Finite Group Databasebeta

← #27299248

Subgroups — #27299248

Order 13568.

456 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 109 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#20283421C211yesmaximal, characteristic
2#20276354C212yesmaximal
3#20274517C212yesmaximal, complemented
4#20272838C211yesmaximal, characteristic, complemented
5C53×#1391C211yesmaximal, characteristic, complemented
6C2×#13133224C2211yescharacteristic
7C22×#11719912C411yescharacteristic, complemented
8#16811130C2211yescharacteristic
9#16810698—22no—
10#16810691—22no—
11#16810649C411yescharacteristic, complemented
12#16810607C2212yes—
13#16810586C411yescharacteristic, complemented
14#16810581C411yescharacteristic, complemented
15#16810579—22no—
16#16810412C414yescomplemented
17#16810409C414yescomplemented
18#16810393—22no—
19C53×#410C2212yes—
20C53×#408C2211yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22#13133224—42no—
23C2×#11719917—22no—
24C2×#11719917C2×C411yescharacteristic
25C2×#11719917D412yes—
26C2×#11719912—21no—
27C2×#11719912—21no—
28C2×#11719912—42no—
29C22×#88322C2×C411yescharacteristic
30C22×#88322D412yes—
31#13133198C2×C412yes—
32#13133193—21no—
33#13133188—21no—
34#13133185C2×C412yes—
35C4×C22⋊D53—22no—
36#13133176C2×C411yescharacteristic
37#13133146—22no—
38#13133136—22no—
39#13133134C2×C414yes—
40#13133129—22no—
41#13133128—42no—
42#13133127—21no—
43C4×#88322—42no—
44C8×Dic53—21no—
45C2×#41×C53C2311yescharacteristic
46C2×M16×C53C2×C411yescharacteristic
47C22×C8×C53C2×C411yescharacteristic
48#117×C53—22no—
49#99×C53—22no—
50#99×C53C2×C414yes—
51C22⋊C8×C53—22no—
52C23×D53C4211yescharacteristic
53#41×C53—41no—
54#41×C53—41no—
55C2×Q8×C53#4111yescharacteristic
56C2×D4×C53C22×C412yes—
57C2×D4×C53#4111yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D412yes—
60C2×C22⋊D53—22no—
61C22×Dic53C4211yescharacteristic
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—41no—
65C2×C4×D53—41no—
66M16×C53—41no—
67C2×C8×C53—21no—
68C2×C8×C53—21no—
69C2×C8×C53—41no—
70C4⋊C4×C53—42no—
71C22⋊C4×C53—24no—
72C42×C53—42no—
73C22⋊Dic53—22no—
74C22⋊Dic53C4212yes—
75#11719921—22no—
76C4⋊Dic53—21no—
77#11719919—22no—
78C4×Dic53—21no—
79#11719917—24no—
80#11719917—41no—
81#11719917—42no—
82#11719917—42no—
83C2×#88322—22no—
84C2×#88322—22no—
85C2×#88322—22no—
86C2×#88322—42no—
87C2×#88322C22⋊C411yescharacteristic
88C2×#88322C22⋊C412yes—
89C2×#88322C22⋊C411yescharacteristic
90#11719912—42no—
91#11719912—42no—
92C23×C53C2×C4211yescharacteristic
93C23×C53C4×D412yes—
94C22×D53—21no—
95C22×D53—41no—
96C22×D53—81no—
97Q8×C53—41no—
98D4×C53—42no—
99D4×C53—41no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—44no—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53#11711yescharacteristic
107C2×C4×C53C4×D412yes—
108C22⋊D53—82no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—41no—
112C4×D53—42no—
113C4×D53—41no—
114C4×D53—41no—
115C8×C53—41no—
116C8×C53—41no—
117#88322—21no—
118#88322—22no—
119#88322—44no—
120#88322#9912yes—
121#12624—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—42no—
125C22×C53—41no—
126C22×C53C4×C22⋊C411yescharacteristic
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—82no—
131C4×C53—22no—
132C4×C53—21no—
133C4×C53—84no—
134C4×C53C2×#9911yescharacteristic
135C4×C53#41011yescharacteristic
136Dic53—42no—
137Dic53—41no—
138#2930—531no—
139#2120—532no—
140#1757—532no—
141#1392—531no—
142#1391D5311yescharacteristic, complemented
143C2×C53—21no—
144C2×C53—42no—
145C2×C53—41no—
146C2×C53C4×#9911yescharacteristic
147D53—81no—
148C2×#131—531no—
149C22×M16—531no—
150#525—531no—
151#436—1062no—
152#435—1062no—
153#424—531no—
154#413—532no—
155#410C2×D5312yes—
156#408C2×D5311yescharacteristic
157#407—531no—
158#405—531no—
159#405—1062no—
160#352—534no—
161AΣL(1,𝔽16)—534no—
162#344—1062no—
163C53#1262411yescharacteristic, complemented, Sylow
164C2×#41C22×D5311yescharacteristic
165C23×C4—531no—
166#131—2122no—
167C2×M16—531no—
168C2×M16—532no—
169C2×M16—1061no—
170C2×M16—1061no—
171C2×M16—1062no—
172C2×M16—2122no—
173C2×M16C4×D5311yescharacteristic
174C22×C8—531no—
175C22×C8—532no—
176C22×C8C4×D5311yescharacteristic
177#123—532no—
178C22⋊Q8—1061no—
179#121—1061no—
180#120—532no—
181C4×D4—1062no—
182#117—22no—
183#117—531no—
184C4⋊C8—1062no—
185#101—1062no—
186#100—534no—
187#99—22no—
188#99C4×D5314yes—
189C22⋊C8—22no—
190C22⋊C8—1062no—
191#97—1061no—
192#97—2122no—
193C4×C8—1061no—
194C4×C8—2122no—
195C24—531no—
196#41—41no—
197#41—41no—
198C2×Q8#1171994911yescharacteristic
199C2×D4—1062no—
200C2×D4C2×C4×D5312yes—
201C2×D4#1171994711yescharacteristic
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—2121no—
206C22×C4—2121no—
207C22×C4C2×C4×D5311yescharacteristic
208C22×C4D4×D5312yes—
209M16—41no—
210M16—1064no—
211M16—2122no—
212M16—2122no—
213M16—2121no—
214M16—2122no—
215M16—2122no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—531no—
220C2×C8—531no—
221C2×C8—532no—
222C2×C8—1062no—
223C2×C8—1062no—
224C2×C8—1062no—
225C2×C8—2122no—
226C4⋊C4—42no—
227C4⋊C4—1061no—
228C4⋊C4—1062no—
229C22⋊C4—24no—
230C22⋊C4—532no—
231C22⋊C4—1062no—
232C22⋊C4—1062no—
233C42—42no—
234C42—1061no—
235C23—1061no—
236C23—4241no—
237C23—2121no—
238C23C42×D5311yescharacteristic
239C23#1313317712yes—
240Q8—41no—
241D4—42no—
242D4—41no—
243D4—4242no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—44no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2122no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4C22⋊C4×D5311yescharacteristic
256C2×C4#1313317511yescharacteristic
257C2×C4#1313317912yes—
258C8—41no—
259C8—41no—
260C8—532no—
261C8—1062no—
262C8—1061no—
263C8—2124no—
264C22—22no—
265C22—21no—
266C22—42no—
267C22—41no—
268C22—4242no—
269C22—2121no—
270C22—2121no—
271C22—2121no—
272C22#1681047011yescharacteristic
273C4—22no—
274C4—21no—
275C4—84no—
276C4—2122no—
277C4—2121no—
278C4#99×D5311yescharacteristic
279C4#1681059511yescharacteristic
280C2—21no—
281C2—42no—
282C2—41no—
283C2—4241no—
284C2#2027450211yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC23×C53C2×C42
FrattiniC2×#41C22×D53
FittingC53×#1391C2
socleC2×C53C4×#99
prime radical#16811130C22
cosocleC2×#41×C53C23
hypercenter#1391D53
hypocenterC53#12624
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #20274517C2semidirect
4. #20272838C2semidirect
5. C53×#1391C2semidirect
7. C22×#11719912C4semidirect
11. #16810649C4semidirect
13. #16810586C4semidirect
14. #16810581C4semidirect
16. #16810412C4semidirect
17. #16810409C4semidirect
142. #1391D53semidirect
163. C53#12624semidirect

Maximal subgroups

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

subgroupindex
#202834212
#202763542
#202745172
#202728382
C53×#13912
#1262453