Finite Group Databasebeta

← #27279132

Subgroups — #27279132

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277333C211yesmaximal, characteristic
2#20274805C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272594C212yesmaximal
5C2×C53×#400C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×C22⋊#88322C2212yes—
8#16810522—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810385—24no—
12#16810349—22no—
13C22×C4⋊C4×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400—22no—
16C53×#400C414yescomplemented
17C2×#95×C53—21no—
18C2×#95×C53—21no—
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21#13133176—42no—
22#13133176—42no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53D414yes—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—22no—
32C23×C4×C53—21no—
33C23×C4×C53—21no—
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D412yes—
45#95×C53—24no—
46#95×C53—24no—
47#95×C53—21no—
48#95×C53C2×C412yes—
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—44no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53SD1612yes—
70C22×C4×C53Q1612yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—42no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C4⋊Dic53—44no—
83C4⋊Dic53—44no—
84C4⋊Dic53—82no—
85C4×Dic53—44no—
86C2×#88322—44no—
87C2×#88322—42no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53C2×C22⋊C411yescharacteristic
91C23×C53#12011yescharacteristic
92C23×C53#12311yescharacteristic
93C23×C53C2×SD1611yescharacteristic
94C23×C53C2×Q1611yescharacteristic
95C23×C53AGL(1,ℤ/8)11yescharacteristic
96C23×C53#13711yescharacteristic
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—24no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—44no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53—82no—
117C2×C4×C53—82no—
118C2×C4×C53—82no—
119C2×C4×C53—82no—
120C2×C4×C53#10314yes—
121C2×C4×C53C4×D412yes—
122C2×C4×C53#12112yes—
123C2×Dic53—44no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126#88322—82no—
127#9023—531nomaximal, Sylow
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53#38611yescharacteristic
137C22×C53C2×#10311yescharacteristic
138C22×C53#41711yescharacteristic
139C22×C53#45011yescharacteristic
140C22×C53#45111yescharacteristic
141C22×C53#45412yes—
142C22×C53#45812yes—
143C22×C53#48311yescharacteristic
144C22×C53#48411yescharacteristic
145C4×C53—24no—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—42no—
149C4×C53—82no—
150C4×C53—82no—
151Dic53—84no—
152C2×#400D5311yescharacteristic
153#2316—531no—
154#1824—532no—
155C2×#324—531no—
156#1294—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53#161612yes—
162C2×C53#161712yes—
163C2×C53#187211yescharacteristic
164C2×C53#188811yescharacteristic
165C2×C53#189011yescharacteristic
166C2×#117—1061no—
167C22×C4⋊C4C2×D5311yescharacteristic
168C22×C4⋊C4C2×D5311yescharacteristic
169C2×C22⋊C8—532no—
170#400—22no—
171#400Dic5314yescomplemented
172#389—1064no—
173#380—532no—
174#380—1061no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95C2×D5311yescharacteristic
178#340—1064no—
179#324—1062no—
180C53#902311yescharacteristic, complemented, Sylow
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊D5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22×D5311yescharacteristic
186C22×C8—1062no—
187#117—2122no—
188#117—2122no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—534no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4C2×Dic5312yes—
198C2×C4⋊C4C2×Dic5312yes—
199C2×C4⋊C4C22⋊D5312yes—
200C2×C22⋊C4—1062no—
201C2×C42—1062no—
202C22⋊C8—1062no—
203C22⋊C8—1062no—
204#95—21no—
205#95—24no—
206#95—24no—
207#95—1062no—
208#95—1062no—
209#95C2×Dic5312yes—
210C24C2×C22⋊D5311yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—1064no—
229C22×C4#1171992412yes—
230C22×C4#1171992512yes—
231C22×C4C22⋊Dic5312yes—
232C22×C4C22⋊Dic5312yes—
233C22×C4D4×D5312yes—
234C22×C4#1171994712yes—
235C22×C4C22×Dic5311yescharacteristic
236C22×C4C2×C22⋊D5311yescharacteristic
237C2×C8—2124no—
238C2×C8—2122no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—42no—
243C4⋊C4—4242no—
244C4⋊C4—2124no—
245C4⋊C4—2124no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C42—2124no—
249C23—22no—
250C23—22no—
251C23C2×C22⋊Dic5311yescharacteristic
252C23#1313318511yescharacteristic
253C23#1313319811yescharacteristic
254C23C2×#1171992411yescharacteristic
255C23C2×#1171992511yescharacteristic
256C23#1313324211yescharacteristic
257C23#1313324711yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—2124no—
282C2×C4—2124no—
283C2×C4—2124no—
284C2×C4#1313314114yes—
285C2×C4D4×Dic5312yes—
286C2×C4#1313318912yes—
287C8—4242no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#1681050311yescharacteristic
297C22C2×#1313314111yescharacteristic
298C22#1681062411yescharacteristic
299C22#1681077311yescharacteristic
300C22#1681077811yescharacteristic
301C22#1681079512yes—
302C22#1681081212yes—
303C22#1681095111yescharacteristic
304C22#1681095611yescharacteristic
305C4—24no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—82no—
310C4—82no—
311C4—4244no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2027375112yes—
317C2#2027375612yes—
318C2#2027502311yescharacteristic
319C2#2027511911yescharacteristic
320C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#400C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#400D53
hypocenterC53#9023
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
16. C53×#400C4semidirect
171. #400Dic53semidirect
180. C53#9023semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202748052
C2×#168103492
#202725942
C2×C53×#4002
#902353