Finite Group Databasebeta

← #28303072

Subgroups — #28303072

Order 13568.

550 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290591C212yesmaximal, complemented
2#20290575C212yesmaximal
3#20289578C212yesmaximal, complemented
4C2×#16810686C211yesmaximal, characteristic
5C2×#13133285—22no—
6C2×#13133275—22no—
7#16811565C2214yes—
8#16811501—22no—
9#16811497—22no—
10C23×#88322C411yescharacteristic, complemented
11C22×C8×D53C2211yescharacteristic
12C2×C4×C22⋊D53C2211yescharacteristic
13#16810686—22no—
14#16810686—22no—
15C2×#13133146C411yescharacteristic, complemented
16C2×#13133129C2211yescharacteristic
17C2×C8×Dic53C411yescharacteristic, complemented
18C2×C22⋊C8×C53C411yescharacteristic, complemented
19C2×C4⋊#6157—24no—
20C2×C22⋊#6157—24no—
21#13133285—44no—
22#13133275—44no—
23(C2×#6157)⋊C4—24no—
24C22×C22⋊D53D411yescharacteristic
25C22×C4×D53C2311yescharacteristic
26C22×#88322—21no—
27C22×#88322—21no—
28C22×#88322—42no—
29C22×#88322C2×C411yescharacteristic
30C22×#88322C2×C411yescharacteristic
31C2×C8×D53—42no—
32C2×C8×D53D412yes—
33C2×C8×D53D412yes—
34C4×C22⋊D53—22no—
35C4×C22⋊D53—22no—
36C2×#11719919Q811yescharacteristic
37C2×C22⋊Dic53Q811yescharacteristic
38C2×#11719921D411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40#13133146—22no—
41#13133129—22no—
42C8×Dic53—21no—
43C8×Dic53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C22×C8×C53C2×C411yescharacteristic
46C22×C8×C53C2×C411yescharacteristic
47C22⋊C8×C53—22no—
48C24×C53C4⋊C411yescharacteristic
49C23×D53C2×D411yescharacteristic
50C22×#6157—44no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53C4⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C2×C22⋊D53—22no—
58C2×C22⋊D53—22no—
59C2×C22⋊D53SD1614yes—
60C22×Dic53C4⋊C411yescharacteristic
61C22×Dic53C4⋊C411yescharacteristic
62C22×Dic53C2×Q811yescharacteristic
63C2×C4×D53—41no—
64C2×C4×D53—41no—
65C2×C4×D53C2×D412yes—
66C2×C4×D53#4112yes—
67C22⋊#6157—44no—
68C22⋊#6157—44no—
69C4⋊#6157—44no—
70C4⋊#6157—44no—
71C2×C8×C53—21no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C2×C8×C53C22⋊C412yes—
75C22⋊Dic53—22no—
76#11719921—22no—
77#11719919—22no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C2×#88322—22no—
81C2×#88322—21no—
82C2×#88322—42no—
83C2×#88322—42no—
84C2×#88322—41no—
85C2×#88322C22⋊C411yescharacteristic
86C2×#88322C22⋊C411yescharacteristic
87C8×D53—24no—
88C8×D53—42no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#10612yescomplemented
94C23×C53C2×C4⋊C411yescharacteristic
95C22×D53—41no—
96C22×D53—41no—
97C22×D53#12112yes—
98C22×D53C2×SD1612yes—
99C22×D53#13512yes—
100C2×#6157—84no—
101C2×#6157—84no—
102C2×#6157—84no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53#11711yescharacteristic
112C22⋊D53—28no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53#10612yescomplemented
119C2×Dic53C22⋊Q812yes—
120C4×D53—22no—
121C4×D53—22no—
122C4×D53—42no—
123C8×C53—22no—
124C8×C53—41no—
125#88322—21no—
126#88322—41no—
127#88322#9912yescomplemented
128#8608—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53#38011yescharacteristic
137C22×C53C2×#10611yescharacteristic
138C22×C53#42711yescharacteristic
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53#46514yes—
144#6157—164no—
145C4×C53—21no—
146C4×C53—41no—
147C4×C53C2×#9911yescharacteristic
148C4×C53#41011yescharacteristic
149Dic53—22no—
150Dic53—22no—
151Dic53—41no—
152C2×C8×D4—531no—
153#1807—532no—
154#1799—532no—
155#1289—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53#179911yescharacteristic
160C2×C53#180711yescharacteristic
161C2×C53#181211yescharacteristic
162D53—24no—
163C23×C8—531no—
164C23×C8—531no—
165C2×C4×D4—531no—
166C8×D4—1062no—
167C8×D4—1062no—
168C2×#106—1062no—
169C2×C4⋊C8—531no—
170C2×#99—1062no—
171C2×C22⋊C8—531no—
172C2×C22⋊C8#615711yescharacteristic, complemented
173C2×C4×C8—531no—
174#380—534no—
175#342—1062no—
176#340—1062no—
177C53#860811yescharacteristic, complemented, Sylow
178C22×D4—531no—
179C23×C4—531no—
180C23×C4C2×#615711yescharacteristic
181C22×C8—531no—
182C22×C8—532no—
183C22×C8—532no—
184C22×C8—531no—
185C22×C8—1061no—
186C22×C8—1061no—
187C22×C8—2122no—
188C22×C8—2122no—
189C22×C8C2×#615711yescharacteristic
190C22×C8C2×#615711yescharacteristic
191C4×D4—1062no—
192C4×D4—1062no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—1064no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—1064no—
198C2×C42—531no—
199#106—2124no—
200C4⋊C8—1062no—
201#99—2124no—
202C22⋊C8—22no—
203C22⋊C8—1062no—
204C4×C8—1061no—
205C4×C8—1061no—
206#95—1064no—
207C24—531no—
208C24C4⋊#615711yescharacteristic
209C2×D4—534no—
210C2×D4—1062no—
211C2×D4—1062no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—531no—
221C22×C4—2121no—
222C22×C4—2121no—
223C22×C4—2124no—
224C22×C4C4⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226C2×C8—21no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—531no—
230C2×C8—531no—
231C2×C8—1064no—
232C2×C8—1062no—
233C2×C8—1061no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2121no—
238C2×C8C22⋊#615712yes—
239C4⋊C4—1062no—
240C4⋊C4—2124no—
241C4⋊C4—2124no—
242C22⋊C4—1062no—
243C22⋊C4—1062no—
244C22⋊C4—2124no—
245C22⋊C4—2124no—
246C42—1061no—
247C42—1061no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—532no—
253C23—532no—
254C23—532no—
255C23—2121no—
256C23—2121no—
257C23#1313328512yescomplemented
258C23C2×C4⋊#615711yescharacteristic
259D4—1068no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—4244no—
275C2×C4—4244no—
276C2×C4—4244no—
277C2×C4—2122no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4C2×C22⋊#615711yescharacteristic
281C2×C4#1313329611yescharacteristic
282C8—22no—
283C8—41no—
284C8—532no—
285C8—1061no—
286C8—2121no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—534no—
295C22—1062no—
296C22—1062no—
297C22—2122no—
298C22—2122no—
299C22#1681156511yescharacteristic
300C22C2×#1313328511yescharacteristic
301C22#1681163311yescharacteristic
302C4—21no—
303C4—41no—
304C4—8484no—
305C4—1062no—
306C4—1062no—
307C4—2121no—
308C4D53⋊#9911yescharacteristic
309C4#1681160311yescharacteristic
310C2—21no—
311C2—21no—
312C2—42no—
313C2—1064no—
314C2#2029057711yescharacteristic
315C2#2029059111yescharacteristic
316C2#2029059811yescharacteristic
subgroupG/H
centerC22#16811565
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊C8×C53C4
socleC22×C53#380
prime radicalC22×C22⋊D53D4
cosocleC22×C4×D53C23
hypercenterC2×C22⋊C8#6157
hypocenterC53#8608
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #20290591C2semidirect
3. #20289578C2semidirect
10. C23×#88322C4semidirect
15. C2×#13133146C4semidirect
17. C2×C8×Dic53C4semidirect
18. C2×C22⋊C8×C53C4semidirect
93. C23×C53#106semidirect
118. C2×Dic53#106semidirect
127. #88322#99semidirect
172. C2×C22⋊C8#6157semidirect
177. C53#8608semidirect
257. C23#13133285semidirect

Maximal subgroups

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

subgroupindex
#202905912
#202905752
#202895782
C2×#168106862
#860853