Finite Group Databasebeta

← #28307907

Subgroups — #28307907

Order 13568.

526 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 90 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#20289947C212yesmaximal
2#20289642C212yesmaximal
3#20289641C212yesmaximal
4#20277300C211yesmaximal, characteristic
5#16811640—22no—
6#16811469—22no—
7#16811466C2214yes—
8C23×Dic106C411yescharacteristic
9C2×#13133173C411yescharacteristic
10C2×#13133173C2211yescharacteristic
11#16810527—22no—
12#16810527—22no—
13#16810486C2211yescharacteristic
14#16810484C411yescharacteristic
15#16810484C2211yescharacteristic
16C53×#381C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20C22×Dic106—21no—
21C22×Dic106—21no—
22C22×Dic106—22no—
23C22×Dic106—42no—
24C22×Dic106—42no—
25C22×Dic106C814yescomplemented
26C23×Dic53C2×C411yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28#13133173—42no—
29#13133173—42no—
30C2×#11719919—21no—
31C2×#11719919D412yes—
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2311yescharacteristic
34C2×C4×Dic53—21no—
35C2×C4×Dic53D412yes—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39C23×C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53C2×C411yescharacteristic
42#95×C53—22no—
43C24×C53C22×C411yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C2×C812yes—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53—21no—
56C22×Dic53—41no—
57C22×Dic53—41no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×C812yes—
61C22×Dic53C2×C812yes—
62C22×Dic53M1612yes—
63C22×Dic53C2×D412yes—
64C2×Dic106—24no—
65C2×Dic106—24no—
66C2×Dic106—42no—
67C2×Dic106—42no—
68C2×Dic106—48no—
69C2×Dic106—42no—
70C2×Dic106—42no—
71C2×Dic106—48no—
72C2×#88324—44no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53—41no—
77#11719919—42no—
78#11719919—42no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—42no—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53C22×C811yescharacteristic
87C23×C53C2×M1611yescharacteristic
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—24no—
100C2×Dic53—24no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—42no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—41no—
110C2×Dic53C22⋊C812yes—
111C2×Dic53C22⋊C812yes—
112C2×Dic53#9912yes—
113C2×Dic53#10012yes—
114Dic106—48no—
115Dic106—48no—
116#88324—84no—
117#11428—531nomaximal, Sylow
118C22×C53—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53C2×C22⋊C811yescharacteristic
126C22×C53C2×#9911yescharacteristic
127C22×C53C2×#10011yescharacteristic
128C4×C53—42no—
129C4×C53—42no—
130C4×C53—81no—
131Dic53—22no—
132Dic53—42no—
133Dic53—42no—
134Dic53—81no—
135Dic53#32314yes—
136#2310—531no—
137#1458—532no—
138#1317—532no—
139#1317—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—41no—
143C2×C53—41no—
144C2×C53C2×#32311yescharacteristic
145C2×C53#212312yes—
146C23×Q8—531no—
147C2×#117—531no—
148C2×#117—531no—
149#431—1062no—
150#391—1062no—
151#391—1062no—
152#381—531no—
153#381—531no—
154#381—531no—
155#381#615711yescharacteristic
156#324—1062no—
157#323—534no—
158C53#1142811yescharacteristic, complemented, Sylow
159C22×Q8—534no—
160C22×Q8—1061no—
161C22×Q8—1061no—
162C22×Q8—1062no—
163C22×Q8—2122no—
164C22×Q8—2122no—
165C23×C4—531no—
166C23×C4—531no—
167C23×C4C2×#615711yescharacteristic
168#117—2122no—
169#117—2122no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1061no—
172C2×C22⋊C4—531no—
173C2×C22⋊C4—531no—
174C2×C22⋊C4C2×#615711yescharacteristic
175C2×C22⋊C4C2×#615711yescharacteristic
176C2×C42—532no—
177C2×C42—1061no—
178C4⋊C8—2122no—
179C22⋊C8—1064no—
180C4×C8—2122no—
181#95—22no—
182#95—1062no—
183#95—1062no—
184#95—1062no—
185C24C22×#615711yescharacteristic
186C2×Q8—1064no—
187C2×Q8—1064no—
188C2×Q8—2122no—
189C2×Q8—2122no—
190C2×Q8—2128no—
191C2×Q8—2128no—
192C2×Q8—2122no—
193C2×Q8—2122no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—532no—
204C22×C4—532no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—2121no—
211C22×C4—2121no—
212C22×C4D53⋊C812yes—
213C22×C4C22⋊#615712yes—
214C2×C8—2124no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—2121no—
220C22⋊C4—2121no—
221C42—1062no—
222C42—1062no—
223C42—2122no—
224C23—21no—
225C23—21no—
226C23—42no—
227C23C2×C22⋊#615711yescharacteristic
228C23C2×D53⋊C811yescharacteristic
229C23D53⋊M1611yescharacteristic
230Q8—2128no—
231Q8—2128no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—532no—
242C2×C4—532no—
243C2×C4—532no—
244C2×C4—532no—
245C2×C4—1064no—
246C2×C4—1064no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—2122no—
252C2×C4—2121no—
253C2×C4—2122no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2122no—
258C8—4244no—
259C22—22no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22D53⋊(C22⋊C8)11yescharacteristic
267C22D53⋊#9911yescharacteristic
268C22D53⋊#10011yescharacteristic
269C4—42no—
270C4—42no—
271C4—81no—
272C4—534no—
273C4—1062no—
274C4—4241no—
275C4—2122no—
276C4—2122no—
277C2—21no—
278C2—21no—
279C2—41no—
280C2—41no—
281C2D53⋊#32311yescharacteristic
282C2#2029097812yes—
subgroupG/H
centerC22D53⋊#99
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC53×#381C4
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊Dic53C23
hypercenter#381#6157
hypocenterC53#11428
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
25. C22×Dic106C8semidirect
158. C53#11428semidirect

Maximal subgroups

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

subgroupindex
#202899472
#202896422
#202896412
#202773002
#1142853