Finite Group Databasebeta

← #27275089

Subgroups — #27275089

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283405C211yesmaximal, characteristic
2#20274494C211yesmaximal, characteristic, complemented
3#20274489C211yesmaximal, characteristic, complemented
4#20273207C211yesmaximal, characteristic
5#20273207C211yesmaximal, characteristic
6#20272600C211yesmaximal, characteristic, complemented
7C53×#1298C211yesmaximal, characteristic, complemented
8C2×#13133225C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16811134C2211yescharacteristic
11#16810691—21no—
12#16810691—21no—
13#16810686—21no—
14#16810686—21no—
15#16810651C411yescharacteristic, complemented
16#16810591C411yescharacteristic, complemented
17#16810581C411yescharacteristic, complemented
18C2×#13133127—21no—
19C2×C8×Dic53—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21C4×C22⋊Dic53C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24#16810361C412yescomplemented
25#16810361C412yescomplemented
26#16810358C412yescomplemented
27#16810358C412yescomplemented
28C53×#407C2211yescharacteristic
29C2×C22⋊C8×C53C2211yescharacteristic
30C2×C22⋊C8×C53C2211yescharacteristic
31C2×#11719945C2×C411yescharacteristic
32#13133225—42no—
33C2×#11719912—21no—
34C22×#88322—42no—
35C22×#88322C2×C411yescharacteristic
36C22×#88322C2×C411yescharacteristic
37C22×#88322D411yescharacteristic
38C22×#88322D411yescharacteristic
39C22×#88322D411yescharacteristic
40C22×#88322D411yescharacteristic
41C2×C8×D53—21no—
42#13133200—21no—
43#13133196C2×C411yescharacteristic
44#13133196C2×C411yescharacteristic
45#13133191C2×C411yescharacteristic
46#13133191C2×C411yescharacteristic
47#13133186—21no—
48C4×C22⋊D53—21no—
49C4×C22⋊D53—21no—
50#13133176C2×C411yescharacteristic
51C2×C22⋊Dic53—21no—
52C2×C22⋊Dic53—21no—
53C2×C4×Dic53—21no—
54C2×C4×Dic53—21no—
55#13133146—21no—
56#13133146—21no—
57#13133129—21no—
58#13133129—21no—
59#13133127—21no—
60#13133127—42no—
61C8×Dic53—21no—
62C8×Dic53—42no—
63#13133124C2×C412yes—
64#13133124C2×C412yes—
65C23×C4×C53C2311yescharacteristic
66C2×M16×C53C2×C411yescharacteristic
67C2×M16×C53C2×C411yescharacteristic
68C22×C8×C53—21no—
69C22×C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53C2×C412yes—
75C22⋊C8×C53C2×C412yes—
76C24×C53#4111yescharacteristic
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C2×C22⋊D53—21no—
86C2×C22⋊D53—21no—
87C2×C22⋊D53—21no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90#11719945—41no—
91#11719945—41no—
92C2×D212C4211yescharacteristic
93C2×C4×D53—21no—
94C2×Dic106C4211yescharacteristic
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C22⋊Dic53—24no—
106C22⋊Dic53—21no—
107C22⋊Dic53—21no—
108C22⋊Dic53—21no—
109C22⋊Dic53—42no—
110#11719921—21no—
111#11719921—21no—
112C4⋊Dic53C4211yescharacteristic
113C4⋊Dic53C4211yescharacteristic
114#11719919—21no—
115#11719919—21no—
116C4×Dic53—21no—
117C4×Dic53—41no—
118C4×Dic53—41no—
119C4×Dic53—41no—
120C4×Dic53—41no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322C22⋊C411yescharacteristic
128C2×#88322C22⋊C411yescharacteristic
129C2×#88322C22⋊C411yescharacteristic
130C2×#88322C22⋊C411yescharacteristic
131#11719912—42no—
132C8×D53—42no—
133C23×C53—22no—
134C23×C53—22no—
135C23×C53—41no—
136C23×C53#11711yescharacteristic
137C23×C53C4×D411yescharacteristic
138C23×C53C4×D411yescharacteristic
139C22×D53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53C2×C4211yescharacteristic
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C22⋊D53—42no—
154C22⋊D53—42no—
155C22⋊D53—42no—
156C2×Dic53—24no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165D212—41no—
166C4×D53—41no—
167C4×D53—41no—
168Dic106—41no—
169C8×C53—42no—
170C8×C53—42no—
171C8×C53—41no—
172C8×C53—41no—
173#88322—21no—
174#88322—21no—
175#88322—21no—
176#88322#9912yes—
177#8240—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—44no—
186C22×C53C4×C22⋊C411yescharacteristic
187C22×C53#44312yes—
188C22×C53#44412yes—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53C2×#9911yescharacteristic
196C4×C53#41011yescharacteristic
197Dic53—44no—
198Dic53—41no—
199Dic53—41no—
200Dic53—41no—
201Dic53—82no—
202#2929—531no—
203#1752—531no—
204C8×C22⋊C4—531no—
205#1501—531no—
206#1501—531no—
207#1298D5311yescharacteristic, complemented
208#1297—531no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53—41no—
212C2×C53C4×#9911yescharacteristic
213C2×C53#176411yescharacteristic
214C2×C53#176511yescharacteristic
215D53—81no—
216C2×#131—531no—
217C23×C8—531no—
218#525—531no—
219#435—1061no—
220#435—1061no—
221C8×D4—1061no—
222C8×D4—1061no—
223#424—531no—
224#408—531no—
225#407C2×D5311yescharacteristic
226C2×C22⋊C8C2×D5311yescharacteristic
227C2×C22⋊C8C2×D5311yescharacteristic
228#405—531no—
229C2×#97—1061no—
230C2×C4×C8—1061no—
231C4×C22⋊C4—531no—
232C4×C22⋊C4—531no—
233#336—1061no—
234#336—1061no—
235#328—532no—
236#328—532no—
237#327—532no—
238#327—532no—
239C53#824011yescharacteristic, complemented, Sylow
240C2×#41—531no—
241C23×C4C22×D5311yescharacteristic
242#131—2122no—
243C2×M16—1061no—
244C2×M16C4×D5311yescharacteristic
245C2×M16C4×D5311yescharacteristic
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—531no—
249C22×C8—531no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8—531no—
253C22×C8—531no—
254C22×C8—1061no—
255C22×C8—2122no—
256#123—1061no—
257C22⋊Q8—531no—
258C22⋊Q8—531no—
259#121—531no—
260#121—531no—
261#120—1061no—
262C4×D4—1061no—
263C4×D4—1061no—
264#117—531no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C42—1061no—
268C2×C42—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—1061no—
276C22⋊C8—1061no—
277C22⋊C8C4×D5312yes—
278C22⋊C8C4×D5312yes—
279#97—1061no—
280#97—2122no—
281C4×C8—1061no—
282C4×C8—2122no—
283#95—532no—
284#95—532no—
285C24#1171994911yescharacteristic
286#41—2121no—
287#41—2121no—
288C2×Q8—531no—
289C2×D4—531no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1061no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—1062no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4C2×C4×D5311yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4#1171994711yescharacteristic
304M16—41no—
305M16—41no—
306M16—2122no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—1062no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8—2122no—
323C2×C8—2121no—
324C2×C8—2121no—
325C2×C8—2121no—
326C4⋊C4—531no—
327C4⋊C4—531no—
328C4⋊C4—1061no—
329C4⋊C4—1061no—
330C22⋊C4—1064no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—2122no—
337C42—1061no—
338C42—2121no—
339C42—2121no—
340C42—2121no—
341C42—2121no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—1061no—
346C23#1313317511yescharacteristic
347C23#1313317911yescharacteristic
348C23#1313317911yescharacteristic
349Q8—2121no—
350D4—2121no—
351D4—2122no—
352D4—2122no—
353D4—2122no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1064no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4C42×D5311yescharacteristic
375C2×C4C22⋊C4×D5311yescharacteristic
376C2×C4#1313317711yescharacteristic
377C2×C4#1313317711yescharacteristic
378C8—42no—
379C8—42no—
380C8—41no—
381C8—41no—
382C8—532no—
383C8—1061no—
384C8—1061no—
385C8—1061no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22#1681047011yescharacteristic
398C22#1681073912yes—
399C22#1681074412yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—4242no—
404C4—2124no—
405C4—2121no—
406C4—2121no—
407C4—2121no—
408C4#99×D5311yescharacteristic
409C4#1681059511yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—4241no—
414C2#2027450211yescharacteristic
415C2#2027455111yescharacteristic
416C2#2027455811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1298C2
socleC22×C53C4×C22⋊C4
prime radical#16811134C22
cosocleC23×C4×C53C23
hypercenter#1298D53
hypocenterC53#8240
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #20274494C2semidirect
3. #20274489C2semidirect
6. #20272600C2semidirect
7. C53×#1298C2semidirect
8. C2×#13133225C4semidirect
15. #16810651C4semidirect
16. #16810591C4semidirect
17. #16810581C4semidirect
24. #16810361C4semidirect
25. #16810361C4semidirect
26. #16810358C4semidirect
27. #16810358C4semidirect
207. #1298D53semidirect
239. C53#8240semidirect

Maximal subgroups

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

subgroupindex
#202834052
#202744942
#202744892
#202732072
#202726002
C53×#12982
#824053