Finite Group Databasebeta

← #27275082

Subgroups — #27275082

Order 13568.

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

normal subgroups only

This group has 510 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#20283405C211yesmaximal, characteristic
2#20274493C212yesmaximal, complemented
3#20273204C212yesmaximal
4#20272602C211yesmaximal, characteristic, complemented
5C53×#1297C211yesmaximal, characteristic, complemented
6C2×#13133225C411yescharacteristic, complemented
7C23×#88322C2211yescharacteristic
8#16811134C2211yescharacteristic
9#16810691—22no—
10#16810686—22no—
11#16810651C411yescharacteristic, complemented
12#16810591C411yescharacteristic, complemented
13C2×C22⋊#88322C2212yes—
14#16810581C411yescharacteristic, complemented
15C2×#13133128—22no—
16#16810378—22no—
17#16810359C414yescomplemented
18#16810356C414yescomplemented
19C53×#407C2211yescharacteristic
20C4×C22⋊C4×C53C2212yes—
21C2×#11719945C2×C411yescharacteristic
22#13133225—42no—
23C2×#11719912—21no—
24C22×#88322—22no—
25C22×#88322—42no—
26C22×#88322C2×C411yescharacteristic
27C22×#88322C2×C411yescharacteristic
28C22×#88322D412yes—
29C22×#88322D412yes—
30C2×C8×D53—21no—
31#13133200—21no—
32#13133196C2×C412yes—
33#13133191C2×C412yes—
34#13133186—21no—
35C4×C22⋊D53—22no—
36#13133176C2×C411yescharacteristic
37#13133146—22no—
38C22⋊#88322—22no—
39C22⋊#88322C2×C414yes—
40#13133129—22no—
41#13133128—44no—
42#13133127—21no—
43C8×Dic53—21no—
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53C2×C411yescharacteristic
46C2×M16×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C42×C53—22no—
49C22⋊C8×C53—22no—
50#95×C53C2×C414yes—
51C24×C53#4111yescharacteristic
52C22×C4×C53—24no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C22×C412yes—
57C22×C4×C53C2×D412yes—
58C22×C4×C53#4111yescharacteristic
59C2×C22⋊D53—21no—
60C2×C22⋊D53—22no—
61#11719945—41no—
62#11719945—41no—
63C2×D212C4211yescharacteristic
64C2×C4×D53—21no—
65C2×Dic106C4211yescharacteristic
66M16×C53—41no—
67M16×C53—41no—
68C2×C8×C53—21no—
69C2×C8×C53—21no—
70C22⋊C4×C53—44no—
71C42×C53—42no—
72C42×C53—42no—
73C22⋊Dic53—21no—
74C22⋊Dic53—22no—
75#11719921—22no—
76C4⋊Dic53C4212yes—
77#11719919—22no—
78C4×Dic53—21no—
79C2×#88322—24no—
80C2×#88322—22no—
81C2×#88322—24no—
82C2×#88322—41no—
83C2×#88322—42no—
84C2×#88322—42no—
85C2×#88322—42no—
86C2×#88322C22⋊C411yescharacteristic
87C2×#88322C22⋊C412yes—
88C2×#88322C22⋊C411yescharacteristic
89#11719912—42no—
90C8×D53—42no—
91C23×C53—24no—
92C23×C53—41no—
93C23×C53#11711yescharacteristic
94C23×C53C4×D412yes—
95C22×D53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—44no—
103C2×C4×C53—44no—
104C2×C4×C53—44no—
105C2×C4×C53C2×C4211yescharacteristic
106C2×C4×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53C4×D412yes—
108C22⋊D53—42no—
109C22⋊D53—44no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112D212—41no—
113C4×D53—41no—
114C4×D53—41no—
115Dic106—41no—
116C8×C53—41no—
117C8×C53—41no—
118#88322—21no—
119#88322—22no—
120#88322—44no—
121#88322#9912yes—
122#8239—531nomaximal, Sylow
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—21no—
126C22×C53—22no—
127C22×C53—44no—
128C22×C53C4×C22⋊C411yescharacteristic
129C22×C53#44414yes—
130C2×D53—41no—
131C2×D53—42no—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—84no—
135C4×C53C2×#9911yescharacteristic
136C4×C53#41011yescharacteristic
137Dic53—41no—
138Dic53—42no—
139#2929—531no—
140#1752—532no—
141#1501—532no—
142#1298—531no—
143#1297D5311yescharacteristic, complemented
144C2×C53—24no—
145C2×C53—41no—
146C2×C53C4×#9911yescharacteristic
147C2×C53#176512yes—
148D53—81no—
149C2×#131—531no—
150C23×C8—531no—
151#525—531no—
152#435—1062no—
153C8×D4—1062no—
154#424—531no—
155#408—531no—
156#407C2×D5311yescharacteristic
157C2×C22⋊C8—532no—
158#405—531no—
159C2×#97—1062no—
160C4×C22⋊C4C2×D5312yes—
161#336—1062no—
162#328—534no—
163#327—534no—
164C53#823911yescharacteristic, complemented, Sylow
165C2×#41—531no—
166C23×C4C22×D5311yescharacteristic
167#131—2122no—
168C2×M16—1061no—
169C2×M16C4×D5311yescharacteristic
170C2×M16C4×D5311yescharacteristic
171C22×C8—531no—
172C22×C8—532no—
173C22×C8—531no—
174C22×C8—532no—
175C22×C8—1061no—
176C22×C8—1062no—
177C22×C8—2122no—
178#123—1061no—
179C22⋊Q8—532no—
180#121—532no—
181#120—1061no—
182C4×D4—1062no—
183#117—531no—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186C4⋊C8—1062no—
187C22⋊C8—22no—
188C22⋊C8—534no—
189C22⋊C8—1062no—
190C22⋊C8—1062no—
191#97—1061no—
192#97—2124no—
193C4×C8—1061no—
194#95C4×D5314yes—
195C24#1171994711yescharacteristic
196#41—2121no—
197#41—2121no—
198C2×Q8—531no—
199C2×D4—531no—
200C2×D4—1061no—
201C2×D4—1062no—
202C22×C4—24no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—1061no—
206C22×C4C2×C4×D5311yescharacteristic
207C22×C4C2×C4×D5312yes—
208C22×C4D4×D5312yes—
209C22×C4#1171994911yescharacteristic
210M16—41no—
211M16—41no—
212M16—2122no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—531no—
216C2×C8—531no—
217C2×C8—532no—
218C2×C8—1064no—
219C2×C8—1062no—
220C2×C8—1064no—
221C2×C8—2122no—
222C2×C8—2121no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C4⋊C4—532no—
227C4⋊C4—1062no—
228C22⋊C4—44no—
229C22⋊C4—1061no—
230C22⋊C4—1062no—
231C22⋊C4—1062no—
232C42—42no—
233C42—42no—
234C42—1061no—
235C23—24no—
236C23—41no—
237C23—1061no—
238C23#1313317511yescharacteristic
239C23#1313317712yes—
240Q8—2121no—
241D4—2121no—
242D4—2122no—
243D4—2124no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4C42×D5311yescharacteristic
258C2×C4C22⋊C4×D5311yescharacteristic
259C2×C4#1313317912yes—
260C8—41no—
261C8—41no—
262C8—532no—
263C8—1062no—
264C8—1061no—
265C8—2124no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—44no—
271C22—2121no—
272C22—2122no—
273C22#1681047011yescharacteristic
274C22#1681074214yes—
275C4—22no—
276C4—21no—
277C4—84no—
278C4—2121no—
279C4—2122no—
280C4#99×D5311yescharacteristic
281C4#1681059511yescharacteristic
282C2—24no—
283C2—41no—
284C2—4241no—
285C2#2027450211yescharacteristic
286C2#2027455612yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1297C2
socleC22×C53C4×C22⋊C4
prime radical#16811134C22
cosocleC23×C4×C53C23
hypercenter#1297D53
hypocenterC53#8239
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
2. #20274493C2semidirect
4. #20272602C2semidirect
5. C53×#1297C2semidirect
6. C2×#13133225C4semidirect
11. #16810651C4semidirect
12. #16810591C4semidirect
14. #16810581C4semidirect
17. #16810359C4semidirect
18. #16810356C4semidirect
143. #1297D53semidirect
164. C53#8239semidirect

Maximal subgroups

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

subgroupindex
#202834052
#202744932
#202732042
#202726022
C53×#12972
#823953