Finite Group Databasebeta

← #28299837

Subgroups — #28299837

Order 13568.

432 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290501C211yesmaximal, characteristic
2#20290430C211yesmaximal, characteristic
3#20289534C214yesmaximal
4#20283076C211yesmaximal, characteristic
5C2×C4⋊#88324—21no—
6C2×#13133272—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133208—21no—
12#16810522—21no—
13#16810522C412yes—
14#16810519—21no—
15#16810519C2212yes—
16#16810493C411yescharacteristic
17#16810488—21no—
18C42×Dic53C2211yescharacteristic
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C4⋊#88324—24no—
23#13133272—42no—
24C4×#88324—24no—
25#13133208—42no—
26#13133208—42no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38#13133124—22no—
39#13133124—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C42×C53C2×C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C22⋊C411yescharacteristic
51C22×Dic53C2×D411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C2×#88324—24no—
54C2×#88324—24no—
55C2×#88324—42no—
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—22no—
63C4⋊Dic53—24no—
64C4⋊Dic53—42no—
65C4⋊Dic53—42no—
66C4⋊Dic53—44no—
67#11719919—42no—
68#11719919—42no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—41no—
74C4×Dic53D812yes—
75C4×Dic53SD1612yes—
76C23×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53#10112yes—
85C2×C4×C53#10212yes—
86C2×C4×C53#10412yes—
87C2×C4×C53#10412yes—
88C2×C4×C53#13112yes—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53#10212yes—
102C2×Dic53C2×D811yescharacteristic
103C2×Dic53C2×SD1611yescharacteristic
104C2×Dic53#13712yes—
105#88324—44no—
106#88324—44no—
107#7680—531nomaximal, Sylow
108C22×C53#40811yescharacteristic
109C22×C53#41011yescharacteristic
110C22×C53C2×#10111yescharacteristic
111C22×C53C2×#10211yescharacteristic
112C22×C53#41911yescharacteristic
113C22×C53C2×#10411yescharacteristic
114C22×C53C2×#10411yescharacteristic
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—82no—
118C4×C53#33014yes—
119Dic53—22no—
120Dic53—21no—
121Dic53—21no—
122Dic53—22no—
123Dic53—41no—
124Dic53—82no—
125#2859—531no—
126#1767—531no—
127#1744—531no—
128#1271—534no—
129C2×C53#147011yescharacteristic
130C2×C53#148511yescharacteristic
131C2×C53#149211yescharacteristic
132C2×C53#150211yescharacteristic
133C2×C53#150611yescharacteristic
134C2×C53C2×#33011yescharacteristic
135C2×C53#154611yescharacteristic
136C2×#125—1061no—
137C2×C4⋊C8—1061no—
138C2×#97—1061no—
139C2×C4×C8—1061no—
140#389—532no—
141#389—532no—
142#389—1061no—
143#389—1061no—
144#384—531no—
145#384#615711yescharacteristic
146#382—1061no—
147C43—531no—
148#336—532no—
149#336—532no—
150#336—1061no—
151C53#768011yescharacteristic, complemented, Sylow
152C22×C8—1062no—
153C22×C8—1062no—
154#125—2122no—
155#125—2122no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4C2×#615712yes—
164C2×C42—531no—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42C2×#615711yescharacteristic
171C4⋊C8—1064no—
172#97—2122no—
173C4×C8—1064no—
174#95—1062no—
175#95—1062no—
176C22×C4—22no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—1062no—
182C22×C4C22⋊#615711yescharacteristic
183C22×C4C22⋊#615711yescharacteristic
184C22×C4C22×#615711yescharacteristic
185C2×C8—1064no—
186C2×C8—1064no—
187C2×C8—2122no—
188C2×C8—2122no—
189C2×C8—2122no—
190C2×C8—2122no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—1064no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2124no—
198C4⋊C4—2122no—
199C4⋊C4—2122no—
200C42—22no—
201C42—532no—
202C42—532no—
203C42—1062no—
204C42—1062no—
205C42—1062no—
206C42—1062no—
207C42—2121no—
208C23C2×C22⋊#615711yescharacteristic
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—532no—
217C2×C4—531no—
218C2×C4—531no—
219C2×C4—532no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4#1313327812yes—
233C2×C4#1313327912yes—
234C2×C4#1313328112yes—
235C2×C4#1313328212yes—
236C2×C4#1313330512yes—
237C8—2124no—
238C8—2124no—
239C22#1681159811yescharacteristic
240C22#1681160311yescharacteristic
241C22C2×#1313327811yescharacteristic
242C22C2×#1313327911yescharacteristic
243C22#1681161611yescharacteristic
244C22C2×#1313328111yescharacteristic
245C22C2×#1313328211yescharacteristic
246C4—22no—
247C4—22no—
248C4—82no—
249C4—1062no—
250C4—1061no—
251C4—1061no—
252C4—1062no—
253C4—4242no—
254C4—2121no—
255C4#1681148014yes—
256C2#2028998911yescharacteristic
257C2#2029003911yescharacteristic
258C2#2029006211yescharacteristic
259C2#2029009511yescharacteristic
260C2#2029010911yescharacteristic
261C2C2×#1681148011yescharacteristic
262C2#2029022411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7680
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
151. C53#7680semidirect

Maximal subgroups

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

subgroupindex
#202905012
#202904302
#202895342
#202830762
#768053