Finite Group Databasebeta

← #27279152

Subgroups — #27279152

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277333C211yesmaximal, characteristic
2#20274801C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272594C212yesmaximal
5C2×C53×#398C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×C22⋊#88322C2212yes—
8#16810522—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810385—24no—
12#16810349—22no—
13C22×C4⋊C4×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#398—22no—
17C53×#398C414yescomplemented
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—24no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—42no—
44C2×C4⋊C4×C53—42no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C4⋊C4×C53D412yes—
48#95×C53—24no—
49#95×C53—21no—
50#95×C53C2×C412yes—
51C24×C53C2×D411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—24no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—42no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53D812yes—
75C22×C4×C53SD1612yes—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4112yes—
80C22×Dic53—24no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—48no—
85C4⋊C4×C53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—44no—
89C4⋊Dic53—44no—
90C4⋊Dic53—82no—
91C4×Dic53—44no—
92C2×#88322—44no—
93C2×#88322—42no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53#12011yescharacteristic
98C23×C53#12311yescharacteristic
99C23×C53C2×D811yescharacteristic
100C23×C53C2×SD1611yescharacteristic
101C23×C53AGL(1,ℤ/8)11yescharacteristic
102C23×C53#13711yescharacteristic
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—24no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—24no—
111C2×C4×C53—22no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53#10214yes—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12112yes—
132C2×Dic53—44no—
133C2×Dic53—44no—
134C2×Dic53—44no—
135#88322—82no—
136#9027—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53#38611yescharacteristic
146C22×C53C2×#10211yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#44711yescharacteristic
149C22×C53#44811yescharacteristic
150C22×C53#45612yes—
151C22×C53#45812yes—
152C22×C53#48011yescharacteristic
153C22×C53#48111yescharacteristic
154C4×C53—24no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—44no—
158C4×C53—42no—
159C4×C53—82no—
160Dic53—84no—
161C2×#398D5311yescharacteristic
162#2316—531no—
163#1823—532no—
164C2×#324—531no—
165#1294—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53#161012yes—
171C2×C53#161112yes—
172C2×C53#187411yescharacteristic
173C2×C53#188611yescharacteristic
174C2×C53#189011yescharacteristic
175C2×#117—1061no—
176C22×C4⋊C4—21no—
177C22×C4⋊C4C2×D5311yescharacteristic
178C22×C4⋊C4C2×D5311yescharacteristic
179C2×C22⋊C8—532no—
180#398—22no—
181#398Dic5314yescomplemented
182#389—1064no—
183#380—532no—
184#380—1061no—
185C2×#95—21no—
186C2×#95C2×D5311yescharacteristic
187#340—1064no—
188#324—1062no—
189C53#902711yescharacteristic, complemented, Sylow
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C22⋊D5311yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C22×C8—1062no—
196#117—2122no—
197#117—2122no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—534no—
207C2×C4⋊C4—1062no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4C2×Dic5312yes—
210C2×C4⋊C4C2×Dic5312yes—
211C2×C4⋊C4C22⋊D5312yes—
212C2×C22⋊C4—1062no—
213C2×C42—1062no—
214C22⋊C8—1062no—
215C22⋊C8—1062no—
216#95—21no—
217#95—24no—
218#95—1062no—
219#95—1062no—
220#95C2×Dic5312yes—
221C24C2×C22⋊D5311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1064no—
243C22×C4#1171992212yes—
244C22×C4#1171992312yes—
245C22×C4C22⋊Dic5312yes—
246C22×C4C22⋊Dic5312yes—
247C22×C4D4×D5312yes—
248C22×C4#1171994712yes—
249C22×C4C22×Dic5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C2×C8—2124no—
252C2×C8—2122no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—42no—
258C4⋊C4—4242no—
259C4⋊C4—2124no—
260C4⋊C4—2124no—
261C22⋊C4—2122no—
262C22⋊C4—2122no—
263C42—2124no—
264C23—22no—
265C23—22no—
266C23C2×C22⋊Dic5311yescharacteristic
267C23#1313318511yescharacteristic
268C23#1313319811yescharacteristic
269C23C2×#1171992211yescharacteristic
270C23C2×#1171992311yescharacteristic
271C23#1313324211yescharacteristic
272C23#1313324711yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2124no—
302C2×C4#1313313814yes—
303C2×C4D4×Dic5312yes—
304C2×C4#1313318912yes—
305C8—4242no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22#1681050311yescharacteristic
315C22C2×#1313313811yescharacteristic
316C22#1681062411yescharacteristic
317C22#1681075611yescharacteristic
318C22#1681076111yescharacteristic
319C22#1681080512yes—
320C22#1681081212yes—
321C22#1681093411yescharacteristic
322C22#1681093911yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—42no—
328C4—82no—
329C4—4244no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2027371712yes—
335C2#2027372212yes—
336C2#2027503311yescharacteristic
337C2#2027510911yescharacteristic
338C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#398C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#398D53
hypocenterC53#9027
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
17. C53×#398C4semidirect
181. #398Dic53semidirect
189. C53#9027semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202748012
C2×#168103492
#202725942
C2×C53×#3982
#902753