Finite Group Databasebeta

← #27277429

Subgroups — #27277429

Order 13568.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16810487C211yesmaximal, characteristic
2#20274781C212yesmaximal
3#20272558C211yesmaximal, characteristic
4#20272557C212yesmaximal
5C53×#1280C211yesmaximal, characteristic
6C22×#11719919C412yes—
7C22×C4×Dic53C2211yescharacteristic
8C2×C22⋊#88322C2212yes—
9#16810487—22no—
10#16810487—22no—
11C2×#13133124C412yes—
12C2×#13133124C2211yescharacteristic
13#16810377—22no—
14#16810377—22no—
15C2×C22⋊C8×C53C2212yes—
16C2×#95×C53C2211yescharacteristic
17C23×Dic53C2×C411yescharacteristic
18C23×Dic53C2×C411yescharacteristic
19C23×Dic53C2×C412yes—
20C22×#88322—22no—
21C2×#11719919—22no—
22C2×#11719919—22no—
23C2×#11719919—44no—
24C2×C4×Dic53—22no—
25C2×C4×Dic53—21no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28C2×C4×Dic53D412yes—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—24no—
33C23×C4×C53C2×C412yes—
34C23×C4×C53C2311yescharacteristic
35C22×C8×C53—22no—
36C22⋊C8×C53—22no—
37C22⋊C8×C53—22no—
38#95×C53—22no—
39C24×C53C22×C411yescharacteristic
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—42no—
44C22×C4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53C2×D412yes—
47C22×C4×C53#4112yes—
48C22×C4×C53#4111yescharacteristic
49C22×C4×C53#4111yescharacteristic
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—21no—
55C22×Dic53—21no—
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—42no—
62C22×Dic53—42no—
63C22×Dic53C22⋊C412yes—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C2×C8×C53—44no—
67C2×C8×C53—42no—
68#11719919—44no—
69#11719919—44no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C2×#88322—44no—
75C2×#88322—42no—
76C23×C53—22no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53C2×C22⋊C411yescharacteristic
80C23×C53#11711yescharacteristic
81C23×C53#11711yescharacteristic
82C23×C53#13112yes—
83C23×C53#13112yes—
84C2×C4×C53—24no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—82no—
92C2×C4×C53—82no—
93C2×C4×C53#12312yes—
94C2×C4×C53#12312yes—
95C2×Dic53—24no—
96C2×Dic53—24no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—24no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—82no—
110C2×Dic53—82no—
111C2×Dic53#10012yes—
112C2×Dic53#10112yes—
113C8×C53—82no—
114#88322—82no—
115#8703—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53#38111yescharacteristic
125C22×C53#40812yes—
126C22×C53#41012yes—
127C22×C53C2×#10011yescharacteristic
128C22×C53C2×#10111yescharacteristic
129C22×C53#43311yescharacteristic
130C22×C53#43312yes—
131C22×C53#43311yescharacteristic
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—82no—
135Dic53—22no—
136Dic53—22no—
137Dic53—42no—
138Dic53—82no—
139C2×#382—531no—
140#1817—532no—
141#1280—532no—
142#1280—531no—
143#1280D5311yescharacteristic
144C2×C53—21no—
145C2×C53—22no—
146C2×C53—21no—
147C2×C53#146912yes—
148C2×C53#147012yes—
149C2×C53#181811yescharacteristic
150C2×C53#182011yescharacteristic
151C2×C53#182111yescharacteristic
152C22×C4⋊C4—532no—
153C22×C42—531no—
154C2×C22⋊C8—532no—
155C2×C22⋊C8C2×D5312yes—
156#382—1062no—
157#382—1062no—
158C2×#95—532no—
159C2×#95—531no—
160C2×#95C2×D5311yescharacteristic
161#336—1062no—
162#336—1062no—
163C53#870311yescharacteristic, complemented, Sylow
164C23×C4—531no—
165C23×C4—531no—
166C23×C4—532no—
167C23×C4C4×D5312yes—
168C23×C4C22×D5311yescharacteristic
169C22×C8—22no—
170C22×C8—1062no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—2124no—
174C2×C42—532no—
175C2×C42—532no—
176C2×C42—1062no—
177C2×C42—1061no—
178C2×C42—1061no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—1062no—
182C22⋊C8—1062no—
183#95—22no—
184#95—1064no—
185#95—1062no—
186C24C2×C4×D5311yescharacteristic
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—532no—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4—1061no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—1061no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—2122no—
208C22×C4—2122no—
209C22×C4#1171994512yes—
210C22×C4D4×D5312yes—
211C22×C4#1171994711yescharacteristic
212C22×C4#1171994911yescharacteristic
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—2124no—
216C2×C8—2122no—
217C4⋊C4—2124no—
218C4⋊C4—2124no—
219C42—1062no—
220C42—1062no—
221C42—1062no—
222C42—1062no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23C22⋊C4×D5311yescharacteristic
227C23#1313317411yescharacteristic
228C23#1313317511yescharacteristic
229C23#1313322312yes—
230C23#1313322512yes—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—532no—
241C2×C4—532no—
242C2×C4—1064no—
243C2×C4—1064no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—1064no—
249C2×C4—1062no—
250C2×C4—4242no—
251C2×C4—4242no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2121no—
258C2×C4#1313319912yes—
259C2×C4#1313320212yes—
260C8—82no—
261C8—4242no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22#1681048511yescharacteristic
271C22#1681058812yes—
272C22#1681059512yes—
273C22#100×D5311yescharacteristic
274C22#101×D5311yescharacteristic
275C22#1681068211yescharacteristic
276C22#1681068312yes—
277C22#1681068511yescharacteristic
278C4—22no—
279C4—22no—
280C4—82no—
281C4—1062no—
282C4—1062no—
283C4—4242no—
284C4—2122no—
285C2—21no—
286C2—22no—
287C2—21no—
288C2#2027306512yes—
289C2#2027306912yes—
290C2#2027478511yescharacteristic
291C2#2027479111yescharacteristic
292C2#2027479411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenter#1280D53
hypocenterC53#8703
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
163. C53#8703semidirect

Maximal subgroups

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

subgroupindex
C2×#168104872
#202747812
#202725582
#202725572
C53×#12802
#870353