Finite Group Databasebeta

← #27276692

Subgroups — #27276692

Order 13568.

718 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274798C211yesmaximal, characteristic, complemented
2#20274786C211yesmaximal, characteristic, complemented
3#20273063C214yesmaximal, complemented
4C53×#3491C211yesmaximal, characteristic
5#16810587C2212yes—
6C2×C22⋊#88322C2212yes—
7#16810392—22no—
8#16810378—22no—
9#16810349—24no—
10#16810347—24no—
11C53×#552—24no—
12C53×#551—24no—
13C53×#543C411yescharacteristic
14C53×#539—22no—
15C53×#534C411yescharacteristic
16C53×#529—22no—
17C2×#123×C53C2212yes—
18C53×#518C412yes—
19C2×#117×C53C2211yescharacteristic
20C2×#11719917—22no—
21C22×#88322—22no—
22C22⋊#88322—24no—
23C22⋊#88322—24no—
24C22×D4×C53C2×C412yes—
25C23×C4×C53C2311yescharacteristic
26#126×C53—24no—
27#126×C53—24no—
28#125×C53—22no—
29#125×C53—24no—
30#124×C53—22no—
31#123×C53—24no—
32#123×C53—22no—
33#123×C53—22no—
34#123×C53—22no—
35#123×C53—24no—
36C22⋊Q8×C53—24no—
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—24no—
39#121×C53—22no—
40#120×C53—22no—
41C4×Q8×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—24no—
44C4×D4×C53—24no—
45#117×C53—24no—
46#117×C53—22no—
47#117×C53—24no—
48C2×C4⋊C4×C53D412yes—
49C2×C4⋊C4×C53D414yes—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2×C412yes—
54C2×C22⋊C4×C53D414yes—
55C2×C42×C53D412yes—
56C24×C53C22⋊C412yes—
57C24×C53C22×C411yescharacteristic
58C2×Q8×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—44no—
63C22×C4×C53—24no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22⋊C414yes—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D414yes—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—24no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C42×C53—24no—
88C42×C53—22no—
89C42×C53—22no—
90#11719917—44no—
91C2×#88322—44no—
92C2×#88322—42no—
93C2×#88322—42no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—44no—
99C23×C53C2×C22⋊C412yes—
100C23×C53C2×C22⋊C411yescharacteristic
101Q8×C53—82no—
102D4×C53—44no—
103D4×C53—44no—
104D4×C53—82no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—84no—
117C2×C4×C53#12012yes—
118C2×C4×C53#12012yes—
119#88322—82no—
120#88322—82no—
121#8540—531nomaximal, Sylow
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53—42no—
126C22×C53—44no—
127C22×C53—82no—
128C22×C53—162no—
129C22×C53#37911yescharacteristic
130C22×C53#41012yes—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134C4×C53—42no—
135C4×C53—44no—
136C4×C53—42no—
137#3491D5311yescharacteristic
138#1822—531no—
139#1818—531no—
140#1468—534no—
141C2×C53—22no—
142C2×C53—42no—
143C2×C53—82no—
144C2×C53#178211yescharacteristic
145C2×C53#179112yes—
146#552—24no—
147#551—24no—
148#543Dic5311yescharacteristic
149#539—22no—
150#534Dic5311yescharacteristic
151#529—22no—
152C2×#123C2×D5312yes—
153#518Dic5312yes—
154C2×#117C2×D5311yescharacteristic
155#407—532no—
156C2×C22⋊C8—532no—
157#343—1062no—
158#336—1062no—
159#324—1064no—
160#323—1064no—
161C53#854011yescharacteristic, complemented, Sylow
162C22×D4C2×Dic5312yes—
163C23×C4C22×D5311yescharacteristic
164C2×M16—1062no—
165C22×C8—1062no—
166#126—24no—
167#126—24no—
168#125—22no—
169#125—24no—
170#124—22no—
171#123—24no—
172#123—22no—
173#123—22no—
174#123—24no—
175#123—22no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—24no—
179#121—22no—
180#120—22no—
181C4×Q8—22no—
182C4×D4—22no—
183C4×D4—24no—
184C4×D4—24no—
185#117—24no—
186#117—24no—
187#117—22no—
188C2×C4⋊C4C22⋊D5312yes—
189C2×C4⋊C4C22⋊D5314yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×Dic5311yescharacteristic
192C2×C22⋊C4C2×Dic5311yescharacteristic
193C2×C22⋊C4C2×Dic5312yes—
194C2×C22⋊C4C22⋊D5314yes—
195C2×C42C22⋊D5312yes—
196C22⋊C8—1064no—
197C22⋊C8—1064no—
198C24C22⋊Dic5312yes—
199C24C22×Dic5311yescharacteristic
200C2×Q8—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4C22⋊Dic5312yes—
208C22×C4C22⋊Dic5312yes—
209C22×C4C22⋊Dic5314yes—
210C22×C4C22⋊Dic5312yes—
211C22×C4C2×C22⋊D5311yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213C22×C4C2×C22⋊D5314yes—
214M16—2124no—
215C2×C8—2124no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23C2×C22⋊Dic5312yes—
242C23C2×C22⋊Dic5311yescharacteristic
243Q8—82no—
244D4—44no—
245D4—44no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4#1313318612yes—
260C2×C4#1313318612yes—
261C8—4242no—
262C8—4242no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—44no—
268C22—82no—
269C22—162no—
270C22#1681047911yescharacteristic
271C22#1681059712yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2#2027463311yescharacteristic
282C2#2027466612yes—
subgroupG/H
centerC22#16810479
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3491C2
socleC22×C53#379
prime radicalC53×#534C4
cosocleC23×C4×C53C23
hypercenter#3491D53
hypocenterC53#8540
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20274786C2semidirect
3. #20273063C2semidirect
161. C53#8540semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202747862
#202730632
C53×#34912
#854053