Finite Group Databasebeta

← #28308209

Subgroups — #28308209

Order 13568.

418 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289969C212yesmaximal
2#20289658C212yesmaximal, complemented
3#20289657C212yesmaximal, complemented
4#20277340C211yesmaximal, characteristic
5#16811644—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133175C2211yescharacteristic
9C2×#13133174C411yescharacteristic
10C2×C4⋊C4×D53C411yescharacteristic
11#16810520—22no—
12#16810497—22no—
13#16810493C2211yescharacteristic
14#16810492C2211yescharacteristic
15#16810488C411yescharacteristic
16C53×#382C411yescharacteristic
17C4⋊#88324—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C22×C4×D53—21no—
21C22×C4×D53C2×C411yescharacteristic
22#13133175—42no—
23#13133174—22no—
24#13133174—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2311yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—22no—
31C4⋊C4×D53—42no—
32C2×#11719921—21no—
33C2×#11719921D412yes—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D412yes—
36#13133124—22no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40#95×C53—22no—
41C23×D53C22⋊C411yescharacteristic
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53C22⋊C412yes—
45C22×C4×C53C22×C411yescharacteristic
46C22×Dic53—21no—
47C22×Dic53—21no—
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C2×D412yes—
50C2×C4×D53—22no—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—22no—
54C2×C4×D53—42no—
55C2×C4×D53—82no—
56C2×#88324—44no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—41no—
59C42×C53—21no—
60C42×C53M1612yes—
61#11719921—24no—
62#11719921—42no—
63C4⋊Dic53—41no—
64C4⋊Dic53—41no—
65C4⋊Dic53—82no—
66#11719919—24no—
67#11719919—42no—
68#11719919—82no—
69C4×Dic53—21no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53M1612yes—
73C23×C53C2×C22⋊C411yescharacteristic
74C22×D53—21no—
75C22×D53—21no—
76C22×D53—41no—
77C22×D53—41no—
78C22×D53#10112yes—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53C2×M1611yescharacteristic
89C2×C4×C53C2×M1611yescharacteristic
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—41no—
99C2×Dic53#9912yes—
100C4×D53—24no—
101C4×D53—24no—
102C4×D53—84no—
103#88324—84no—
104#11516—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#40711yescharacteristic
108C22×C53C2×#9911yescharacteristic
109C22×C53C2×#10111yescharacteristic
110C2×D53—22no—
111C2×D53—42no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—42no—
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—42no—
118C4×C53—42no—
119C4×C53—81no—
120Dic53—22no—
121Dic53—42no—
122Dic53—42no—
123Dic53—81no—
124#2317—531no—
125#1464—532no—
126#1325—532no—
127#1325—532no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#146311yescharacteristic
131C2×C53#212811yescharacteristic
132C2×C53#213511yescharacteristic
133D53—42no—
134C2×#117—531no—
135C2×#117—531no—
136C22×C4⋊C4—531no—
137#432—1062no—
138#389—1062no—
139#385—1062no—
140#384—531no—
141#384—531no—
142#382—531no—
143#382#615711yescharacteristic
144#324—534no—
145#323—1062no—
146C53#1151611yescharacteristic, complemented, Sylow
147C23×C4—531no—
148C23×C4—1061no—
149#117—1062no—
150#117—1062no—
151#117—2122no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—1061no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—2122no—
159C2×C4⋊C4C2×#615711yescharacteristic
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C22⋊C4—532no—
162C2×C22⋊C4—1061no—
163C2×C42—531no—
164C2×C42—532no—
165C2×C42C2×#615711yescharacteristic
166C4⋊C8—2122no—
167C22⋊C8—1064no—
168#97—2122no—
169#95—22no—
170#95—1062no—
171C24—531no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—531no—
175C22×C4—532no—
176C22×C4—1062no—
177C22×C4—1062no—
178C22×C4—1061no—
179C22×C4—1061no—
180C22×C4—1061no—
181C22×C4—1061no—
182C22×C4—4242no—
183C22×C4—2122no—
184C22×C4C22⋊#615712yes—
185C22×C4C22×#615711yescharacteristic
186C2×C8—2124no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—1064no—
190C4⋊C4—4242no—
191C4⋊C4—4242no—
192C4⋊C4—2121no—
193C4⋊C4—2121no—
194C4⋊C4—2122no—
195C22⋊C4—1064no—
196C22⋊C4—2122no—
197C42—21no—
198C42—532no—
199C42—1062no—
200C42—1062no—
201C42—1061no—
202C42#1171993612yes—
203C23—532no—
204C23—1061no—
205C23—1061no—
206C23—2121no—
207C23—2121no—
208C23C2×C22⋊#615711yescharacteristic
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—532no—
219C2×C4—1064no—
220C2×C4—1064no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—4244no—
226C2×C4—2122no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4—2121no—
230C2×C4—2122no—
231C2×C4D53⋊M1611yescharacteristic
232C2×C4C2×#1171993611yescharacteristic
233C8—4244no—
234C22—21no—
235C22—21no—
236C22—1062no—
237C22—2122no—
238C22—2122no—
239C22—2121no—
240C22—2121no—
241C22#1681159411yescharacteristic
242C22D53⋊#9911yescharacteristic
243C22D53⋊#10111yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—1062no—
250C4—4241no—
251C4—2122no—
252C4—2122no—
253C2—21no—
254C2—21no—
255C2—2122no—
256C2#2028996411yescharacteristic
257C2#2029099511yescharacteristic
258C2#2029101811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenter#382#6157
hypocenterC53#11516
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #20289658C2semidirect
3. #20289657C2semidirect
146. C53#11516semidirect

Maximal subgroups

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

subgroupindex
#202899692
#202896582
#202896572
#202773402
#1151653