Finite Group Databasebeta

← #27293522

Subgroups — #27293522

Order 13568.

330 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 75 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#20283346C211yesmaximal, characteristic
2#20276527C211yesmaximal, characteristic, complemented
3#20272665C212yesmaximal, complemented
4#20272650C212yesmaximal
5C53×#2165C211yesmaximal, characteristic, complemented
6C2×C42×D53C411yescharacteristic
7#16810678—21no—
8#16810678—21no—
9#16810674—21no—
10#16810674—21no—
11#16810646C411yescharacteristic
12#16810646C2211yescharacteristic
13#16810586C411yescharacteristic
14#16810586C2211yescharacteristic
15C4×#11719917C411yescharacteristic
16#16810444—21no—
17#16810405C2212yes—
18#16810401—21no—
19C4×M16×C53C2211yescharacteristic
20C53×#364—21no—
21C53×#349C2212yes—
22C53×#347—21no—
23C22×C4×D53—21no—
24C22×C4×D53C2×C411yescharacteristic
25C2×#11719917C2×C411yescharacteristic
26C2×#11719917C2×C411yescharacteristic
27C2×C4×Dic53—21no—
28C2×C4×Dic53C2×C411yescharacteristic
29C42×D53—21no—
30C42×D53—21no—
31C42×D53—21no—
32C42×D53—21no—
33#13133157—22no—
34#13133146—21no—
35#13133146—21no—
36#13133146—21no—
37#13133146—21no—
38#13133129—21no—
39#13133129—21no—
40#13133129—21no—
41#13133129—21no—
42#13133128—21no—
43C4×#88322—21no—
44C2×M16×C53C2×C411yescharacteristic
45C2×M16×C53C2311yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47M32×C53—22no—
48#97×C53D411yescharacteristic
49#97×C53D411yescharacteristic
50C4×C8×C53D411yescharacteristic
51C4×C8×C53D411yescharacteristic
52C23×D53M1611yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53M1611yescharacteristic
57C2×C4×D53—21no—
58C2×C4×D53—21no—
59C2×C4×D53—21no—
60C2×C4×D53—21no—
61C2×C4×D53—21no—
62C2×C4×D53—21no—
63C2×C4×D53—44no—
64C2×C4×D53—41no—
65C2×C4×D53—41no—
66M16×C53—41no—
67M16×C53—41no—
68C2×C8×C53C22⋊C411yescharacteristic
69C2×C8×C53C22⋊C411yescharacteristic
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53C2×D411yescharacteristic
72C42×C53—21no—
73C42×C53C22⋊C411yescharacteristic
74C42×C53C22⋊C411yescharacteristic
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53M1611yescharacteristic
81C4×Dic53M1611yescharacteristic
82#11719917—41no—
83#11719917—41no—
84C2×#88322—21no—
85C2×#88322—21no—
86C16×C53—42no—
87#11719908—42no—
88C23×C53C2×M1611yescharacteristic
89C22×D53—21no—
90C22×D53—42no—
91C22×D53—41no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C4×D53—24no—
105C4×D53—42no—
106C4×D53—42no—
107C4×D53—42no—
108C4×D53—42no—
109C4×D53—41no—
110C4×D53—41no—
111C8×C53—21no—
112C8×C53—21no—
113C8×C53—21no—
114C8×C53—21no—
115#88322—41no—
116#88322—41no—
117#11521—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—41no—
120C22×C53#40711yescharacteristic
121C2×D53—22no—
122C2×D53—41no—
123C2×D53—42no—
124C2×D53—41no—
125C2×D53—42no—
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53#41011yescharacteristic
130C4×C53#41311yescharacteristic
131Dic53—22no—
132Dic53—41no—
133Dic53—41no—
134#2920—531no—
135#2165—531no—
136#2165D5311yescharacteristic, complemented
137#1327—532no—
138#1320—532no—
139C2×C53—21no—
140C2×C53—41no—
141C2×C53#146411yescharacteristic
142D53—42no—
143C22×C42—531no—
144#432—1061no—
145#432—1061no—
146#431—1061no—
147#431—1061no—
148#423—531no—
149#423—531no—
150#407—531no—
151#407—531no—
152C4×M16—531no—
153C4×M16C2×D5311yescharacteristic
154#364—21no—
155#364—1061no—
156#349—532no—
157#349C2×D5312yes—
158#347—21no—
159#347—1061no—
160C53#1152111yescharacteristic, complemented, Sylow
161C23×C4—531no—
162C23×C4—1061no—
163C2×M16—531no—
164C2×M16—531no—
165C2×M16C4×D5311yescharacteristic
166C2×M16C22×D5311yescharacteristic
167C2×C42—531no—
168C2×C42—1061no—
169C2×C42—1061no—
170C2×C42—1061no—
171C2×C42—1061no—
172C2×C42—1061no—
173C2×C42C4×D5311yescharacteristic
174M32—22no—
175M32—1062no—
176C4⋊C8—1061no—
177C4⋊C8—1061no—
178C4⋊C8—1061no—
179C4⋊C8—1061no—
180C22⋊C8—1061no—
181C22⋊C8—1061no—
182C22⋊C8—1061no—
183C22⋊C8—1061no—
184#97—1061no—
185#97D21211yescharacteristic
186#97C22⋊D5311yescharacteristic
187C4×C8—1061no—
188C4×C8D21211yescharacteristic
189C4×C8C22⋊D5311yescharacteristic
190C24—531no—
191C22×C4—21no—
192C22×C4—531no—
193C22×C4—1061no—
194C22×C4—1061no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—1061no—
200C22×C4—2121no—
201C22×C4—2121no—
202C22×C4—2124no—
203C22×C4C2×C4×D5311yescharacteristic
204M16—41no—
205M16—41no—
206M16—2121no—
207M16—2121no—
208C2×C8—1061no—
209C2×C8—1061no—
210C2×C8#1171992111yescharacteristic
211C2×C8#1171992111yescharacteristic
212C2×C8C2×D21211yescharacteristic
213C2×C8C2×C22⋊D5311yescharacteristic
214C42—21no—
215C42—531no—
216C42—531no—
217C42—1061no—
218C42—1061no—
219C42—1061no—
220C42—1061no—
221C42—1061no—
222C42#1171992111yescharacteristic
223C42#1171992111yescharacteristic
224C16—42no—
225C16—2122no—
226C23—1061no—
227C23—2121no—
228C23—2122no—
229C23M16×D5311yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1064no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4—2122no—
243C2×C4—2122no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2121no—
247C2×C4C2×#1171992111yescharacteristic
248C2×C4M16×D5311yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—2121no—
254C8—2121no—
255C22—21no—
256C22—41no—
257C22—1062no—
258C22—2121no—
259C22—2122no—
260C22—2122no—
261C22—2121no—
262C22#1681058611yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—1062no—
267C4—2121no—
268C4—2121no—
269C4#1681059611yescharacteristic
270C4#1681060511yescharacteristic
271C2—21no—
272C2—41no—
273C2—2122no—
274C2#2027304711yescharacteristic
subgroupG/H
centerC4#16810596
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#2165C2
socleC2×C53#1464
prime radicalC23×D53M16
cosocleC2×M16×C53C23
hypercenter#2165D53
hypocenterC53#11521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #20276527C2semidirect
3. #20272665C2semidirect
5. C53×#2165C2semidirect
136. #2165D53semidirect
160. C53#11521semidirect

Maximal subgroups

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

subgroupindex
#202833462
#202765272
#202726652
#202726502
C53×#21652
#1152153