Finite Group Databasebeta

← #27412235

Subgroups — #27412235

Order 13568.

1118 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288120C211yesmaximal, characteristic
2#20277338C212yesmaximal
3#20277335C211yesmaximal, characteristic
4#20276069C218yesmaximal
5#20274447C212yesmaximal
6C53×#1740C211yesmaximal, characteristic
7#16811254—21no—
8#16811254—21no—
9#16811251—22no—
10#16811251—22no—
11#16811249C2214yes—
12#16811180—22no—
13#16811176—24no—
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2212yes—
16C22×C4⋊Dic53C2211yescharacteristic
17#16810522—24no—
18#16810521—22no—
19#16810498C2218yes—
20#16810495C2214yes—
21#16810493—22no—
22#16810492—21no—
23#16810489—22no—
24#16810488—21no—
25#16810389C2218yes—
26C2×#117×C53C2212yes—
27C2×#117×C53C2211yescharacteristic
28C53×#342C2214yes—
29C23×Dic53C2312yes—
30#13133217—22no—
31#13133217—22no—
32#13133216—24no—
33#13133216—24no—
34#13133209—24no—
35#13133208—24no—
36#13133208—24no—
37#13133196—24no—
38#13133196—24no—
39#13133194—28no—
40#13133194—28no—
41C4×Dic106—24no—
42C4×Dic106—24no—
43#13133176—22no—
44#13133176—22no—
45#13133176—22no—
46#13133176—22no—
47#13133176—22no—
48#13133176Q814yescomplemented
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—28no—
55C2×C4⋊Dic53D412yes—
56C2×C4⋊Dic53D412yes—
57C2×#11719919—24no—
58C2×C22⋊Dic53C2312yes—
59C2×C22⋊Dic53C2314yes—
60C2×C4×Dic53—21no—
61C2×C4×Dic53D414yes—
62#13133124—24no—
63#13133124—28no—
64C23×C4×C53C2311yescharacteristic
65#117×C53—22no—
66#117×C53—22no—
67#117×C53—24no—
68#117×C53Q818yescomplemented
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53C2314yes—
73C2×C22⋊C4×C53C2312yes—
74C2×C42×C53—22no—
75C2×C42×C53D412yes—
76C24×C53C2411yescharacteristic
77C22×C4×C53—24no—
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D412yes—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×Q814yes—
85C22×C4×C53C2×Q812yes—
86C22×C4×C53#4112yes—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—24no—
90C22×Dic53—82no—
91C22×Dic53C2×D414yes—
92C22×Dic53C2×D414yes—
93C2×Dic106—24no—
94C2×Dic106—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—28no—
98C22⋊C4×C53—24no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53C2×Q818yes—
101C42×C53—22no—
102C42×C53—22no—
103C42×C53—28no—
104C22⋊Dic53—24no—
105C22⋊Dic53—24no—
106C22⋊Dic53—22no—
107C22⋊Dic53C2×Q814yes—
108C4⋊Dic53—24no—
109C4⋊Dic53—24no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112C4⋊Dic53—22no—
113C4⋊Dic53—22no—
114C4⋊Dic53—24no—
115C4⋊Dic53—41no—
116C4⋊Dic53—42no—
117C4⋊Dic53—41no—
118C4⋊Dic53—88no—
119#11719919—28no—
120#11719919—28no—
121#11719919—84no—
122C4×Dic53—24no—
123C4×Dic53—24no—
124C4×Dic53—24no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C22×D412yes—
128C23×C53C22×D411yescharacteristic
129C23×C53C22×Q811yescharacteristic
130C23×C53C22×Q812yes—
131C23×C53C2×#4111yescharacteristic
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—28no—
135C2×C4×C53—44no—
136C2×C4×C53—44no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12012yes—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53C22⋊Q814yes—
143C2×C4×C53C22⋊Q814yes—
144C2×C4×C53C22⋊Q812yes—
145C2×C4×C53C22⋊Q812yes—
146C2×C4×C53C4⋊Q814yes—
147C2×C4×C53C4⋊Q814yes—
148C2×C4×C53C4⋊Q812yes—
149C2×C4×C53C4⋊Q812yes—
150C2×Dic53—24no—
151C2×Dic53—24no—
152C2×Dic53—24no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—44no—
156C2×Dic53—44no—
157C2×Dic53—44no—
158C2×Dic53—48no—
159C2×Dic53—42no—
160Dic106—84no—
161Dic106—84no—
162#23566—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—42no—
169C22×C53—41no—
170C22×C53C2×#12011yescharacteristic
171C22×C53C2×C22⋊Q812yes—
172C22×C53C2×C22⋊Q811yescharacteristic
173C22×C53C2×C4⋊Q812yes—
174C22×C53C2×C4⋊Q811yescharacteristic
175C4×C53—48no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53#39512yes—
179C4×C53#39514yes—
180C4×C53#39512yes—
181Dic53—44no—
182Dic53—44no—
183Dic53—44no—
184Dic53—82no—
185#3458—531no—
186#2317—531no—
187#2317—532no—
188#2062—538no—
189#1740—532no—
190#1740D5311yescharacteristic
191C2×C53—21no—
192C2×C53—22no—
193C2×C53—22no—
194C2×C53—21no—
195C2×C53C2×#39511yescharacteristic
196C2×C53#302711yescharacteristic
197C2×C53#302811yescharacteristic
198#544—1062no—
199#544—1062no—
200#544—1061no—
201#544—1061no—
202#543—534no—
203#533—1064no—
204#533—1062no—
205C2×#117—532no—
206C2×#117—531no—
207C2×#117C2×D5312yes—
208C2×#117C2×D5311yescharacteristic
209C22×C4⋊C4—531no—
210#389—1064no—
211#389—1062no—
212#385—538no—
213#385—534no—
214#384—1062no—
215#384—1061no—
216#382—1062no—
217#382—1061no—
218#342—538no—
219#342C2×D5314yes—
220C53#2356611yescharacteristic, complemented, Sylow
221C23×C4—532no—
222C23×C4C22×D5311yescharacteristic
223C4⋊Q8—1062no—
224C4⋊Q8—1062no—
225C4⋊Q8—1064no—
226C4⋊Q8—1064no—
227#125—1064no—
228#125—1064no—
229#125—1064no—
230C22⋊Q8—1068no—
231C22⋊Q8—1068no—
232C22⋊Q8—1064no—
233C22⋊Q8—1064no—
234C4×Q8—1064no—
235C4×Q8—1064no—
236#117—22no—
237#117—22no—
238#117—24no—
239#117—534no—
240#117—1062no—
241#117—1062no—
242#117—1062no—
243#117—1062no—
244#117—1062no—
245#117Dic10618yescomplemented
246C2×C4⋊C4—22no—
247C2×C4⋊C4—532no—
248C2×C4⋊C4—532no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1068no—
251C2×C4⋊C4—1064no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1062no—
256C2×C4⋊C4D21211yescharacteristic
257C2×C4⋊C4D21211yescharacteristic
258C2×C22⋊C4—532no—
259C2×C22⋊C4—534no—
260C2×C22⋊C4C22×D5312yes—
261C2×C22⋊C4C22×D5314yes—
262C2×C42—22no—
263C2×C42—534no—
264C2×C42—1061no—
265C2×C42D21212yes—
266#95—1068no—
267#95—1064no—
268C24C23×D5311yescharacteristic
269C2×Q8—1064no—
270C2×Q8—1064no—
271C22×C4—24no—
272C22×C4—534no—
273C22×C4—534no—
274C22×C4—1062no—
275C22×C4—1064no—
276C22×C4—1062no—
277C22×C4—4242no—
278C22×C4C2×Dic10614yes—
279C22×C4C2×D21212yes—
280C22×C4C2×D21212yes—
281C22×C4C2×D21211yescharacteristic
282C22×C4C2×D21211yescharacteristic
283C22×C4D4×D5312yes—
284C22×C4D4×D5312yes—
285C22×C4#1171994712yes—
286C22×C4Q8×D5312yes—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—28no—
290C4⋊C4—1068no—
291C4⋊C4—1068no—
292C4⋊C4—1064no—
293C4⋊C4—1064no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—1062no—
297C4⋊C4—1062no—
298C4⋊C4—1064no—
299C4⋊C4—4244no—
300C4⋊C4—4248no—
301C4⋊C4—2121no—
302C4⋊C4—2121no—
303C4⋊C4—2122no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—534no—
307C22⋊C4—1064no—
308C22⋊C4—1064no—
309C22⋊C4—1062no—
310C22⋊C4C2×Dic10618yes—
311C42—22no—
312C42—22no—
313C42—28no—
314C42—1064no—
315C42—1064no—
316C42—1064no—
317C23—41no—
318C23—41no—
319C23C2×D4×D5312yes—
320C23C22×D21211yescharacteristic
321C23C2×Q8×D5311yescharacteristic
322C23C22×Dic10612yes—
323C23C2×#1171994711yescharacteristic
324Q8—4244no—
325Q8—4244no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—1064no—
334C2×C4—1064no—
335C2×C4—1064no—
336C2×C4—2128no—
337C2×C4—2124no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2124no—
342C2×C4—2124no—
343C2×C4#1313318711yescharacteristic
344C2×C4#1313318711yescharacteristic
345C2×C4#1313318712yes—
346C2×C4#1313319414yes—
347C2×C4#1313319414yes—
348C2×C4#1313319712yes—
349C2×C4#1313319712yes—
350C2×C4#1313321614yes—
351C2×C4#1313321614yes—
352C2×C4#1313321712yes—
353C2×C4#1313321712yes—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22C2×#1313318711yescharacteristic
362C22C2×#1313319412yes—
363C22C2×#1313319711yescharacteristic
364C22C2×#1313321612yes—
365C22C2×#1313321711yescharacteristic
366C4—48no—
367C4—42no—
368C4—42no—
369C4—4242no—
370C4—2124no—
371C4—2124no—
372C4—2124no—
373C4#1681054412yes—
374C4#1681054412yes—
375C4#1681054414yes—
376C2—22no—
377C2—21no—
378C2—22no—
379C2—21no—
380C2C2×#1681054411yescharacteristic
381C2#2028411811yescharacteristic
382C2#2028412211yescharacteristic
subgroupG/H
centerC22C2×#13133187
commutator, prime radical, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#1740C2
socleC22×C53C2×#120
hypercenter#1740D53
hypocenterC53#23566
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
48. #13133176Q8semidirect
68. #117×C53Q8semidirect
220. C53#23566semidirect
245. #117Dic106semidirect

Maximal subgroups

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

subgroupindex
#202881202
#202773382
#202773352
#202760692
#202744472
C53×#17402
#2356653