Finite Group Databasebeta

← #28308384

Subgroups — #28308384

Order 13568.

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

normal subgroups only

This group has 444 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
1#20289970C211yesmaximal, characteristic
2#20289954C211yesmaximal, characteristic
3#20289641C214yesmaximal, complemented
4#20279017C211yesmaximal, characteristic
5#16811640—21no—
6#16811594—21no—
7#16811469—21no—
8#16811467—21no—
9#16811466C2212yes—
10#16811466C2212yes—
11C2×#13133200C412yes—
12C2×C4×C22⋊D53C2211yescharacteristic
13#16810554—21no—
14#16810549—21no—
15#16810519—21no—
16#16810486C2212yes—
17#16810483C411yescharacteristic
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C2×#11719936—41no—
21#13133284—41no—
22#13133274—41no—
23C22⋊#88324—22no—
24C22⋊#88324—22no—
25C4×#88324—41no—
26C22×C22⋊D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28#13133200—44no—
29C4×C22⋊D53—41no—
30C4×C22⋊D53—41no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—22no—
33C2×#11719919D411yescharacteristic
34C2×C22⋊Dic53C2×C412yes—
35C2×C22⋊Dic53C2311yescharacteristic
36C2×#11719921—22no—
37C2×#11719921D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42C23×C4×C53C2×C412yes—
43C23×C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53—41no—
48C24×C53C22×C411yescharacteristic
49C23×D53C22⋊C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—24no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
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×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71#11719936—82no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53—42no—
77#11719921—41no—
78#11719921—44no—
79C4⋊Dic53—42no—
80#11719919—41no—
81#11719919—44no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53#13112yes—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—41no—
94C22×D53—41no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—84no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C22⋊D53—44no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53#10012yes—
117C4×D53—42no—
118C4×D53—81no—
119#88324—82no—
120#88324—82no—
121#11562—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53#40811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10011yescharacteristic
132C2×D53—42no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53—81no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—82no—
140C4×C53#35212yes—
141Dic53—22no—
142Dic53—41no—
143Dic53—41no—
144Dic53—82no—
145#2486—531no—
146#1465—531no—
147#1460—531no—
148#1317—534no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—42no—
152C2×C53#147111yescharacteristic
153C2×C53C2×#35211yescharacteristic
154C2×C53#212311yescharacteristic
155D53—81no—
156C2×#123—532no—
157C2×C4×D4—531no—
158C22×C4⋊C4#615711yescharacteristic
159#431—1061no—
160#407—1061no—
161#398—1061no—
162#397—1061no—
163#389—1061no—
164#381—532no—
165#380—531no—
166#380—1061no—
167#324—1061no—
168#323—532no—
169#323—532no—
170#323—1061no—
171C53#1156211yescharacteristic, complemented, Sylow
172C22×D4—531no—
173C23×C4—531no—
174C23×C4C2×#615712yes—
175C23×C4C2×#615711yescharacteristic
176C2×M16—2121no—
177#123—2124no—
178C4×D4—2121no—
179C4×D4—2121no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—531no—
188C2×C22⋊C4—531no—
189C2×C22⋊C4—532no—
190C2×C22⋊C4—1062no—
191C2×C42—531no—
192C4⋊C8—2121no—
193C22⋊C8—1062no—
194C22⋊C8—1062no—
195C22⋊C8—2121no—
196C4×C8—2121no—
197#95—1062no—
198#95—1062no—
199#95—1061no—
200C24—531no—
201C24C22×#615711yescharacteristic
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—2121no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1062no—
218C22×C4—2122no—
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22⋊#615711yescharacteristic
221M16—4242no—
222C2×C8—2122no—
223C2×C8—2122no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—2121no—
228C4⋊C4—2122no—
229C4⋊C4—2124no—
230C22⋊C4—2121no—
231C22⋊C4—2124no—
232C22⋊C4—2121no—
233C22⋊C4—2122no—
234C42—1061no—
235C42—1061no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—1061no—
241C23—1061no—
242C23—1061no—
243C23—2121no—
244C23—2121no—
245C23C2×C22⋊#615711yescharacteristic
246C23#1313330412yes—
247D4—2124no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—532no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—4241no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4#1313327511yescharacteristic
270C2×C4#1313327511yescharacteristic
271C8—4242no—
272C8—4242no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—4241no—
281C22—2122no—
282C22—2121no—
283C22—2121no—
284C22—2121no—
285C22#1681159711yescharacteristic
286C22C2×#1313327511yescharacteristic
287C22D53⋊#10011yescharacteristic
288C4—21no—
289C4—41no—
290C4—82no—
291C4—1062no—
292C4—4242no—
293C4—2121no—
294C4—2121no—
295C4#1681152512yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—4241no—
300C2#2028999011yescharacteristic
301C2C2×#1681152511yescharacteristic
302C2#2029097711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11562
LCSG ▷ C24×C53 ▷ C22×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. #20289641C2semidirect
171. C53#11562semidirect

Maximal subgroups

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

subgroupindex
#202899702
#202899542
#202896412
#202790172
#1156253