Finite Group Databasebeta

← #27293410

Subgroups — #27293410

Order 13568.

418 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16810686C211yesmaximal, characteristic
2#20276515C212yesmaximal, complemented
3#20272639C211yesmaximal, characteristic, complemented
4#20272638C212yesmaximal, complemented
5C53×#1314C211yesmaximal, characteristic, complemented
6C23×#88322C411yescharacteristic
7C22×C8×D53C411yescharacteristic
8C2×C4×C22⋊D53C411yescharacteristic
9#16810686—22no—
10#16810686—22no—
11C2×#13133146C411yescharacteristic
12C2×#13133129C2211yescharacteristic
13C2×C8×Dic53C2211yescharacteristic
14#16810441—22no—
15C22⋊#11719908C2212yes—
16#16810398—22no—
17C2×C22⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×C22⋊D53C811yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21C22×#88322—21no—
22C22×#88322—21no—
23C22×#88322—42no—
24C22×#88322C2×C411yescharacteristic
25C22×#88322C2×C411yescharacteristic
26C2×C8×D53—21no—
27C2×C8×D53—21no—
28C2×C8×D53—42no—
29C4×C22⋊D53—22no—
30C4×C22⋊D53—22no—
31C2×#11719919C811yescharacteristic
32C2×C22⋊Dic53C811yescharacteristic
33C2×#11719921C811yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C2×#11719908—22no—
36#13133146—22no—
37#13133129—22no—
38C8×Dic53D412yes—
39C8×Dic53D412yes—
40C23×C4×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42C22×C8×C53C2311yescharacteristic
43C2×C16×C53—22no—
44C22⋊C8×C53—22no—
45C24×C53C2×C811yescharacteristic
46C23×D53C2×C811yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53C2×C811yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C22×Dic53C2×C811yescharacteristic
57C22×Dic53C2×C811yescharacteristic
58C22×Dic53C2×C811yescharacteristic
59C2×C4×D53—21no—
60C2×C4×D53—21no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×C8×C53—21no—
64C2×C8×C53—41no—
65C2×C8×C53—41no—
66C2×C8×C53C2×D412yes—
67C22⋊Dic53—22no—
68#11719921—22no—
69#11719919—22no—
70C4×Dic53C22⋊C412yes—
71C4×Dic53M1612yes—
72C2×#88322—22no—
73C2×#88322—21no—
74C2×#88322—42no—
75C2×#88322—42no—
76C2×#88322—41no—
77C2×#88322C22⋊C411yescharacteristic
78C2×#88322C22⋊C411yescharacteristic
79C8×D53—42no—
80C8×D53—42no—
81C16×C53—42no—
82#11719908—42no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53M3212yes—
88C23×C53C22×C811yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53M3212yes—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C22⋊D53—28no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53C22⋊C812yes—
110C2×Dic53C22⋊C812yes—
111C4×D53—42no—
112C4×D53—41no—
113C4×D53—41no—
114C8×C53—22no—
115C8×C53—41no—
116#88322—21no—
117#88322—41no—
118#88322#10012yes—
119#11488—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—22no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53C2×C22⋊C811yescharacteristic
128C22×C53C2×M3211yescharacteristic
129C22×C53C2×M3211yescharacteristic
130C2×D53—22no—
131C2×D53—42no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—42no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53#41011yescharacteristic
138C4×C53C2×#10011yescharacteristic
139Dic53—22no—
140Dic53—22no—
141Dic53—41no—
142C2×C8×D4—531no—
143#2163—532no—
144#1314—531no—
145#1314—532no—
146#1314D5311yescharacteristic, complemented
147C2×C53—21no—
148C2×C53—21no—
149C2×C53—42no—
150C2×C53#146011yescharacteristic
151C2×C53#211211yescharacteristic
152C2×C53#211511yescharacteristic
153D53—42no—
154C23×C8—531no—
155C23×C8—531no—
156C2×C4×D4—531no—
157C8×D4—1062no—
158C8×D4—1062no—
159C2×C4⋊C8—531no—
160C2×C22⋊C8—531no—
161C2×C22⋊C8C2×D5311yescharacteristic
162C2×C4×C8—531no—
163C4⋊C16—1062no—
164C22⋊C16—532no—
165C22⋊C16C2×D5312yes—
166#346—1062no—
167C53#1148811yescharacteristic, complemented, Sylow
168C22×D4—531no—
169C23×C4—531no—
170C23×C4C4×D5311yescharacteristic
171C22×C8—531no—
172C22×C8—531no—
173C22×C8—1061no—
174C22×C8—1061no—
175C22×C8—1061no—
176C22×C8—1061no—
177C22×C8—2122no—
178C22×C8—2122no—
179C22×C8C4×D5311yescharacteristic
180C22×C8C22×D5311yescharacteristic
181C4×D4—1062no—
182C4×D4—1062no—
183C2×C4⋊C4—531no—
184C2×C22⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C42—531no—
187C2×C16—22no—
188C2×C16—1062no—
189C4⋊C8—1062no—
190C22⋊C8—22no—
191C22⋊C8—1062no—
192C4×C8—532no—
193C4×C8—532no—
194C24—531no—
195C24C8×D5311yescharacteristic
196C2×D4—1062no—
197C2×D4—1062no—
198C2×D4—1062no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4—2121no—
209C22×C4—2121no—
210C22×C4C8×D5311yescharacteristic
211C22×C4C2×C4×D5311yescharacteristic
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—531no—
216C2×C8—531no—
217C2×C8—1062no—
218C2×C8—1061no—
219C2×C8—2122no—
220C2×C8—2122no—
221C2×C8—2122no—
222C2×C8—2122no—
223C2×C8—2121no—
224C2×C8D4×D5312yes—
225C4⋊C4—1062no—
226C22⋊C4—1062no—
227C22⋊C4—1062no—
228C42—532no—
229C42—532no—
230C16—42no—
231C16—2122no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—532no—
237C23—1061no—
238C23—1061no—
239C23—2121no—
240C23—2121no—
241C23#1313315612yes—
242C23C2×C8×D5311yescharacteristic
243D4—1068no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—532no—
252C2×C4—532no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—2122no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4C22⋊C4×D5311yescharacteristic
262C2×C4M16×D5311yescharacteristic
263C8—22no—
264C8—41no—
265C8—532no—
266C8—1061no—
267C8—2121no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—1062no—
276C22—2122no—
277C22—2122no—
278C22—2121no—
279C22—2121no—
280C22C22⋊C8×D5311yescharacteristic
281C22M32×D5311yescharacteristic
282C22C2×#1313315611yescharacteristic
283C4—21no—
284C4—41no—
285C4—1062no—
286C4—1062no—
287C4—2121no—
288C4#1681059511yescharacteristic
289C4#100×D5311yescharacteristic
290C2—21no—
291C2—21no—
292C2—42no—
293C2—2122no—
294C2#2027303411yescharacteristic
295C2#2027632411yescharacteristic
296C2#2027633411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1314C2
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×C8×C53C23
hypercenter#1314D53
hypocenterC53#11488
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20276515C2semidirect
3. #20272639C2semidirect
4. #20272638C2semidirect
5. C53×#1314C2semidirect
146. #1314D53semidirect
167. C53#11488semidirect

Maximal subgroups

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

subgroupindex
C2×#168106862
#202765152
#202726392
#202726382
C53×#13142
#1148853