Finite Group Databasebeta

← #27293500

Subgroups — #27293500

Order 13568.

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

normal subgroups only

This group has 434 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#20273045C212yesmaximal
2#20272661C212yesmaximal, complemented
3#20272659C212yesmaximal, complemented
4C53×#2317C211yesmaximal, characteristic
5#16810587—22no—
6#16810349—22no—
7#16810349C2214yes—
8C2×#117×C53C411yescharacteristic
9C2×#117×C53C411yescharacteristic
10C22×C4⋊C4×C53C2211yescharacteristic
11C53×#389—22no—
12C53×#385—22no—
13C53×#384C411yescharacteristic
14C53×#384C2211yescharacteristic
15C53×#382C411yescharacteristic
16C53×#382C2211yescharacteristic
17C2×#11719917—42no—
18C22⋊#88322—24no—
19C22⋊#88322—42no—
20C23×C4×C53C2×C411yescharacteristic
21C23×C4×C53D412yes—
22#117×C53—22no—
23#117×C53—22no—
24#117×C53—42no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53—22no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—42no—
29C2×C4⋊C4×C53C2×C411yescharacteristic
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53C2311yescharacteristic
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—21no—
36C2×C42×C53—21no—
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#95×C53—22no—
40#95×C53—22no—
41C24×C53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—82no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D412yes—
57C4⋊C4×C53—24no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—82no—
64C4⋊C4×C53—82no—
65C22⋊C4×C53—24no—
66C22⋊C4×C53—42no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—42no—
70C42×C53M1612yescomplemented
71C42×C53M1612yescomplemented
72#11719917—84no—
73C2×#88322—44no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—41no—
77C23×C53—41no—
78C23×C53#9912yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—44no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—82no—
102C2×C4×C53#10112yes—
103C2×C4×C53C2×M1611yescharacteristic
104C2×C4×C53C2×M1611yescharacteristic
105#88322—84no—
106#11514—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53#40711yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—42no—
123C4×C53—42no—
124C4×C53—81no—
125C4×C53—81no—
126#2317D5311yescharacteristic
127#1463—532no—
128#1325—532no—
129#1324—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53—42no—
133C2×C53#146311yescharacteristic
134C2×C53#212711yescharacteristic
135C2×C53#213311yescharacteristic
136C2×#117Dic5311yescharacteristic
137C2×#117Dic5311yescharacteristic
138C22×C4⋊C4C2×D5311yescharacteristic
139#407—1062no—
140#389—22no—
141#385—22no—
142#384Dic5311yescharacteristic
143#384C2×D5311yescharacteristic
144#382Dic5311yescharacteristic
145#382C2×D5311yescharacteristic
146#324—534no—
147#324—1062no—
148C53#1151411yescharacteristic, complemented, Sylow
149C23×C4C2×Dic5311yescharacteristic
150C23×C4C22⋊D5312yes—
151C2×M16—2122no—
152#117—22no—
153#117—22no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×Dic5311yescharacteristic
160C2×C4⋊C4C2×Dic5311yescharacteristic
161C2×C4⋊C4C2×Dic5311yescharacteristic
162C2×C4⋊C4C22⋊D5312yes—
163C2×C4⋊C4C22×D5311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42C2×Dic5311yescharacteristic
168C2×C42C2×Dic5311yescharacteristic
169C22⋊C8—1064no—
170C22⋊C8—2122no—
171#95—22no—
172#95—22no—
173C24C22⋊Dic5311yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—82no—
185C22×C4C22⋊Dic5311yescharacteristic
186C22×C4C22⋊Dic5312yes—
187C22×C4C22×Dic5311yescharacteristic
188C22×C4C2×C22⋊D5312yes—
189M16—4244no—
190C2×C8—2124no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42#1171991712yescomplemented
205C42#1171991712yescomplemented
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#1313313212yes—
211C23C2×C22⋊Dic5311yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4#1313313612yes—
235C2×C4C2×#1171991711yescharacteristic
236C2×C4C2×#1171991711yescharacteristic
237C8—4244no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#1681058711yescharacteristic
246C22C2×#1313313211yescharacteristic
247C22C2×#1313313611yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2027304511yescharacteristic
261C2#2027638411yescharacteristic
262C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2317C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2317D53
hypocenterC53#11514
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #20272661C2semidirect
3. #20272659C2semidirect
70. C42×C53M16semidirect
71. C42×C53M16semidirect
148. C53#11514semidirect
204. C42#11719917semidirect
205. C42#11719917semidirect

Maximal subgroups

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

subgroupindex
#202730452
#202726612
#202726592
C53×#23172
#1151453