Finite Group Databasebeta

← #27279550

Subgroups — #27279550

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283800C211yesmaximal, characteristic, complemented
2C2×#16810721C211yesmaximal, characteristic
3#20273105C211yesmaximal, characteristic
4#20273035C211yesmaximal, characteristic
5#20272559C211yesmaximal, characteristic, complemented
6#20272545C211yesmaximal, characteristic, complemented
7C53×#1479C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810754—22no—
11#16810754—22no—
12#16810721—22no—
13#16810721—22no—
14#16810680—22no—
15#16810676C2211yescharacteristic
16#16810676C2211yescharacteristic
17C2×#13133150C411yescharacteristic
18C2×#13133141C411yescharacteristic
19C2×#13133137C411yescharacteristic
20C2×#13133128C2211yescharacteristic
21#16810350—22no—
22#16810346—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719924—22no—
30C2×#11719924—22no—
31C2×#11719924—22no—
32C22×#88322C2×C411yescharacteristic
33C22×#88322C2×C411yescharacteristic
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719921—21no—
40#13133150—22no—
41C4⋊#88322—24no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133141—22no—
45#13133137—22no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48#13133128—24no—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52C4×#88322D412yes—
53C22×Q8×C53C2×C411yescharacteristic
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53D412yes—
57C4×Q8×C53D412yes—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53C22⋊C412yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212M1612yes—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—82no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53C22⋊C412yes—
96C42×C53—22no—
97C42×C53—22no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102#11719924—28no—
103#11719921—41no—
104#11719921—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—22no—
111C2×#88322—22no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322M1612yes—
115C23×C53C4×D411yescharacteristic
116C22×D53—22no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—81no—
120C22×D53—81no—
121Q8×C53—22no—
122Q8×C53—22no—
123Q8×C53—41no—
124Q8×C53—41no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—81no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C2×M1611yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53#13111yescharacteristic
141C2×C4×C53#13111yescharacteristic
142C2×Dic53—41no—
143C2×Dic53—81no—
144D212—22no—
145D212—22no—
146D212—41no—
147D212—41no—
148C4×D53—82no—
149C4×D53—82no—
150C8×C53—81no—
151C8×C53—81no—
152#88322—22no—
153#88322—42no—
154#88322—42no—
155#88322—41no—
156#9094—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#38611yescharacteristic
160C22×C53#43511yescharacteristic
161C22×C53#43611yescharacteristic
162C2×D53—42no—
163C2×D53—82no—
164C2×D53—81no—
165C2×D53—81no—
166C2×D53—82no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53#40711yescharacteristic
174C4×C53#40811yescharacteristic
175C4×C53#43511yescharacteristic
176C4×C53#43611yescharacteristic
177Dic53—81no—
178#2983—531no—
179C2×#440—531no—
180#1479D5311yescharacteristic, complemented
181#1478—531no—
182#1460—531no—
183#1281—531no—
184#1275—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#187011yescharacteristic
188C2×C53#190311yescharacteristic
189C2×C53#190511yescharacteristic
190D53—82no—
191C22×SD16—531no—
192C2×C4×Q8C2×D5311yescharacteristic
193C2×C4×D4—531no—
194#446—1062no—
195#446—1062no—
196#440—1062no—
197#440—1062no—
198#432—1062no—
199#431—531no—
200#431—531no—
201#431C2×D5311yescharacteristic
202#431C2×D5311yescharacteristic
203C2×#107—531no—
204C2×#103—531no—
205C2×#102—531no—
206C2×#97—531no—
207Q8⋊C8—22no—
208D4⋊C8—1062no—
209#323—1062no—
210C53#909411yescharacteristic, complemented, Sylow
211C22×Q8C4×D5311yescharacteristic
212C22×D4—531no—
213C23×C4—1061no—
214C2×SD16—1062no—
215C2×SD16—1062no—
216C2×SD16—1062no—
217C22×C8—531no—
218C22×C8—531no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8D21212yes—
222C4×Q8C22⋊D5312yes—
223C4×D4—1061no—
224C4×D4—1061no—
225C4×D4—1061no—
226C4×D4—1061no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4C4×D5311yescharacteristic
230C2×C22⋊C4—1061no—
231C2×C42—21no—
232C2×C42C22×D5311yescharacteristic
233#107—1062no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—1064no—
237C4⋊C8—1061no—
238C4⋊C8—1061no—
239#103—1062no—
240#102—1062no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—532no—
244C22⋊C8—1061no—
245#97—1064no—
246#97—1061no—
247#97—1061no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—532no—
251C4×C8—1061no—
252C24—1061no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8#1171992112yes—
259C2×D4—532no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—1061no—
268C22×C4—4242no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4C2×C4×D5311yescharacteristic
272C22×C4D4×D5311yescharacteristic
273C22×C4#1171994711yescharacteristic
274SD16—1068no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—532no—
278C2×C8—1061no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1062no—
282C2×C8—1062no—
283C2×C8—1061no—
284C2×C8—1061no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4#1171992112yes—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C42—22no—
294C42—22no—
295C42C2×D21211yescharacteristic
296C42D4×D5311yescharacteristic
297C42#1171994711yescharacteristic
298C42C2×C22⋊D5311yescharacteristic
299C23—1062no—
300C23—4241no—
301C23—4241no—
302C23—2121no—
303C23—2121no—
304C23#1313317711yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—41no—
308Q8—41no—
309D4—1062no—
310D4—1062no—
311D4—2121no—
312D4—2121no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—4242no—
321C2×C4—4242no—
322C2×C4—4241no—
323C2×C4—2121no—
324C2×C4C2×#1171992111yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4#1313318711yescharacteristic
327C2×C4#1313318911yescharacteristic
328C2×C4#1313318911yescharacteristic
329C2×C4#1313320111yescharacteristic
330C2×C4M16×D5311yescharacteristic
331C2×C4M16×D5311yescharacteristic
332C2×C4#1313322311yescharacteristic
333C2×C4#1313322311yescharacteristic
334C8—81no—
335C8—81no—
336C8—1062no—
337C8—2122no—
338C8—2122no—
339C8—2121no—
340C22—21no—
341C22—21no—
342C22—4242no—
343C22—4242no—
344C22—4241no—
345C22—4241no—
346C22—2122no—
347C22#1681050011yescharacteristic
348C22#1681069711yescharacteristic
349C22#1681069811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—4241no—
357C4#1681058611yescharacteristic
358C4#1681058911yescharacteristic
359C4#1681069711yescharacteristic
360C4#1681069811yescharacteristic
361C2—21no—
362C2—21no—
363C2—4242no—
364C2#2027501011yescharacteristic
365C2#2027519311yescharacteristic
366C2#2027520711yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1479C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1479D53
hypocenterC53#9094
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
5. #20272559C2semidirect
6. #20272545C2semidirect
7. C53×#1479C2semidirect
180. #1479D53semidirect
210. C53#9094semidirect

Maximal subgroups

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

subgroupindex
#202838002
C2×#168107212
#202731052
#202730352
#202725592
#202725452
C53×#14792
#909453