Finite Group Databasebeta

← #27283367

Subgroups — #27283367

Order 13568.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic
3C4⋊C4×#88322C211yesmaximal, characteristic, complemented
4C2×#16810375C211yesmaximal, characteristic, complemented
5C2×#16810369C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C53×#1981C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810971—22no—
10#16810971C414yescomplemented
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133137—21no—
14C2×#13133137—21no—
15C2×#13133137—22no—
16C2×C4×#88322—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810378—21no—
19#16810375—22no—
20#16810369—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#103×C53—22no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31C22×#88322D411yescharacteristic
32C22×#88322D411yescharacteristic
33#13133214—21no—
34#13133214—21no—
35#13133214—21no—
36#13133214C2×C412yes—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D412yes—
41#13133137—24no—
42#13133137—24no—
43#13133137—44no—
44C4×#88322—42no—
45C4×#88322C2×C412yes—
46C4×#88322D412yes—
47C22×Q8×C53—21no—
48C22×C8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53C2×C412yes—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53D412yes—
56C2×C42×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60#103×C53—44no—
61#95×C53—21no—
62C23×D53—41no—
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C2×Q8×C53—42no—
66C2×Q8×C53—42no—
67C22×C4×C53—22no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C2×D212—22no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—42no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—42no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—82no—
85C2×C8×C53—41no—
86C2×C8×C53—42no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53D814yes—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53C22×C411yescharacteristic
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53#4111yescharacteristic
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322—22no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322—42no—
108C2×#88322C22⋊C412yes—
109C2×#88322C22⋊C412yes—
110C23×C53#12111yescharacteristic
111C22×D53—42no—
112C22×D53—81no—
113C22×D53—81no—
114C22×D53—82no—
115C22×D53—82no—
116Q8×C53—44no—
117Q8×C53—81no—
118Q8×C53—81no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—82no—
125C2×C4×C53C2×C22⋊C411yescharacteristic
126C2×C4×C53C4×D411yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12411yescharacteristic
130C2×C4×C53#12711yescharacteristic
131C2×C4×C53C2×D812yes—
132C2×C4×C53#13512yes—
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137D212—42no—
138D212—42no—
139D212—44no—
140D212—82no—
141D212—82no—
142D212—81no—
143D212—81no—
144D212—81no—
145D212—81no—
146C8×C53—81no—
147#88322—22no—
148#88322—22no—
149#88322—42no—
150#88322—81no—
151#9738—531nomaximal, Sylow
152C22×C53#39011yescharacteristic
153C22×C53#46311yescharacteristic
154C22×C53#46411yescharacteristic
155C22×C53#46612yes—
156C22×C53#47112yes—
157C2×D53—82no—
158C2×D53—82no—
159C2×D53—82no—
160C2×D53—162no—
161C2×D53—162no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—44no—
165C4×C53—82no—
166C4×C53#41611yescharacteristic
167C4×C53#41711yescharacteristic
168C4×C53C4×D812yes—
169C4×C53#46311yescharacteristic
170C4×C53#46411yescharacteristic
171C4×C53#49311yescharacteristic
172C4×C53#49511yescharacteristic
173C2×#486—531no—
174#1982—531no—
175#1981D5311yescharacteristic, complemented
176C8×C4⋊C4—531no—
177C2×#335—531no—
178C2×#332—531no—
179C2×#331—531no—
180C2×C53#168112yes—
181C2×C53#168212yes—
182C2×C53#197011yescharacteristic
183C2×C53#197711yescharacteristic
184C2×C53#198311yescharacteristic
185D53—162no—
186C2×C4⋊Q8C2×D5311yescharacteristic
187C2×#127—531no—
188#486—534no—
189#486—1062no—
190C2×C4⋊C8—531no—
191C2×C4⋊C8—531no—
192C2×C4⋊C8C2×D5311yescharacteristic
193C2×#103—22no—
194C2×#102—1061no—
195C2×#102—1061no—
196C2×#102—1062no—
197C2×C4×C8—531no—
198C2×C4×C8—1061no—
199C4×C4⋊C4C2×D5311yescharacteristic
200#336—1061no—
201#335—1062no—
202#332—1062no—
203#331—1062no—
204C53#973811yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×D4—1061no—
207C22×D4—1061no—
208C22×D4—1061no—
209C22×C8—21no—
210C22×C8—531no—
211C22×C8—531no—
212C22×C8—1061no—
213C22×C8—1061no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C4×D5312yes—
218#127—532no—
219#127—1061no—
220#127—1061no—
221#127—1061no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D5312yes—
225C2×C42—21no—
226C2×C42C22×D5311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—532no—
230C4⋊C8—1061no—
231C4⋊C8—1061no—
232C4⋊C8—1061no—
233#103—44no—
234#102—1064no—
235#102—1064no—
236#102—2124no—
237C4×C8—532no—
238C4×C8—532no—
239C4×C8—2122no—
240#95—21no—
241C24—2121no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—1062no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—4242no—
250C2×D4—2121no—
251C2×D4—2121no—
252C2×D4—2121no—
253C2×D4—2122no—
254C2×D4—2122no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2121no—
258C2×D4—2121no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4D4×D5311yescharacteristic
262C22×C4#1171994711yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—532no—
267C2×C8—532no—
268C2×C8—1061no—
269C2×C8—1061no—
270C2×C8—1061no—
271C2×C8—1061no—
272C2×C8—1062no—
273C2×C8—2121no—
274C2×C8—2122no—
275C2×C8—2121no—
276C2×C8—2121no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4#1171992214yes—
283C42—41no—
284C42—41no—
285C42C2×C4×D5311yescharacteristic
286C42#1171994511yescharacteristic
287C42D4×D5311yescharacteristic
288C42D4×D5311yescharacteristic
289C23—4241no—
290C23—4241no—
291C23—4242no—
292C23—4242no—
293C23—2122no—
294C23#1313318911yescharacteristic
295Q8—44no—
296Q8—81no—
297Q8—81no—
298D4—4242no—
299D4—4242no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—2122no—
305D4—2122no—
306D4—2124no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4C22⋊C4×D5311yescharacteristic
314C2×C4#1313317711yescharacteristic
315C2×C4C4×C22⋊D5311yescharacteristic
316C2×C4#1313319011yescharacteristic
317C2×C4#1313320311yescharacteristic
318C2×C4#1313321311yescharacteristic
319C2×C4C2×#1171992212yes—
320C2×C4#1313323712yes—
321C2×C4#1313323811yescharacteristic
322C2×C4#1313324011yescharacteristic
323C2×C4#1313324411yescharacteristic
324C2×C4#1313324911yescharacteristic
325C8—81no—
326C8—1062no—
327C8—1062no—
328C8—4241no—
329C8—2122no—
330C22—8482no—
331C22—8482no—
332C22—4242no—
333C22—4242no—
334C22—4242no—
335C22#1681052311yescharacteristic
336C22#1681084011yescharacteristic
337C22#1681084711yescharacteristic
338C22#1681085712yes—
339C22#1681088812yes—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—82no—
344C4#1681061811yescharacteristic
345C4#1681062311yescharacteristic
346C4C4×#1171992212yes—
347C4#1681084111yescharacteristic
348C4#1681084811yescharacteristic
349C4#1681099911yescharacteristic
350C4#1681100511yescharacteristic
351C2—8482no—
352C2#2027415412yes—
353C2#2027416112yes—
354C2#2027558011yescharacteristic
355C2#2027561711yescharacteristic
356C2#2027564411yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1981C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1981D53
hypocenterC53#9738
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×#88322C2semidirect
4. C2×#16810375C2semidirect
5. C2×#16810369C2semidirect
7. C53×#1981C2semidirect
10. #16810971C4semidirect
175. #1981D53semidirect
204. C53#9738semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202756422
C4⋊C4×#883222
C2×#168103752
C2×#168103692
C2×#168103662
C53×#19812
#973853