Finite Group Databasebeta

← #28302210

Subgroups — #28302210

Order 13568.

1482 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 164 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#20290605C212yesmaximal, complemented
2C2×#16811467C214yesmaximal, complemented
3C22×#13133179C211yesmaximal, characteristic
4C2×C22⋊#88324C2214yes—
5#16811489—24no—
6#16811467—28no—
7C23×D212C411yescharacteristic
8C23×C4×D53C412yes—
9C2×#13133179—28no—
10C2×#13133179—24no—
11C22×#11719921C2212yes—
12C22×C4×Dic53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16C24×D53C2×C412yes—
17C22×D212—24no—
18C22×D212—24no—
19C22×D212—22no—
20C22×D212—22no—
21C22×D212C814yescomplemented
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2311yescharacteristic
24C22×C4×D53—24no—
25C22×C4×D53—24no—
26C22×C4×D53—24no—
27C22×C4×D53—48no—
28#13133179—232no—
29C2×#11719921—24no—
30C2×#11719921—24no—
31C2×#11719921D418yes—
32C2×C4×Dic53—24no—
33C2×C4×Dic53D414yes—
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53C2×C412yes—
36C2×C4⋊C4×C53—24no—
37C2×C4⋊C4×C53—22no—
38C24×C53C22×C411yescharacteristic
39C23×D53—22no—
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—44no—
44C23×D53—44no—
45C23×D53C22⋊C414yes—
46C23×D53C2×C814yes—
47C23×D53M1614yes—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—44no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22⋊C414yes—
55C22×C4×C53C2×C812yes—
56C22×C4×C53M1612yes—
57C22×Dic53—22no—
58C22×Dic53—24no—
59C22×Dic53—42no—
60C22×Dic53C22⋊C412yes—
61C22×Dic53C2×D412yes—
62C22×Dic53C2×D414yes—
63C2×D212—28no—
64C2×D212—28no—
65C2×D212—216no—
66C2×D212—216no—
67C2×D212—28no—
68C2×C4×D53—216no—
69C2×C4×D53—416no—
70C2×C4×D53—416no—
71C2×C4×D53—48no—
72C2×C4×D53—48no—
73C2×#88324—48no—
74C2×#88324—44no—
75C4⋊C4×C53—28no—
76#11719921—216no—
77C4×Dic53—28no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53C2×C22⋊C412yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C23×C53C22×C811yescharacteristic
83C23×C53C2×M1612yes—
84C23×C53C2×M1611yescharacteristic
85C22×D53—24no—
86C22×D53—24no—
87C22×D53—24no—
88C22×D53—24no—
89C22×D53—24no—
90C22×D53—24no—
91C22×D53—44no—
92C22×D53—48no—
93C22×D53—48no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53—44no—
97C22×D53—44no—
98C22×D53—44no—
99C22×D53—48no—
100C22×D53—48no—
101C22×D53C22⋊C818yes—
102C2×C4×C53—22no—
103C2×C4×C53—24no—
104C2×C4×C53—22no—
105C2×C4×C53—24no—
106C2×C4×C53—24no—
107C2×C4×C53—44no—
108C2×C4×C53—44no—
109C2×C4×C53—44no—
110C2×C4×C53C22⋊C814yes—
111C2×Dic53—28no—
112C2×Dic53—22no—
113C2×Dic53—24no—
114C2×Dic53—22no—
115C2×Dic53—44no—
116C2×Dic53#10014yes—
117C2×Dic53#12014yes—
118D212—232no—
119C4×D53—432no—
120#88324—84no—
121#8302—531nomaximal, Sylow
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53#37911yescharacteristic
128C22×C53C2×C22⋊C812yes—
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53#40712yes—
131C22×C53#40711yescharacteristic
132C22×C53#41012yes—
133C22×C53C2×#10012yes—
134C2×D53—28no—
135C2×D53—48no—
136C2×D53—48no—
137C2×D53—48no—
138C2×D53—48no—
139C2×D53—48no—
140C2×D53—44no—
141C2×D53—44no—
142C2×D53—48no—
143C4×C53—24no—
144C4×C53—44no—
145Dic53—24no—
146Dic53—24no—
147C22×C4×D4—531no—
148#1817—532no—
149C2×#323—534no—
150C2×C53—22no—
151C2×C53—22no—
152C2×C53#146012yes—
153C2×C53#146612yes—
154C2×C53#177911yescharacteristic
155C2×C53#178211yescharacteristic
156C2×C53#178311yescharacteristic
157D53—48no—
158C23×D4—531no—
159C24×C4—532no—
160C2×C4×D4—1064no—
161C2×C4×D4—1068no—
162C22×C4⋊C4#615711yescharacteristic
163C22×C22⋊C4—532no—
164C22×C42—531no—
165C2×C22⋊C8—534no—
166#336—1064no—
167#323—1068no—
168C53#830211yescharacteristic, complemented, Sylow
169C25—532no—
170C22×D4—534no—
171C22×D4—1064no—
172C22×D4—1064no—
173C22×D4—1062no—
174C22×D4—1062no—
175C23×C4—531no—
176C23×C4—531no—
177C23×C4—1064no—
178C23×C4—1064no—
179C23×C4—1064no—
180C23×C4—2128no—
181C23×C4C2×#615712yes—
182C23×C4C2×#615711yescharacteristic
183C22×C8—1064no—
184C4×D4—10632no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C22⋊C4—538no—
188C2×C22⋊C4—1064no—
189C2×C22⋊C4—1064no—
190C2×C42—534no—
191C2×C42—1064no—
192C22⋊C8—1068no—
193C24—534no—
194C24—534no—
195C24—534no—
196C24—1062no—
197C24—1062no—
198C24—1062no—
199C24—1062no—
200C24—2124no—
201C24—2124no—
202C24C22×#615711yescharacteristic
203C2×D4—10616no—
204C2×D4—1068no—
205C2×D4—10616no—
206C2×D4—1068no—
207C2×D4—1068no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—44no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4—534no—
216C22×C4—10616no—
217C22×C4—1064no—
218C22×C4—1062no—
219C22×C4—2128no—
220C22×C4—21216no—
221C22×C4—2128no—
222C22×C4—21216no—
223C22×C4—2122no—
224C22×C4D53⋊C812yes—
225C22×C4#1171993612yes—
226C22×C4C22⋊#615712yes—
227C22×C4C22⋊#615714yes—
228C2×C8—2124no—
229C2×C8—2128no—
230C4⋊C4—28no—
231C22⋊C4—10616no—
232C42—1068no—
233C23—22no—
234C23—22no—
235C23—538no—
236C23—1064no—
237C23—1064no—
238C23—1064no—
239C23—1064no—
240C23—1064no—
241C23—1064no—
242C23—2124no—
243C23—2128no—
244C23—2128no—
245C23—2124no—
246C23—2124no—
247C23—2128no—
248C23—2128no—
249C23—2124no—
250C23—2124no—
251C23—2124no—
252C23C2×C22⋊#615712yes—
253C23C2×C22⋊#615711yescharacteristic
254C23C2×D53⋊C811yescharacteristic
255C23D53⋊M1612yes—
256C23C2×#1171993611yescharacteristic
257D4—10632no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—534no—
267C2×C4—534no—
268C2×C4—1068no—
269C2×C4—1062no—
270C2×C4—1064no—
271C2×C4—1062no—
272C2×C4—21232no—
273C2×C4—2124no—
274C2×C4#1313327414yes—
275C8—4244no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—1068no—
282C22—2128no—
283C22—2128no—
284C22—2128no—
285C22—2128no—
286C22—2128no—
287C22—2128no—
288C22—2124no—
289C22—2124no—
290C22#1681156411yescharacteristic
291C22D53⋊(C22⋊C8)12yes—
292C22C2×#1313327411yescharacteristic
293C22#1681159412yes—
294C22#1681159511yescharacteristic
295C22#1681160412yes—
296C22D53⋊#10012yes—
297C4—24no—
298C4—44no—
299C4—1064no—
300C4—1064no—
301C2—22no—
302C2—22no—
303C2—2128no—
304C2#2028995612yes—
305C2#2028997612yes—
306C2#2029053811yescharacteristic
307C2#2029054611yescharacteristic
308C2#2029054811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D212C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8302
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290605C2semidirect
2. C2×#16811467C2semidirect
21. C22×D212C8semidirect
168. C53#8302semidirect

Maximal subgroups

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

subgroupindex
#202906052
C2×#168114672
C22×#131331792
#830253