Finite Group Databasebeta

← #27367708

Subgroups — #27367708

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286057C211yesmaximal, characteristic, complemented
2#20285653C211yesmaximal, characteristic
3#20285090C211yesmaximal, characteristic
4#20274283C212yesmaximal, complemented
5#20274255C212yesmaximal, complemented
6#20274159C212yesmaximal, complemented
7#20274131C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273373C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic
11C53×#2962C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13C2×#13133214C2211yescharacteristic
14#16810997—21no—
15#16810988—22no—
16#16810971—21no—
17#16810970C2212yes—
18#16810970C2212yes—
19#16810918C2212yescomplemented
20#16810915—21no—
21#16810862—22no—
22#16810833C2212yescomplemented
23#16810830—21no—
24#16810823C2212yes—
25#16810823C2212yes—
26#16810773—22no—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810629—21no—
30#16810629—21no—
31#16810376C2212yescomplemented
32#16810374C2212yescomplemented
33#16810371C2211yescharacteristic, complemented
34#16810371C2211yescharacteristic, complemented
35#16810368C2211yescharacteristic, complemented
36#16810368C2211yescharacteristic, complemented
37#16810366C2212yes—
38#16810366C2212yes—
39C53×#533C2211yescharacteristic
40C53×#446—22no—
41C53×#446C2212yescomplemented
42C8×Q8×C53C2212yescomplemented
43C53×#424—21no—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#405—21no—
46C4×M16×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50C2×#11719925—22no—
51C2×#11719924—22no—
52C2×#11719924—22no—
53C22×#88322—21no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133214—21no—
57#13133214—21no—
58#13133214—21no—
59#13133214C2312yes—
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65#13133176—21no—
66#13133150—22no—
67C4⋊#88322—21no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2312yes—
71#13133141—22no—
72#13133140—22no—
73#13133139—22no—
74#13133139—22no—
75#13133139—22no—
76#13133139—22no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—21no—
80C4×#88322—21no—
81C2×M16×C53D411yescharacteristic
82C2×M16×C53D411yescharacteristic
83C22×C8×C53—21no—
84C4⋊Q8×C53D411yescharacteristic
85C4⋊Q8×C53D411yescharacteristic
86#125×C53—21no—
87C22⋊Q8×C53—21no—
88C22⋊Q8×C53D412yes—
89C4×Q8×C53—21no—
90C4×Q8×C53C2312yes—
91#117×C53—21no—
92C2×C42×C53C2311yescharacteristic
93C4⋊C8×C53—22no—
94C4⋊C8×C53—22no—
95C4⋊C8×C53—21no—
96C4⋊C8×C53C2311yescharacteristic
97C4⋊C8×C53C2311yescharacteristic
98#97×C53—22no—
99#97×C53D412yes—
100C4×C8×C53—22no—
101C4×C8×C53C2312yes—
102C23×D53—41no—
103C2×Q8×C53—21no—
104C2×Q8×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C2×D212—22no—
109C2×D212—22no—
110C2×D212—22no—
111C2×D212—42no—
112C2×D212—42no—
113C2×D212—41no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—81no—
117C2×D212—81no—
118C2×D212—82no—
119C2×Dic106—21no—
120C2×Dic106—21no—
121M16×C53—22no—
122M16×C53—22no—
123C2×C8×C53—21no—
124C2×C8×C53—21no—
125C2×C8×C53—42no—
126C2×C8×C53C2×D412yes—
127C2×C8×C53C2×D412yes—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53C2×D412yes—
134C22⋊C4×C53—22no—
135C42×C53—22no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140C22⋊Dic53—41no—
141#11719925—82no—
142#11719924—82no—
143#11719924—82no—
144C4⋊Dic53—21no—
145C4⋊Dic53—21no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148#11719919—41no—
149#11719919—41no—
150C4×Dic53—41no—
151C2×#88322—21no—
152C2×#88322—21no—
153C2×#88322—22no—
154C2×#88322—81no—
155C23×C53#12111yescharacteristic
156C22×D53—42no—
157C22×D53—82no—
158C22×D53—82no—
159C22×D53—82no—
160Q8×C53—22no—
161Q8×C53—22no—
162Q8×C53—81no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—41no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#13512yes—
172C2×C4×C53#13712yes—
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C2×Dic53—41no—
177C2×Dic53—41no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—82no—
183D212—82no—
184D212—82no—
185D212—82no—
186Dic106—41no—
187Dic106—41no—
188Dic106—81no—
189C8×C53—22no—
190C8×C53—22no—
191C8×C53—41no—
192C8×C53—41no—
193#88322—42no—
194#88322—41no—
195#88322—41no—
196#19242—531nomaximal, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53#46412yes—
200C22×C53C2×#12111yescharacteristic
201C2×D53—82no—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—164no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—42no—
210C4×C53#46412yes—
211C4×C53C2×#13511yescharacteristic
212C4×C53C2×#13711yescharacteristic
213C4×C53#57611yescharacteristic
214C4×C53#57711yescharacteristic
215Dic53—81no—
216Dic53—81no—
217#3249—531no—
218#3208—531no—
219#3138—531no—
220#2962D5311yescharacteristic, complemented
221#1702—532no—
222#1698—532no—
223#1682—532no—
224#1678—532no—
225#1565—531no—
226#1542—531no—
227C2×#331—531no—
228C2×C53—22no—
229C2×C53C2×#46411yescharacteristic
230C2×C53#306211yescharacteristic
231C2×C53#306811yescharacteristic
232D53—162no—
233#533—531no—
234#533C2×D5311yescharacteristic
235C2×#127—531no—
236#492—1061no—
237#490—1062no—
238#486—532no—
239#486—532no—
240#486—1061no—
241#477—532no—
242#477—1061no—
243#467—1062no—
244#462—532no—
245#462—1061no—
246#460—532no—
247#460—532no—
248#450—1062no—
249#446—22no—
250#446C2×D5312yescomplemented
251C8×Q8C2×D5312yescomplemented
252#431—531no—
253#424—21no—
254C2×C4⋊C8—531no—
255C2×C4⋊C8C2×D5311yescharacteristic
256#418—1061no—
257#418—1061no—
258#405—21no—
259C4×M16C2×D5311yescharacteristic
260#335—532no—
261#334—532no—
262#333—531no—
263#333—531no—
264#332—531no—
265#332—531no—
266#331—532no—
267#331—532no—
268C53#1924211yescharacteristic, complemented, Sylow
269C22×D4—1061no—
270C22×D4—1061no—
271C22×D4—1061no—
272C2×Q16—1062no—
273C2×SD16—1062no—
274C2×SD16—1062no—
275C2×M16C22⋊D5311yescharacteristic
276C2×M16C22⋊D5311yescharacteristic
277C22×C8—21no—
278C22×C8—1061no—
279C4⋊Q8—531no—
280C4⋊Q8—531no—
281C4⋊Q8D21211yescharacteristic
282C4⋊Q8D21211yescharacteristic
283#127—532no—
284#127—1061no—
285#127—1061no—
286#127—1061no—
287#125—21no—
288#125—1061no—
289C22⋊Q8—21no—
290C22⋊Q8—1061no—
291C22⋊Q8—1061no—
292C22⋊Q8D21212yes—
293C4×Q8—21no—
294C4×Q8—1061no—
295C4×Q8—1061no—
296C4×Q8C22×D5312yes—
297#117—21no—
298#117—1061no—
299C2×C42C22×D5311yescharacteristic
300#107—1062no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—21no—
304C4⋊C8—532no—
305C4⋊C8—531no—
306C4⋊C8—531no—
307C4⋊C8—1061no—
308C4⋊C8C22×D5311yescharacteristic
309C4⋊C8C22×D5311yescharacteristic
310#103—1062no—
311#103—1062no—
312#102—1062no—
313#102—1062no—
314#102—1062no—
315#102—1062no—
316#102—1062no—
317#102—1062no—
318C22⋊C8—1061no—
319#97—22no—
320#97C22⋊D5312yes—
321C4×C8—22no—
322C4×C8—1061no—
323C4×C8C22×D5312yes—
324C24—2121no—
325C2×Q8—21no—
326C2×Q8—1061no—
327C2×Q8—1061no—
328C2×Q8C2×D21212yes—
329C2×D4—1062no—
330C2×D4—1062no—
331C2×D4—1062no—
332C2×D4—4241no—
333C2×D4—4241no—
334C2×D4—4242no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2122no—
339C2×D4—2122no—
340C22×C4C2×D21211yescharacteristic
341C22×C4#1171994511yescharacteristic
342C22×C4C2×C22⋊D5311yescharacteristic
343Q16—4242no—
344SD16—4242no—
345SD16—4242no—
346M16—22no—
347M16—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—42no—
351C2×C8—1061no—
352C2×C8—1061no—
353C2×C8—1062no—
354C2×C8—4241no—
355C2×C8C2×C22⋊D5312yes—
356C2×C8C2×C22⋊D5312yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—1061no—
363C4⋊C4—1061no—
364C4⋊C4—1061no—
365C4⋊C4—1061no—
366C4⋊C4—2121no—
367C4⋊C4—2121no—
368C4⋊C4C2×D21212yes—
369C22⋊C4—22no—
370C22⋊C4—2121no—
371C42—22no—
372C42—2121no—
373C42C2×D21211yescharacteristic
374C42#1171994511yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C42C23×D5311yescharacteristic
377C23—4242no—
378C23—4242no—
379C23—4242no—
380C23—2122no—
381C23#1313319111yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—81no—
385Q8—4241no—
386Q8—2121no—
387Q8—2121no—
388D4—4242no—
389D4—4242no—
390D4—4242no—
391D4—4242no—
392D4—2122no—
393D4—2122no—
394D4—2122no—
395D4—2122no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4#1313319111yescharacteristic
404C2×C4#1313319111yescharacteristic
405C2×C4#1313319111yescharacteristic
406C2×C4#1313323712yes—
407C2×C4#1313324812yes—
408C2×C4C22×C22⋊D5311yescharacteristic
409C2×C4C22×D21211yescharacteristic
410C2×C4C2×#1171994511yescharacteristic
411C8—22no—
412C8—22no—
413C8—41no—
414C8—41no—
415C8—2122no—
416C8—2121no—
417C8—2121no—
418C22—21no—
419C22—21no—
420C22—8484no—
421C22—4242no—
422C22—4242no—
423C22—4242no—
424C22#1681084512yes—
425C22C2×#1313319111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—4241no—
432C4—4241no—
433C4#1681084512yes—
434C4C2×#1313323711yescharacteristic
435C4C2×#1313324811yescharacteristic
436C4#1681142811yescharacteristic
437C4#1681143511yescharacteristic
438C2—22no—
439C2—8482no—
440C2C2×#1681084511yescharacteristic
441C2#2028439511yescharacteristic
442C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2962C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#2962D53
hypocenterC53#19242
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #20286057C2semidirect
4. #20274283C2semidirect
5. #20274255C2semidirect
6. #20274159C2semidirect
7. #20274131C2semidirect
8. #20273467C2semidirect
9. #20273373C2semidirect
11. C53×#2962C2semidirect
19. #16810918C22semidirect
22. #16810833C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810371C22semidirect
34. #16810371C22semidirect
35. #16810368C22semidirect
36. #16810368C22semidirect
41. C53×#446C22semidirect
42. C8×Q8×C53C22semidirect
220. #2962D53semidirect
250. #446C2×D53semidirect
251. C8×Q8C2×D53semidirect
268. C53#19242semidirect

Maximal subgroups

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

subgroupindex
#202860572
#202856532
#202850902
#202742832
#202742552
#202741592
#202741312
#202734672
#202733732
C2×#168103662
C53×#29622
#1924253