Finite Group Databasebeta

← #28308401

Subgroups — #28308401

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
1#20289977C211yesmaximal, characteristic, complemented
2#20289957C211yesmaximal, characteristic, complemented
3#20289643C214yesmaximal, complemented
4#20279718C211yesmaximal, characteristic
5#16811644—21no—
6#16811640—21no—
7#16811469—21no—
8#16811469—21no—
9#16811468C2212yes—
10#16811468C2212yes—
11C2×#13133204C412yes—
12C2×C4×Dic106C2211yescharacteristic
13#16810537—21no—
14#16810537—21no—
15#16810526—21no—
16#16810526—21no—
17#16810526C2212yes—
18C4×C4⋊Dic53C411yescharacteristic
19C2×#127×C53C411yescharacteristic
20#13133284—41no—
21C4⋊#88324—41no—
22C22⋊#88324—22no—
23C22⋊#88324—22no—
24#13133272—41no—
25C4×#88324—41no—
26C22×Dic106C2×C411yescharacteristic
27#13133204—44no—
28C4×Dic106—41no—
29C4×Dic106—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2311yescharacteristic
32C2×#11719919D411yescharacteristic
33C2×#11719919D411yescharacteristic
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C4×Dic53C2×C412yes—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42C22×D4×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53C2×C412yes—
45#127×C53—41no—
46#127×C53—41no—
47C2×C42×C53C2×C411yescharacteristic
48C24×C53—22no—
49C2×D4×C53—24no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—82no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×Dic106—22no—
67C2×Dic106—22no—
68C2×Dic106—41no—
69C2×#88324—42no—
70C2×#88324—42no—
71C42×C53—21no—
72C42×C53—21no—
73C22⋊Dic53—44no—
74C22⋊Dic53—44no—
75C22⋊Dic53—82no—
76C22⋊Dic53—82no—
77C4⋊Dic53—41no—
78C4⋊Dic53—42no—
79#11719919—41no—
80#11719919—41no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94D4×C53—84no—
95D4×C53—82no—
96D4×C53—84no—
97D4×C53—82no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#13112yes—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53#10012yes—
119Dic106—44no—
120#88324—82no—
121#88324—82no—
122#11567—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53—82no—
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53#40811yescharacteristic
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10011yescharacteristic
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—41no—
138C4×C53#35512yes—
139Dic53—22no—
140Dic53—41no—
141Dic53—41no—
142Dic53—41no—
143Dic53—82no—
144#2548—531no—
145#1467—531no—
146#1461—531no—
147#1318—534no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—162no—
151C2×C53#147111yescharacteristic
152C2×C53#212811yescharacteristic
153C2×C53C2×#35511yescharacteristic
154C2×#127#615711yescharacteristic
155C2×#124—532no—
156C2×C4×Q8—531no—
157#432—1061no—
158#431—1061no—
159#394—1061no—
160#394—1061no—
161#390—532no—
162#390—1061no—
163#390—1061no—
164C4×C4⋊C4—531no—
165#324—532no—
166#324—532no—
167#324—1061no—
168#324—1061no—
169C53#1156711yescharacteristic, complemented, Sylow
170C22×Q8—531no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×#615712yes—
174#127—41no—
175#127—41no—
176#124—2124no—
177C4×Q8—2121no—
178C4×Q8—2121no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C22⋊C4—1062no—
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1062no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—532no—
190C2×C42C2×#615711yescharacteristic
191C4⋊C8—2121no—
192C4⋊C8—2121no—
193C22⋊C8—1062no—
194C22⋊C8—1062no—
195#97—2121no—
196C4×C8—2121no—
197#95—1061no—
198C24—22no—
199C2×Q8—1062no—
200C2×Q8—1062no—
201C2×Q8—2121no—
202C2×D4—24no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—82no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1062no—
216C22×C4C22⋊#615711yescharacteristic
217C22×C4C22⋊#615711yescharacteristic
218C22×C4C22×#615711yescharacteristic
219C2×C8—2122no—
220C2×C8—2122no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2122no—
225C22⋊C4—4242no—
226C22⋊C4—4242no—
227C22⋊C4—2124no—
228C22⋊C4—2124no—
229C42—21no—
230C42—21no—
231C42—1061no—
232C42—1061no—
233C42—1061no—
234C42—1061no—
235C42—2122no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23C2×C22⋊#615711yescharacteristic
244Q8—2124no—
245D4—84no—
246D4—82no—
247D4—84no—
248D4—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—532no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313327511yescharacteristic
268C2×C4#1313327511yescharacteristic
269C2×C4#1313330412yes—
270C8—4242no—
271C8—4242no—
272C22—21no—
273C22—21no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#1681159711yescharacteristic
282C22C2×#1313327511yescharacteristic
283C22D53⋊#10011yescharacteristic
284C4—21no—
285C4—22no—
286C4—41no—
287C4—1062no—
288C4—4242no—
289C4—2121no—
290C4—2121no—
291C4—2121no—
292C4#1681153112yes—
293C2—21no—
294C2—21no—
295C2—162no—
296C2#2028999011yescharacteristic
297C2#2029099411yescharacteristic
298C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC2×#127#6157
hypocenterC53#11567
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20289977C2semidirect
2. #20289957C2semidirect
3. #20289643C2semidirect
169. C53#11567semidirect

Maximal subgroups

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

subgroupindex
#202899772
#202899572
#202896432
#202797182
#1156753