Finite Group Databasebeta

← #27270764

Subgroups — #27270764

Order 13568.

338 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 49 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#20275713C211yesmaximal, characteristic
2#20272639C211yesmaximal, characteristic, complemented
3C53×#1289C211yesmaximal, characteristic, complemented
4C22×D424C411yescharacteristic, complemented
5C2×#13133191—21no—
6C2×#13133139—21no—
7C22⋊#11719908—21no—
8#16810386C411yescharacteristic, complemented
9C2×C22⋊C8×C53C2211yescharacteristic
10C53×#380—21no—
11C22×D212—21no—
12C22×C22⋊D53—41no—
13C2×D424—42no—
14C2×D424—41no—
15#13133191—42no—
16#13133191—42no—
17C2×C4⋊Dic53—21no—
18C2×#11719921—41no—
19C2×#11719908—41no—
20#13133139—42no—
21C23×C4×C53D411yescharacteristic
22C22×C8×C53C2×C411yescharacteristic
23C22×C8×C53D411yescharacteristic
24C2×C4⋊C4×C53—41no—
25C2×C22⋊C4×C53—41no—
26C22⋊C8×C53—41no—
27#95×C53—22no—
28C24×C53D811yescharacteristic
29C23×D53—41no—
30C22×C4×C53—21no—
31C22×C4×C53—21no—
32C22×C4×C53—41no—
33C22×C4×C53—41no—
34C22×C4×C53—42no—
35C22×C4×C53C22⋊C411yescharacteristic
36C22×C4×C53SD1611yescharacteristic
37C2×C22⋊D53—44no—
38C2×C22⋊D53—82no—
39C2×C22⋊D53—81no—
40C2×C22⋊D53—81no—
41C22×Dic53—41no—
42C2×D212—22no—
43C2×D212—41no—
44C2×D212—41no—
45C2×D212—41no—
46C2×D212—41no—
47C2×D212—41no—
48C2×C8×C53—21no—
49C2×C8×C53—21no—
50C2×C8×C53—41no—
51C2×C8×C53—41no—
52C4⋊C4×C53—81no—
53C4⋊C4×C53—81no—
54C22⋊C4×C53—81no—
55C22⋊C4×C53—81no—
56#11719921—44no—
57C4⋊Dic53—22no—
58C4⋊Dic53—41no—
59D424—42no—
60D424—42no—
61#11719908—81no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—41no—
65C23×C53—41no—
66C23×C53#10211yescharacteristic
67C23×C53D1611yescharacteristic
68C23×C53SD3211yescharacteristic
69C22×D53—42no—
70C22×D53—81no—
71C22×D53—81no—
72C22×D53—81no—
73C22×D53—81no—
74C22×D53—81no—
75C22×D53—81no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—81no—
83C2×C4×C53—82no—
84C2×C4×C53—82no—
85C2×C4×C53—82no—
86C2×C4×C53#9911yescharacteristic
87C2×C4×C53#10411yescharacteristic
88C22⋊D53—84no—
89C22⋊D53—84no—
90C2×Dic53—42no—
91C2×Dic53—81no—
92C2×Dic53—81no—
93D212—42no—
94D212—42no—
95D212—81no—
96D212—81no—
97C8×C53—41no—
98C8×C53—41no—
99#7076—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—81no—
109C22×C53#32711yescharacteristic
110C22×C53#35711yescharacteristic
111C22×C53#36011yescharacteristic
112C2×D53—82no—
113C2×D53—82no—
114C2×D53—82no—
115C2×D53—161no—
116C2×D53—161no—
117C2×D53—161no—
118C2×D53—161no—
119C4×C53—21no—
120C4×C53—41no—
121C4×C53—162no—
122C4×C53AΣL(1,𝔽16)11yescharacteristic
123C4×C53#35311yescharacteristic
124Dic53—82no—
125#1999—531no—
126#1314—531no—
127#1289D5311yescharacteristic, complemented
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—41no—
131C2×C53—41no—
132C2×C53#133911yescharacteristic
133C2×C53#134311yescharacteristic
134C2×C53#134711yescharacteristic
135D53—162no—
136C22×D8—531no—
137C2×#121—1061no—
138C2×#102—1061no—
139C2×C22⋊C8C2×D5311yescharacteristic
140#380—21no—
141C22⋊C16—1061no—
142#340—531no—
143C53#707611yescharacteristic, complemented, Sylow
144C22×D4—1061no—
145C22×D4—2121no—
146C23×C4C22⋊D5311yescharacteristic
147C2×D8—2121no—
148C2×D8—2122no—
149C22×C8C4×D5311yescharacteristic
150C22×C8D21211yescharacteristic
151#121—2122no—
152#121—2122no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—1061no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—2121no—
157C2×C16—2121no—
158#102—2122no—
159C22⋊C8—41no—
160#95—22no—
161C24—2121no—
162C24#1171992211yescharacteristic
163C2×D4—1062no—
164C2×D4—4242no—
165C2×D4—4241no—
166C2×D4—4241no—
167C2×D4—2121no—
168C2×D4—2121no—
169C2×D4—2121no—
170C2×D4—2124no—
171C2×D4—2121no—
172C2×D4—2121no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—2121no—
179C22×C4#1171992111yescharacteristic
180C22×C4#1171992411yescharacteristic
181D8—2122no—
182D8—2122no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—1062no—
190C4⋊C4—2121no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—2124no—
194C16—4241no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—4241no—
200C23—4241no—
201C23—4241no—
202C23—4241no—
203C23—4241no—
204C23—4241no—
205C23—2122no—
206C23#1313313711yescharacteristic
207C23#1313315811yescharacteristic
208C23#1313316011yescharacteristic
209D4—4241no—
210D4—4244no—
211D4—4244no—
212D4—4241no—
213D4—2122no—
214D4—2122no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—4241no—
226C2×C4—4241no—
227C2×C4—2122no—
228C2×C4#1313313111yescharacteristic
229C2×C4#1313314511yescharacteristic
230C8—41no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—8481no—
242C22—8481no—
243C22—8481no—
244C22—8481no—
245C22—4242no—
246C22—4242no—
247C22—4242no—
248C22#1681035611yescharacteristic
249C22#1681042311yescharacteristic
250C22#1681043211yescharacteristic
251C4—21no—
252C4—41no—
253C4—162no—
254C4—4242no—
255C4#1681040911yescharacteristic
256C4#1681041511yescharacteristic
257C2—21no—
258C2—21no—
259C2—41no—
260C2—41no—
261C2—8482no—
262C2#2027269911yescharacteristic
263C2#2027271111yescharacteristic
264C2#2027272311yescharacteristic
subgroupG/H
centerC22#16810356
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#1289C2
socleC22×C53#327
prime radical#20275713C2
cosocleC2×C22⋊C8×C53C22
hypercenter#1289D53
hypocenterC53#7076
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #20272639C2semidirect
3. C53×#1289C2semidirect
4. C22×D424C4semidirect
8. #16810386C4semidirect
127. #1289D53semidirect
143. C53#7076semidirect

Maximal subgroups

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

subgroupindex
#202757132
#202726392
C53×#12892
#707653