Finite Group Databasebeta

← #27288506

Subgroups — #27288506

Order 13568.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280829C211yesmaximal, characteristic
2#20274905C212yesmaximal
3#20272971C211yesmaximal, characteristic
4#20272595C212yesmaximal
5C53×#1280C211yesmaximal, characteristic
6C2×#13133196—21no—
7C22×C4⋊Dic53C2211yescharacteristic
8C2×#13133151—22no—
9#16810555—22no—
10#16810544—22no—
11#16810483—21no—
12#16810483C2212yes—
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—22no—
17C2×C22⋊C8×C53C2212yes—
18C2×#95×C53C2211yescharacteristic
19C22×Dic106—21no—
20C23×Dic53—21no—
21C23×Dic53—21no—
22#13133196—42no—
23#13133196—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—42no—
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53D412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33#13133151—44no—
34#13133124—22no—
35#13133124—22no—
36#13133124—21no—
37#13133124—42no—
38#13133124—42no—
39#13133124Q812yescomplemented
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C22×C8×C53—22no—
44C22⋊C8×C53—22no—
45C22⋊C8×C53Q814yescomplemented
46#95×C53—22no—
47C24×C53C2×D411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53D812yes—
59C22×C4×C53Q1612yes—
60C22×C4×C53C2×D412yes—
61C22×C4×C53C2×Q812yes—
62C22×C4×C53C2×Q811yescharacteristic
63C22×C4×C53#4111yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C2×Dic106—42no—
81C2×Dic106—81no—
82C2×Dic106—81no—
83C2×C8×C53—24no—
84C2×C8×C53—42no—
85C2×C8×C53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—24no—
89C4⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—81no—
94C4⋊Dic53—81no—
95#11719919—44no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53#12011yescharacteristic
101C23×C53C22⋊Q811yescharacteristic
102C23×C53C4⋊Q811yescharacteristic
103C23×C53C2×D811yescharacteristic
104C23×C53C2×Q1611yescharacteristic
105C23×C53AGL(1,ℤ/8)11yescharacteristic
106C23×C53#13711yescharacteristic
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53C22⋊Q812yes—
123C2×C4×C53C4⋊Q812yes—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144Dic106—84no—
145C8×C53—44no—
146#10569—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#39511yescharacteristic
160C22×C53#44711yescharacteristic
161C22×C53#45111yescharacteristic
162C22×C53#45411yescharacteristic
163C22×C53#45511yescharacteristic
164C22×C53B(4,𝔽2)11yescharacteristic
165C22×C53#45811yescharacteristic
166C22×C53#47411yescharacteristic
167C22×C53#47711yescharacteristic
168C22×C53#50011yescharacteristic
169C22×C53#50111yescharacteristic
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—42no—
173C4×C53—42no—
174Dic53—82no—
175Dic53—82no—
176Dic53—82no—
177Dic53—82no—
178#2659—531no—
179#1852—532no—
180#1442—531no—
181#1294—532no—
182#1280D5311yescharacteristic
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#159811yescharacteristic
188C2×C53#160211yescharacteristic
189C2×C53#160711yescharacteristic
190C2×C53#161711yescharacteristic
191C2×C53#205711yescharacteristic
192C2×C53#206111yescharacteristic
193C2×C53#206211yescharacteristic
194C2×C22⋊Q8—1061no—
195C22×C4⋊C4—531no—
196C2×#107—1062no—
197C2×C22⋊C8C2×D5312yes—
198#398—1062no—
199#395—1062no—
200#380—532no—
201#380—1061no—
202C2×#95—531no—
203C2×#95—1061no—
204C2×#95—1061no—
205C2×#95C2×D5311yescharacteristic
206#340—1062no—
207C53#1056911yescharacteristic, complemented, Sylow
208C22×Q8—1061no—
209C23×C4—1061no—
210C23×C4—1061no—
211C23×C4D21211yescharacteristic
212C23×C4D21211yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C22×C8—22no—
215C22⋊Q8—2122no—
216C22⋊Q8—2122no—
217C2×C4⋊C4—532no—
218C2×C4⋊C4—532no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—2122no—
225C2×C22⋊C4—1062no—
226#107—2124no—
227C22⋊C8—22no—
228C22⋊C8Dic10614yescomplemented
229#95—22no—
230#95—532no—
231#95—1062no—
232#95—1062no—
233#95—1061no—
234#95—2122no—
235#95—2122no—
236C24C2×D21211yescharacteristic
237C2×Q8—4241no—
238C2×Q8—4241no—
239C2×Q8—2122no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—2121no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4D42412yes—
267C22×C4Dic21212yes—
268C22×C4C2×Dic10612yes—
269C22×C4D4×D5312yes—
270C22×C4#1171994711yescharacteristic
271C22×C4Q8×D5311yescharacteristic
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—1064no—
276C4⋊C4—4241no—
277C4⋊C4—4241no—
278C4⋊C4—2122no—
279C4⋊C4—2122no—
280C4⋊C4—2122no—
281C4⋊C4—2124no—
282C4⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1313318711yescharacteristic
290C23#1313319711yescharacteristic
291C23#1313321711yescharacteristic
292C23C2×D42411yescharacteristic
293C23C2×Dic21211yescharacteristic
294C23#1313324511yescharacteristic
295C23#1313325011yescharacteristic
296Q8—4244no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4#1313319412yes—
333C2×C4#1313321612yes—
334C8—44no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1681054411yescharacteristic
348C22#1681075911yescharacteristic
349C22#1681078111yescharacteristic
350C22#1681079811yescharacteristic
351C22#1681080311yescharacteristic
352C22#1681081111yescharacteristic
353C22#1681081411yescharacteristic
354C22#1681090411yescharacteristic
355C22#1681091911yescharacteristic
356C22#1681102811yescharacteristic
357C22#1681103211yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—4242no—
363C4—4242no—
364C4—4242no—
365C4—4242no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2027365211yescharacteristic
371C2#2027367411yescharacteristic
372C2#2027370311yescharacteristic
373C2#2027375911yescharacteristic
374C2#2027604411yescharacteristic
375C2#2027606611yescharacteristic
376C2#2027606911yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1280D53
hypocenterC53#10569
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
39. #13133124Q8semidirect
45. C22⋊C8×C53Q8semidirect
207. C53#10569semidirect
228. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202749052
#202729712
#202725952
C53×#12802
#1056953