Finite Group Databasebeta

← #27276727

Subgroups — #27276727

Order 13568.

1154 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#13133206C211yesmaximal, characteristic
2#20274639C211yesmaximal, characteristic, complemented
3#20273041C214yesmaximal, complemented
4C53×#1784C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic
6C2×#13133206—28no—
7C2×#13133206—22no—
8C2×#13133206—22no—
9C22×#11719921C412yes—
10C22×#11719921C2212yes—
11C22×C4×Dic53C411yescharacteristic
12C2×#13133136—22no—
13#16810587C2212yes—
14#16810346—28no—
15C23×Q8×C53C2211yescharacteristic
16C2×#101×C53—22no—
17C53×#407C2212yes—
18C24×D53C2×C412yes—
19C22×D212—24no—
20C22×D212—22no—
21C22×D212—44no—
22C23×Dic53C2×C412yes—
23C2×#11719917—22no—
24#13133206—216no—
25#13133206—216no—
26C2×#11719921—24no—
27C2×#11719921—24no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—24no—
31C2×#11719921—24no—
32C2×#11719921D418yes—
33C2×C4×Dic53—22no—
34C2×C4×Dic53—24no—
35#13133136—44no—
36C22⋊#88322—24no—
37C22×Q8×C53—24no—
38C22×Q8×C53—44no—
39C22×Q8×C53D412yes—
40C22×Q8×C53D412yes—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C2×M16×C53—22no—
45#101×C53—44no—
46C22⋊C8×C53—24no—
47C24×C53C22×C411yescharacteristic
48C23×D53—22no—
49C23×D53—22no—
50C23×D53—44no—
51C23×D53—44no—
52C23×D53—44no—
53C23×D53C22⋊C414yes—
54C2×Q8×C53—24no—
55C2×Q8×C53—24no—
56C2×Q8×C53—24no—
57C2×Q8×C53—24no—
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C2×Q8×C53—416no—
61C22×C4×C53—22no—
62C22×C4×C53—24no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D414yes—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—44no—
74C22×Dic53C22⋊C414yes—
75C2×D212—28no—
76C2×D212—416no—
77C2×D212—44no—
78C2×D212—44no—
79M16×C53—44no—
80C2×C8×C53—42no—
81#11719921—216no—
82#11719921—28no—
83#11719921—28no—
84C4×Dic53—28no—
85#11719917—44no—
86C2×#88322—42no—
87C23×C53—22no—
88C23×C53—24no—
89C23×C53C2×C22⋊C412yes—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—24no—
92C22×D53—28no—
93C22×D53—44no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53—44no—
97C22×D53—44no—
98C22×D53—44no—
99C22×D53—44no—
100C22×D53—44no—
101C22×D53—816no—
102Q8×C53—48no—
103Q8×C53—48no—
104C2×C4×C53—28no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53#9914yes—
115C2×C4×C53#12012yes—
116C2×C4×C53#12012yes—
117C2×Dic53—24no—
118C2×Dic53—44no—
119C2×Dic53—44no—
120C2×Dic53—44no—
121D212—416no—
122C8×C53—82no—
123#88322—82no—
124#8552—531nomaximal, Sylow
125C22×C53—28no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53#37911yescharacteristic
130C22×C53C2×#9912yes—
131C22×C53#41314yes—
132C2×D53—44no—
133C2×D53—44no—
134C2×D53—44no—
135C2×D53—88no—
136C2×D53—88no—
137C2×D53—88no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—24no—
141C4×C53—42no—
142Dic53—44no—
143C22×#124—531no—
144#1784—531no—
145#1784D5311yescharacteristic, complemented
146#1462—534no—
147C2×C53—22no—
148C2×C53—24no—
149C2×C53#178111yescharacteristic
150C2×C53#178512yes—
151D53—84no—
152C23×Q8C2×D5311yescharacteristic
153C23×D4—531no—
154C2×#124—1068no—
155C2×#124—1062no—
156C2×#124—1062no—
157C22×C22⋊C4—532no—
158C22×C22⋊C4—532no—
159C22×C42—531no—
160C2×#101—22no—
161C2×#101—1062no—
162#407—532no—
163#407C2×D5312yes—
164#323—1068no—
165C53#855211yescharacteristic, complemented, Sylow
166C25—532no—
167C22×Q8—24no—
168C22×Q8—44no—
169C22×Q8D21212yes—
170C22×Q8C22⋊D5312yes—
171C22×D4—1064no—
172C22×D4—1062no—
173C22×D4—2124no—
174C23×C4—532no—
175C23×C4C4×D5311yescharacteristic
176C23×C4C4×D5311yescharacteristic
177C23×C4C22×D5311yescharacteristic
178C2×M16—22no—
179C2×M16—1062no—
180#124—10616no—
181#124—10616no—
182C2×C22⋊C4—538no—
183C2×C22⋊C4—1064no—
184C2×C22⋊C4—1064no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1064no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1064no—
189C2×C42—1064no—
190C2×C42—1062no—
191#101—44no—
192#101—2124no—
193C22⋊C8—24no—
194C22⋊C8—1064no—
195C24—534no—
196C24—1062no—
197C24—1062no—
198C24—2124no—
199C24—2124no—
200C24—2124no—
201C24C2×C4×D5311yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—416no—
209C2×D4—1068no—
210C2×D4—21216no—
211C2×D4—2124no—
212C2×D4—2124no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—534no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—2124no—
222C22×C4#1171992112yes—
223C22×C4#1171992112yes—
224C22×C4C2×D21211yescharacteristic
225C22×C4D4×D5314yes—
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—44no—
228M16—2124no—
229C2×C8—42no—
230C2×C8—2122no—
231C22⋊C4—10616no—
232C22⋊C4—1068no—
233C22⋊C4—1068no—
234C42—1068no—
235C23—22no—
236C23—24no—
237C23—1068no—
238C23—1064no—
239C23—42416no—
240C23—2124no—
241C23—2124no—
242C23—2124no—
243C23—2124no—
244C23—2124no—
245C23—2124no—
246C23—2124no—
247C23—2124no—
248C23C22⋊C4×D5312yes—
249C23C2×#1171992111yescharacteristic
250Q8—48no—
251Q8—48no—
252D4—21216no—
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—1064no—
264C2×C4—2124no—
265C2×C4—2124no—
266C2×C4—2124no—
267C2×C4#1313313114yes—
268C2×C4#1313318512yes—
269C2×C4#1313318712yes—
270C8—82no—
271C8—4242no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—4248no—
277C22—4248no—
278C22—4248no—
279C22—2124no—
280C22—2124no—
281C22—2124no—
282C22#1681047811yescharacteristic
283C22C2×#1313313112yes—
284C22#1681060614yes—
285C4—22no—
286C4—22no—
287C4—24no—
288C4—42no—
289C4—2124no—
290C2—22no—
291C2—24no—
292C2—4244no—
293C2#2027462911yescharacteristic
294C2#2027464212yes—
subgroupG/H
centerC22#16810478
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1784C2
socleC22×C53#379
prime radicalC23×D212C4
cosocleC23×C4×C53C23
hypercenter#1784D53
hypocenterC53#8552
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #20274639C2semidirect
3. #20273041C2semidirect
4. C53×#1784C2semidirect
145. #1784D53semidirect
165. C53#8552semidirect

Maximal subgroups

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

subgroupindex
C22×#131332062
#202746392
#202730412
C53×#17842
#855253