Finite Group Databasebeta

← #27288129

Subgroups — #27288129

Order 13568.

390 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285861C211yesmaximal, characteristic
2#20275666C211yesmaximal, characteristic
3#20275659C211yesmaximal, characteristic
4#20275559C211yesmaximal, characteristic
5#20273403C211yesmaximal, characteristic
6#20272617C211yesmaximal, characteristic
7C53×#1565C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811021—21no—
11#16810974—21no—
12#16810954—22no—
13#16810909—21no—
14#16810909—21no—
15#16810647C2211yescharacteristic
16C2×#13133135—21no—
17C2×#13133135—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810386—22no—
20#16810384—21no—
21#16810384—21no—
22#16810371—21no—
23#16810370—21no—
24C53×#431C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C4×M16×C53C2211yescharacteristic
27C53×#335—21no—
28C53×#334—21no—
29C22×Dic106—21no—
30C2×#11719917—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217D411yescharacteristic
36#13133217D411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42#13133176—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53D412yes—
46C2×C4×Dic53—21no—
47#13133149—22no—
48#13133149—22no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133142—22no—
52#13133142—22no—
53#13133135—42no—
54#13133135—42no—
55#13133124—21no—
56C2×M16×C53D411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C22×C8×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53D411yescharacteristic
63C4⋊C8×C53D411yescharacteristic
64C22⋊C8×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—41no—
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81M16×C53—21no—
82M16×C53—21no—
83M16×C53—21no—
84M16×C53SD1612yes—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—42no—
90C2×C8×C53—41no—
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53#4111yescharacteristic
94C42×C53#4111yescharacteristic
95C22⋊Dic53—41no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—22no—
102C4⋊Dic53—21no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106#11719919—41no—
107#11719919—41no—
108C4×Dic53—41no—
109C4×Dic53—41no—
110C4×Dic53—41no—
111#11719917—41no—
112#11719917—41no—
113C2×#88322—41no—
114C23×C53#12111yescharacteristic
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12111yescharacteristic
120C2×C4×C53#12111yescharacteristic
121C2×C4×C53#12311yescharacteristic
122C2×C4×C53#12311yescharacteristic
123C2×C4×C53#12411yescharacteristic
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53#13511yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C2×Dic53—42no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136Dic106—41no—
137Dic106—41no—
138Dic106—81no—
139Dic106—81no—
140Dic106—81no—
141Dic106—81no—
142Dic106—81no—
143C8×C53—22no—
144C8×C53—41no—
145C8×C53—41no—
146C8×C53—41no—
147C8×C53—81no—
148#88322—81no—
149#10507—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53#39411yescharacteristic
153C22×C53#46111yescharacteristic
154C22×C53#46311yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53#45411yescharacteristic
159C4×C53#45611yescharacteristic
160C4×C53#46011yescharacteristic
161C4×C53#46411yescharacteristic
162C4×C53#46511yescharacteristic
163C4×C53#46811yescharacteristic
164Dic53—82no—
165Dic53—81no—
166Dic53—82no—
167Dic53—81no—
168#3229—531no—
169#1988—531no—
170#1986—531no—
171#1966—531no—
172#1565D5311yescharacteristic
173#1550—531no—
174#1303—531no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53#203811yescharacteristic
178C2×C53#204911yescharacteristic
179C2×C53#205511yescharacteristic
180#533—531no—
181C2×C4⋊Q8—531no—
182#498—1061no—
183#487—1061no—
184#483—1062no—
185#475—1061no—
186#475—1061no—
187#431C2×D5311yescharacteristic
188#423—531no—
189C2×C4⋊C8C2×D5311yescharacteristic
190C2×#101—1061no—
191C2×#101—1061no—
192C4×M16C2×D5311yescharacteristic
193C4×C4⋊C4—531no—
194#340—1062no—
195#339—1061no—
196#339—1061no—
197#335—21no—
198#334—21no—
199#333—1061no—
200#332—1061no—
201C53#1050711yescharacteristic, complemented, Sylow
202C22×Q8—1061no—
203C2×M16—1061no—
204C2×M16D21211yescharacteristic
205C2×M16D21211yescharacteristic
206C22×C8—21no—
207C4⋊Q8—531no—
208C4⋊Q8—531no—
209C4⋊Q8—1061no—
210C4⋊Q8—1061no—
211C4⋊Q8—1061no—
212C4⋊Q8—1061no—
213#125—1061no—
214C22⋊Q8—1061no—
215C22⋊Q8—1061no—
216C4×Q8—1061no—
217C4×Q8—1061no—
218#117—1061no—
219C2×C4⋊C4—532no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C42—1061no—
223C2×C42C22×D5311yescharacteristic
224#106—1062no—
225#106—1062no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—1061no—
229C4⋊C8—1061no—
230C4⋊C8C22⋊D5311yescharacteristic
231C4⋊C8C22⋊D5311yescharacteristic
232#103—1062no—
233#103—1062no—
234#101—2122no—
235#101—2122no—
236C22⋊C8—21no—
237#97—21no—
238C4×C8—21no—
239C4×C8—21no—
240#95—1061no—
241C2×Q8—1061no—
242C2×Q8—1061no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C2×Q8—2121no—
248C2×Q8—2121no—
249C22×C4—1062no—
250C22×C4—2121no—
251C22×C4C2×D21211yescharacteristic
252C22×C4D4×D5311yescharacteristic
253C22×C4#1171994911yescharacteristic
254M16—21no—
255M16—21no—
256M16—21no—
257M16—2121no—
258M16—2121no—
259M16#1171991312yes—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—2121no—
267C4⋊C4—1062no—
268C4⋊C4—1062no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1062no—
273C4⋊C4—1061no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C22⋊C4—2121no—
280C42—2121no—
281C42—2121no—
282C42—2121no—
283C42#1171994511yescharacteristic
284C42D4×D5311yescharacteristic
285C42#1171994911yescharacteristic
286C42C2×C22⋊D5311yescharacteristic
287C23#1313319211yescharacteristic
288Q8—4241no—
289Q8—4241no—
290Q8—4241no—
291Q8—4241no—
292Q8—4241no—
293Q8—2121no—
294Q8—2121no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—4241no—
299C2×C4—4241no—
300C2×C4—2122no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4#1313318511yescharacteristic
307C2×C4#1313319111yescharacteristic
308C2×C4#1313319211yescharacteristic
309C2×C4#1313320211yescharacteristic
310C2×C4#1313320211yescharacteristic
311C2×C4#1313320611yescharacteristic
312C2×C4C2×#1171991311yescharacteristic
313C2×C4#1313323911yescharacteristic
314C2×C4#1313324511yescharacteristic
315C2×C4#1313325011yescharacteristic
316C8—22no—
317C8—41no—
318C8—41no—
319C8—41no—
320C8—81no—
321C8—4241no—
322C22—21no—
323C22—21no—
324C22#1681054011yescharacteristic
325C22#1681082911yescharacteristic
326C22#1681083911yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—4242no—
331C4—4241no—
332C4—4242no—
333C4—4241no—
334C4#1681079411yescharacteristic
335C4#1681080411yescharacteristic
336C4#1681082411yescharacteristic
337C4#1681084611yescharacteristic
338C4#1681085211yescharacteristic
339C4#1681086911yescharacteristic
340C2—21no—
341C2—21no—
342C2#2027593411yescharacteristic
343C2#2027599711yescharacteristic
344C2#2027603311yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1565C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#1565D53
hypocenterC53#10507
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
201. C53#10507semidirect

Maximal subgroups

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

subgroupindex
#202858612
#202756662
#202756592
#202755592
#202734032
#202726172
C53×#15652
#1050753