Finite Group Databasebeta

← #27281946

Subgroups — #27281946

Order 13568.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#20279620C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic, complemented
3#20273141C212yesmaximal
4#20272574C212yesmaximal, complemented
5C53×#2982C211yesmaximal, characteristic, complemented
6C2×#13133193—21no—
7C2×C4×D212C2211yescharacteristic
8#16810647C2212yes—
9C2×#13133136—21no—
10#16810553C412yes—
11#16810540C412yes—
12#16810514—22no—
13#16810489C2211yescharacteristic
14#16810480—21no—
15#16810392—21no—
16#16810352—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C8×Q8×C53—24no—
19C53×#431—22no—
20C53×#431C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23C2×#11719917—22no—
24#13133193—42no—
25#13133193—42no—
26C4×D212—22no—
27C4×D212—22no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—21no—
34C4⋊#88322—22no—
35C4⋊#88322—22no—
36#13133136—42no—
37#13133124—22no—
38#13133124—21no—
39#13133124C2×C412yes—
40C22×Q8×C53D411yescharacteristic
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C42×C53D412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—24no—
48C4⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50C4×C8×C53—24no—
51C4×C8×C53D414yes—
52C23×D53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—22no—
55C2×Q8×C53D812yes—
56C2×Q8×C53SD1612yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—21no—
64C2×D212—22no—
65C2×D212—42no—
66C2×C4×D53—42no—
67C2×C4×D53—41no—
68C2×C4×D53—41no—
69C2×C4×D53—42no—
70C2×C8×C53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—22no—
73C42×C53—22no—
74C42×C53—22no—
75C42×C53C2×D412yes—
76C42×C53#4112yes—
77#11719921—44no—
78#11719921—44no—
79#11719921—41no—
80#11719921—41no—
81C4⋊Dic53—22no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84#11719919—44no—
85#11719917—42no—
86#11719917—42no—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—41no—
96Q8×C53—24no—
97Q8×C53—24no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53#10214yes—
105C2×C4×C53C4×D412yes—
106C2×C4×C53#12011yescharacteristic
107C2×C4×C53#12112yes—
108C2×C4×C53#12311yescharacteristic
109C2×C4×C53C2×D811yescharacteristic
110C2×C4×C53C2×SD1611yescharacteristic
111C2×C4×C53#13512yes—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—41no—
118D212—24no—
119D212—42no—
120C4×D53—82no—
121C4×D53—82no—
122C8×C53—44no—
123#88322—82no—
124#9486—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#38611yescharacteristic
128C22×C53C2×#10211yescharacteristic
129C22×C53#41611yescharacteristic
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—81no—
134C2×D53—81no—
135C2×D53—81no—
136C2×D53—81no—
137C4×C53—22no—
138C4×C53—24no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—42no—
142C4×C53#46512yes—
143C4×C53#46612yes—
144Dic53—82no—
145Dic53—81no—
146#2982D5311yescharacteristic, complemented
147#2536—531no—
148#1801—531no—
149#1486—532no—
150#1286—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#189111yescharacteristic
154C2×C53#189311yescharacteristic
155C2×C53#191211yescharacteristic
156D53—82no—
157C2×C22⋊Q8—1061no—
158C2×C4×Q8C2×D5311yescharacteristic
159C2×C4×D4—531no—
160C8×Q8—24no—
161#431—22no—
162#431C2×D5312yes—
163#423—532no—
164C2×#101—1061no—
165#398—532no—
166#394—532no—
167#388—1062no—
168#382—531no—
169#380—1061no—
170#343—1061no—
171D4⋊C8—1064no—
172C53#948611yescharacteristic, complemented, Sylow
173C22×Q8D21211yescharacteristic
174C22×D4—531no—
175C23×C4—1061no—
176C2×M16—1062no—
177C22⋊Q8—2122no—
178C22⋊Q8—2122no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1062no—
182C4×D4—1062no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4C4×D5312yes—
187C2×C4⋊C4D21211yescharacteristic
188C2×C22⋊C4—1061no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C42C22⋊D5312yes—
192C2×C42C22×D5311yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—1062no—
196C4⋊C8—1062no—
197#101—2122no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8C22⋊D5314yes—
201#95—532no—
202#95—1061no—
203#95—1062no—
204C24—1061no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1171991312yes—
208C2×Q8D42412yes—
209C2×D4—1062no—
210C2×D4—2122no—
211C22×C4—1062no—
212C22×C4—1061no—
213C22×C4—2122no—
214C22×C4—2121no—
215C22×C4—2121no—
216C22×C4—2122no—
217C22×C4#1171992112yes—
218C22×C4#1171992112yes—
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4C2×D21211yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222M16—2122no—
223M16—2122no—
224C2×C8—24no—
225C2×C8—2122no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—1062no—
229C4⋊C4—2124no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C22⋊C4—2124no—
233C22⋊C4—2124no—
234C22⋊C4—2121no—
235C22⋊C4—2121no—
236C42—22no—
237C42—22no—
238C42#1171994512yes—
239C42C2×C22⋊D5312yes—
240C23—1062no—
241C23—2121no—
242C23—2121no—
243C23—2121no—
244C23—2121no—
245C23—2121no—
246C23—2121no—
247C23C2×#1171992111yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—1064no—
251D4—2122no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—4242no—
259C2×C4—4242no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4—2122no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4#1313313914yes—
267C2×C4C4×C22⋊D5312yes—
268C2×C4#1313318611yescharacteristic
269C2×C4#1313319112yes—
270C2×C4#1313320011yescharacteristic
271C2×C4C2×D42411yescharacteristic
272C2×C4C2×#1171991311yescharacteristic
273C2×C4#1313323912yes—
274C8—44no—
275C8—4242no—
276C22—21no—
277C22—21no—
278C22—4241no—
279C22—4241no—
280C22—4241no—
281C22—4241no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22#1681050211yescharacteristic
286C22C2×#1313313911yescharacteristic
287C22#1681062011yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—24no—
292C4—42no—
293C4—4242no—
294C4—4241no—
295C4#1681085212yes—
296C4#1681085912yes—
297C2—21no—
298C2—21no—
299C2—4242no—
300C2#2027513311yescharacteristic
301C2#2027514311yescharacteristic
302C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2982C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#2982D53
hypocenterC53#9486
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #20274711C2semidirect
4. #20272574C2semidirect
5. C53×#2982C2semidirect
146. #2982D53semidirect
172. C53#9486semidirect

Maximal subgroups

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

subgroupindex
#202796202
#202747112
#202731412
#202725742
C53×#29822
#948653