Finite Group Databasebeta

← #27293520

Subgroups — #27293520

Order 13568.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277335C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272664C214yesmaximal
4C53×#1464C211yesmaximal, characteristic
5C2×#13133176C411yescharacteristic
6C22×C4⋊Dic53C411yescharacteristic
7#16810680—21no—
8#16810521—21no—
9#16810521—21no—
10#16810495C412yes—
11#16810495C2212yes—
12#16810492—21no—
13#16810488—21no—
14#16810349—21no—
15#16810347C2212yes—
16C2×#117×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#324—21no—
19C53×#323C2212yes—
20C23×Dic53C2×C412yes—
21#13133176—24no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×C22⋊Dic53C2×C412yes—
30C2×C4×Dic53—21no—
31C4⋊#88322—41no—
32C22⋊#88322—22no—
33#13133128—41no—
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53C2×C411yescharacteristic
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53C2311yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47#97×C53—41no—
48C24×C53C22×C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—42no—
62C22×Dic53—82no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—41no—
67C22⋊C4×C53—41no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C22⋊Dic53—22no—
73C22⋊Dic53M1614yes—
74C4⋊Dic53—24no—
75C4⋊Dic53—42no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78#11719919—82no—
79#11719919—82no—
80C4×Dic53—24no—
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—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53#9911yescharacteristic
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#10112yes—
105C2×Dic53—24no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—84no—
112C8×C53—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—41no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132Dic53—44no—
133Dic53—82no—
134#2317—531no—
135#1464—531no—
136#1464D5311yescharacteristic
137#1326—534no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—42no—
141C2×C53#146311yescharacteristic
142C2×C53#212211yescharacteristic
143C2×C53#212311yescharacteristic
144C2×#117—531no—
145C2×#117C2×D5311yescharacteristic
146C22×C4⋊C4—531no—
147#432—21no—
148#432—1061no—
149#389—1061no—
150#389—1061no—
151#385—532no—
152#385—532no—
153#384—1061no—
154#382—1061no—
155#324—21no—
156#324—1061no—
157#323—532no—
158#323C2×D5312yes—
159C53#1152011yescharacteristic, complemented, Sylow
160C23×C4—532no—
161C23×C4C4×D5311yescharacteristic
162#117—41no—
163#117—41no—
164#117—1064no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4—2121no—
171C2×C4⋊C4—2121no—
172C2×C4⋊C4D21211yescharacteristic
173C2×C4⋊C4C22⋊D5311yescharacteristic
174C2×C22⋊C4—532no—
175C2×C22⋊C4C4×D5311yescharacteristic
176C2×C22⋊C4C22×D5311yescharacteristic
177C2×C42—1061no—
178C2×C42D21211yescharacteristic
179C2×C42C22⋊D5311yescharacteristic
180C4⋊C8—41no—
181C4⋊C8—2121no—
182C22⋊C8—22no—
183C22⋊C8—1062no—
184#97—41no—
185#97—2121no—
186#95—1062no—
187#95—1062no—
188C24C2×C4×D5311yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—1062no—
194C22×C4—1062no—
195C22×C4—4242no—
196C22×C4—2122no—
197C22×C4#1171992111yescharacteristic
198C22×C4#1171992111yescharacteristic
199C22×C4#1171992111yescharacteristic
200C22×C4#1171992111yescharacteristic
201C22×C4C2×D21211yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203C2×C8—42no—
204C2×C8—2122no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—1064no—
208C4⋊C4—4242no—
209C4⋊C4—4242no—
210C4⋊C4—2122no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—534no—
216C22⋊C4—1062no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—1064no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23C2×#1171992111yescharacteristic
227C23M16×D5312yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—1064no—
243C2×C4—4244no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4#1313313111yescharacteristic
250C2×C4#1313313111yescharacteristic
251C2×C4#1313313512yes—
252C8—82no—
253C8—4242no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22#1681058611yescharacteristic
262C22C2×#1313313111yescharacteristic
263C22C2×#1313313511yescharacteristic
264C4—21no—
265C4—21no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—4242no—
272C4—2124no—
273C2—21no—
274C2—21no—
275C2—42no—
276C2#2027304411yescharacteristic
277C2#2027636611yescharacteristic
278C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutator, prime radicalC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11520
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

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
#202773352
#202730492
#202726642
C53×#14642
#1152053