Finite Group Databasebeta

← #28303359

Subgroups — #28303359

Order 13568.

526 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290442C211yesmaximal, characteristic
2#20290442C211yesmaximal, characteristic
3#20289591C214yesmaximal, complemented
4#20277120C211yesmaximal, characteristic
5C2×#13133274—21no—
6C2×#13133274—21no—
7C2×#13133272C2212yes—
8C2×#13133272C2212yes—
9#16811489—21no—
10#16811489—21no—
11C2×#13133175C412yes—
12C2×C42×D53C2211yescharacteristic
13#16810493C411yescharacteristic
14#16810488C2212yes—
15#16810480—21no—
16#16810480—21no—
17C4⋊C4×Dic53—21no—
18C4⋊C4×Dic53—21no—
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133274—24no—
23#13133274—24no—
24#13133272—24no—
25#13133272—24no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133175—44no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719921—22no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38C42×D53—24no—
39#13133124—22no—
40#13133124—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45C23×D53C22⋊C411yescharacteristic
46C22×C4×C53—22no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—22no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×#88324—24no—
64C2×#88324—24no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C42×C53—22no—
73#11719921—44no—
74C4⋊Dic53—42no—
75C4⋊Dic53—42no—
76C4⋊Dic53—42no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C4×Dic53—44no—
80C4×Dic53#4114yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C22×D53—21no—
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53#10012yes—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53#10412yes—
99C2×C4×C53#10412yes—
100C2×C4×C53#11712yes—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53#10112yes—
111C2×Dic53#12312yes—
112C2×Dic53#12312yes—
113C4×D53—28no—
114C4×D53—44no—
115C4×D53—44no—
116#88324—44no—
117#88324—44no—
118#8722—531nomaximal, Sylow
119C22×C53#38111yescharacteristic
120C22×C53C2×#10011yescharacteristic
121C22×C53C2×#10111yescharacteristic
122C22×C53C2×#10411yescharacteristic
123C22×C53C2×#10411yescharacteristic
124C22×C53#42111yescharacteristic
125C22×C53#42111yescharacteristic
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—41no—
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—82no—
137Dic53—22no—
138Dic53—22no—
139Dic53—22no—
140Dic53—82no—
141#2285—531no—
142#1749—531no—
143#1749—531no—
144#1295—534no—
145C2×C53#152611yescharacteristic
146C2×C53#153011yescharacteristic
147C2×C53#153011yescharacteristic
148C2×C53#153711yescharacteristic
149C2×C53#182111yescharacteristic
150C2×C53#182611yescharacteristic
151C2×C53#182611yescharacteristic
152D53—42no—
153C2×#117—532no—
154C22×C42—531no—
155C2×C22⋊C8—1061no—
156C2×C22⋊C8—1061no—
157C2×#97—532no—
158C2×#97—532no—
159#384—531no—
160#384#615711yescharacteristic
161#382—532no—
162#380—1061no—
163#380—1061no—
164C4×C4⋊C4—1061no—
165C4×C4⋊C4—1061no—
166#336—1061no—
167#336—1061no—
168C53#872211yescharacteristic, complemented, Sylow
169C23×C4—531no—
170C23×C4—531no—
171C23×C4—531no—
172C22×C8—1062no—
173C22×C8—1062no—
174#117—2124no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—532no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4C2×#615712yes—
181C2×C22⋊C4—1062no—
182C2×C42—531no—
183C2×C42—531no—
184C2×C42—531no—
185C2×C42—1064no—
186C2×C42—1062no—
187C2×C42C2×#615711yescharacteristic
188C22⋊C8—1064no—
189C22⋊C8—1064no—
190#97—1064no—
191#97—1064no—
192#95—1062no—
193#95—1062no—
194C24—531no—
195C22×C4—22no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212C2×C8—1064no—
213C2×C8—1064no—
214C2×C8—2122no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—2122no—
222C4⋊C4—2122no—
223C4⋊C4—2122no—
224C22⋊C4—2124no—
225C42—22no—
226C42—534no—
227C42—1062no—
228C42—1062no—
229C42—2124no—
230C23—532no—
231C23—1061no—
232C23—1061no—
233C23—1061no—
234C23—1061no—
235C23—1061no—
236C23—1061no—
237C23C2×C22⋊#615711yescharacteristic
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—532no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—1068no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—2124no—
258C2×C4—2124no—
259C2×C4—2122no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4#1313328112yes—
263C2×C4#1313328212yes—
264C2×C4#1313329612yes—
265C8—2124no—
266C8—2124no—
267C22—1062no—
268C22—1062no—
269C22—1062no—
270C22—2121no—
271C22—2121no—
272C22—2121no—
273C22—2121no—
274C22#1681156611yescharacteristic
275C22D53⋊#10011yescharacteristic
276C22D53⋊#10111yescharacteristic
277C22C2×#1313328111yescharacteristic
278C22C2×#1313328211yescharacteristic
279C22#1681162111yescharacteristic
280C22#1681162211yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—82no—
285C4—1062no—
286C4—1062no—
287C4—1062no—
288C4—4242no—
289C2—2122no—
290C2#2029017311yescharacteristic
291C2#2029018511yescharacteristic
292C2#2029018611yescharacteristic
293C2#2029020711yescharacteristic
294C2#2029061511yescharacteristic
295C2#2029062111yescharacteristic
296C2#2029062211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#8722
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
3. #20289591C2semidirect
168. C53#8722semidirect

Maximal subgroups

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

subgroupindex
#202904422
#202895912
#202771202
#872253