Finite Group Databasebeta

← #28299764

Subgroups — #28299764

Order 13568.

426 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 120 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#20290632C211yesmaximal, characteristic
2C4×#13133284C211yesmaximal, characteristic
3#20289533C214yesmaximal
4#20283076C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×C4⋊#88324—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—
22#13133284—24no—
23C4⋊#88324—42no—
24C4×#88324—42no—
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—42no—
55C2×#88324—42no—
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C4⋊C4×C53—24no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—22no—
63C4⋊Dic53—42no—
64C4⋊Dic53—42no—
65C4⋊Dic53—42no—
66C4⋊Dic53—42no—
67C4⋊Dic53—42no—
68#11719919—44no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—41no—
72C4×Dic53—41no—
73C4×Dic53D812yes—
74C4×Dic53SD1612yes—
75C4×Dic53SD1612yes—
76C4×Dic53Q1612yes—
77C23×C53C2×C22⋊C411yescharacteristic
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53#10112yes—
86C2×C4×C53#10212yes—
87C2×C4×C53#10312yes—
88C2×C4×C53#10412yes—
89C2×C4×C53#13112yes—
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×Dic53#10312yes—
103C2×Dic53C2×D811yescharacteristic
104C2×Dic53C2×SD1611yescharacteristic
105C2×Dic53C2×SD1611yescharacteristic
106C2×Dic53C2×Q1611yescharacteristic
107#88324—44no—
108#88324—82no—
109#7660—531nomaximal, Sylow
110C22×C53#40811yescharacteristic
111C22×C53#41011yescharacteristic
112C22×C53C2×#10111yescharacteristic
113C22×C53C2×#10211yescharacteristic
114C22×C53C2×#10311yescharacteristic
115C22×C53C2×#10411yescharacteristic
116C22×C53#42111yescharacteristic
117C4×C53—22no—
118C4×C53—22no—
119C4×C53—82no—
120C4×C53#33214yes—
121Dic53—22no—
122Dic53—22no—
123Dic53—21no—
124Dic53—21no—
125Dic53—21no—
126Dic53—21no—
127Dic53—82no—
128#2859—531no—
129#1831—531no—
130C4×C4⋊C8—531no—
131#1271—534no—
132C2×C53#147011yescharacteristic
133C2×C53#148111yescharacteristic
134C2×C53#149211yescharacteristic
135C2×C53#150311yescharacteristic
136C2×C53#151711yescharacteristic
137C2×C53#152811yescharacteristic
138C2×C53C2×#33211yescharacteristic
139C2×#125—1061no—
140C2×C4⋊C8—1061no—
141C2×C4⋊C8—1061no—
142C2×C4×C8—1061no—
143#389—532no—
144#389—532no—
145#389—1061no—
146#389—1061no—
147#384—531no—
148#384#615711yescharacteristic
149#382—1061no—
150C43—531no—
151#336—532no—
152#336—532no—
153#336—1061no—
154C53#766011yescharacteristic, complemented, Sylow
155C22×C8—1062no—
156C22×C8—1062no—
157#125—2122no—
158#125—2122no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4C2×#615712yes—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42C2×#615711yescharacteristic
174C4⋊C8—1064no—
175C4⋊C8—2122no—
176C4×C8—2122no—
177#95—1062no—
178#95—1062no—
179C22×C4—22no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—1062no—
185C22×C4C22⋊#615711yescharacteristic
186C22×C4C22⋊#615711yescharacteristic
187C22×C4C22×#615711yescharacteristic
188C2×C8—1064no—
189C2×C8—2122no—
190C2×C8—2122no—
191C2×C8—2122no—
192C2×C8—2122no—
193C2×C8—2122no—
194C4⋊C4—24no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—2122no—
198C4⋊C4—2122no—
199C4⋊C4—2124no—
200C4⋊C4—2122no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C42—22no—
204C42—532no—
205C42—532no—
206C42—532no—
207C42—532no—
208C42—1062no—
209C42—1062no—
210C42—2121no—
211C42—2121no—
212C23C2×C22⋊#615711yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—532no—
221C2×C4—531no—
222C2×C4—531no—
223C2×C4—532no—
224C2×C4—531no—
225C2×C4—531no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4#1313327812yes—
238C2×C4#1313327912yes—
239C2×C4#1313328012yes—
240C2×C4#1313328112yes—
241C2×C4#1313330412yes—
242C8—4242no—
243C8—2124no—
244C22#1681159711yescharacteristic
245C22#1681160411yescharacteristic
246C22C2×#1313327811yescharacteristic
247C22C2×#1313327911yescharacteristic
248C22C2×#1313328011yescharacteristic
249C22C2×#1313328111yescharacteristic
250C22#1681162211yescharacteristic
251C4—22no—
252C4—22no—
253C4—82no—
254C4—1062no—
255C4—1061no—
256C4—1061no—
257C4—1061no—
258C4—1061no—
259C4—1062no—
260C4—4242no—
261C4#1681148214yes—
262C2#2028998811yescharacteristic
263C2#2029002411yescharacteristic
264C2#2029006111yescharacteristic
265C2#2029009811yescharacteristic
266C2#2029014611yescharacteristic
267C2#2029017711yescharacteristic
268C2C2×#1681148211yescharacteristic
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#7660
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
154. C53#7660semidirect

Maximal subgroups

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

subgroupindex
#202906322
C4×#131332842
#202895332
#202830762
#766053