Finite Group Databasebeta

← #28307945

Subgroups — #28307945

Order 13568.

450 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290943C212yesmaximal
2#20289652C212yesmaximal, complemented
3#20289651C212yesmaximal, complemented
4C2×C4⋊C8×D53C211yesmaximal, characteristic
5C2×#13133292—22no—
6#16811518—22no—
7C22⋊#11719910C2214yes—
8C22×C8×D53C411yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×C42×D53C411yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133146C411yescharacteristic
13C2×#13133146C2211yescharacteristic
14C4⋊C8×D53—22no—
15C4⋊C8×D53—22no—
16C2×C4⋊C8×C53C411yescharacteristic
17#13133292—44no—
18C2×#11719910—24no—
19C22×C4×D53C811yescharacteristic
20C22×C4×D53C811yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22C22×#88322C2×C411yescharacteristic
23C22×#88322C2311yescharacteristic
24C2×C8×D53—21no—
25C2×C8×D53—21no—
26C2×C8×D53—42no—
27C2×C8×D53—42no—
28C2×C8×D53D412yes—
29C2×C8×D53D412yes—
30C2×C4×Dic53C811yescharacteristic
31C2×C4×Dic53C811yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C42×D53—21no—
34C42×D53—21no—
35C42×D53—21no—
36C42×D53—21no—
37C4⋊#88322—22no—
38#13133146—22no—
39#13133146—22no—
40C22×C8×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C4⋊C8×C53—22no—
44C23×D53C2×C811yescharacteristic
45C22×C4×C53C2×C811yescharacteristic
46C22×C4×C53C2×C811yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53C2×C811yescharacteristic
49C22×Dic53C2×C811yescharacteristic
50C22×Dic53C2×C811yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—22no—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—21no—
58C2×C4×D53—21no—
59C2×C4×D53—42no—
60C2×C4×D53C22⋊C411yescharacteristic
61C2×C4×D53C22⋊C411yescharacteristic
62C2×C4×D53M1611yescharacteristic
63C2×C4×D53M1611yescharacteristic
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C2×C8×C53C22⋊C412yes—
68C42×C53—21no—
69C42×C53—21no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C2×#88322—21no—
77C2×#88322—41no—
78C2×#88322—41no—
79C2×#88322C2×D412yes—
80C8×D53—24no—
81C8×D53—42no—
82C8×D53—42no—
83C8×D53—42no—
84#11719910—44no—
85C23×C53C22×C811yescharacteristic
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53C22⋊C811yescharacteristic
91C22×D53C22⋊C811yescharacteristic
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53M3212yescomplemented
99C2×C4×C53C2×C22⋊C411yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53C22⋊C811yescharacteristic
107C2×Dic53C22⋊C811yescharacteristic
108C2×Dic53M3212yescomplemented
109C4×D53—24no—
110C4×D53—24no—
111C4×D53—22no—
112C4×D53—21no—
113C4×D53—21no—
114C4×D53—22no—
115C4×D53—42no—
116C4×D53—42no—
117C4×D53#9912yes—
118C4×D53#10112yes—
119C8×C53—22no—
120C8×C53—41no—
121#88322—22no—
122#88322—41no—
123#11440—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53C2×C22⋊C811yescharacteristic
127C22×C53C2×M3211yescharacteristic
128C22×C53C2×M3211yescharacteristic
129C2×D53—24no—
130C2×D53—24no—
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—21no—
135C2×D53#32412yes—
136C2×D53#34912yes—
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—41no—
140C4×C53C2×#9911yescharacteristic
141C4×C53C2×#10111yescharacteristic
142Dic53—22no—
143Dic53—21no—
144Dic53—41no—
145Dic53#32412yes—
146C22×C4⋊C8—531no—
147#2112—532no—
148#1322—532no—
149#1322—532no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53C2×#32411yescharacteristic
153C2×C53#211211yescharacteristic
154C2×C53C2×#34911yescharacteristic
155D53—22no—
156D53—22no—
157C23×C8—531no—
158C23×C8—531no—
159C22×C42—531no—
160C2×M32—1062no—
161C2×C4⋊C8—531no—
162C2×C4⋊C8—531no—
163C2×C4⋊C8—531no—
164C2×C4⋊C8—1062no—
165C2×C4⋊C8—1062no—
166C2×C4⋊C8#615711yescharacteristic
167C22⋊C16—534no—
168C22⋊C16—1062no—
169C53#1144011yescharacteristic, complemented, Sylow
170C23×C4—531no—
171C23×C4—531no—
172C23×C4—531no—
173C22×C8—531no—
174C22×C8—532no—
175C22×C8—532no—
176C22×C8—531no—
177C22×C8—1061no—
178C22×C8—1061no—
179C22×C8—2122no—
180C22×C8—2122no—
181C22×C8C2×#615711yescharacteristic
182C22×C8C2×#615711yescharacteristic
183C2×C42—531no—
184C2×C42—531no—
185C2×C42—531no—
186C2×C42—1061no—
187C2×C42—1061no—
188C2×C42—1061no—
189C2×C42—1061no—
190C2×C42C2×#615711yescharacteristic
191M32—2124no—
192C2×C16—1064no—
193C4⋊C8—22no—
194C4⋊C8—1062no—
195C4⋊C8—1062no—
196C4⋊C8—1062no—
197C24—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—1062no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—1062no—
213C22×C4—2122no—
214C22×C4D53⋊C811yescharacteristic
215C22×C4D53⋊C811yescharacteristic
216C22×C4C22×#615711yescharacteristic
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—532no—
221C2×C8—1064no—
222C2×C8—1061no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2121no—
227C2×C8—2121no—
228C2×C8C22⋊#615712yes—
229C42—21no—
230C42—21no—
231C42—1061no—
232C42—1061no—
233C42—1061no—
234C42—1061no—
235C42—1061no—
236C42—1061no—
237C16—2124no—
238C23—531no—
239C23—531no—
240C23—1062no—
241C23—1062no—
242C23—1061no—
243C23—1061no—
244C23C2×D53⋊C811yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—531no—
252C2×C4—532no—
253C2×C4—532no—
254C2×C4—531no—
255C2×C4—532no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1064no—
259C2×C4—1064no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2121no—
269C2×C4#1313329212yescomplemented
270C2×C4C2×C22⋊#615711yescharacteristic
271C2×C4D53⋊M1611yescharacteristic
272C8—22no—
273C8—41no—
274C8—1062no—
275C8—2121no—
276C22—21no—
277C22—21no—
278C22—532no—
279C22—532no—
280C22—1064no—
281C22—1064no—
282C22—1061no—
283C22—1061no—
284C22—1061no—
285C22—1061no—
286C22D53⋊(C22⋊C8)11yescharacteristic
287C22D53⋊M3211yescharacteristic
288C22C2×#1313329211yescharacteristic
289C4—22no—
290C4—21no—
291C4—41no—
292C4—532no—
293C4—1062no—
294C4—1061no—
295C4—2121no—
296C4D53⋊#9911yescharacteristic
297C4D53⋊#10111yescharacteristic
298C2—21no—
299C2—21no—
300C2—1062no—
301C2—1062no—
302C2D53⋊#32411yescharacteristic
303C2#2029094311yescharacteristic
304C2D53⋊#34911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4⋊C8×C53C4
socleC22×C53C2×C22⋊C8
prime radicalC23×D53C2×C8
cosocleC22×#88322C23
hypercenterC2×C4⋊C8#6157
hypocenterC53#11440
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20289652C2semidirect
3. #20289651C2semidirect
98. C2×C4×C53M32semidirect
108. C2×Dic53M32semidirect
169. C53#11440semidirect
269. C2×C4#13133292semidirect

Maximal subgroups

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

subgroupindex
#202909432
#202896522
#202896512
C2×C4⋊C8×D532
#1144053