Finite Group Databasebeta

← #27358145

Subgroups — #27358145

Order 13568.

908 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285856C212yesmaximal, complemented
2#20280481C211yesmaximal, characteristic
3#20273723C214yesmaximal, complemented
4#20273706C214yesmaximal
5#20273279C212yesmaximal, complemented
6#20273029C211yesmaximal, characteristic
7C53×#3458C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133196—21no—
10#16811019—22no—
11#16810973—22no—
12#16810951C2218yes—
13#16810908—24no—
14#16810676C2212yes—
15#16810553—21no—
16#16810553C2212yes—
17#16810548—21no—
18#16810548C2212yes—
19#16810529—21no—
20#16810489C2211yescharacteristic
21#16810360C2218yes—
22#16810349C2212yes—
23#16810347C2212yes—
24C53×#544—22no—
25C53×#544—22no—
26C53×#544—21no—
27C53×#544C2212yes—
28C53×#543—21no—
29C53×#543C2212yes—
30C53×#533—24no—
31C53×#533—21no—
32C53×#533C2212yes—
33C2×#117×C53—21no—
34C2×#117×C53C2211yescharacteristic
35C22×Dic106—21no—
36#13133217—22no—
37#13133217—22no—
38#13133196—44no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2312yes—
42C2×#11719919—22no—
43C2×C22⋊Dic53—22no—
44#13133148—28no—
45C4⋊#88322—22no—
46#13133141—28no—
47C22⋊#88322C2314yes—
48C4×#88322—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124—21no—
52#13133124C2312yes—
53C23×C4×C53D411yescharacteristic
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—24no—
57C4⋊Q8×C53—24no—
58C4⋊Q8×C53D412yes—
59C4⋊Q8×C53D412yes—
60#125×C53—24no—
61#125×C53—22no—
62#125×C53—24no—
63#125×C53—22no—
64C22⋊Q8×C53—24no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53—24no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53C2314yes—
72C4×Q8×C53—22no—
73C4×Q8×C53—22no—
74C4×Q8×C53—22no—
75C4×Q8×C53D414yes—
76#117×C53—22no—
77#117×C53—22no—
78#117×C53—22no—
79#117×C53—22no—
80#117×C53—22no—
81#117×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53D411yescharacteristic
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C22⋊C4×C53—22no—
87C2×C22⋊C4×C53D411yescharacteristic
88C2×C22⋊C4×C53C2311yescharacteristic
89C2×C42×C53—22no—
90C2×C42×C53D411yescharacteristic
91C2×C42×C53C2311yescharacteristic
92C24×C53C2×D411yescharacteristic
93C2×Q8×C53—22no—
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53C2×D414yes—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53SD1614yes—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2411yescharacteristic
109C22×Dic53—22no—
110C22×Dic53—22no—
111C2×Dic106—41no—
112C2×Dic106—41no—
113C2×Dic106—42no—
114C2×Dic106—42no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—24no—
119C4⋊C4×C53—24no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—24no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—24no—
128C4⋊C4×C53C2×D414yes—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—22no—
133C42×C53—24no—
134C42×C53—24no—
135C42×C53—21no—
136C42×C53—21no—
137C42×C53C2×D412yes—
138C42×C53C2×D412yes—
139C22⋊Dic53—44no—
140C4⋊Dic53—24no—
141C4⋊Dic53—42no—
142C4⋊Dic53—42no—
143#11719919—44no—
144C2×#88322—24no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53C2×SD1612yes—
150C23×C53C22×D411yescharacteristic
151Q8×C53—44no—
152Q8×C53—82no—
153Q8×C53—82no—
154Q8×C53—82no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—24no—
158C2×C4×C53—24no—
159C2×C4×C53—24no—
160C2×C4×C53—24no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—82no—
168C2×C4×C53—82no—
169C2×C4×C53#12012yes—
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53C2×SD1612yes—
173C2×C4×C53C2×SD1612yes—
174C2×C4×C53#13712yes—
175C2×C4×C53C22×D412yes—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182Dic106—82no—
183Dic106—82no—
184#88322—44no—
185#18310—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—22no—
189C22×C53—42no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—81no—
193C22×C53C2×#12011yescharacteristic
194C22×C53C22×SD1611yescharacteristic
195C22×C53C2×#13711yescharacteristic
196C4×C53—22no—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—44no—
201C4×C53—41no—
202C4×C53—44no—
203C4×C53#44812yes—
204C4×C53#44812yes—
205Dic53—82no—
206Dic53—82no—
207#3458D5311yescharacteristic
208#3229—532no—
209#2621—531no—
210#1611—534no—
211#1608—534no—
212#1518—532no—
213#1458—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—42no—
217C2×C53C2×#44811yescharacteristic
218C2×C53#302211yescharacteristic
219C2×C53#302811yescharacteristic
220#544—22no—
221#544—22no—
222#544—21no—
223#544C2×D5312yes—
224#543—21no—
225#543C2×D5312yes—
226#533—24no—
227#533—21no—
228#533C2×D5312yes—
229C2×C4⋊Q8—531no—
230C2×C22⋊Q8—1061no—
231C2×#117—21no—
232C2×#117C2×D5311yescharacteristic
233#498—1062no—
234#487—1062no—
235#483—538no—
236#475—1064no—
237#431—532no—
238#398—532no—
239#398—1061no—
240#397—532no—
241#397—1061no—
242#391—1061no—
243#382—531no—
244#328—538no—
245#324—532no—
246#323—532no—
247C53#1831011yescharacteristic, complemented, Sylow
248C22×Q8—1061no—
249C23×C4C22⋊D5311yescharacteristic
250C4⋊Q8—24no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—24no—
254C4⋊Q8—1062no—
255C4⋊Q8—1062no—
256C4⋊Q8C22⋊D5312yes—
257C4⋊Q8C22⋊D5312yes—
258#125—24no—
259#125—22no—
260#125—24no—
261#125—22no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—2124no—
270C22⋊Q8C22×D5314yes—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8C22⋊D5314yes—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—532no—
284C2×C4⋊C4—1062no—
285C2×C4⋊C4—1061no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4C22⋊D5311yescharacteristic
288C2×C4⋊C4C22×D5311yescharacteristic
289C2×C22⋊C4—22no—
290C2×C22⋊C4—1062no—
291C2×C22⋊C4C22⋊D5311yescharacteristic
292C2×C22⋊C4C22×D5311yescharacteristic
293C2×C42—22no—
294C2×C42C22⋊D5311yescharacteristic
295C2×C42C22×D5311yescharacteristic
296#106—1068no—
297C4⋊C8—1062no—
298#103—1068no—
299C22⋊C8—534no—
300C4×C8—1062no—
301#95—532no—
302#95—1061no—
303#95—1062no—
304#95—1062no—
305C24C2×C22⋊D5311yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—2121no—
310C2×Q8—2121no—
311C2×Q8—2122no—
312C2×Q8—2122no—
313C2×Q8C2×C22⋊D5314yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4#1171992414yes—
322C22×C4C2×C22⋊D5311yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326C22×C4C2×C22⋊D5311yescharacteristic
327C22×C4C23×D5311yescharacteristic
328C2×C8—1064no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—1064no—
343C4⋊C4—2122no—
344C4⋊C4—2124no—
345C4⋊C4—2122no—
346C4⋊C4C2×C22⋊D5314yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—2124no—
352C42—24no—
353C42—24no—
354C42—21no—
355C42—21no—
356C42C2×C22⋊D5312yes—
357C42C2×C22⋊D5312yes—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—41no—
362C23C2×#1171992412yes—
363C23C22×C22⋊D5311yescharacteristic
364Q8—44no—
365Q8—82no—
366Q8—82no—
367Q8—82no—
368Q8—4242no—
369Q8—4242no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4#1313318612yes—
391C2×C4#1313318611yescharacteristic
392C2×C4#1313318611yescharacteristic
393C2×C4C2×#1171992412yes—
394C2×C4C2×#1171992412yes—
395C2×C4#1313324812yes—
396C2×C4C22×C22⋊D5312yes—
397C8—2124no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22C2×#1313318611yescharacteristic
406C22C22×#1171992411yescharacteristic
407C22C2×#1313324811yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—44no—
413C4—41no—
414C4—44no—
415C4—4242no—
416C4—4242no—
417C4#1681076212yes—
418C4#1681076212yes—
419C2—21no—
420C2—21no—
421C2—42no—
422C2C2×#1681076211yescharacteristic
423C2#2028409411yescharacteristic
424C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3458C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#3458D53
hypocenterC53#18310
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #20285856C2semidirect
3. #20273723C2semidirect
5. #20273279C2semidirect
247. C53#18310semidirect

Maximal subgroups

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

subgroupindex
#202858562
#202804812
#202737232
#202737062
#202732792
#202730292
C53×#34582
#1831053