Finite Group Databasebeta

← #28299846

Subgroups — #28299846

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#20289533C214yesmaximal
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#16810522C2212yes—
14#16810519—21no—
15#16810519C412yes—
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—44no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—22no—
38#13133124—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C42×C53C2×C411yescharacteristic
43C22×C4×C53—22no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47C22×Dic53—22no—
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C2×D411yescharacteristic
51C22×Dic53C2×D411yescharacteristic
52C2×#88324—24no—
53C2×#88324—24no—
54C2×#88324—42no—
55C2×#88324—42no—
56C2×#88324—42no—
57C2×#88324—42no—
58C4⋊C4×C53—24no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C42×C53—22no—
62C4⋊Dic53—42no—
63C4⋊Dic53—42no—
64C4⋊Dic53—44no—
65C4⋊Dic53—42no—
66#11719919—44no—
67C4×Dic53—22no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—41no—
72C4×Dic53SD1612yes—
73C4×Dic53Q1612yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53#10112yes—
83C2×C4×C53#10312yes—
84C2×C4×C53#10412yes—
85C2×C4×C53#10412yes—
86C2×C4×C53#13112yes—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53#10312yes—
100C2×Dic53C2×SD1611yescharacteristic
101C2×Dic53C2×Q1611yescharacteristic
102C2×Dic53AGL(1,ℤ/8)12yes—
103#88324—44no—
104#88324—44no—
105#7683—531nomaximal, Sylow
106C22×C53#40811yescharacteristic
107C22×C53#41011yescharacteristic
108C22×C53C2×#10111yescharacteristic
109C22×C53C2×#10311yescharacteristic
110C22×C53#41811yescharacteristic
111C22×C53C2×#10411yescharacteristic
112C22×C53C2×#10411yescharacteristic
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—82no—
116C4×C53#33014yes—
117Dic53—22no—
118Dic53—21no—
119Dic53—21no—
120Dic53—22no—
121Dic53—41no—
122Dic53—82no—
123#2859—531no—
124#1767—531no—
125#1744—531no—
126#1271—534no—
127C2×C53#147011yescharacteristic
128C2×C53#148411yescharacteristic
129C2×C53#149311yescharacteristic
130C2×C53#150311yescharacteristic
131C2×C53#150511yescharacteristic
132C2×C53C2×#33011yescharacteristic
133C2×C53#154911yescharacteristic
134C2×#125—1061no—
135C2×C4⋊C8—1061no—
136C2×#97—1061no—
137C2×C4×C8—1061no—
138#389—532no—
139#389—532no—
140#389—1061no—
141#389—1061no—
142#384—531no—
143#384#615711yescharacteristic
144#382—1061no—
145C43—531no—
146#336—532no—
147#336—532no—
148#336—1061no—
149C53#768311yescharacteristic, complemented, Sylow
150C22×C8—1062no—
151C22×C8—1062no—
152#125—2124no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4C2×#615712yes—
161C2×C42—531no—
162C2×C42—531no—
163C2×C42—531no—
164C2×C42—531no—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42C2×#615711yescharacteristic
168C4⋊C8—1064no—
169#97—2122no—
170C4×C8—1064no—
171#95—1062no—
172#95—1062no—
173C22×C4—22no—
174C22×C4—531no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—1062no—
179C22×C4C22⋊#615711yescharacteristic
180C22×C4C22⋊#615711yescharacteristic
181C22×C4C22×#615711yescharacteristic
182C2×C8—1064no—
183C2×C8—1064no—
184C2×C8—2122no—
185C2×C8—2122no—
186C2×C8—2122no—
187C2×C8—2122no—
188C4⋊C4—24no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—2122no—
192C4⋊C4—2122no—
193C4⋊C4—2124no—
194C4⋊C4—2124no—
195C4⋊C4—2122no—
196C42—22no—
197C42—532no—
198C42—532no—
199C42—1062no—
200C42—1062no—
201C42—1062no—
202C42—1062no—
203C42—2121no—
204C23C2×C22⋊#615711yescharacteristic
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—532no—
213C2×C4—531no—
214C2×C4—531no—
215C2×C4—532no—
216C2×C4—1061no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4#1313327812yes—
229C2×C4#1313328012yes—
230C2×C4#1313328112yes—
231C2×C4#1313328212yes—
232C2×C4#1313330412yes—
233C8—2124no—
234C8—2124no—
235C22#1681159711yescharacteristic
236C22#1681160411yescharacteristic
237C22C2×#1313327811yescharacteristic
238C22C2×#1313328011yescharacteristic
239C22#1681161511yescharacteristic
240C22C2×#1313328111yescharacteristic
241C22C2×#1313328211yescharacteristic
242C4—22no—
243C4—22no—
244C4—82no—
245C4—1062no—
246C4—1061no—
247C4—1061no—
248C4—1062no—
249C4—4242no—
250C4—2121no—
251C4#1681148014yes—
252C2#2028998811yescharacteristic
253C2#2029003511yescharacteristic
254C2#2029006411yescharacteristic
255C2#2029009811yescharacteristic
256C2#2029010611yescharacteristic
257C2C2×#1681148011yescharacteristic
258C2#2029023011yescharacteristic
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#7683
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
149. C53#7683semidirect

Maximal subgroups

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

subgroupindex
#202905012
#202904302
#202895332
#202830762
#768353