Finite Group Databasebeta

← #28309707

Subgroups — #28309707

Order 13568.

296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 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#20291078C211yesmaximal, characteristic
2#20290263C211yesmaximal, characteristic
3#20289734C212yesmaximal, complemented
4#20289730C212yesmaximal, complemented
5#20285298C211yesmaximal, characteristic
6#16811642—21no—
7#16811544C2212yes—
8#16811518—21no—
9#16811511—21no—
10#16811487C2212yes—
11#16811484—21no—
12#16811177—21no—
13#16811027C2211yescharacteristic
14#16811001C2211yescharacteristic
15#16810972—21no—
16#16810865C411yescharacteristic
17#16810863—21no—
18#16810861C411yescharacteristic
19C4×Dic212C411yescharacteristic
20C4×#11719925—21no—
21C4×C8×D53C2211yescharacteristic
22C4×Q16×C53C411yescharacteristic
23C2×#11719910—22no—
24C4⋊#88324—22no—
25#13133274—41no—
26C4×#88324—41no—
27C2×Dic212C2×C411yescharacteristic
28C2×#11719925—21no—
29C2×C8×D53C2×C411yescharacteristic
30C2×C8×D53D411yescharacteristic
31#13133216—21no—
32#13133215—21no—
33#13133207—41no—
34#13133197—21no—
35#13133195—41no—
36#13133193—21no—
37C4×Dic106—21no—
38C4×Dic106—41no—
39#13133174—41no—
40C42×D53D411yescharacteristic
41#13133151C2×C411yescharacteristic
42#13133150—21no—
43#13133142—21no—
44#13133141—21no—
45#13133140—21no—
46C4×#88322D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53C2311yescharacteristic
49C2×Q16×C53C2×C411yescharacteristic
50C4×Q8×C53—21no—
51#107×C53C2×C411yescharacteristic
52#103×C53—21no—
53C4×C8×C53C2×C411yescharacteristic
54C2×Q8×C53—21no—
55C2×C4×D53C22⋊C411yescharacteristic
56C2×C4×D53D811yescharacteristic
57C2×C4×D53SD1611yescharacteristic
58C2×Dic106—21no—
59C2×Dic106—41no—
60C2×#88324—42no—
61Q16×C53—41no—
62C2×C8×C53C22⋊C411yescharacteristic
63C2×C8×C53C22×C411yescharacteristic
64C4⋊C4×C53—21no—
65C4⋊C4×C53—41no—
66C42×C53—41no—
67C42×C53C22⋊C411yescharacteristic
68#11719925—42no—
69#11719921—41no—
70#11719921—41no—
71C4⋊Dic53—21no—
72C4⋊Dic53—41no—
73#11719919—41no—
74#11719919—41no—
75#11719919—41no—
76#11719919—41no—
77#11719919—41no—
78C4×Dic53—41no—
79C4×Dic53D811yescharacteristic
80C4×Dic53SD1611yescharacteristic
81C4×Dic53C2×D411yescharacteristic
82C2×#88322C22⋊C411yescharacteristic
83C2×#88322C2×D411yescharacteristic
84Dic212—41no—
85C8×D53—22no—
86C8×D53—41no—
87#11719910—42no—
88C22×D53#10211yescharacteristic
89Q8×C53—41no—
90Q8×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53#10211yescharacteristic
94C2×C4×C53#10211yescharacteristic
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53#10211yescharacteristic
99C2×Dic53C2×D811yescharacteristic
100C2×Dic53C2×SD1611yescharacteristic
101C4×D53—21no—
102C4×D53—21no—
103C4×D53—22no—
104C4×D53—41no—
105Dic106—41no—
106Dic106—41no—
107Dic106—82no—
108#88324—82no—
109C8×C53—21no—
110C8×C53—21no—
111#88322—21no—
112#88322#13512yes—
113#11953—531nomaximal, Sylow
114C22×C53C2×#10211yescharacteristic
115C2×D53—21no—
116C2×D53—21no—
117C2×D53—21no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—81no—
121C4×C53—81no—
122C4×C53#41311yescharacteristic
123C4×C53#41611yescharacteristic
124Dic53—21no—
125Dic53—21no—
126Dic53—81no—
127Dic53—81no—
128Dic53#51012yes—
129#3170—531no—
130#2163—531no—
131#1564—531no—
132#1365—532no—
133#1362—532no—
134C2×C53#154011yescharacteristic
135C2×C53#214111yescharacteristic
136C2×C53#214811yescharacteristic
137D53—41no—
138#533—1061no—
139#500—531no—
140#494—531no—
141#487—1061no—
142#467—531no—
143#467—531no—
144#467—1061no—
145C4×Q16—531no—
146C4×Q16—1061no—
147C4×Q16#615711yescharacteristic
148#431—1061no—
149C2×C4×C8—531no—
150C4⋊C16—532no—
151C22⋊C16—1061no—
152#346—1061no—
153#335—532no—
154#334—1061no—
155C53#1195311yescharacteristic, complemented, Sylow
156C2×Q16—531no—
157C2×Q16—1061no—
158C2×Q16C2×#615711yescharacteristic
159C22×C8—531no—
160C22×C8—531no—
161C4⋊Q8—1061no—
162C4⋊Q8—1061no—
163#125—2121no—
164C22⋊Q8—1061no—
165C22⋊Q8—1061no—
166C22⋊Q8—2121no—
167C4×Q8—21no—
168C4×Q8—1061no—
169C4×Q8—2121no—
170#117—2121no—
171C2×C42—531no—
172C2×C16—1062no—
173#107—531no—
174#107—1061no—
175#107C2×#615711yescharacteristic
176C4⋊C8—1062no—
177#103—21no—
178#103—1061no—
179#103—1061no—
180#103—1061no—
181C22⋊C8—2121no—
182C4×C8—531no—
183C4×C8—531no—
184C4×C8—531no—
185C4×C8—2121no—
186C4×C8C2×#615711yescharacteristic
187C2×Q8—21no—
188C2×Q8—1061no—
189C2×Q8—2121no—
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—531no—
193Q16—41no—
194Q16—2121no—
195Q16—2122no—
196C2×C8—531no—
197C2×C8—531no—
198C2×C8—1062no—
199C2×C8—2121no—
200C2×C8—2122no—
201C2×C8C22⋊#615711yescharacteristic
202C2×C8C22×#615711yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—1061no—
206C4⋊C4—2121no—
207C4⋊C4—2121no—
208C4⋊C4—2121no—
209C4⋊C4—2121no—
210C4⋊C4—2121no—
211C4⋊C4—2121no—
212C22⋊C4—2121no—
213C22⋊C4—2121no—
214C42—41no—
215C42—531no—
216C42—531no—
217C42—531no—
218C42—2121no—
219C42C22⋊#615711yescharacteristic
220C16—2122no—
221C23—531no—
222Q8—41no—
223Q8—41no—
224Q8—4242no—
225Q8—2121no—
226Q8—2121no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—531no—
230C2×C4—531no—
231C2×C4—531no—
232C2×C4—1062no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4#1313327911yescharacteristic
239C2×C4#1313327911yescharacteristic
240C2×C4C2×C22⋊#615711yescharacteristic
241C8—21no—
242C8—21no—
243C8—532no—
244C8—1061no—
245C8—4242no—
246C22—1061no—
247C22—1061no—
248C22—1061no—
249C22C2×#1313327911yescharacteristic
250C4—21no—
251C4—21no—
252C4—81no—
253C4—81no—
254C4—532no—
255C4—1061no—
256C4—1061no—
257C4—4241no—
258C4—4241no—
259C4#1681160911yescharacteristic
260C4#1681161311yescharacteristic
261C2—2121no—
262C2#2029021211yescharacteristic
263C2#2029102511yescharacteristic
264C2#2029104311yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC4×Q16×C53C4
socleC22×C53C2×#102
prime radicalC22×D53#102
cosocleC8×Dic53C23
hypercenterC4×Q16#6157
hypocenterC53#11953
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #20289734C2semidirect
4. #20289730C2semidirect
155. C53#11953semidirect

Maximal subgroups

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

subgroupindex
#202910782
#202902632
#202897342
#202897302
#202852982
#1195353