Finite Group Databasebeta

← #28299923

Subgroups — #28299923

Order 13568.

500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290607C211yesmaximal, characteristic
2#20290442C211yesmaximal, characteristic
3#20289533C214yesmaximal, complemented
4#20283068C211yesmaximal, characteristic
5C2×#13133274—21no—
6C2×#13133274—21no—
7C2×#13133272—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133197—21no—
13C2×C42×D53C2211yescharacteristic
14#16810528—21no—
15#16810522—21no—
16#16810522C2212yes—
17#16810480—21no—
18#16810480C412yes—
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133274—24no—
23#13133274—42no—
24#13133272—42no—
25C22×Dic106—21no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133217—42no—
30#13133197—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53C2×C412yes—
34C2×#11719921—22no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38C42×D53—42no—
39#13133124—22no—
40#13133124—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45C23×D53C22⋊C411yescharacteristic
46C22×C4×C53—22no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×Dic106—42no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66C2×#88324—24no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C42×C53—22no—
76#11719921—44no—
77C4⋊Dic53—42no—
78C4⋊Dic53—44no—
79C4⋊Dic53—42no—
80C4×Dic53—22no—
81C4×Dic53—41no—
82C4×Dic53—41no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53#10112yes—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53#9912yes—
101C2×C4×C53#10412yes—
102C2×C4×C53#13112yes—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53AGL(1,ℤ/8)12yes—
114C2×Dic53#13712yes—
115C4×D53—44no—
116C4×D53—44no—
117C4×D53—44no—
118Dic106—84no—
119#88324—44no—
120#88324—82no—
121#7705—531nomaximal, Sylow
122C22×C53#40811yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C22×C53#41811yescharacteristic
126C22×C53#41911yescharacteristic
127C22×C53C2×#10411yescharacteristic
128C22×C53#42111yescharacteristic
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—41no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—82no—
140Dic53—22no—
141Dic53—22no—
142Dic53—41no—
143Dic53—82no—
144#2857—531no—
145#1817—531no—
146#1749—531no—
147#1271—534no—
148C2×C53#147211yescharacteristic
149C2×C53#148511yescharacteristic
150C2×C53#149611yescharacteristic
151C2×C53#150711yescharacteristic
152C2×C53#151111yescharacteristic
153C2×C53#153111yescharacteristic
154C2×C53#155611yescharacteristic
155D53—42no—
156C2×C4⋊Q8—531no—
157C2×C22⋊Q8—1061no—
158C22×C42—531no—
159C2×C22⋊C8—1061no—
160C2×C22⋊C8—1061no—
161C2×#97—1061no—
162#391—1061no—
163#389—532no—
164#389—1061no—
165#384#615711yescharacteristic
166#380—532no—
167#380—1061no—
168#336—532no—
169#336—532no—
170#336—1061no—
171C53#770511yescharacteristic, complemented, Sylow
172C22×Q8—1061no—
173C23×C4—531no—
174C23×C4—531no—
175C23×C4—531no—
176C22×C8—1062no—
177C22×C8—1062no—
178C4⋊Q8—2122no—
179C22⋊Q8—2124no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4C2×#615712yes—
186C2×C22⋊C4—1062no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—2122no—
191C2×C42C2×#615711yescharacteristic
192C22⋊C8—1064no—
193C22⋊C8—2122no—
194#97—2122no—
195#95—1062no—
196#95—1062no—
197C24—531no—
198C2×Q8—2122no—
199C2×Q8—2122no—
200C2×Q8—2122no—
201C22×C4—22no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—1062no—
208C22×C4—1062no—
209C22×C4—1062no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4—2121no—
213C22×C4—2121no—
214C22×C4—2121no—
215C22×C4C22⋊#615711yescharacteristic
216C22×C4C22⋊#615711yescharacteristic
217C22×C4C22×#615711yescharacteristic
218C2×C8—1064no—
219C2×C8—2122no—
220C2×C8—2122no—
221C2×C8—2122no—
222C2×C8—2122no—
223C2×C8—2122no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—2122no—
228C4⋊C4—2124no—
229C4⋊C4—2122no—
230C22⋊C4—2124no—
231C42—22no—
232C42—1062no—
233C42—2121no—
234C42—2121no—
235C23—532no—
236C23—1061no—
237C23—1061no—
238C23—1061no—
239C23—1061no—
240C23—1061no—
241C23—1061no—
242C23C2×C22⋊#615711yescharacteristic
243Q8—4244no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—532no—
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—2124no—
263C2×C4—2124no—
264C2×C4—2124no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4#1313327612yes—
269C2×C4#1313328212yes—
270C2×C4#1313330412yes—
271C8—4242no—
272C8—2124no—
273C22—1062no—
274C22—1062no—
275C22—1062no—
276C22—2121no—
277C22—2121no—
278C22—2121no—
279C22—2121no—
280C22#1681159711yescharacteristic
281C22C2×#1313327611yescharacteristic
282C22D53⋊#10111yescharacteristic
283C22#1681161511yescharacteristic
284C22#1681161611yescharacteristic
285C22C2×#1313328211yescharacteristic
286C22#1681162111yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—82no—
291C4—1062no—
292C4—1062no—
293C4—4242no—
294C4—2121no—
295C2—2122no—
296C2#2028999311yescharacteristic
297C2#2029003811yescharacteristic
298C2#2029007511yescharacteristic
299C2#2029011211yescharacteristic
300C2#2029012311yescharacteristic
301C2#2029018911yescharacteristic
302C2#2029024511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7705
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #20289533C2semidirect
171. C53#7705semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202904422
#202895332
#202830682
#770553