Finite Group Databasebeta

← #27300355

Subgroups — #27300355

Order 13568.

642 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274633C212yesmaximal
2#20272840C214yesmaximal, complemented
3C53×#2311C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810486—22no—
7C22×#41×C53C2211yescharacteristic
8C2×#117×C53C411yescharacteristic, complemented
9C22×C4⋊C4×C53C411yescharacteristic, complemented
10C53×#391C412yescomplemented
11C53×#390C2212yes—
12C53×#386—22no—
13C53×#380—22no—
14C2×C22⋊Dic53—24no—
15#13133132—28no—
16#13133132—44no—
17#13133124—44no—
18C2×#41×C53—24no—
19C2×#41×C53—42no—
20C2×#41×C53—42no—
21C22×Q8×C53C2×C411yescharacteristic
22C22×D4×C53D412yes—
23C22×D4×C53C2311yescharacteristic
24C23×C4×C53—21no—
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53D412yes—
27#117×C53—42no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—42no—
31C2×C4⋊C4×C53C2×C412yes—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—24no—
34C2×C42×C53C2×C412yes—
35#95×C53—24no—
36C24×C53C22⋊C411yescharacteristic
37C24×C53C2×D412yes—
38#41×C53—416no—
39C2×Q8×C53—22no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—42no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53#4114yes—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—24no—
52C22×C4×C53—24no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—82no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—44no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—84no—
67C42×C53—42no—
68C22⋊Dic53—44no—
69C22⋊Dic53—44no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53—42no—
77C23×C53—42no—
78C23×C53#9912yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C23×C53#12314yes—
81Q8×C53—44no—
82D4×C53—44no—
83D4×C53—48no—
84C2×C4×C53—28no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—48no—
93C2×C4×C53—42no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—84no—
99C2×C4×C53#9912yes—
100C2×C4×C53#11712yes—
101C2×Dic53—84no—
102C2×Dic53—84no—
103C2×Dic53—84no—
104#12859—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—24no—
108C22×C53—22no—
109C22×C53—42no—
110C22×C53—44no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—41no—
116C22×C53—82no—
117C22×C53#38111yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53C2×#9911yescharacteristic
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—82no—
124C4×C53—82no—
125Dic53—164no—
126#2311D5311yescharacteristic
127#1782—532no—
128#1393—534no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—44no—
132C2×C53—42no—
133C2×C53#181911yescharacteristic
134C2×C53#212811yescharacteristic
135C2×C53#212911yescharacteristic
136C22×#41C2×D5311yescharacteristic
137C2×#117Dic5311yescharacteristic, complemented
138C22×C4⋊C4Dic5311yescharacteristic, complemented
139C2×#99—534no—
140C2×#99—1062no—
141#391Dic5312yescomplemented
142#390C2×D5312yes—
143#386—22no—
144#381—1062no—
145#380—22no—
146C53#1285911yescharacteristic, complemented, Sylow
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8C2×Dic5311yescharacteristic
151C22×D4C22⋊D5312yes—
152C22×D4C22×D5311yescharacteristic
153C23×C4—21no—
154C23×C4C2×Dic5311yescharacteristic
155C23×C4C22⋊D5312yes—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C2×Dic5312yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—1064no—
164C2×C42C2×Dic5312yes—
165#99—1068no—
166#99—2124no—
167#95—24no—
168#95—2124no—
169C24C22⋊Dic5311yescharacteristic
170C24C2×C22⋊D5312yes—
171#41—416no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#1171994514yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—2124no—
192C22×C4C22⋊Dic5312yes—
193C22×C4C22⋊Dic5311yescharacteristic
194C22×C4C22×Dic5311yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—84no—
200C22⋊C4—2124no—
201C22⋊C4—2124no—
202C42—42no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23#1313313212yes—
212C23C2×C22⋊Dic5311yescharacteristic
213C23#1313320014yes—
214Q8—44no—
215D4—44no—
216D4—48no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4—4244no—
233C2×C4—4244no—
234C2×C4—4244no—
235C2×C4#1313313212yes—
236C2×C4#1313317612yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—82no—
249C22#1681048611yescharacteristic
250C22C2×#1313313211yescharacteristic
251C22C2×#1313313211yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—8484no—
258C2—21no—
259C2—21no—
260C2—44no—
261C2—42no—
262C2#2027478911yescharacteristic
263C2#2027638711yescharacteristic
264C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2311C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#2311D53
hypocenterC53#12859
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #20272840C2semidirect
8. C2×#117×C53C4semidirect
9. C22×C4⋊C4×C53C4semidirect
10. C53×#391C4semidirect
137. C2×#117Dic53semidirect
138. C22×C4⋊C4Dic53semidirect
141. #391Dic53semidirect
146. C53#12859semidirect

Maximal subgroups

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

subgroupindex
#202746332
#202728402
C53×#23112
#1285953