Finite Group Databasebeta

← #28309550

Subgroups — #28309550

Order 13568.

452 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290575C211yesmaximal, characteristic
2#20289954C211yesmaximal, characteristic
3#20289712C214yesmaximal, complemented
4#20279026C211yesmaximal, characteristic
5#16811640—21no—
6C2×#13133275—21no—
7#16811565—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×C4⋊#6157—41no—
21C2×C22⋊#6157—22no—
22#13133284—41no—
23#13133275—42no—
24C22⋊#88324—22no—
25C4×#88324—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—41no—
60C2×C22⋊D53—41no—
61C2×C22⋊D53SD1612yes—
62C2×C22⋊D53Q1612yes—
63C22×Dic53—22no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—42no—
70C22⋊#6157—42no—
71C22⋊#6157—42no—
72C2×#88324—42no—
73C4⋊#6157—82no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—42no—
79#11719921—41no—
80#11719921—82no—
81C4⋊Dic53—42no—
82#11719919—41no—
83#11719919—82no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53#10312yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53C2×SD1611yescharacteristic
97C22×D53C2×Q1611yescharacteristic
98C2×#6157—82no—
99C2×#6157—82no—
100C2×#6157—82no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—84no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C22⋊D53—42no—
113C22⋊D53—42no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53#10312yes—
124C4×D53—42no—
125C4×D53—81no—
126#88324—82no—
127#11899—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53C2×#9911yescharacteristic
136C22×C53C2×#10311yescharacteristic
137C22×C53#42111yescharacteristic
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—41no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—81no—
145#6157—162no—
146C4×C53—21no—
147C4×C53—41no—
148C4×C53—82no—
149C4×C53#35212yes—
150Dic53—22no—
151Dic53—41no—
152Dic53—41no—
153Dic53—82no—
154#2487—531no—
155#1799—531no—
156#1460—531no—
157#1352—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—42no—
161C2×C53#152011yescharacteristic
162C2×C53C2×#35211yescharacteristic
163C2×C53#212811yescharacteristic
164D53—41no—
165D53—41no—
166C2×C4×D4—531no—
167C22×C4⋊C4#615711yescharacteristic, complemented
168#431—1061no—
169C2×#99—1061no—
170#400—532no—
171#399—532no—
172#389—1061no—
173#386—1061no—
174#385—1061no—
175#380—531no—
176#380—1061no—
177#380—1061no—
178#342—532no—
179#323—532no—
180#323—1061no—
181C53#1189911yescharacteristic, complemented, Sylow
182C22×D4—531no—
183C23×C4—21no—
184C23×C4—531no—
185C23×C4C2×#615711yescharacteristic
186C4×D4—2121no—
187C4×D4—2121no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—2121no—
195C2×C4⋊C4C2×#615712yes—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4—1062no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1062no—
201C2×C42—531no—
202C4⋊C8—2121no—
203#99—2122no—
204C22⋊C8—1062no—
205C4×C8—2121no—
206#95—532no—
207#95—1062no—
208#95—1062no—
209#95—2121no—
210C24—531no—
211C24C22⋊#615711yescharacteristic
212C2×D4—532no—
213C2×D4—532no—
214C2×D4—2121no—
215C2×D4—2121no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4—1062no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232C2×C8—2122no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—4242no—
237C4⋊C4—4242no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C22⋊C4—4242no—
241C22⋊C4—2121no—
242C22⋊C4—2121no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—1061no—
247C42—1061no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—531no—
253C23—531no—
254C23—1061no—
255C23—1061no—
256C23—2121no—
257C23—2121no—
258C23#1313328012yes—
259C23C2×C22⋊#615711yescharacteristic
260D4—2122no—
261D4—2122no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—532no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—4241no—
277C2×C4—4242no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4#1313327511yescharacteristic
287C2×C4#1313327511yescharacteristic
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—1061no—
297C22—1061no—
298C22—4241no—
299C22—2121no—
300C22—2121no—
301C22—2121no—
302C22—2121no—
303C22C2×#1313327511yescharacteristic
304C22C2×#1313328011yescharacteristic
305C22#1681162211yescharacteristic
306C4—21no—
307C4—41no—
308C4—82no—
309C4—8482no—
310C4—1062no—
311C4—4242no—
312C4—2121no—
313C4—2121no—
314C4#1681152512yes—
315C2—21no—
316C2—21no—
317C2—42no—
318C2—2121no—
319C2—2121no—
320C2#2029015511yescharacteristic
321C2C2×#1681152511yescharacteristic
322C2#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#11899
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
167. C22×C4⋊C4#6157semidirect
181. C53#11899semidirect

Maximal subgroups

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

subgroupindex
#202905752
#202899542
#202897122
#202790262
#1189953