Finite Group Databasebeta

← #27293483

Subgroups — #27293483

Order 13568.

670 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 99 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#20273049C212yesmaximal, complemented
2#20272647C214yesmaximal, complemented
3C53×#2309C211yesmaximal, characteristic
4#16810680—22no—
5#16810349C2214yescomplemented
6#16810347—22no—
7C23×D4×C53C411yescharacteristic
8C2×#117×C53C411yescharacteristic
9C2×#117×C53C2211yescharacteristic
10C53×#390—22no—
11C53×#390C412yes—
12C53×#390C2212yes—
13C53×#379—22no—
14C4⋊#88322—42no—
15C22⋊#88322—24no—
16#13133128—42no—
17C25×C53—21no—
18C22×D4×C53—24no—
19C22×D4×C53—21no—
20C22×D4×C53—42no—
21C22×D4×C53—42no—
22C22×D4×C53C2×C412yes—
23C23×C4×C53C2×C411yescharacteristic
24#117×C53—42no—
25#117×C53—42no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53C2×C411yescharacteristic
28C2×C4⋊C4×C53C2311yescharacteristic
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53—24no—
31C2×C22⋊C4×C53—24no—
32C2×C22⋊C4×C53D412yes—
33C2×C42×C53C2×C412yes—
34C2×C42×C53D412yes—
35C24×C53—24no—
36C24×C53—22no—
37C24×C53—42no—
38C24×C53—42no—
39C24×C53—82no—
40C24×C53—82no—
41C24×C53C22⋊C411yescharacteristic
42C2×D4×C53—24no—
43C2×D4×C53—22no—
44C2×D4×C53—416no—
45C2×D4×C53—42no—
46C2×D4×C53—42no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53M1614yes—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D412yes—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C22⋊C4×C53—42no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—84no—
64C22⋊C4×C53—84no—
65C42×C53—22no—
66C42×C53—22no—
67C42×C53—42no—
68C2×#88322—44no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—24no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53—48no—
75C23×C53—44no—
76C23×C53—44no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—88no—
80C23×C53—88no—
81C23×C53—84no—
82C23×C53—84no—
83C23×C53#10012yes—
84C23×C53C2×C22⋊C411yescharacteristic
85D4×C53—44no—
86D4×C53—48no—
87D4×C53—44no—
88C2×C4×C53—28no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53#9912yes—
101C2×C4×C53C2×M1612yes—
102#88322—84no—
103#11509—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—22no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—48no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—44no—
113C22×C53—84no—
114C22×C53—816no—
115C22×C53—82no—
116C22×C53—82no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53C2×#10011yescharacteristic
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—42no—
124C4×C53—82no—
125#2309D5311yescharacteristic
126#1464—532no—
127#1318—534no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—42no—
131C2×C53—84no—
132C2×C53#146211yescharacteristic
133C2×C53#212811yescharacteristic
134C2×C53#213411yescharacteristic
135C23×D4Dic5311yescharacteristic
136C2×#117Dic5311yescharacteristic
137C2×#117C2×D5311yescharacteristic
138#432—1062no—
139#390—22no—
140#390Dic5312yes—
141#390C2×D5312yes—
142#379—22no—
143#324—534no—
144#323—1062no—
145C53#1150911yescharacteristic, complemented, Sylow
146C25—21no—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C2×Dic5312yes—
152C23×C4C2×Dic5311yescharacteristic
153#117—42no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C2×Dic5311yescharacteristic
157C2×C4⋊C4C22×D5311yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊D5312yes—
162C2×C42C2×Dic5312yes—
163C2×C42C22⋊D5312yes—
164C4⋊C8—2122no—
165C22⋊C8—1064no—
166#97—2122no—
167C24—24no—
168C24—22no—
169C24—42no—
170C24—42no—
171C24—82no—
172C24—82no—
173C24C22⋊Dic5311yescharacteristic
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—416no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4#1171991714yes—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4C22⋊Dic5312yes—
187C22×C4C22⋊Dic5311yescharacteristic
188C22×C4C22×Dic5311yescharacteristic
189C22×C4C2×C22⋊D5312yes—
190C2×C8—2124no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C42—22no—
199C42—22no—
200C42—42no—
201C23—21no—
202C23—21no—
203C23—24no—
204C23—41no—
205C23—41no—
206C23—44no—
207C23—44no—
208C23—48no—
209C23—42no—
210C23—42no—
211C23—88no—
212C23—88no—
213C23—84no—
214C23—84no—
215C23#1313313412yes—
216C23C2×C22⋊Dic5311yescharacteristic
217D4—44no—
218D4—48no—
219D4—44no—
220C2×C4—28no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4#1313313212yes—
233C2×C4C2×#1171991712yes—
234C8—4244no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—42no—
239C22—42no—
240C22—48no—
241C22—41no—
242C22—41no—
243C22—44no—
244C22—84no—
245C22—816no—
246C22—82no—
247C22—82no—
248C22#1681058711yescharacteristic
249C22C2×#1313313211yescharacteristic
250C22C2×#1313313411yescharacteristic
251C4—22no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—82no—
256C2—21no—
257C2—21no—
258C2—42no—
259C2—84no—
260C2#2027304211yescharacteristic
261C2#2027638711yescharacteristic
262C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2309C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2309D53
hypocenterC53#11509
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. #20273049C2semidirect
2. #20272647C2semidirect
5. #16810349C22semidirect
145. C53#11509semidirect

Maximal subgroups

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

subgroupindex
#202730492
#202726472
C53×#23092
#1150953