Finite Group Databasebeta

← #27272509

Subgroups — #27272509

Order 13568.

384 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283097C211yesmaximal, characteristic
2#20274858C211yesmaximal, characteristic
3#20272540C212yesmaximal
4#20272537C212yesmaximal
5C4×C4⋊C8×C53C211yesmaximal, characteristic
6#16810529C411yescharacteristic
7#16810529C411yescharacteristic
8#16810529C2211yescharacteristic
9#16810521C411yescharacteristic
10#16810521C411yescharacteristic
11#16810521C2211yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810507—21no—
15#16810491—21no—
16#16810378—21no—
17#16810378—21no—
18#16810378C2212yes—
19C2×C4⋊C8×C53—21no—
20C2×C4×C8×C53—21no—
21C43×C53C2211yescharacteristic
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31#13133124—21no—
32#13133124—21no—
33#13133124—21no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40C22×C8×C53—22no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—24no—
49C4×C8×C53—42no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×Dic53—21no—
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53—21no—
61C2×Dic106—22no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×C8×C53—24no—
65C2×C8×C53—42no—
66C2×C8×C53—42no—
67C42×C53—22no—
68C42×C53—22no—
69C42×C53—22no—
70C42×C53—41no—
71C42×C53—41no—
72C42×C53—41no—
73C42×C53SD1612yes—
74C42×C53Q1612yes—
75C4⋊Dic53—22no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78#11719919—42no—
79#11719919—42no—
80#11719919—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
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—21no—
99C2×C4×C53—21no—
100C2×C4×C53#10312yes—
101C2×C4×C53#10312yes—
102C2×C4×C53#10412yes—
103C2×C4×C53#13112yes—
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53C2×Q1611yescharacteristic
106C2×C4×C53#13512yes—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119Dic106—82no—
120C8×C53—44no—
121#88322—82no—
122#7673—531nomaximal, Sylow
123C22×C53#40811yescharacteristic
124C22×C53#41011yescharacteristic
125C22×C53#41311yescharacteristic
126C22×C53C2×#10311yescharacteristic
127C22×C53#41611yescharacteristic
128C22×C53C2×#10411yescharacteristic
129C22×C53#42111yescharacteristic
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—41no—
138Dic53—81no—
139Dic53—81no—
140Dic53—81no—
141Dic53—81no—
142#2863—531no—
143#1839—531no—
144C4×C4⋊C8D5311yescharacteristic
145#1272—532no—
146#1271—532no—
147C2×C53#147311yescharacteristic
148C2×C53#148111yescharacteristic
149C2×C53#149411yescharacteristic
150C2×C53#150311yescharacteristic
151C2×C53#152111yescharacteristic
152C2×C53#152911yescharacteristic
153C2×C53#154211yescharacteristic
154C2×C4⋊C8—21no—
155C2×C4×C8—21no—
156#391—531no—
157#391—531no—
158#391—531no—
159#389—531no—
160#389—531no—
161#389—531no—
162#387—1061no—
163#387—1061no—
164#387—1061no—
165#383—1061no—
166C43C2×D5311yescharacteristic
167#336—532no—
168#336—1061no—
169#336—1061no—
170#336C2×D5312yes—
171C53#767311yescharacteristic, complemented, Sylow
172C22×Q8—531no—
173C22×C8—22no—
174C22×C8—1062no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—1061no—
181C2×C42C4×D5311yescharacteristic
182C2×C42C4×D5311yescharacteristic
183C2×C42D21211yescharacteristic
184C2×C42D21211yescharacteristic
185C2×C42C22⋊D5311yescharacteristic
186C2×C42C22⋊D5311yescharacteristic
187C2×C42C22×D5311yescharacteristic
188C4⋊C8—24no—
189C4×C8—42no—
190#95—1061no—
191#95—1061no—
192#95—1061no—
193#95—1061no—
194#95—1061no—
195#95—1061no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199C2×Q8—1062no—
200C2×Q8—2121no—
201C2×Q8—2121no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4#1171992111yescharacteristic
207C22×C4#1171992111yescharacteristic
208C22×C4#1171992111yescharacteristic
209C22×C4#1171992111yescharacteristic
210C22×C4C2×C4×D5311yescharacteristic
211C22×C4C2×D21211yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—2122no—
217C2×C8—2122no—
218C2×C8—2122no—
219C4⋊C4—1062no—
220C4⋊C4—2121no—
221C4⋊C4—2121no—
222C4⋊C4—2122no—
223C4⋊C4—2122no—
224C4⋊C4—2122no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42#1171992312yes—
232C42#1171992512yes—
233C23C2×#1171992111yescharacteristic
234Q8—4242no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4#1313314012yes—
263C2×C4#1313314012yes—
264C2×C4#1313314512yes—
265C2×C4#1313322312yes—
266C2×C4C2×#1171992311yescharacteristic
267C2×C4C2×#1171992511yescharacteristic
268C2×C4#1313323912yes—
269C8—44no—
270C8—4242no—
271C22#1681058911yescharacteristic
272C22#1681059611yescharacteristic
273C22#1681060511yescharacteristic
274C22C2×#1313314011yescharacteristic
275C22#1681062011yescharacteristic
276C22C2×#1313314511yescharacteristic
277C22#1681063911yescharacteristic
278C4—22no—
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—4241no—
287C4—4241no—
288C4—4241no—
289C4—4241no—
290C2#2027308211yescharacteristic
291C2#2027311711yescharacteristic
292C2#2027317611yescharacteristic
293C2#2027321311yescharacteristic
294C2#2027329311yescharacteristic
295C2#2027332011yescharacteristic
296C2#2027337311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×C4⋊C8×C53C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×C4⋊C8D53
hypocenterC53#7673
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
171. C53#7673semidirect

Maximal subgroups

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

subgroupindex
#202830972
#202748582
#202725402
#202725372
C4×C4⋊C8×C532
#767353