Finite Group Databasebeta

← #27283361

Subgroups — #27283361

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#20275641C211yesmaximal, characteristic
3#20275639C211yesmaximal, characteristic, complemented
4#20274580C211yesmaximal, characteristic, complemented
5C2×#16810375C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810971—22no—
10#16810971C414yescomplemented
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133141—22no—
13C2×#13133139—22no—
14C2×#13133137—21no—
15C2×#13133137—21no—
16C2×C4×#88322C2211yescharacteristic
17C2×C8×Dic53—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810377—21no—
20#16810375—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#332—22no—
26C22×D212—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×#88322—21no—
30C22×#88322D411yescharacteristic
31C22×#88322D411yescharacteristic
32#13133214—21no—
33#13133214—21no—
34#13133214—21no—
35#13133214C2×C412yes—
36C2×C4⋊Dic53D412yes—
37C2×C4×Dic53—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40#13133141—44no—
41#13133139—44no—
42#13133137—24no—
43#13133137—24no—
44C4×#88322C2×C412yes—
45C4×#88322D412yes—
46C8×Dic53—42no—
47#13133124—21no—
48C22×Q8×C53—21no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53C2×C412yes—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D412yes—
62C23×D53—41no—
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C2×Q8×C53—42no—
66C2×Q8×C53—42no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×Dic53—22no—
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—22no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—42no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C42×C53C22×C411yescharacteristic
95C42×C53C2×D411yescharacteristic
96C42×C53C2×D411yescharacteristic
97C42×C53#4111yescharacteristic
98C4⋊Dic53—22no—
99C4⋊Dic53D814yes—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C2×#88322—21no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
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—42no—
121C2×C4×C53—82no—
122C2×C4×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53C4×D411yescharacteristic
124C2×C4×C53C4×D411yescharacteristic
125C2×C4×C53#12111yescharacteristic
126C2×C4×C53#12411yescharacteristic
127C2×C4×C53#12711yescharacteristic
128C2×C4×C53C2×D812yes—
129C2×C4×C53#13511yescharacteristic
130C2×C4×C53#13512yes—
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×Dic53—24no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137D212—42no—
138D212—42no—
139D212—44no—
140D212—82no—
141D212—82no—
142D212—81no—
143D212—81no—
144D212—81no—
145D212—81no—
146C8×C53—42no—
147C8×C53—81no—
148#88322—22no—
149#88322—22no—
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—82no—
165C4×C53#41611yescharacteristic
166C4×C53#41711yescharacteristic
167C4×C53C4×D812yes—
168C4×C53#46311yescharacteristic
169C4×C53#46411yescharacteristic
170C4×C53#49311yescharacteristic
171C4×C53#49511yescharacteristic
172Dic53—44no—
173C2×#486—531no—
174#1982—531no—
175#1981—531no—
176C8×C4⋊C4—531no—
177C2×#335—531no—
178C2×#332D5311yescharacteristic, complemented
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⋊C8C2×D5311yescharacteristic
192C2×C4⋊C8C2×D5311yescharacteristic
193C2×#103—1062no—
194C2×#102—1061no—
195C2×#102—1061no—
196C2×#102—1062no—
197C2×C4×C8—531no—
198C2×C4×C8—1061no—
199C4×C4⋊C4—531no—
200#336—1061no—
201#335—1062no—
202#332—22no—
203#331—1062no—
204C53#973811yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×D4—1061no—
207C22×D4—1061no—
208C22×D4—1061no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—531no—
212C22×C8—531no—
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⋊C4—532no—
225C2×C42—1061no—
226C2×C42C22×D5311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—1061no—
231C4⋊C8—1061no—
232C4⋊C8C22⋊D5312yes—
233#103—2124no—
234#102—1064no—
235#102—1064no—
236#102—2124no—
237C4×C8—532no—
238C4×C8—532no—
239C4×C8—2122no—
240#95—1061no—
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—2121no—
255C2×D4—2122no—
256C2×D4—2121no—
257C2×D4—2121no—
258C2×D4—2121no—
259C22×C4—41no—
260C22×C4—1062no—
261C22×C4#1171994511yescharacteristic
262C22×C4D4×D5311yescharacteristic
263C22×C4D4×D5311yescharacteristic
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—532no—
270C2×C8—532no—
271C2×C8—1061no—
272C2×C8—1061no—
273C2×C8—1061no—
274C2×C8—1061no—
275C2×C8—2121no—
276C2×C8—2122no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—534no—
282C4⋊C4—1062no—
283C42—2121no—
284C42—2121no—
285C42C2×C4×D5311yescharacteristic
286C42D4×D5311yescharacteristic
287C42#1171994711yescharacteristic
288C42C2×C22⋊D5311yescharacteristic
289C23—4241no—
290C23—4241no—
291C23—4242no—
292C23—4242no—
293C23—2122no—
294C23#1313319011yescharacteristic
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—42no—
309C2×C4—82no—
310C2×C4—1064no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4C22⋊C4×D5311yescharacteristic
314C2×C4#1313317711yescharacteristic
315C2×C4C4×C22⋊D5311yescharacteristic
316C2×C4#1313318911yescharacteristic
317C2×C4#1313320311yescharacteristic
318C2×C4#1313321311yescharacteristic
319C2×C4D8×D5312yes—
320C2×C4#1313323811yescharacteristic
321C2×C4#1313323812yes—
322C2×C4#1313324011yescharacteristic
323C2×C4#1313324411yescharacteristic
324C2×C4#1313324911yescharacteristic
325C8—42no—
326C8—81no—
327C8—1062no—
328C8—1062no—
329C8—4241no—
330C22—8482no—
331C22—8482no—
332C22—4242no—
333C22—4242no—
334C22—4242no—
335C22#1681052311yescharacteristic
336C22#1681084111yescharacteristic
337C22#1681084811yescharacteristic
338C22#1681085812yes—
339C22#1681088912yes—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—2124no—
344C4#1681061811yescharacteristic
345C4#1681062311yescharacteristic
346C4#1681070312yes—
347C4#1681084011yescharacteristic
348C4#1681084711yescharacteristic
349C4#1681099811yescharacteristic
350C4#1681100411yescharacteristic
351C2—8482no—
352C2#2027415512yes—
353C2#2027416212yes—
354C2#2027557811yescharacteristic
355C2#2027561911yescharacteristic
356C2#2027564411yescharacteristic
subgroupG/H
centerC23#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9738
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #20275639C2semidirect
4. #20274580C2semidirect
5. C2×#16810375C2semidirect
7. C2×C53×#332C2semidirect
10. #16810971C4semidirect
178. C2×#332D53semidirect
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
#202756412
#202756392
#202745802
C2×#168103752
C2×#168103662
C2×C53×#3322
#973853