Finite Group Databasebeta

← #27293315

Subgroups — #27293315

Order 13568.

454 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#16810688C211yesmaximal, characteristic
2C22⋊C16×D53C212yesmaximal, complemented
3#20272655C211yesmaximal, characteristic, complemented
4#20272638C212yesmaximal, complemented
5C53×#1322C211yesmaximal, characteristic, complemented
6C22×C8×D53C411yescharacteristic
7C22×C8×D53C2211yescharacteristic
8C2×C4×D212C411yescharacteristic
9C2×C16×D53—22no—
10#16810688—22no—
11#16810688—22no—
12C2×#13133129C411yescharacteristic
13C2×#13133129C2211yescharacteristic
14C2×C4×#88322C411yescharacteristic
15C22⋊#11719908C2212yes—
16#16810402—22no—
17C2×C4⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×D212C811yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22C22×#88322C2×C411yescharacteristic
23C22×#88322C2×C411yescharacteristic
24C2×C8×D53—21no—
25C2×C8×D53—21no—
26C2×C8×D53—42no—
27C2×C8×D53—42no—
28C2×C8×D53D412yes—
29C2×C8×D53D412yes—
30C4×D212—22no—
31C4×D212—22no—
32C2×C4⋊Dic53C811yescharacteristic
33C2×#11719921C811yescharacteristic
34C2×#11719921C811yescharacteristic
35C2×#11719908—22no—
36C16×D53—44no—
37#13133129—22no—
38#13133129—22no—
39C4×#88322—21no—
40C4×#88322—21no—
41C22×C8×C53C2×C411yescharacteristic
42C22×C8×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C16×C53—22no—
45C4⋊C8×C53—22no—
46C23×D53C2×C811yescharacteristic
47C23×D53C2×C811yescharacteristic
48C22×C4×C53C2×C811yescharacteristic
49C22×C4×C53C2×C811yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53C2×C811yescharacteristic
52C22×Dic53C2×C811yescharacteristic
53C2×D212—22no—
54C2×D212—22no—
55C2×D212C1614yescomplemented
56C2×C4×D53—21no—
57C2×C4×D53—21no—
58C2×C4×D53—41no—
59C2×C4×D53—41no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53C22⋊C412yes—
63C2×C4×D53M1612yes—
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C2×C8×C53C2×D412yes—
68C42×C53—21no—
69C42×C53—21no—
70#11719921—22no—
71#11719921—22no—
72C4⋊Dic53—22no—
73C2×#88322—22no—
74C2×#88322—21no—
75C2×#88322—41no—
76C2×#88322C22⋊C411yescharacteristic
77C2×#88322C22⋊C411yescharacteristic
78C8×D53—24no—
79C8×D53—42no—
80C8×D53—42no—
81C8×D53—42no—
82C16×C53—42no—
83#11719908—42no—
84C23×C53C22×C811yescharacteristic
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—41no—
88C22×D53—41no—
89C22×D53—41no—
90C22×D53—41no—
91C22×D53C22⋊C812yes—
92C22×D53C2×C1612yes—
93C22×D53C2×C1612yescomplemented
94C22×D53M3212yes—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53C2×C1612yescomplemented
102C2×C4×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53C22⋊C812yes—
108D212—28no—
109C4×D53—22no—
110C4×D53—22no—
111C4×D53—42no—
112C4×D53—42no—
113C4×D53—41no—
114C4×D53—41no—
115C8×C53—22no—
116C8×C53—41no—
117#88322—21no—
118#88322—41no—
119#88322#9912yes—
120#11465—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53C2×C22⋊C811yescharacteristic
124C22×C53C22×C1611yescharacteristic
125C22×C53C2×M3211yescharacteristic
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—42no—
135C2×D53C22⋊C1614yes—
136C4×C53—22no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53C2×#9911yescharacteristic
140C4×C53#41311yescharacteristic
141Dic53—22no—
142Dic53—41no—
143C2×C8×D4—531no—
144C2×C22⋊C16—532no—
145#1322—531no—
146#1322D5311yescharacteristic, complemented
147#1314—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53#145911yescharacteristic
151C2×C53C2×C22⋊C1611yescharacteristic
152C2×C53#211511yescharacteristic
153D53—24no—
154D53—42no—
155C23×C8—531no—
156C23×C8—531no—
157C2×C4×D4—531no—
158C22×C16—1062no—
159C8×D4—1062no—
160C8×D4—1062no—
161C2×C4⋊C8C2×D5311yescharacteristic
162C2×C22⋊C8—531no—
163C2×C22⋊C8—531no—
164C2×C4×C8—531no—
165C22⋊C16—532no—
166C22⋊C16—1062no—
167C22⋊C16C2×D5312yes—
168C53#1146511yescharacteristic, complemented, Sylow
169C22×D4—531no—
170C23×C4—531no—
171C23×C4—531no—
172C22×C8—531no—
173C22×C8—532no—
174C22×C8—532no—
175C22×C8—531no—
176C22×C8—1061no—
177C22×C8—1061no—
178C22×C8—2122no—
179C22×C8—2122no—
180C22×C8C4×D5311yescharacteristic
181C22×C8C22×D5311yescharacteristic
182C4×D4—1062no—
183C4×D4—1062no—
184C2×C4⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—531no—
187C2×C42C4×D5311yescharacteristic
188C2×C16—22no—
189C2×C16—1062no—
190C2×C16—2124no—
191C4⋊C8—22no—
192C22⋊C8—1062no—
193C22⋊C8—1062no—
194C4×C8—1061no—
195C4×C8—1061no—
196C24—531no—
197C24—531no—
198C2×D4—534no—
199C2×D4—1062no—
200C2×D4—1062no—
201C22×C4—531no—
202C22×C4—532no—
203C22×C4—532no—
204C22×C4—531no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4—2121no—
210C22×C4—2121no—
211C22×C4C8×D5311yescharacteristic
212C22×C4C8×D5311yescharacteristic
213C22×C4C2×C4×D5311yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—531no—
218C2×C8—531no—
219C2×C8—1064no—
220C2×C8—1062no—
221C2×C8—1061no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2121no—
226C2×C8D4×D5312yes—
227C4⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C42—21no—
231C42—21no—
232C16—42no—
233C16—2122no—
234C23—532no—
235C23—532no—
236C23—532no—
237C23—532no—
238C23—1061no—
239C23—1061no—
240C23—2121no—
241C23—2121no—
242C23—2121no—
243C23—2121no—
244C23C2×C8×D5311yescharacteristic
245D4—1068no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—532no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1061no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4C16×D5312yescomplemented
263C2×C4C22⋊C4×D5311yescharacteristic
264C2×C4M16×D5311yescharacteristic
265C8—22no—
266C8—41no—
267C8—532no—
268C8—1061no—
269C8—2121no—
270C22—21no—
271C22—21no—
272C22—534no—
273C22—1062no—
274C22—1062no—
275C22—1062no—
276C22—2122no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2121no—
281C22—2121no—
282C22C22⋊C8×D5311yescharacteristic
283C22C2×C16×D5311yescharacteristic
284C22M32×D5311yescharacteristic
285C4—22no—
286C4—21no—
287C4—41no—
288C4—1062no—
289C4—2121no—
290C4#99×D5311yescharacteristic
291C4#1681060611yescharacteristic
292C2—21no—
293C2—21no—
294C2—1064no—
295C2—2122no—
296C2#2027303211yescharacteristic
297C2C22⋊C16×D5311yescharacteristic
298C2#2027633411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC22×C53C22×C16
FrattiniC22×C8C22×D53
FittingC53×#1322C2
socleC22×C53C2×C22⋊C8
prime radicalC22×D212C8
cosocleC22×C8×C53C23
hypercenter#1322D53
hypocenterC53#11465
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D53C2semidirect
3. #20272655C2semidirect
4. #20272638C2semidirect
5. C53×#1322C2semidirect
55. C2×D212C16semidirect
93. C22×D53C2×C16semidirect
101. C2×C4×C53C2×C16semidirect
146. #1322D53semidirect
168. C53#11465semidirect
262. C2×C4C16×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106882
C22⋊C16×D532
#202726552
#202726382
C53×#13222
#1146553