacteristic
232C2×C53#306911yescharacteristic
233D53—161no—
234C2×#137—1061no—
235C2×#137—1061no—
236#533C2×D5311yescharacteristic
237#532—531no—
238C2×C4⋊Q8C2×D5311yescharacteristic
239#498—21no—
240#497—1062no—
241#494—532no—
242#492—532no—
243#487—21no—
244#483—22no—
245#476—1061no—
246#475—532no—
247#475C2×D5312yes—
248#475C2×D5312yes—
249#465—1062no—
250#461—1061no—
251#460—532no—
252#446—532no—
253#446—1062no—
254C8×Q8—532no—
255#431C2×D5311yescharacteristic
256#424—1061no—
257C2×C4⋊C8—531no—
258C2×C4⋊C8—531no—
259#405—1061no—
260C4×M16—531no—
261#335—532no—
262#334—532no—
263#332—532no—
264#332—532no—
265#332—531no—
266#332—531no—
267#331—531no—
268#331—531no—
269C53#1923911yescharacteristic, complemented, Sylow
270C2×#41—1061no—
271C22×Q8—21no—
272#137—2124no—
273#137—2124no—
274C2×Q16—1062no—
275C2×Q16—1062no—
276C2×SD16—1062no—
277C2×SD16—1062no—
278C2×SD16—1062no—
279C2×M16—531no—
280C2×M16—531no—
281C22×C8—1061no—
282C22×C8—1061no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—531no—
287C4⋊Q8C22⋊D5311yescharacteristic
288C4⋊Q8C22⋊D5311yescharacteristic
289C4⋊Q8C22×D5312yes—
290#127—531no—
291#125—21no—
292#124—1061no—
293C22⋊Q8—21no—
294C22⋊Q8C22⋊D5312yes—
295#121—1061no—
296#121—1061no—
297C4×Q8—21no—
298C4×Q8C22×D5312yes—
299C4×D4—1061no—
300C4×D4—1061no—
301#117—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C42C22×D5311yescharacteristic
305#106—22no—
306#106—22no—
307#106—1062no—
308C4⋊C8—531no—
309C4⋊C8—531no—
310C4⋊C8—532no—
311C4⋊C8—1062no—
312C4⋊C8—1062no—
313C4⋊C8—1061no—
314C4⋊C8—1061no—
315C4⋊C8C22×D5311yescharacteristic
316C4⋊C8C22×D5311yescharacteristic
317#103—22no—
318#103—22no—
319#103—1062no—
320#102—1062no—
321C22⋊C8—21no—
322#97—532no—
323#97—1062no—
324C4×C8—21no—
325C4×C8—532no—
326C4×C8—1062no—
327#41—2122no—
328#41—2122no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—1061no—
335C2×Q8C2×C22⋊D5312yes—
336C2×D4—1061no—
337C2×D4—1061no—
338C2×D4—1061no—
339C2×D4—2121no—
340C22×C4—41no—
341C22×C4—2121no—
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345Q16—2124no—
346Q16—2124no—
347SD16—4242no—
348SD16—2124no—
349SD16—2124no—
350M16—534no—
351M16—1062no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—532no—
355C2×C8—532no—
356C2×C8—1061no—
357C2×C8—1061no—
358C2×C8—1062no—
359C2×C8—4241no—
360C2×C8—2122no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—1061no—
371C4⋊C4—1061no—
372C4⋊C4C2×C22⋊D5312yes—
373C22⋊C4—22no—
374C22⋊C4—2121no—
375C22⋊C4—2121no—
376C42—22no—
377C42D4×D5311yescharacteristic
378C42#1171994711yescharacteristic
379C42C2×C22⋊D5311yescharacteristic
380C42C23×D5311yescharacteristic
381C23—2121no—
382C23#1313318911yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—42no—
386Q8—42no—
387Q8—81no—
388Q8—82no—
389Q8—2121no—
390Q8—2121no—
391D4—4241no—
392D4—4242no—
393D4—2121no—
394D4—2121no—
395D4—2121no—
396D4—2121no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—4242no—
405C2×C4—2121no—
406C2×C4#1313318911yescharacteristic
407C2×C4#1313318911yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4SD16×D5312yes—
410C2×C4#1313324412yes—
411C2×C4C2×D4×D5311yescharacteristic
412C2×C4C22×C22⋊D5311yescharacteristic
413C2×C4C2×#1171994711yescharacteristic
414C8—41no—
415C8—41no—
416C8—534no—
417C8—1062no—
418C8—2121no—
419C8—2121no—
420C8—2122no—
421C22—21no—
422C22—21no—
423C22—4241no—
424C22—4241no—
425C22—4241no—
426C22#1681082112yes—
427C22C2×#1313318911yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—4241no—
435C4#1681082112yes—
436C4C2×SD16×D5311yescharacteristic
437C4C2×#1313324411yescharacteristic
438C4#1681143111yescharacteristic
439C4#1681143811yescharacteristic
440C2—22no—
441C2—8481no—
442C2C2×#1681082111yescharacteristic
443C2#2028440211yescharacteristic
444C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3229C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3229D53
hypocenterC53#19239
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
3. #20283664C2semidirect
4. #20274207C2semidirect
5. #20274189C2semidirect
6. #20274083C2semidirect
7. #20274065C2semidirect
8. #20273468C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3229C2semidirect
150. #11719917SD16semidirect
219. #3229D53semidirect
269. C53#19239semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202851722
#202836642
#202742072
#202741892
#202740832
#202740652
#202734682
#202733622
C2×#168103692
C53×#32292
#1923953