Finite Group Databasebeta

← #27272364

Subgroups — #27272364

Order 13568.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277284C211yesmaximal, characteristic
2#20274462C212yesmaximal
3#20272541C211yesmaximal, characteristic
4#20272540C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133182C411yescharacteristic
9C2×#13133127—22no—
10#16810529C2211yescharacteristic
11#16810520—22no—
12#16810492C411yescharacteristic
13#16810474—22no—
14C42×Dic53C2211yescharacteristic
15#16810378C2212yes—
16#16810377—22no—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×#88322—22no—
21#13133215—22no—
22#13133215—22no—
23Q8×Dic53—42no—
24#13133182—42no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×#11719919—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C2×C4×Dic53D412yes—
36#13133127—28no—
37#13133124—22no—
38C22×Q8×C53C2×C411yescharacteristic
39C22×C8×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2311yescharacteristic
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53M1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C412yes—
56C2×Dic106—21no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106M1612yes—
60C2×C8×C53—24no—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C4⋊C4×C53—41no—
64C42×C53—41no—
65C4⋊Dic53—42no—
66C4⋊Dic53—41no—
67#11719919—28no—
68#11719919—42no—
69#11719919—44no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—24no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C2×#88322—24no—
79C2×#88322—42no—
80C2×#88322—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—42no—
83Q8×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—42no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×C4×C53#13712yes—
96C2×C4×C53#13712yes—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53#10012yes—
108C2×Dic53#10414yes—
109Dic106—42no—
110Dic106—42no—
111C8×C53—44no—
112#88322—44no—
113#7639—531nomaximal, Sylow
114C22×C53#40711yescharacteristic
115C22×C53#41011yescharacteristic
116C22×C53C2×#10011yescharacteristic
117C22×C53#41912yes—
118C22×C53C2×#10412yes—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—42no—
124C4×C53—81no—
125Dic53—22no—
126Dic53—24no—
127Dic53—42no—
128Dic53—41no—
129Dic53—81no—
130C4×C4⋊Q8—531no—
131#1745—532no—
132#1272—531no—
133#1272—532no—
134#1272D5311yescharacteristic
135C2×C53#146611yescharacteristic
136C2×C53#149112yes—
137C2×C53#150612yes—
138C2×C53#152511yescharacteristic
139C2×C53#155811yescharacteristic
140C2×C4⋊Q8—531no—
141C2×C4×Q8—531no—
142C2×C4×Q8—531no—
143C2×#97—1062no—
144#391—531no—
145#391C2×D5311yescharacteristic
146#389—1062no—
147#384—531no—
148C4×C4⋊C4—1062no—
149C43—531no—
150#336—532no—
151#336—1062no—
152#336C2×D5312yes—
153C53#763911yescharacteristic, complemented, Sylow
154C22×Q8—531no—
155C22×Q8C4×D5311yescharacteristic
156C22×C8—22no—
157C22×C8—1062no—
158C4⋊Q8—1062no—
159C4⋊Q8—1062no—
160C4×Q8—2122no—
161C4×Q8—2122no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4C4×D5311yescharacteristic
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—532no—
171C2×C42—532no—
172C2×C42—1061no—
173C2×C42—1061no—
174C2×C42C22×D5311yescharacteristic
175#97—1068no—
176#95—22no—
177#95—1062no—
178C2×Q8—21no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—532no—
182C2×Q8—1061no—
183C2×Q8—2121no—
184C2×Q8—2121no—
185C2×Q8#1171991212yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—531no—
189C22×C4—532no—
190C22×C4—531no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4C2×C4×D5311yescharacteristic
194C22×C4D4×D5312yes—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—1064no—
199C2×C8—2122no—
200C2×C8—2122no—
201C4⋊C4—41no—
202C4⋊C4—1068no—
203C4⋊C4—2121no—
204C4⋊C4—2124no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C42—41no—
208C42—1064no—
209C42—1062no—
210C42—1061no—
211C42—1061no—
212C42—1062no—
213C42—1062no—
214C42—2122no—
215C42—2122no—
216C23C22⋊C4×D5311yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—2122no—
220Q8—2122no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—532no—
231C2×C4—534no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1061no—
237C2×C4—1062no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4M16×D5311yescharacteristic
243C2×C4C2×#1171991211yescharacteristic
244C2×C4#1313324612yes—
245C2×C4#1313324912yes—
246C8—44no—
247C8—2124no—
248C22#1681058511yescharacteristic
249C22#1681059511yescharacteristic
250C22#100×D5311yescharacteristic
251C22#1681063012yes—
252C22#104×D5312yes—
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—81no—
259C4—1064no—
260C4—1062no—
261C4—4241no—
262C4—2121no—
263C4—2122no—
264C2#2027305411yescharacteristic
265C2#2027316012yes—
266C2#2027322512yes—
267C2#2027330511yescharacteristic
268C2#2027343911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7639
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
153. C53#7639semidirect

Maximal subgroups

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

subgroupindex
#202772842
#202744622
#202725412
#202725402
C53×#12722
#763953