Finite Group Databasebeta

← #28309152

Subgroups — #28309152

Order 13568.

462 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13133275C211yesmaximal, characteristic
2#20289974C211yesmaximal, characteristic
3#20289712C214yesmaximal, complemented
4#20279026C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133275—21no—
7C4×C22⋊#6157—21no—
8#16811501C2212yes—
9#16811467C2212yes—
10#16811466—21no—
11C2×C4×C22⋊D53C2211yescharacteristic
12#16810561C412yescomplemented
13#16810558C2212yes—
14#16810519—21no—
15#16810502—21no—
16#16810496—21no—
17#16810483C411yescharacteristic, complemented
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic, complemented
20C2×C22⋊#6157—22no—
21C2×C4×#6157—41no—
22C4⋊#88324—41no—
23#13133275—24no—
24C22⋊#88324—22no—
25#13133272—41no—
26(C2×#6157)⋊C4—41no—
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C4×C22⋊D53—41no—
30C4×C22⋊D53—41no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—22no—
33C2×#11719919C2×C411yescharacteristic
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53D411yescharacteristic
36C2×#11719921—22no—
37C2×#11719921C2311yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C23×D53C2×D411yescharacteristic
50C22×#6157—42no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53—82no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C2×C22⋊D53—22no—
60C2×C22⋊D53—22no—
61C2×C22⋊D53—41no—
62C22×Dic53—22no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—42no—
69C22⋊#6157—24no—
70C22⋊#6157—42no—
71C2×#88324—42no—
72C4×#6157—82no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C22⋊Dic53—41no—
77C22⋊Dic53—42no—
78#11719921—41no—
79#11719921—82no—
80C4⋊Dic53—42no—
81#11719919—41no—
82#11719919—82no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53#10412yes—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53#13712yes—
96C2×#6157—44no—
97C2×#6157—82no—
98C2×#6157—82no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—84no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C22⋊D53—44no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C4×D53—42no—
122C4×D53—81no—
123#88324—82no—
124#11786—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—22no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53C2×#9911yescharacteristic
133C22×C53#41911yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C2×D53—22no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—42no—
139C2×D53—81no—
140#6157—84no—
141C4×C53—21no—
142C4×C53—41no—
143C4×C53—82no—
144C4×C53#35212yes—
145Dic53—41no—
146Dic53—41no—
147Dic53—41no—
148Dic53—82no—
149#2487—531no—
150C4×#99—531no—
151#1466—531no—
152#1352—534no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53—42no—
156C2×C53#150811yescharacteristic
157C2×C53C2×#35211yescharacteristic
158C2×C53#212811yescharacteristic
159D53—42no—
160C2×C4×D4—531no—
161C22×C4⋊C4#615711yescharacteristic, complemented
162#432—1061no—
163C2×#99—1061no—
164#400—532no—
165#399—532no—
166#389—1061no—
167#386—1061no—
168#385—1061no—
169#380—531no—
170#380—1061no—
171C4×C22⋊C4—1061no—
172#342—532no—
173#323—532no—
174#323—1061no—
175C53#1178611yescharacteristic, complemented, Sylow
176C22×D4—531no—
177C23×C4—21no—
178C23×C4—531no—
179C23×C4C2×#615711yescharacteristic
180C4×D4—2121no—
181C4×D4—2121no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C2×#615712yes—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4—1061no—
194C2×C42—531no—
195C2×C42—2121no—
196C4⋊C8—2121no—
197#99—1064no—
198C22⋊C8—1062no—
199#97—2121no—
200#95—532no—
201#95—1062no—
202#95—1062no—
203#95—2121no—
204C24—531no—
205C24C22⋊#615711yescharacteristic
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—2121no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—82no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—1062no—
221C22×C4—2122no—
222C22×C4—2122no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225C2×C8—2122no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—4242no—
230C4⋊C4—2121no—
231C4⋊C4—2122no—
232C22⋊C4—1064no—
233C22⋊C4—4242no—
234C22⋊C4—2121no—
235C22⋊C4—2121no—
236C22⋊C4—2122no—
237C22⋊C4—2122no—
238C42—1061no—
239C42—1061no—
240C42—4242no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—532no—
246C23—1061no—
247C23—1061no—
248C23—2121no—
249C23—2121no—
250C23#1313328212yes—
251C23C2×C22⋊#615711yescharacteristic
252D4—2124no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—4241no—
268C2×C4—4242no—
269C2×C4—4242no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2124no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4#1313327511yescharacteristic
278C2×C4#1313327511yescharacteristic
279C8—4242no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—1062no—
288C22—4241no—
289C22—2122no—
290C22—2121no—
291C22—2121no—
292C22C2×#1313327511yescharacteristic
293C22#1681161611yescharacteristic
294C22C2×#1313328211yescharacteristic
295C4—21no—
296C4—41no—
297C4—82no—
298C4—4242no—
299C4—4244no—
300C4—2121no—
301C4—2121no—
302C4—2121no—
303C4#1681152512yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—2122no—
308C2#2029011511yescharacteristic
309C2C2×#1681152511yescharacteristic
310C2#2029099411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719921C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11786
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289712C2semidirect
12. #16810561C4semidirect
17. #16810483C4semidirect
19. C22×C4⋊C4×C53C4semidirect
161. C22×C4⋊C4#6157semidirect
175. C53#11786semidirect

Maximal subgroups

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

subgroupindex
C4×#131332752
#202899742
#202897122
#202790262
#1178653