Finite Group Databasebeta

← #27489960

Subgroups — #27489960

Order 13568.

956 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×#16811254C211yesmaximal, characteristic
2#20284659C212yesmaximal
3#20276060C214yesmaximal, complemented
4#20275876C214yesmaximal, complemented
5#20274815C212yesmaximal
6#20274795C211yesmaximal, characteristic
7C53×#2502C211yesmaximal, characteristic
8#16811254—24no—
9#16811254—22no—
10#16811254C2214yes—
11C2×#13133216—22no—
12C2×#13133208—22no—
13C2×#13133196—22no—
14C2×#13133176C2212yes—
15#16810961—24no—
16#16810956—22no—
17#16810951—22no—
18#16810663C2214yes—
19#16810632C2214yes—
20#16810587C2212yes—
21#16810392C2214yes—
22#16810383C2218yes—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C22×C4⋊C4×C53C2211yescharacteristic
25C53×#398—22no—
26C53×#391C2212yes—
27C53×#385—22no—
28C53×#384C2212yes—
29C4×C22⋊C4×C53C2211yescharacteristic
30C22×Dic106—22no—
31C2×#11719917C2314yes—
32#13133216—28no—
33#13133216—28no—
34#13133208—28no—
35#13133208—28no—
36#13133196—28no—
37#13133196—28no—
38#13133176—22no—
39#13133176—22no—
40#13133176D414yes—
41#13133176C2314yes—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53D412yes—
44C2×C4⋊Dic53D412yes—
45C2×#11719919—24no—
46C2×C22⋊Dic53—22no—
47C2×C4×Dic53—22no—
48#13133150—24no—
49#13133148—24no—
50#13133141—24no—
51#13133141—24no—
52C22⋊#88322—22no—
53C22⋊#88322—22no—
54C22×Q8×C53C2311yescharacteristic
55C23×C4×C53D412yes—
56C23×C4×C53C2311yescharacteristic
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—22no—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—24no—
63C2×C4⋊C4×C53D412yes—
64C2×C4⋊C4×C53C2311yescharacteristic
65C2×C4⋊C4×C53C2312yes—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2312yes—
69#95×C53—21no—
70#95×C53—24no—
71C24×C53C2×D411yescharacteristic
72C2×Q8×C53—22no—
73C2×Q8×C53—24no—
74C22×C4×C53—24no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—82no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D414yes—
85C22×C4×C53C2411yescharacteristic
86C22×Dic53—22no—
87C22×Dic53—22no—
88C2×Dic106—28no—
89C2×Dic106—84no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—84no—
95C22⋊C4×C53—24no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C42×C53—24no—
99C22⋊Dic53—24no—
100C22⋊Dic53—24no—
101C4⋊Dic53—28no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53C2×D414yes—
105C4⋊Dic53C2×D414yes—
106#11719919—216no—
107C4×Dic53—24no—
108C4×Dic53—24no—
109#11719917—28no—
110C2×#88322—24no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C23×C53#12011yescharacteristic
116C23×C53C22×D411yescharacteristic
117Q8×C53—44no—
118C2×C4×C53—22no—
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—24no—
122C2×C4×C53—24no—
123C2×C4×C53—44no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—44no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—82no—
132C2×C4×C53#12012yes—
133C2×C4×C53C22×D412yes—
134C2×C4×C532−1+414yes—
135C2×Dic53—24no—
136C2×Dic53—24no—
137C2×Dic53—44no—
138C2×Dic53—44no—
139Dic106—88no—
140#88322—44no—
141#29653—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—44no—
150C22×C53#45612yes—
151C22×C53C2×#12011yescharacteristic
152C22×C53#53612yes—
153C4×C53—22no—
154C4×C53—44no—
155C4×C53—44no—
156C4×C53—81no—
157C4×C53—81no—
158C4×C53#53614yes—
159Dic53—44no—
160Dic53—44no—
161C2×#544—531no—
162#3095—532no—
163#2502D5311yescharacteristic
164#2060—534no—
165#2028—534no—
166#1826—532no—
167#1821—531no—
168C2×C53—22no—
169C2×C53—22no—
170C2×C53—41no—
171C2×C53#261411yescharacteristic
172C2×C53C2×#45611yescharacteristic
173C2×C53#301911yescharacteristic
174#544—534no—
175#544—1064no—
176#544—1062no—
177C2×C4⋊Q8—1062no—
178C2×#125—1062no—
179C2×C22⋊Q8—1062no—
180C2×C22⋊Q8C2×D5311yescharacteristic
181C2×#117—532no—
182C22×C4⋊C4C2×D5311yescharacteristic
183#485—1064no—
184#484—1062no—
185#483—1062no—
186#428—534no—
187#419—534no—
188#407—532no—
189#398—22no—
190#391C2×D5312yes—
191#385—22no—
192#384C2×D5312yes—
193C4×C22⋊C4C2×D5311yescharacteristic
194#343—534no—
195#339—538no—
196C53#2965311yescharacteristic, complemented, Sylow
197C22×Q8—1062no—
198C22×Q8C22×D5311yescharacteristic
199C23×C4C22⋊D5312yes—
200C23×C4C22×D5311yescharacteristic
201C2×M16—534no—
202C4⋊Q8—1068no—
203C4⋊Q8—1068no—
204#125—1068no—
205#125—1068no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—1068no—
209C22⋊Q8—1068no—
210#117—534no—
211#117—534no—
212#117—1062no—
213#117—1062no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—532no—
219C2×C4⋊C4—532no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1064no—
222C2×C4⋊C4C22⋊D5312yes—
223C2×C4⋊C4C22×D5311yescharacteristic
224C2×C4⋊C4C22×D5312yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1062no—
228C2×C42—1062no—
229C2×C42C22×D5312yes—
230#107—1064no—
231#106—1064no—
232#103—1064no—
233#103—1064no—
234C22⋊C8—1062no—
235C22⋊C8—1062no—
236#95—21no—
237#95—24no—
238C24C2×C22⋊D5311yescharacteristic
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—1068no—
242C2×Q8—4244no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—82no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4D4×D5312yes—
253C22×C4D4×D5312yes—
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5314yes—
256C22×C4C23×D5311yescharacteristic
257M16—1068no—
258C2×C8—1064no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4—534no—
265C4⋊C4—534no—
266C4⋊C4—1068no—
267C4⋊C4—10616no—
268C4⋊C4—1062no—
269C4⋊C4—1062no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—1064no—
274C22⋊C4—1064no—
275C42—24no—
276C42—1064no—
277C42—1064no—
278C23—22no—
279C23—22no—
280C23—41no—
281C23#1313318511yescharacteristic
282C23#1313318511yescharacteristic
283C23C22×C22⋊D5311yescharacteristic
284Q8—44no—
285Q8—4248no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—1064no—
301C2×C4—1064no—
302C2×C4—2124no—
303C2×C4—2124no—
304C2×C4#1313318512yes—
305C2×C4C2×D4×D5312yes—
306C2×C4#1313326414yes—
307C8—2124no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—44no—
316C22#1681080512yes—
317C22C2×#1313318511yescharacteristic
318C22#1681119412yes—
319C4—22no—
320C4—44no—
321C4—44no—
322C4—81no—
323C4—81no—
324C4—2124no—
325C4—2124no—
326C4#1681119514yes—
327C2—22no—
328C2—22no—
329C2—41no—
330C2#2028043611yescharacteristic
331C2C2×#1681080511yescharacteristic
332C2#2028407111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2502C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#2502D53
hypocenterC53#29653
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
3. #20276060C2semidirect
4. #20275876C2semidirect
196. C53#29653semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202846592
#202760602
#202758762
#202748152
#202747952
C53×#25022
#2965353