Finite Group Databasebeta

← #27276872

Subgroups — #27276872

Order 13568.

540 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 113 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#20277254C211yesmaximal, characteristic
2#20274569C212yesmaximal, complemented
3C2×#16810362C212yesmaximal
4#20272596C211yesmaximal, characteristic, complemented
5C53×#1295C211yesmaximal, characteristic, complemented
6C2×#13133205C2211yescharacteristic
7C2×#13133182C411yescharacteristic
8C2×#13133179C411yescharacteristic
9C2×#13133146—22no—
10C2×#13133128C2212yes—
11#16810528C411yescharacteristic
12#16810524C411yescharacteristic
13#16810512—22no—
14#16810470—22no—
15C42×Dic53C2211yescharacteristic
16#16810377—22no—
17#16810362—24no—
18C2×#97×C53C2212yes—
19C53×#382C2211yescharacteristic
20C22×Dic106C2×C411yescharacteristic
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23C22×#88322—22no—
24#13133205—22no—
25#13133205D412yes—
26#13133205Q812yes—
27#13133182—42no—
28#13133179—42no—
29C2×#11719919—21no—
30C2×#11719921—22no—
31C2×#11719921—21no—
32C2×#11719921—22no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37C2×C4×Dic53D412yes—
38#13133146—28no—
39#13133128—22no—
40#13133128—22no—
41#13133124—22no—
42C22×C8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C42×C53C2311yescharacteristic
46#97×C53—22no—
47#97×C53—22no—
48#95×C53—22no—
49C23×D53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22×C411yescharacteristic
53C22×C4×C53C2×D412yes—
54C22×Dic53—21no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C412yes—
58C2×D212—21no—
59C2×D212—41no—
60C2×D212—41no—
61C2×D212C4⋊C412yes—
62C2×C4×D53—44no—
63C2×C4×D53—42no—
64C2×Dic106—21no—
65C2×Dic106—41no—
66C2×Dic106—41no—
67C2×Dic106C4⋊C412yes—
68C2×C8×C53—24no—
69C2×C8×C53—42no—
70C2×C8×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C42×C53C2×D411yescharacteristic
74C42×C53C2×Q811yescharacteristic
75C42×C53#4112yes—
76#11719921—24no—
77#11719921—24no—
78#11719921—42no—
79#11719921—44no—
80#11719919—42no—
81C4×Dic53—22no—
82C4×Dic53—24no—
83C4×Dic53—41no—
84C4×Dic53—42no—
85C4×Dic53—42no—
86C2×#88322—24no—
87C2×#88322—42no—
88C2×#88322—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—22no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—42no—
94C22×D53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53C2×C4⋊C411yescharacteristic
104C2×C4×C53#11711yescharacteristic
105C2×C4×C53#12112yes—
106C2×C4×C53C22⋊Q812yes—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53#10012yes—
116C2×Dic53#10414yes—
117D212—42no—
118D212—42no—
119C4×D53—84no—
120Dic106—42no—
121Dic106—42no—
122C8×C53—44no—
123#88322—44no—
124#8583—531nomaximal, Sylow
125C22×C53#38011yescharacteristic
126C22×C53C2×#10011yescharacteristic
127C22×C53#41311yescharacteristic
128C22×C53C2×#10412yes—
129C22×C53#42112yes—
130C2×D53—42no—
131C2×D53—44no—
132C2×D53—81no—
133C2×D53—82no—
134C2×D53—81no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—81no—
140C4×C53—81no—
141Dic53—24no—
142Dic53—22no—
143Dic53—42no—
144Dic53—41no—
145C4×#124—531no—
146#1767—532no—
147C2×#329—532no—
148#1295—531no—
149#1295D5311yescharacteristic, complemented
150C2×C53#152511yescharacteristic
151C2×C53#153212yes—
152C2×C53#153311yescharacteristic
153C2×C53#180011yescharacteristic
154C2×C53#180612yes—
155D53—82no—
156C2×#124—531no—
157C2×C4×Q8—531no—
158C2×C4×D4—531no—
159C2×C4⋊C8—1062no—
160C2×#97—532no—
161C2×#97C2×D5312yes—
162#391—531no—
163#390—531no—
164#388—1062no—
165#382C2×D5311yescharacteristic
166C4×C22⋊C4—1062no—
167C43—531no—
168#336—1062no—
169#329—1064no—
170C53#858311yescharacteristic, complemented, Sylow
171C22×Q8—531no—
172C22×D4—531no—
173C23×C4—1061no—
174C22×C8—22no—
175C22×C8—1062no—
176#124—532no—
177#124—532no—
178#124—1062no—
179C4×Q8—2122no—
180C4×D4—2122no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4C4×D5311yescharacteristic
183C2×C4⋊C4C4×D5311yescharacteristic
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1061no—
186C2×C22⋊C4—1062no—
187C2×C42—531no—
188C2×C42—532no—
189C2×C42—531no—
190C2×C42—532no—
191C2×C42—1061no—
192C2×C42C22×D5311yescharacteristic
193C4⋊C8—1068no—
194#97—22no—
195#97—22no—
196#97—1062no—
197#97—1062no—
198#95—22no—
199#95—1062no—
200C24—1061no—
201C2×Q8—532no—
202C2×Q8—1061no—
203C2×Q8—2121no—
204C2×Q8—2121no—
205C2×D4—532no—
206C2×D4—1061no—
207C2×D4—2121no—
208C2×D4—2121no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—532no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—1061no—
215C22×C4—2122no—
216C22×C4—2124no—
217C22×C4C2×C4×D5311yescharacteristic
218C22×C4D4×D5312yes—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—1064no—
223C2×C8—2122no—
224C2×C8—2122no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—2122no—
228C22⋊C4—1064no—
229C22⋊C4—1064no—
230C22⋊C4—2124no—
231C22⋊C4—2122no—
232C42—1064no—
233C42—1062no—
234C42—2121no—
235C42—2122no—
236C42—2122no—
237C42#1171994512yes—
238C42D4×D5311yescharacteristic
239C42Q8×D5311yescharacteristic
240C23—1062no—
241C23—2121no—
242C23—2121no—
243C23—2122no—
244C23—2122no—
245C23C22⋊C4×D5311yescharacteristic
246Q8—2122no—
247Q8—2122no—
248D4—2122no—
249D4—2122no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—532no—
259C2×C4—534no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—4244no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4C4⋊C4×D5311yescharacteristic
270C2×C4#1313317411yescharacteristic
271C2×C4#1313319012yes—
272C2×C4#1313319512yes—
273C8—44no—
274C8—2124no—
275C22—4241no—
276C22—4242no—
277C22—4241no—
278C22—2122no—
279C22—2124no—
280C22#1681048211yescharacteristic
281C22#100×D5311yescharacteristic
282C22#1681060611yescharacteristic
283C22#104×D5312yes—
284C22#1681063812yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—1064no—
292C4—1062no—
293C4—2122no—
294C4—2121no—
295C2—4242no—
296C2#2027330511yescharacteristic
297C2#2027333412yes—
298C2#2027333711yescharacteristic
299C2#2027470611yescharacteristic
300C2#2027473412yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1295D53
hypocenterC53#8583
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #20274569C2semidirect
4. #20272596C2semidirect
5. C53×#1295C2semidirect
149. #1295D53semidirect
170. C53#8583semidirect

Maximal subgroups

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

subgroupindex
#202772542
#202745692
C2×#168103622
#202725962
C53×#12952
#858353