Finite Group Databasebeta

← #28303986

Subgroups — #28303986

Order 13568.

510 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 98 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#20291611C211yesmaximal, characteristic
2#20290019C212yesmaximal
3#20289958C211yesmaximal, characteristic
4#20289543C212yesmaximal
5#20277284C211yesmaximal, characteristic
6#16811674—24no—
7#16811643—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10#16811473—24no—
11#16811468—22no—
12C2×#13133215C411yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×#13133182C2211yescharacteristic
15#16810529C411yescharacteristic
16#16810520—22no—
17#16810492C411yescharacteristic
18#16810474—22no—
19C42×Dic53C2211yescharacteristic
20C53×#391C411yescharacteristic
21#13133284—24no—
22C4⋊#88324—22no—
23C4⋊#88324D414yes—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26#13133272—24no—
27C4×#88324—22no—
28C4×#88324—22no—
29C22×Dic106C2×C411yescharacteristic
30#13133215—22no—
31#13133215—22no—
32Q8×Dic53—22no—
33Q8×Dic53—22no—
34#13133182—22no—
35#13133182—22no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D412yes—
46C2×C4×Dic53C2311yescharacteristic
47#13133124—22no—
48C22×Q8×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—21no—
54C2×Q8×C53—21no—
55C2×Q8×C53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C412yes—
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—21no—
68C2×Dic106M1612yes—
69C2×#88324—24no—
70C2×#88324—42no—
71C4⋊C4×C53—22no—
72C42×C53—22no—
73C4⋊Dic53—22no—
74C4⋊Dic53—24no—
75#11719919—28no—
76#11719919—44no—
77#11719919—42no—
78C4×Dic53—24no—
79C4×Dic53—21no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—42no—
83C4×Dic53—41no—
84C4×Dic53—41no—
85C4×Dic53M1612yes—
86C4×Dic53C2×D412yes—
87C4×Dic53#4112yes—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—42no—
90Q8×C53—42no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53C4×D412yes—
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53#13112yes—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53#12011yescharacteristic
114C2×Dic53#12112yes—
115C2×Dic53#12311yescharacteristic
116Dic106—24no—
117Dic106—42no—
118#88324—44no—
119#88324—82no—
120#8962—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#38611yescharacteristic
124C22×C53#40711yescharacteristic
125C22×C53#40811yescharacteristic
126C4×C53—42no—
127C4×C53—42no—
128C4×C53—81no—
129C4×C53#43512yes—
130C4×C53#43612yes—
131Dic53—24no—
132Dic53—22no—
133Dic53—21no—
134Dic53—42no—
135Dic53—42no—
136Dic53#45612yes—
137#2983—531no—
138C4×C4⋊Q8—531no—
139#1479—532no—
140#1461—531no—
141#1276—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53#187011yescharacteristic
145C2×C53#188411yescharacteristic
146C2×C53#188611yescharacteristic
147C2×C4⋊Q8—531no—
148C2×C4×Q8—531no—
149C2×C4×Q8—531no—
150#446—1064no—
151#432—1062no—
152#431—532no—
153#431—532no—
154#391—531no—
155#391#615711yescharacteristic
156#389—1062no—
157#384—531no—
158C4×C4⋊C4—1062no—
159C43—531no—
160Q8⋊C8—1064no—
161#324—1062no—
162C53#896211yescharacteristic, complemented, Sylow
163C22×Q8—531no—
164C22×Q8C2×#615711yescharacteristic
165C4⋊Q8—1062no—
166C4⋊Q8—1062no—
167C4×Q8—1062no—
168C4×Q8—1062no—
169C4×Q8—1062no—
170C4×Q8—1062no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4C2×#615711yescharacteristic
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—532no—
180C2×C42—1061no—
181C2×C42—1061no—
182C2×C42—1061no—
183C2×C42C2×#615711yescharacteristic
184C4⋊C8—534no—
185C4⋊C8—1062no—
186C4⋊C8—1064no—
187C22⋊C8—1062no—
188C22⋊C8—1062no—
189#97—1064no—
190C4×C8—1062no—
191C4×C8—1062no—
192#95—22no—
193#95—1062no—
194C2×Q8—22no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—22no—
198C2×Q8—532no—
199C2×Q8—1061no—
200C2×Q8—1062no—
201C2×Q8—1062no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—532no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4C22⋊#615712yes—
210C22×C4C22×#615711yescharacteristic
211C2×C8—1064no—
212C2×C8—2122no—
213C4⋊C4—22no—
214C4⋊C4—1064no—
215C4⋊C4—1068no—
216C4⋊C4—1062no—
217C4⋊C4—2122no—
218C4⋊C4—2124no—
219C42—22no—
220C42—532no—
221C42—532no—
222C42—532no—
223C42—1062no—
224C42—1062no—
225C42—1064no—
226C42—1061no—
227C42—2121no—
228C42—2121no—
229C42—2122no—
230C23C2×C22⋊#615711yescharacteristic
231Q8—42no—
232Q8—42no—
233Q8—1064no—
234Q8—2122no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—531no—
244C2×C4—532no—
245C2×C4—531no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2122no—
256C2×C4D4×#615712yes—
257C2×C4D53⋊M1611yescharacteristic
258C2×C4D53⋊M1611yescharacteristic
259C2×C4#1313330412yes—
260C8—4242no—
261C8—2124no—
262C22—21no—
263C22—21no—
264C22#1681157111yescharacteristic
265C22#1681159511yescharacteristic
266C22#1681159611yescharacteristic
267C4—42no—
268C4—42no—
269C4—81no—
270C4—532no—
271C4—1064no—
272C4—1062no—
273C4—1061no—
274C4—2122no—
275C4—2122no—
276C4#1681165512yes—
277C4#1681165612yes—
278C2—21no—
279C2—21no—
280C2#2029069411yescharacteristic
281C2#2029071911yescharacteristic
282C2#2029072111yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#8962
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
162. C53#8962semidirect

Maximal subgroups

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

subgroupindex
#202916112
#202900192
#202899582
#202895432
#202772842
#896253