Finite Group Databasebeta

← #27293391

Subgroups — #27293391

Order 13568.

382 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 83 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#20276334C212yesmaximal, complemented
3#20272667C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic, complemented
5C53×#1320C211yesmaximal, characteristic, complemented
6C2×#13133225C2211yescharacteristic
7C23×#88322C411yescharacteristic
8#16811134C411yescharacteristic
9M32×D53—22no—
10#16810691—22no—
11#16810686—22no—
12#16810651C2211yescharacteristic
13#16810591C411yescharacteristic
14#16810581C411yescharacteristic
15#16810405C2212yes—
16#16810404—22no—
17C53×#407C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719945C2×C411yescharacteristic
20#13133225—41no—
21#13133225—41no—
22C2×#11719912D412yes—
23C22×#88322—21no—
24C22×#88322—21no—
25C22×#88322—41no—
26C22×#88322—41no—
27C22×#88322C2×C411yescharacteristic
28C22×#88322C2×C411yescharacteristic
29C2×C8×D53D412yes—
30#13133200C812yes—
31#13133196—21no—
32#13133191—21no—
33#13133186C812yes—
34C4×C22⋊D53—22no—
35#13133176C2×C411yescharacteristic
36#13133157—22no—
37#13133156—42no—
38C16×D53—42no—
39#13133146—22no—
40#13133129—22no—
41#13133127—21no—
42C8×Dic53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C2×M16×C53C2×C411yescharacteristic
45C2×M16×C53C2311yescharacteristic
46M32×C53—22no—
47C22⋊C8×C53—22no—
48C24×C53C2×C811yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53C2×C811yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C2×C22⋊D53—22no—
56C2×C22⋊D53C2×C812yes—
57#11719945—41no—
58#11719945—41no—
59C2×D212M1611yescharacteristic
60C2×C4×D53C22⋊C412yes—
61C2×Dic106M1611yescharacteristic
62M16×C53—41no—
63M16×C53—41no—
64C2×C8×C53—21no—
65C2×C8×C53C2×D412yes—
66C22⋊Dic53—22no—
67C22⋊Dic53C2×C812yes—
68#11719921—22no—
69C4⋊Dic53—21no—
70#11719919—22no—
71C4×Dic53—21no—
72C2×#88322—21no—
73C2×#88322—21no—
74C2×#88322—21no—
75C2×#88322—42no—
76C2×#88322—41no—
77C2×#88322—41no—
78C2×#88322—41no—
79C2×#88322C22⋊C411yescharacteristic
80C2×#88322C22⋊C411yescharacteristic
81#11719912—42no—
82C8×D53—22no—
83C8×D53—22no—
84C16×C53—42no—
85#11719908—42no—
86C23×C53—21no—
87C23×C53—41no—
88C23×C53—42no—
89C23×C53C22×C811yescharacteristic
90C22×D53C22⋊C812yes—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C22⋊D53—42no—
101C22⋊D53—44no—
102C2×Dic53—22no—
103C2×Dic53C22⋊C812yes—
104D212—41no—
105C4×D53—22no—
106C4×D53—22no—
107Dic106—41no—
108C8×C53—22no—
109C8×C53—41no—
110#88322—21no—
111#88322—41no—
112#88322#9911yescharacteristic
113#88322#9911yescharacteristic
114#11483—531nomaximal, Sylow
115C22×C53—22no—
116C22×C53—21no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53C2×C22⋊C811yescharacteristic
123C2×D53—22no—
124C2×D53—42no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53C2×#9911yescharacteristic
128C4×C53#41311yescharacteristic
129Dic53—22no—
130Dic53—42no—
131#2929—531no—
132#2115—532no—
133#1328—532no—
134#1320—531no—
135#1320D5311yescharacteristic, complemented
136C2×C53—21no—
137C2×C53—42no—
138C2×C53—41no—
139C2×C53#145911yescharacteristic
140D53—42no—
141C2×#131—531no—
142C23×C8—531no—
143#525—531no—
144C2×M32—1062no—
145#435—1062no—
146C8×D4—1062no—
147#424—531no—
148#408—531no—
149#407C2×D5311yescharacteristic
150#405—531no—
151#349—532no—
152#349—1062no—
153#349C2×D5312yes—
154C53#1148311yescharacteristic, complemented, Sylow
155C2×#41—531no—
156C23×C4C4×D5311yescharacteristic
157#131—2121no—
158#131—2121no—
159C2×M16—532no—
160C2×M16C4×D5311yescharacteristic
161C2×M16C22×D5311yescharacteristic
162C22×C8—531no—
163C22×C8—532no—
164C22×C8—531no—
165C22×C8—1061no—
166C22×C8—1061no—
167C22×C8—2121no—
168C22×C8—2121no—
169#123—532no—
170C22⋊Q8—1061no—
171#121—1061no—
172#120—532no—
173C4×D4—1062no—
174#117—531no—
175M32—22no—
176M32—1062no—
177M32—2122no—
178C2×C16—2122no—
179C4⋊C8—1062no—
180C22⋊C8—22no—
181C22⋊C8—1062no—
182#97—1061no—
183C4×C8—1061no—
184C24C8×D5311yescharacteristic
185#41—2121no—
186#41—2121no—
187C2×Q8—531no—
188C2×D4—532no—
189C2×D4—531no—
190C2×D4—1062no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—532no—
196C22×C4C8×D5311yescharacteristic
197C22×C4C2×C4×D5311yescharacteristic
198M16—41no—
199M16—41no—
200M16—2122no—
201C2×C8—21no—
202C2×C8—531no—
203C2×C8—531no—
204C2×C8—1062no—
205C2×C8—1062no—
206C2×C8—1061no—
207C2×C8—1061no—
208C2×C8—1061no—
209C2×C8—2121no—
210C2×C8—2121no—
211C2×C8—2122no—
212C2×C8—2121no—
213C2×C8D4×D5312yes—
214C4⋊C4—1062no—
215C4⋊C4—1061no—
216C22⋊C4—532no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C42—1061no—
220C16—42no—
221C16—2122no—
222C23—21no—
223C23—41no—
224C23—42no—
225C23—532no—
226C23C2×C8×D5311yescharacteristic
227Q8—2121no—
228D4—2121no—
229D4—2122no—
230D4—2124no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—532no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4C22⋊C4×D5311yescharacteristic
243C2×C4M16×D5311yescharacteristic
244C8—22no—
245C8—41no—
246C8—531no—
247C8—531no—
248C8—1061no—
249C8—2121no—
250C22—22no—
251C22—21no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—1062no—
258C22—2122no—
259C22C22⋊C8×D5311yescharacteristic
260C4—21no—
261C4—41no—
262C4—1062no—
263C4—2122no—
264C4#99×D5311yescharacteristic
265C4#1681060411yescharacteristic
266C2—21no—
267C2—42no—
268C2—41no—
269C2—2122no—
270C2#2027303011yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC23×C53C22×C8
FrattiniC2×M16C22×D53
FittingC53×#1320C2
socleC2×C53#1459
prime radical#16811134C4
cosocleC2×M16×C53C23
hypercenter#1320D53
hypocenterC53#11483
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #20276334C2semidirect
3. #20272667C2semidirect
4. #20272651C2semidirect
5. C53×#1320C2semidirect
135. #1320D53semidirect
154. C53#11483semidirect

Maximal subgroups

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

subgroupindex
#202834052
#202763342
#202726672
#202726512
C53×#13202
#1148353