Finite Group Databasebeta

← #27294325

Subgroups — #27294325

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276326C211yesmaximal, characteristic, complemented
2C2×#16810386C211yesmaximal, characteristic, complemented
3#20272727C212yesmaximal, complemented
4#20272724C212yesmaximal, complemented
5C53×#3050C211yesmaximal, characteristic
6C22×C4⋊Dic53—21no—
7C2×#13133157—21no—
8C22⋊#11719908—21no—
9C22⋊#11719908C2212yes—
10#16810386—21no—
11#16810386C2212yes—
12C2×#135×C53C411yescharacteristic, complemented
13C23×C8×C53C2211yescharacteristic
14C53×#525—21no—
15C53×#467C411yescharacteristic, complemented
16C53×#467C2211yescharacteristic
17C53×#466C411yescharacteristic, complemented
18C53×#466C2211yescharacteristic
19C53×#465—21no—
20C53×#465—21no—
21C53×#427C411yescharacteristic, complemented
22C53×#416—21no—
23C23×Dic53—41no—
24C2×C4⋊Dic53—24no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—42no—
27C2×C4⋊Dic53—41no—
28#13133157—42no—
29C2×#11719908—22no—
30C2×#41×C53—21no—
31C23×C4×C53D411yescharacteristic
32#135×C53—42no—
33C2×Q16×C53C2×C411yescharacteristic, complemented
34C2×SD16×C53—21no—
35C2×D8×C53C2×C411yescharacteristic, complemented
36C22×C8×C53—42no—
37C22×C8×C53C2×C411yescharacteristic
38C22×C8×C53C2×C411yescharacteristic
39C22×C8×C53D411yescharacteristic
40C22×C8×C53D411yescharacteristic
41C22×C8×C53D411yescharacteristic
42C22×C8×C53C2311yescharacteristic
43#123×C53—41no—
44C22⋊Q8×C53—21no—
45C22⋊Q8×C53—21no—
46#121×C53—21no—
47#121×C53—21no—
48#120×C53—41no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51#117×C53—21no—
52#107×C53C2×C411yescharacteristic, complemented
53#107×C53C2×C411yescharacteristic, complemented
54#106×C53—21no—
55#103×C53—21no—
56#103×C53—21no—
57#102×C53—21no—
58#102×C53—21no—
59C24×C53D811yescharacteristic
60#41×C53—42no—
61#41×C53—42no—
62C2×Q8×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53D811yescharacteristic
73C22×C4×C53SD1611yescharacteristic
74C22×C4×C53SD1611yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—44no—
77C22×Dic53—42no—
78C22×Dic53—82no—
79C22×Dic53—82no—
80Q16×C53—41no—
81SD16×C53—42no—
82D8×C53—41no—
83C2×C8×C53—22no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53C22⋊C411yescharacteristic
90C2×C8×C53C22⋊C411yescharacteristic
91C2×C8×C53C22×C411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C42×C53—41no—
103C4⋊Dic53—24no—
104C4⋊Dic53—44no—
105C4⋊Dic53—42no—
106#11719908—42no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53#10211yescharacteristic
113C23×C53#10211yescharacteristic
114C23×C53C2×D811yescharacteristic
115Q8×C53—41no—
116Q8×C53—41no—
117D4×C53—41no—
118D4×C53—41no—
119D4×C53—82no—
120D4×C53—82no—
121D4×C53—81no—
122D4×C53—81no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—82no—
134C2×C4×C53#9912yes—
135C2×C4×C53#10211yescharacteristic
136C2×C4×C53#10211yescharacteristic
137C2×C4×C53#10412yes—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×Dic53—44no—
141C2×Dic53—88no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144C8×C53—21no—
145C8×C53—21no—
146C8×C53—21no—
147C8×C53—21no—
148#11704—531nomaximal, Sylow
149C22×C53—24no—
150C22×C53—24no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#32712yes—
161C22×C53#36012yes—
162C22×C53C2×#10211yescharacteristic
163C22×C53#50911yescharacteristic
164C22×C53#51011yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—81no—
168C4×C53—81no—
169C4×C53—81no—
170C4×C53#32712yes—
171C4×C53C2×#9911yescharacteristic
172C4×C53C2×#10411yescharacteristic
173Dic53—84no—
174#3050D5311yescharacteristic
175#2112—531no—
176C2×#340—531no—
177#1348—532no—
178#1347—532no—
179C2×C53—21no—
180C2×C53—22no—
181C2×C53—22no—
182C2×C53—21no—
183C2×C53—161no—
184C2×C53C2×#32711yescharacteristic
185C2×C53#213911yescharacteristic
186C2×C53C2×#36011yescharacteristic
187C2×#135Dic5311yescharacteristic, complemented
188C23×C8C2×D5311yescharacteristic
189#525—21no—
190C22×C4⋊C4—1061no—
191C2×M32—1061no—
192#467Dic5311yescharacteristic, complemented
193#467C2×D5311yescharacteristic
194#466Dic5311yescharacteristic, complemented
195#466C2×D5311yescharacteristic
196#465—21no—
197#465—21no—
198#427Dic5311yescharacteristic, complemented
199#416—21no—
200C22⋊C16—532no—
201C22⋊C16—1061no—
202#340—532no—
203#340—1061no—
204C53#1170411yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C23×C4—2121no—
207C23×C4C22⋊D5311yescharacteristic
208#135—42no—
209C2×Q16C2×Dic5311yescharacteristic, complemented
210C2×SD16—21no—
211C2×D8C2×Dic5311yescharacteristic, complemented
212C22×C8—42no—
213C22×C8C2×Dic5311yescharacteristic
214C22×C8C2×Dic5311yescharacteristic
215C22×C8C22⋊D5311yescharacteristic
216C22×C8C22⋊D5311yescharacteristic
217C22×C8C22⋊D5311yescharacteristic
218C22×C8C22×D5311yescharacteristic
219#123—41no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224#120—41no—
225C4×D4—41no—
226C4×D4—41no—
227#117—21no—
228C2×C4⋊C4—1064no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—2122no—
231C2×C4⋊C4—2121no—
232M32—2122no—
233C2×C16—1062no—
234#107C2×Dic5311yescharacteristic, complemented
235#107C2×Dic5311yescharacteristic, complemented
236#106—21no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C24#1171992211yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—4242no—
253C22×C4—4242no—
254C22×C4—2124no—
255C22×C4—2122no—
256C22×C4#1171992211yescharacteristic
257C22×C4#1171992311yescharacteristic
258C22×C4#1171992311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262Q16—41no—
263SD16—42no—
264D8—41no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8C22⋊Dic5311yescharacteristic
272C2×C8C22⋊Dic5311yescharacteristic
273C2×C8C22×Dic5311yescharacteristic
274C2×C8C2×C22⋊D5311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—1064no—
280C4⋊C4—2122no—
281C4⋊C4—2124no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—41no—
288C16—2122no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23#1313313811yescharacteristic
295C23#1313313811yescharacteristic
296C23C2×#1171992211yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—41no—
300D4—41no—
301D4—81no—
302D4—81no—
303D4—82no—
304D4—82no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—4248no—
317C2×C4—4242no—
318C2×C4—4242no—
319C2×C4—2124no—
320C2×C4#1313313212yes—
321C2×C4#1313313811yescharacteristic
322C2×C4#1313313811yescharacteristic
323C2×C4#1313314412yes—
324C2×C4C2×C22⋊Dic5311yescharacteristic
325C2×C4C2×#1171992311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1681035712yes—
342C22#1681043312yes—
343C22C2×#1313313811yescharacteristic
344C22#1681105811yescharacteristic
345C22#1681106311yescharacteristic
346C4—21no—
347C4—21no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—4244no—
352C4#1681035712yes—
353C4C2×#1313313211yescharacteristic
354C4C2×#1313314411yescharacteristic
355C2—21no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—161no—
360C2C2×#1681035711yescharacteristic
361C2#2027642711yescharacteristic
362C2C2×#1681043311yescharacteristic
subgroupG/H
centerC22C2×#13133138
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3050C2
socleC22×C53C2×#102
cosocleC22×C8×C53C23
hypercenter#3050D53
hypocenterC53#11704
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #20276326C2semidirect
2. C2×#16810386C2semidirect
3. #20272727C2semidirect
4. #20272724C2semidirect
12. C2×#135×C53C4semidirect
15. C53×#467C4semidirect
17. C53×#466C4semidirect
21. C53×#427C4semidirect
33. C2×Q16×C53C2×C4semidirect
35. C2×D8×C53C2×C4semidirect
52. #107×C53C2×C4semidirect
53. #107×C53C2×C4semidirect
187. C2×#135Dic53semidirect
192. #467Dic53semidirect
194. #466Dic53semidirect
198. #427Dic53semidirect
204. C53#11704semidirect
209. C2×Q16C2×Dic53semidirect
211. C2×D8C2×Dic53semidirect
234. #107C2×Dic53semidirect
235. #107C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202763262
C2×#168103862
#202727272
#202727242
C53×#30502
#1170453