Finite Group Databasebeta

← #27439983

Subgroups — #27439983

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288525C211yesmaximal, characteristic
2#20285264C212yesmaximal
3#20276065C212yesmaximal
4#20276060C212yesmaximal
5#20275658C212yesmaximal
6#20274929C212yesmaximal
7#20274929C212yesmaximal
8C4×#13133136C211yesmaximal, characteristic
9C53×#2565C211yesmaximal, characteristic
10Q8×Dic106—22no—
11#16811352—22no—
12#16811349—24no—
13#16811254—22no—
14#16811254C2212yes—
15#16811180—22no—
16#16811180C2212yes—
17#16811101C2212yes—
18#16811033—22no—
19#16811033—22no—
20#16811022—24no—
21#16810663C2214yes—
22#16810663C2214yes—
23C2×#13133136C2212yes—
24C4×#11719917C2212yes—
25#16810392C2212yes—
26#16810383C2214yes—
27#16810383C2214yes—
28C2×#125×C53C2211yescharacteristic
29C2×C4×Q8×C53C2211yescharacteristic
30C53×#398—22no—
31C53×#391C2211yescharacteristic
32C53×#391C2211yescharacteristic
33C53×#387—22no—
34C53×#384C2211yescharacteristic
35C53×#382C2211yescharacteristic
36C4×C4⋊C4×C53C2211yescharacteristic
37C2×#11719917C2314yes—
38#13133217—22no—
39#13133217—22no—
40#13133217D412yes—
41#13133217D412yes—
42#13133216—28no—
43#13133216—24no—
44#13133209—24no—
45#13133209D414yes—
46#13133208—28no—
47#13133208—24no—
48#13133196—22no—
49#13133196—24no—
50#13133196—22no—
51Q8×Dic53—24no—
52Q8×Dic53—24no—
53C4×Dic106—24no—
54C4×Dic106—24no—
55C4×Dic106—22no—
56C4×Dic106—22no—
57#13133176—24no—
58#13133176C2314yes—
59#13133150—24no—
60#13133150—24no—
61#13133148—24no—
62#13133148—24no—
63#13133136Q818yescomplemented
64#13133128—22no—
65C4×#88322—22no—
66C22×Q8×C53C2311yescharacteristic
67#125×C53—22no—
68#125×C53—22no—
69C4×Q8×C53—22no—
70C4×Q8×C53—22no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53D412yes—
76C2×C4⋊C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53C2311yescharacteristic
79C2×C42×C53D412yes—
80C2×C42×C53C2311yescharacteristic
81C2×C42×C53C2311yescharacteristic
82C2×C42×C53C2311yescharacteristic
83#95×C53—21no—
84#95×C53—22no—
85#95×C53—22no—
86#95×C53—22no—
87C2×Q8×C53—22no—
88C2×Q8×C53—22no—
89C2×Q8×C53C2×Q814yes—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2411yescharacteristic
99C2×Dic106—22no—
100C2×Dic106—24no—
101C2×Dic106—22no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—44no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53—41no—
111C42×C53—22no—
112C42×C53—22no—
113C42×C53—22no—
114C42×C53—22no—
115C42×C53C2×D412yes—
116C42×C53C2×D412yes—
117C22⋊Dic53—24no—
118C4⋊Dic53—24no—
119C4⋊Dic53—24no—
120C4⋊Dic53—24no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53C2×D414yes—
124C4⋊Dic53C2×D414yes—
125#11719919—28no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53—28no—
129C4×Dic53—24no—
130#11719917—24no—
131#11719917C2×Q818yes—
132C2×#88322—24no—
133C23×C53C22×D411yescharacteristic
134Q8×C53—28no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12012yes—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53C4⋊Q814yes—
155C2×C4×C53C22×D412yes—
156C2×C4×C53C22×Q812yes—
157C2×C4×C532−1+412yes—
158C2×Dic53—24no—
159C2×Dic53—24no—
160Dic106—44no—
161Dic106—82no—
162Dic106—84no—
163#88322—28no—
164#25827—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53C2×#12011yescharacteristic
168C22×C53C2×C4⋊Q811yescharacteristic
169C22×C53#53611yescharacteristic
170C4×C53—24no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—42no—
175C4×C53—42no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53D4×Q814yes—
180Dic53—44no—
181Dic53—44no—
182#3502—531no—
183#3166—532no—
184#2565D5311yescharacteristic
185#2061—532no—
186#2060—532no—
187#1986—532no—
188#1857—532no—
189#1857—532no—
190C4×#101—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#243411yescharacteristic
194C2×C53#301811yescharacteristic
195C2×C53#302011yescharacteristic
196Q82—1062no—
197#557—1064no—
198#557—1062no—
199#544—532no—
200#544—1062no—
201#533—532no—
202#533—1062no—
203C2×#125C2×D5311yescharacteristic
204#519—532no—
205C2×C4×Q8C2×D5311yescharacteristic
206#501—1062no—
207#501—1062no—
208#499—1064no—
209#428—534no—
210#428—534no—
211C2×#101—532no—
212C4×M16—532no—
213#398—22no—
214#391C2×D5311yescharacteristic
215#391C2×D5311yescharacteristic
216#387—22no—
217#384C2×D5311yescharacteristic
218#382C2×D5311yescharacteristic
219C4×C4⋊C4C2×D5311yescharacteristic
220#343—532no—
221#339—534no—
222#339—534no—
223C53#2582711yescharacteristic, complemented, Sylow
224C22×Q8C22×D5311yescharacteristic
225C2×M16—534no—
226C4⋊Q8—532no—
227C4⋊Q8—532no—
228C4⋊Q8—1062no—
229C4⋊Q8—1062no—
230C4⋊Q8—1064no—
231C4⋊Q8—1068no—
232#125—22no—
233#125—22no—
234#125—534no—
235#125—1064no—
236#125—1064no—
237#125—1068no—
238C22⋊Q8—1062no—
239C22⋊Q8—1062no—
240C22⋊Q8—1064no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—1064no—
244C4×Q8—1064no—
245C4×Q8—1064no—
246C4×Q8—1064no—
247C4×Q8—1062no—
248C4×Q8—1062no—
249#117—534no—
250#117—1064no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C22⋊D5312yes—
256C2×C4⋊C4C22×D5311yescharacteristic
257C2×C4⋊C4C22×D5311yescharacteristic
258C2×C4⋊C4C22×D5311yescharacteristic
259C2×C42C22⋊D5312yes—
260C2×C42C22×D5311yescharacteristic
261C2×C42C22×D5311yescharacteristic
262C2×C42C22×D5311yescharacteristic
263#107—1064no—
264#107—1064no—
265#106—1064no—
266#106—1064no—
267#101—538no—
268#97—1062no—
269C4×C8—1062no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—1064no—
277C2×Q8—1062no—
278C2×Q8—1062no—
279C2×Q8Q8×D5314yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4C2×C22⋊D5312yes—
285C22×C4C2×C22⋊D5312yes—
286C22×C4C2×C22⋊D5311yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288C22×C4C23×D5311yescharacteristic
289M16—538no—
290M16—1064no—
291C2×C8—1064no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—534no—
302C4⋊C4—534no—
303C4⋊C4—1064no—
304C4⋊C4—1064no—
305C4⋊C4—1064no—
306C4⋊C4—1068no—
307C4⋊C4—1064no—
308C4⋊C4—1062no—
309C4⋊C4—1064no—
310C4⋊C4—1062no—
311C22⋊C4—1064no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—1068no—
317C42—1064no—
318C42D4×D5312yes—
319C42D4×D5312yes—
320C23C22×C22⋊D5311yescharacteristic
321Q8—28no—
322Q8—4242no—
323Q8—4244no—
324Q8—2124no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—1064no—
342C2×C4—1064no—
343C2×C4#1313318511yescharacteristic
344C2×C4#1313318511yescharacteristic
345C2×C4#1313318512yes—
346C2×C4#1313321514yes—
347C2×C4C2×D4×D5312yes—
348C2×C4C2×Q8×D5312yes—
349C2×C4#1313326412yes—
350C8—1068no—
351C22—21no—
352C22—21no—
353C22C2×#1313318511yescharacteristic
354C22C2×#1313321511yescharacteristic
355C22#1681119411yescharacteristic
356C4—24no—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—42no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—2124no—
366C4—2124no—
367C4#1681128614yes—
368C2—21no—
369C2—21no—
370C2#2027836011yescharacteristic
371C2#2028406311yescharacteristic
372C2#2028407811yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2565C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#2565D53
hypocenterC53#25827
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2565

Complements

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

NH ≅ G/Ntype
63. #13133136Q8semidirect
223. C53#25827semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202852642
#202760652
#202760602
#202756582
#202749292
C4×#131331362
C53×#25652
#2582753