Finite Group Databasebeta

← #27293519

Subgroups — #27293519

Order 13568.

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

normal subgroups only

This group has 418 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#20273049C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272663C214yesmaximal
4C53×#2317C211yesmaximal, characteristic
5#16810680—21no—
6#16810680—21no—
7#16810349—21no—
8#16810349—21no—
9#16810347C2212yes—
10#16810347C2212yes—
11C2×#117×C53C411yescharacteristic
12C2×#117×C53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C53×#389—21no—
15C53×#389—21no—
16C53×#385C412yes—
17C53×#385C2212yes—
18C53×#384—21no—
19C53×#382—21no—
20C4⋊#88322—41no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24#13133128—41no—
25#13133128—41no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53C2×C412yes—
28#117×C53—24no—
29#117×C53—41no—
30#117×C53—41no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53C2×C412yes—
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53C2311yescharacteristic
43C2×C42×C53—21no—
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46#95×C53—22no—
47#95×C53—22no—
48C24×C53C22×C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—82no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C4⋊C4×C53—24no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—82no—
70C4⋊C4×C53—82no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53M1614yes—
75C42×C53—24no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53C2×C22⋊C411yescharacteristic
87C23×C53C2×M1612yes—
88C2×C4×C53—24no—
89C2×C4×C53—24no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—24no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—84no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#10112yes—
112#88322—82no—
113#88322—82no—
114#11520—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—22no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53#40711yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—44no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—82no—
134#2317D5311yescharacteristic
135#1464—531no—
136#1464—531no—
137#1326—534no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—42no—
141C2×C53#146311yescharacteristic
142C2×C53#212211yescharacteristic
143C2×C53#212311yescharacteristic
144C2×#117Dic5311yescharacteristic
145C2×#117C2×D5311yescharacteristic
146C22×C4⋊C4Dic5311yescharacteristic
147#432—1061no—
148#432—1061no—
149#389—21no—
150#389—21no—
151#385Dic5312yes—
152#385C2×D5312yes—
153#384—21no—
154#382—21no—
155#324—1061no—
156#324—1061no—
157#323—532no—
158#323—532no—
159C53#1152011yescharacteristic, complemented, Sylow
160C23×C4C2×Dic5311yescharacteristic
161C23×C4C2×Dic5312yes—
162#117—24no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C22⋊D5311yescharacteristic
173C2×C4⋊C4C22⋊D5311yescharacteristic
174C2×C22⋊C4C2×Dic5312yes—
175C2×C22⋊C4C2×Dic5311yescharacteristic
176C2×C22⋊C4C22×D5311yescharacteristic
177C2×C42—21no—
178C2×C42C22⋊D5311yescharacteristic
179C2×C42C22⋊D5311yescharacteristic
180C4⋊C8—2121no—
181C4⋊C8—2121no—
182C22⋊C8—1062no—
183C22⋊C8—1062no—
184#97—2121no—
185#97—2121no—
186#95—22no—
187#95—22no—
188C24C22×Dic5311yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—82no—
197C22×C4C22⋊Dic5311yescharacteristic
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22⋊Dic5311yescharacteristic
200C22×C4C22⋊Dic5311yescharacteristic
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203C2×C8—2122no—
204C2×C8—2122no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4#1171991714yes—
217C42—24no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23C2×C22⋊Dic5311yescharacteristic
227C23C2×#1171991712yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4#1313313211yescharacteristic
250C2×C4#1313313211yescharacteristic
251C2×C4#1313313612yes—
252C8—4242no—
253C8—4242no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22#1681058711yescharacteristic
262C22C2×#1313313211yescharacteristic
263C22C2×#1313313611yescharacteristic
264C4—21no—
265C4—21no—
266C4—22no—
267C4—44no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C2—21no—
274C2—21no—
275C2—42no—
276C2#2027304511yescharacteristic
277C2#2027636511yescharacteristic
278C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutator, prime radicalC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2317C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2317D53
hypocenterC53#11520
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
159. C53#11520semidirect

Maximal subgroups

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

subgroupindex
#202730492
#202726632
C53×#23172
#1152053