Finite Group Databasebeta

← #28303318

Subgroups — #28303318

Order 13568.

540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290570C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3#20289551C214yesmaximal, complemented
4#20277120C211yesmaximal, characteristic
5C2×C4⋊#88324C2212yes—
6C2×#13133274—21no—
7C2×#13133274—21no—
8C2×C4×#88324C2212yes—
9#16811489—21no—
10#16811489—21no—
11C2×#13133175—21no—
12C2×C42×D53C2211yescharacteristic
13#16810493C411yescharacteristic
14#16810488—21no—
15#16810480—21no—
16#16810480C412yes—
17C4⋊C4×Dic53—21no—
18C4⋊C4×Dic53C2212yes—
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C4⋊#88324—24no—
23#13133274—24no—
24#13133274—42no—
25C4×#88324—24no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133175—42no—
30#13133175—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C412yes—
34C2×#11719921—22no—
35C2×C4×Dic53—22no—
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53C2311yescharacteristic
39C42×D53—24no—
40#13133124—22no—
41#13133124—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C42×C53C2×C411yescharacteristic
46C23×D53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—22no—
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
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—22no—
62C2×C4×D53—41no—
63C2×C4×D53D812yes—
64C2×C4×D53SD1612yes—
65C2×#88324—24no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C42×C53—22no—
75#11719921—44no—
76C4⋊Dic53—24no—
77C4⋊Dic53—42no—
78C4⋊Dic53—42no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53—42no—
83C4×Dic53#4114yes—
84C23×C53C2×C22⋊C411yescharacteristic
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53#10212yes—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53#10212yes—
100C2×C4×C53#10412yes—
101C2×C4×C53#11712yes—
102C2×C4×C53#13112yes—
103C2×C4×C53#13112yes—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53#12312yes—
113C2×Dic53#12312yes—
114C2×Dic53C2×D811yescharacteristic
115C2×Dic53C2×SD1611yescharacteristic
116C2×Dic53#13712yes—
117C4×D53—24no—
118C4×D53—24no—
119C4×D53—44no—
120C4×D53—44no—
121#88324—44no—
122#88324—82no—
123#8709—531nomaximal, Sylow
124C22×C53#38111yescharacteristic
125C22×C53#40811yescharacteristic
126C22×C53#40811yescharacteristic
127C22×C53C2×#10211yescharacteristic
128C22×C53#41911yescharacteristic
129C22×C53C2×#10411yescharacteristic
130C22×C53#42111yescharacteristic
131C2×D53—22no—
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—41no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—82no—
141C4×C53#43314yes—
142Dic53—22no—
143Dic53—41no—
144Dic53—82no—
145Dic53#48012yes—
146Dic53#48312yes—
147#2285—531no—
148#1797—531no—
149C4×C22⋊C8—531no—
150#1279—534no—
151C2×C53#148011yescharacteristic
152C2×C53#148511yescharacteristic
153C2×C53#149211yescharacteristic
154C2×C53#149711yescharacteristic
155C2×C53#181811yescharacteristic
156C2×C53#182311yescharacteristic
157C2×C53#182611yescharacteristic
158D53—42no—
159C2×#117—1061no—
160C22×C42—531no—
161C2×C4⋊C8—532no—
162C2×C22⋊C8—1061no—
163C2×C22⋊C8—1061no—
164C2×C4×C8—532no—
165#384—531no—
166#384#615711yescharacteristic
167#382—1061no—
168#380—532no—
169#380—1061no—
170C4×C4⋊C4—532no—
171C4×C4⋊C4—1061no—
172#336—1061no—
173#336—1061no—
174C53#870911yescharacteristic, complemented, Sylow
175C23×C4—531no—
176C23×C4—531no—
177C23×C4—531no—
178C22×C8—1062no—
179C22×C8—1062no—
180#117—2122no—
181#117—2122no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—532no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4C2×#615712yes—
188C2×C22⋊C4—1062no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1064no—
193C2×C42—1062no—
194C2×C42C2×#615711yescharacteristic
195C4⋊C8—1064no—
196C22⋊C8—1064no—
197C22⋊C8—2122no—
198C4×C8—1064no—
199#95—1062no—
200#95—1062no—
201C24—531no—
202C22×C4—22no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—532no—
206C22×C4—532no—
207C22×C4—531no—
208C22×C4—1062no—
209C22×C4—1062no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—2121no—
217C22×C4C22⋊#615711yescharacteristic
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22×#615711yescharacteristic
220C2×C8—1064no—
221C2×C8—2122no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C4⋊C4—24no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—1064no—
230C4⋊C4—2122no—
231C4⋊C4—2122no—
232C22⋊C4—2124no—
233C42—22no—
234C42—534no—
235C42—1062no—
236C42—1062no—
237C42—2122no—
238C42—2122no—
239C23—532no—
240C23—1061no—
241C23—1061no—
242C23—1061no—
243C23—1061no—
244C23—1061no—
245C23—1061no—
246C23C2×C22⋊#615711yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—532no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—532no—
258C2×C4—532no—
259C2×C4—1064no—
260C2×C4—1064no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—2124no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2124no—
271C2×C4#1313327912yes—
272C2×C4#1313328112yes—
273C2×C4#1313329612yes—
274C2×C4#1313330412yes—
275C2×C4#1313330512yes—
276C8—4242no—
277C8—2124no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—2121no—
282C22—2121no—
283C22—2121no—
284C22—2121no—
285C22#1681156611yescharacteristic
286C22#1681159711yescharacteristic
287C22#1681159811yescharacteristic
288C22C2×#1313327911yescharacteristic
289C22#1681161611yescharacteristic
290C22C2×#1313328111yescharacteristic
291C22#1681162211yescharacteristic
292C4—22no—
293C4—22no—
294C4—82no—
295C4—532no—
296C4—532no—
297C4—1062no—
298C4—4242no—
299C4—2121no—
300C4#1681164814yes—
301C2—2122no—
302C2#2029002211yescharacteristic
303C2#2029003911yescharacteristic
304C2#2029006111yescharacteristic
305C2#2029007811yescharacteristic
306C2#2029060911yescharacteristic
307C2#2029061811yescharacteristic
308C2#2029062111yescharacteristic
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#8709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #20289551C2semidirect
174. C53#8709semidirect

Maximal subgroups

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

subgroupindex
#202905702
C4×#131332742
#202895512
#202771202
#870953