Finite Group Databasebeta

← #27300352

Subgroups — #27300352

Order 13568.

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

normal subgroups only

This group has 786 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#20274630C212yesmaximal
2#20272840C214yesmaximal, complemented
3C53×#2309C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810479—22no—
7C23×D4×C53C2211yescharacteristic
8C2×#117×C53C411yescharacteristic, complemented
9C2×#117×C53C411yescharacteristic, complemented
10C53×#390—22no—
11C53×#390C412yescomplemented
12C53×#390C2212yes—
13C53×#379—22no—
14C2×C22⋊Dic53—24no—
15C2×C22⋊Dic53—44no—
16#13133132—28no—
17#13133132—44no—
18C25×C53D412yes—
19C22×D4×C53—24no—
20C22×D4×C53—42no—
21C22×D4×C53—42no—
22C22×D4×C53C2×C411yescharacteristic
23C22×D4×C53D412yes—
24C22×D4×C53C2311yescharacteristic
25C23×C4×C53C2×C411yescharacteristic
26#117×C53—42no—
27#117×C53—42no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53C2×C412yes—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—21no—
32C2×C22⋊C4×C53—24no—
33C2×C22⋊C4×C53—24no—
34C2×C42×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53—24no—
37C24×C53—24no—
38C24×C53—42no—
39C24×C53—84no—
40C24×C53C22⋊C412yes—
41C24×C53C22⋊C411yescharacteristic
42C24×C53C2×D412yes—
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53—42no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—416no—
52C2×D4×C53#4114yes—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—44no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—84no—
66C22⋊C4×C53—84no—
67C42×C53—42no—
68C42×C53—42no—
69C22⋊Dic53—44no—
70C22⋊Dic53—44no—
71C22⋊Dic53—88no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—42no—
80C23×C53—48no—
81C23×C53—44no—
82C23×C53—44no—
83C23×C53—48no—
84C23×C53—84no—
85C23×C53—88no—
86C23×C53—88no—
87C23×C53#9912yes—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53#12314yes—
90D4×C53—44no—
91D4×C53—48no—
92D4×C53—44no—
93C2×C4×C53—28no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—44no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53#9912yes—
104C2×C4×C53#11712yes—
105C2×Dic53—84no—
106C2×Dic53—88no—
107#12858—531nomaximal, Sylow
108C22×C53—22no—
109C22×C53—24no—
110C22×C53—22no—
111C22×C53—24no—
112C22×C53—44no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—44no—
116C22×C53—44no—
117C22×C53—42no—
118C22×C53—44no—
119C22×C53—82no—
120C22×C53—82no—
121C22×C53—816no—
122C22×C53#38111yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C4×C53—22no—
126C4×C53—22no—
127C4×C53—82no—
128C4×C53—82no—
129Dic53—164no—
130#2309D5311yescharacteristic
131#1781—532no—
132#1393—534no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—44no—
136C2×C53—42no—
137C2×C53—82no—
138C2×C53#181911yescharacteristic
139C2×C53#212712yes—
140C23×D4C2×D5311yescharacteristic
141C2×#117Dic5311yescharacteristic, complemented
142C2×#117Dic5311yescharacteristic, complemented
143C2×#99—534no—
144C2×#99—1062no—
145#390—22no—
146#390Dic5312yescomplemented
147#390C2×D5312yes—
148#379—22no—
149#379—1062no—
150C53#1285811yescharacteristic, complemented, Sylow
151C25C22⋊D5312yes—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C2×Dic5311yescharacteristic
156C22×D4C22⋊D5312yes—
157C22×D4C22×D5311yescharacteristic
158C23×C4C2×Dic5311yescharacteristic
159#117—42no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic5312yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—1064no—
168C2×C22⋊C4—2124no—
169C2×C42—21no—
170C2×C42C2×Dic5312yes—
171#99—1068no—
172#99—2124no—
173C24—24no—
174C24—24no—
175C24—42no—
176C24—84no—
177C24C22⋊Dic5312yes—
178C24C22⋊Dic5311yescharacteristic
179C24C2×C22⋊D5312yes—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—416no—
189C2×D4#1171994514yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2124no—
196C22×C4C22⋊Dic5311yescharacteristic
197C22×C4C22×Dic5311yescharacteristic
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—84no—
203C22⋊C4—84no—
204C22⋊C4—4248no—
205C22⋊C4—2124no—
206C22⋊C4—2124no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—42no—
217C23—48no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—84no—
222C23—88no—
223C23—88no—
224C23#1313313212yes—
225C23C2×C22⋊Dic5311yescharacteristic
226C23#1313320014yes—
227D4—44no—
228D4—48no—
229D4—44no—
230C2×C4—28no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—4244no—
241C2×C4—4248no—
242C2×C4#1313313212yes—
243C2×C4#1313317612yes—
244C22—22no—
245C22—22no—
246C22—24no—
247C22—24no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22—44no—
252C22—44no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—816no—
258C22#1681048611yescharacteristic
259C22C2×#1313313211yescharacteristic
260C22C2×#1313313211yescharacteristic
261C4—22no—
262C4—22no—
263C4—82no—
264C4—82no—
265C4—8484no—
266C2—21no—
267C2—21no—
268C2—44no—
269C2—42no—
270C2—82no—
271C2#2027478911yescharacteristic
272C2#2027638412yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2309C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenter#2309D53
hypocenterC53#12858
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #20272840C2semidirect
8. C2×#117×C53C4semidirect
9. C2×#117×C53C4semidirect
11. C53×#390C4semidirect
141. C2×#117Dic53semidirect
142. C2×#117Dic53semidirect
146. #390Dic53semidirect
150. C53#12858semidirect

Maximal subgroups

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

subgroupindex
#202746302
#202728402
C53×#23092
#1285853