Finite Group Databasebeta

← #27283346

Subgroups — #27283346

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#16810373C211yesmaximal, characteristic, complemented
5C2×#16810372C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C53×#1987C211yesmaximal, 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#16810373—22no—
20#16810372—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C53×#340—22no—
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#95×C53—21no—
61C23×D53—41no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—82no—
65C22×C4×C53—22no—
66C22×C4×C53—41no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C2×D212—22no—
71C2×D212—22no—
72C2×D212—22no—
73C2×D212—42no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—42no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—82no—
83C2×C8×C53—41no—
84C2×C8×C53—42no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53SD1614yes—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53C22×C411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53C2×D411yescharacteristic
96C42×C53#4111yescharacteristic
97C2×#88322—21no—
98C2×#88322—21no—
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322—22no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C2×#88322—42no—
106C2×#88322C22⋊C412yes—
107C2×#88322C22⋊C412yes—
108C23×C53#12111yescharacteristic
109C22×D53—42no—
110C22×D53—81no—
111C22×D53—81no—
112C22×D53—82no—
113C22×D53—82no—
114Q8×C53—82no—
115Q8×C53—82no—
116C2×C4×C53—22no—
117C2×C4×C53—24no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
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×SD1612yes—
129C2×C4×C53#13512yes—
130C2×C4×C53#13511yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134D212—42no—
135D212—42no—
136D212—44no—
137D212—82no—
138D212—82no—
139D212—81no—
140D212—81no—
141D212—81no—
142D212—81no—
143C8×C53—81no—
144#88322—22no—
145#88322—22no—
146#88322—42no—
147#88322—81no—
148#9735—531nomaximal, Sylow
149C22×C53#39011yescharacteristic
150C22×C53#46311yescharacteristic
151C22×C53#46411yescharacteristic
152C22×C53#46512yes—
153C22×C53#47212yes—
154C2×D53—82no—
155C2×D53—82no—
156C2×D53—82no—
157C2×D53—162no—
158C2×D53—162no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—44no—
162C4×C53—82no—
163C4×C53#41611yescharacteristic
164C4×C53#41711yescharacteristic
165C4×C53C4×SD1612yes—
166C4×C53#46311yescharacteristic
167C4×C53#46411yescharacteristic
168C4×C53#49211yescharacteristic
169C4×C53#49511yescharacteristic
170C2×#486—531no—
171#1987D5311yescharacteristic, complemented
172#1982—531no—
173C8×C4⋊C4—531no—
174C2×#334—531no—
175C2×#333—531no—
176C2×#331—531no—
177C2×C53#167712yes—
178C2×C53#167812yes—
179C2×C53#196811yescharacteristic
180C2×C53#197811yescharacteristic
181C2×C53#198311yescharacteristic
182D53—162no—
183C2×C4⋊Q8C2×D5311yescharacteristic
184C2×#127—531no—
185#486—534no—
186#486—1062no—
187C2×C4⋊C8—531no—
188C2×C4⋊C8—531no—
189C2×C4⋊C8C2×D5311yescharacteristic
190C2×#102—1061no—
191C2×#102—1061no—
192C2×#102—1062no—
193C2×C4×C8—531no—
194C2×C4×C8—1061no—
195C4×C4⋊C4C2×D5311yescharacteristic
196#340—22no—
197#336—1061no—
198#334—1062no—
199#333—1062no—
200#331—1062no—
201C53#973511yescharacteristic, complemented, Sylow
202C22×Q8—21no—
203C22×D4—1061no—
204C22×D4—1061no—
205C22×D4—1061no—
206C22×C8—21no—
207C22×C8—531no—
208C22×C8—531no—
209C22×C8—1061no—
210C22×C8—1061no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C4×D5312yes—
215#127—532no—
216#127—1061no—
217#127—1061no—
218#127—1061no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D5312yes—
222C2×C42—21no—
223C2×C42C22×D5311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—532no—
227C4⋊C8—1061no—
228C4⋊C8—1061no—
229C4⋊C8—1061no—
230#102—1064no—
231#102—1064no—
232#102—2124no—
233C4×C8—532no—
234C4×C8—532no—
235C4×C8—2122no—
236#95—21no—
237C24—2121no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×D4—1062no—
242C2×D4—1062no—
243C2×D4—1062no—
244C2×D4—4242no—
245C2×D4—2121no—
246C2×D4—2121no—
247C2×D4—2121no—
248C2×D4—2122no—
249C2×D4—2122no—
250C2×D4—2121no—
251C2×D4—2121no—
252C2×D4—2121no—
253C2×D4—2121no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4D4×D5311yescharacteristic
257C22×C4#1171994711yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—532no—
262C2×C8—532no—
263C2×C8—1061no—
264C2×C8—1061no—
265C2×C8—1061no—
266C2×C8—1061no—
267C2×C8—1062no—
268C2×C8—2121no—
269C2×C8—2122no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4#1171992414yes—
278C42—41no—
279C42—41no—
280C42C2×C4×D5311yescharacteristic
281C42#1171994511yescharacteristic
282C42D4×D5311yescharacteristic
283C42D4×D5311yescharacteristic
284C23—4241no—
285C23—4241no—
286C23—4242no—
287C23—4242no—
288C23—2122no—
289C23#1313318911yescharacteristic
290Q8—82no—
291Q8—82no—
292D4—4242no—
293D4—4242no—
294D4—4241no—
295D4—4241no—
296D4—4241no—
297D4—4241no—
298D4—2122no—
299D4—2122no—
300D4—2124no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4C22⋊C4×D5311yescharacteristic
308C2×C4#1313317711yescharacteristic
309C2×C4C4×C22⋊D5311yescharacteristic
310C2×C4#1313319011yescharacteristic
311C2×C4#1313320311yescharacteristic
312C2×C4#1313321311yescharacteristic
313C2×C4C2×#1171992412yes—
314C2×C4#1313323712yes—
315C2×C4#1313323811yescharacteristic
316C2×C4#1313324011yescharacteristic
317C2×C4#1313324411yescharacteristic
318C2×C4#1313324911yescharacteristic
319C8—81no—
320C8—1062no—
321C8—1062no—
322C8—4241no—
323C8—2122no—
324C22—8482no—
325C22—8482no—
326C22—4242no—
327C22—4242no—
328C22—4242no—
329C22#1681052311yescharacteristic
330C22#1681084011yescharacteristic
331C22#1681084711yescharacteristic
332C22#1681085412yes—
333C22#1681089112yes—
334C4—21no—
335C4—21no—
336C4—44no—
337C4—82no—
338C4#1681061811yescharacteristic
339C4#1681062311yescharacteristic
340C4C4×#1171992412yes—
341C4#1681084111yescharacteristic
342C4#1681084811yescharacteristic
343C4#1681099711yescharacteristic
344C4#1681100511yescharacteristic
345C2—8482no—
346C2#2027412612yes—
347C2#2027413312yes—
348C2#2027557211yescharacteristic
349C2#2027562411yescharacteristic
350C2#2027564411yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1987C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1987D53
hypocenterC53#9735
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×#88322C2semidirect
4. C2×#16810373C2semidirect
5. C2×#16810372C2semidirect
7. C53×#1987C2semidirect
10. #16810971C4semidirect
171. #1987D53semidirect
201. C53#9735semidirect

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×#168103732
C2×#168103722
C2×#168103662
C53×#19872
#973553