Finite Group Databasebeta

← #27388458

Subgroups — #27388458

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810735C211yesmaximal, characteristic
2#20277352C212yesmaximal, complemented
3#20275592C211yesmaximal, characteristic
4#20275456C211yesmaximal, characteristic, complemented
5#20275443C211yesmaximal, characteristic, complemented
6#20275415C212yesmaximal, complemented
7#20275395C212yesmaximal, complemented
8#20274614C211yesmaximal, characteristic, complemented
9#102×Dic53C212yesmaximal, complemented
10#20274401C211yesmaximal, characteristic, complemented
11C53×#1970C211yesmaximal, characteristic
12C2×D4×Dic53C2212yes—
13#16810735C418yescomplemented
14C2×#13133149C2211yescharacteristic
15C2×#13133149C2211yescharacteristic
16C2×#13133138C2212yes—
17C2×#13133138C2212yes—
18C2×#13133128C2211yescharacteristic
19C2×#13133127C2211yescharacteristic
20C2×#13133127C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23#16810526C2212yes—
24#16810521C2212yes—
25#16810519C2212yes—
26#16810517—22no—
27#16810511—22no—
28#16810484—22no—
29#16810481—22no—
30C4⋊C4×Dic53C2212yes—
31#16810469C2211yescharacteristic
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×D8×C53C2211yescharacteristic
36C2×#107×C53C2211yescharacteristic
37C2×#102×C53C2212yes—
38C2×C4×C8×C53C2211yescharacteristic
39C53×#390C2212yes—
40C23×Dic53—22no—
41C22×#88322C2311yescharacteristic
42C22×#88322C2311yescharacteristic
43D4×Dic53—24no—
44D4×Dic53C2×C418yes—
45C2×C4⋊Dic53C2312yes—
46C2×C4⋊Dic53C2312yes—
47C2×#11719919—22no—
48C2×#11719919—22no—
49C2×C22⋊Dic53—22no—
50C2×C22⋊Dic53—22no—
51C2×C22⋊Dic53—22no—
52C2×C4×Dic53—22no—
53C2×C4×Dic53C2311yescharacteristic
54C2×C4×Dic53C2311yescharacteristic
55#13133149—22no—
56#13133149C2×C414yes—
57#13133138—24no—
58#13133138C2×C418yes—
59#13133128—22no—
60#13133127—22no—
61#13133127C2×C414yes—
62C8×Dic53—22no—
63C8×Dic53—22no—
64#13133124—22no—
65#13133124—22no—
66#13133124—22no—
67#13133124D412yes—
68#13133124D412yes—
69C22×D4×C53C2312yes—
70C2×D8×C53—22no—
71C2×D8×C53—22no—
72C2×D8×C53C2×C414yes—
73C22×C8×C53C2311yescharacteristic
74C22×C8×C53C2311yescharacteristic
75C2×C4⋊C4×C53C2312yes—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C42×C53C2311yescharacteristic
79#107×C53—22no—
80#102×C53—24no—
81C4×C8×C53—22no—
82C24×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—22no—
88C2×D4×C53C22×C414yes—
89C22×C4×C53—22no—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2411yescharacteristic
92C22×Dic53—22no—
93C22×Dic53—28no—
94C22×Dic53—22no—
95C22×Dic53—82no—
96C22×Dic53—82no—
97C22×Dic53C2×D411yescharacteristic
98C22×Dic53C2×D411yescharacteristic
99C22×Dic53C2×D412yes—
100D8×C53—28no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53C22×C412yes—
104C2×C8×C53#4112yes—
105C2×C8×C53#4112yes—
106C2×C8×C53#4112yes—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C22⋊C4×C53—82no—
110C22⋊C4×C53—82no—
111C42×C53—21no—
112C42×C53—21no—
113C22⋊Dic53—28no—
114C22⋊Dic53—82no—
115C22⋊Dic53—82no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53C22×C414yes—
120#11719919—82no—
121#11719919—82no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—82no—
125C4×Dic53C22×C412yes—
126C4×Dic53#4112yes—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322C22×C412yes—
132C2×#88322#4112yes—
133C23×C53—24no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53C22×D411yescharacteristic
141D4×C53—24no—
142D4×C53—24no—
143D4×C53—42no—
144D4×C53—42no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53C23×C411yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×C4×C53C2×#4111yescharacteristic
155C2×C4×C53C2×#4111yescharacteristic
156C2×C4×C53C2×#4111yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—24no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—22no—
163C2×Dic53—22no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—88no—
170C2×Dic53C4×D414yes—
171C8×C53—22no—
172C8×C53#11714yes—
173#88322—22no—
174#88322—22no—
175#21290—531nomaximal, Sylow
176C22×C53—44no—
177C22×C53—44no—
178C22×C53—44no—
179C22×C53—82no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53C2×C4×D411yescharacteristic
184C22×C53#53211yescharacteristic
185C22×C53#53211yescharacteristic
186C22×C53#57612yes—
187C22×C53#57712yes—
188C4×C53—42no—
189C4×C53—82no—
190C4×C53C2×#11711yescharacteristic
191C4×C53C4×#4111yescharacteristic
192C4×C53#52911yescharacteristic
193C4×C53#52911yescharacteristic
194Dic53—24no—
195Dic53—41no—
196Dic53—41no—
197Dic53—44no—
198Dic53—82no—
199C2×#442—531no—
200#2319—532no—
201#1972—531no—
202#1970D5311yescharacteristic
203#1944—531no—
204#1941—531no—
205#1936—532no—
206#1934—532no—
207#1776—531no—
208C4×#102—532no—
209#1726—531no—
210C2×C53—84no—
211C2×C53#229311yescharacteristic
212C2×C53#294612yes—
213C2×C53#317311yescharacteristic
214C2×C53#317611yescharacteristic
215C2×C53#317912yes—
216C22×D8C2×D5311yescharacteristic
217C2×C4×D4—532no—
218#442—538no—
219C2×#107C2×D5311yescharacteristic
220C2×#106—531no—
221C2×#106—531no—
222C2×#102—532no—
223C2×#102—532no—
224C2×#102C2×D5312yes—
225C2×#97—531no—
226C2×#97—531no—
227C2×#97—531no—
228C2×C4×C8—531no—
229C2×C4×C8—531no—
230C2×C4×C8C2×D5311yescharacteristic
231#390—532no—
232#390C2×D5312yes—
233#389—532no—
234#389—532no—
235#388—1062no—
236#388—1062no—
237#381—1062no—
238#380—1062no—
239C4×C4⋊C4—532no—
240#376—531no—
241#340—532no—
242#340—532no—
243#340—532no—
244C53#2129011yescharacteristic, complemented, Sylow
245C22×D4C22×D5312yes—
246C23×C4—1062no—
247C2×D8—22no—
248C2×D8—22no—
249C2×D8C4×D5314yes—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8C22×D5311yescharacteristic
253C22×C8C22×D5311yescharacteristic
254C4×D4—538no—
255C4×D4—1064no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—1062no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4C22×D5312yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—1062no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4—1062no—
266C2×C42—531no—
267C2×C42—531no—
268C2×C42—1062no—
269C2×C42C22×D5311yescharacteristic
270#107—22no—
271#106—534no—
272#106—1062no—
273#102—24no—
274#102—538no—
275#102—1064no—
276#97—534no—
277#97—1062no—
278#97—1062no—
279C4×C8—22no—
280C4×C8—1062no—
281C4×C8—1062no—
282#95—532no—
283#95—532no—
284#95—1062no—
285#95—1062no—
286#95—1062no—
287C24—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4C2×C4×D5314yes—
294C22×C4—22no—
295C22×C4—532no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—1068no—
299C22×C4—1062no—
300C22×C4—1062no—
301C22×C4—4242no—
302C22×C4—4242no—
303C22×C4D4×D5312yes—
304C22×C4C23×D5311yescharacteristic
305D8—28no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—532no—
309C2×C8—532no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8C2×C4×D5312yes—
315C2×C8#1171994712yes—
316C2×C8#1171994712yes—
317C2×C8#1171994712yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—534no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—4242no—
325C4⋊C4—4242no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—1068no—
329C22⋊C4—4242no—
330C22⋊C4—4242no—
331C42—21no—
332C42—21no—
333C42—532no—
334C42—532no—
335C42—1061no—
336C42—1061no—
337C42—4242no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23C2×D4×D5311yescharacteristic
346D4—24no—
347D4—24no—
348D4—42no—
349D4—42no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—534no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—1064no—
360C2×C4—1062no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1062no—
364C2×C4—4248no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4C22×C4×D5311yescharacteristic
371C2×C4#41×D5311yescharacteristic
372C2×C4#41×D5311yescharacteristic
373C2×C4C2×#1171994711yescharacteristic
374C2×C4C2×#1171994711yescharacteristic
375C2×C4C2×#1171994711yescharacteristic
376C8—22no—
377C8—1062no—
378C8—1062no—
379C8#1313317314yes—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22C4×D4×D5311yescharacteristic
388C22#1681116811yescharacteristic
389C22#1681117111yescharacteristic
390C22#1681142412yes—
391C22#1681143012yes—
392C4—42no—
393C4—82no—
394C4—1064no—
395C4—4242no—
396C4—2121no—
397C4—2124no—
398C4—2121no—
399C4C2×#1313317311yescharacteristic
400C4C4×#1171994711yescharacteristic
401C4#1681115211yescharacteristic
402C4#1681115411yescharacteristic
403C2—84no—
404C2#2027717711yescharacteristic
405C2#2028354012yes—
406C2#2028532911yescharacteristic
407C2#2028536211yescharacteristic
408C2#2028540112yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1970C2
socleC23×C53C22×D4
prime radicalC22×D8×C53C22
cosocleC22×C4×C53C24
hypercenter#1970D53
hypocenterC53#21290
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. #20277352C2semidirect
4. #20275456C2semidirect
5. #20275443C2semidirect
6. #20275415C2semidirect
7. #20275395C2semidirect
8. #20274614C2semidirect
9. #102×Dic53C2semidirect
10. #20274401C2semidirect
13. #16810735C4semidirect
244. C53#21290semidirect

Maximal subgroups

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

subgroupindex
C2×#168107352
#202773522
#202755922
#202754562
#202754432
#202754152
#202753952
#202746142
#102×Dic532
#202744012
C53×#19702
#2129053