Finite Group Databasebeta

← #27412234

Subgroups — #27412234

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#20277339C212yesmaximal
3#20276068C218yesmaximal
4#20274448C211yesmaximal, characteristic
5#20274447C212yesmaximal
6C53×#2317C211yesmaximal, characteristic
7#16811254—21no—
8#16811254—21no—
9#16811251—22no—
10#16811251—22no—
11#16811249C2214yes—
12#16811180—22no—
13#16811176—24no—
14C2×#13133176C2212yes—
15C2×#13133176C2212yes—
16#16810519—24no—
17#16810496C2218yes—
18#16810492—22no—
19#16810488—22no—
20#16810390C2214yes—
21#16810389C2218yes—
22C2×#117×C53C2211yescharacteristic
23C2×#117×C53C2211yescharacteristic
24C22×C4⋊C4×C53C2211yescharacteristic
25C53×#389—22no—
26C53×#385C2214yes—
27C53×#384—21no—
28C53×#382—21no—
29#13133217—22no—
30#13133217—22no—
31#13133216—24no—
32#13133216—24no—
33#13133209—24no—
34#13133208—24no—
35#13133208—24no—
36#13133196—24no—
37#13133196—24no—
38#13133194—28no—
39#13133194—28no—
40C4×Dic106—24no—
41C4×Dic106—24no—
42#13133176—22no—
43#13133176—22no—
44#13133176—22no—
45#13133176—22no—
46#13133176—24no—
47#13133176Q818yescomplemented
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53D412yes—
50C2×C4⋊Dic53D412yes—
51C2×#11719919—28no—
52C2×C22⋊Dic53C2314yes—
53C2×C22⋊Dic53C2314yes—
54C2×C4×Dic53—22no—
55C2×C4×Dic53D414yes—
56#13133124—28no—
57C23×C4×C53C2312yes—
58C23×C4×C53C2311yescharacteristic
59#117×C53—22no—
60#117×C53—22no—
61#117×C53—22no—
62#117×C53Q814yescomplemented
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—24no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53C2312yes—
72C2×C22⋊C4×C53C2312yes—
73C2×C42×C53—21no—
74C2×C42×C53D412yes—
75#95×C53—24no—
76C24×C53C2411yescharacteristic
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—24no—
80C22×C4×C53—82no—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D412yes—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×Q812yes—
88C22×C4×C53C2×Q814yes—
89C22×C4×C53#4112yes—
90C22×Dic53—24no—
91C22×Dic53C2×D414yes—
92C22×Dic53C2×D414yes—
93C2×Dic106—24no—
94C2×Dic106—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—84no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53C2×Q814yes—
105C42×C53—22no—
106C42×C53—22no—
107C42×C53—24no—
108C22⋊Dic53—24no—
109C22⋊Dic53—24no—
110C22⋊Dic53—24no—
111C22⋊Dic53C2×Q818yes—
112C4⋊Dic53—24no—
113C4⋊Dic53—24no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—28no—
119#11719919—28no—
120#11719919—28no—
121#11719919—88no—
122C4×Dic53—24no—
123C4×Dic53—24no—
124C4×Dic53—28no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C22×D412yes—
128C23×C53C22×D411yescharacteristic
129C23×C53C22×Q812yes—
130C23×C53C22×Q811yescharacteristic
131C23×C53C2×#4111yescharacteristic
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—24no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—48no—
140C2×C4×C53—44no—
141C2×C4×C53—42no—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12012yes—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53C22⋊Q814yes—
146C2×C4×C53C22⋊Q814yes—
147C2×C4×C53C22⋊Q812yes—
148C2×C4×C53C22⋊Q812yes—
149C2×C4×C53C4⋊Q812yes—
150C2×C4×C53C4⋊Q812yes—
151C2×C4×C53C4⋊Q814yes—
152C2×C4×C53C4⋊Q814yes—
153C2×Dic53—24no—
154C2×Dic53—24no—
155C2×Dic53—28no—
156C2×Dic53—44no—
157C2×Dic53—44no—
158C2×Dic53—44no—
159C2×Dic53—44no—
160Dic106—84no—
161Dic106—84no—
162#23566—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—42no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C2×#12011yescharacteristic
171C22×C53C2×C22⋊Q812yes—
172C22×C53C2×C22⋊Q811yescharacteristic
173C22×C53C2×C4⋊Q811yescharacteristic
174C22×C53C2×C4⋊Q812yes—
175C4×C53—44no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53—82no—
179C4×C53#39512yes—
180C4×C53#39514yes—
181C4×C53#39512yes—
182Dic53—48no—
183Dic53—44no—
184Dic53—44no—
185#3458—531no—
186#2317—532no—
187#2317D5311yescharacteristic
188#2062—538no—
189#1740—532no—
190#1740—531no—
191C2×C53—22no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
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—532no—
207C2×#117C2×D5311yescharacteristic
208C2×#117C2×D5311yescharacteristic
209C22×C4⋊C4C2×D5311yescharacteristic
210#389—22no—
211#389—1064no—
212#385—538no—
213#385C2×D5314yes—
214#384—21no—
215#384—1062no—
216#382—21no—
217#382—1062no—
218#342—538no—
219#342—534no—
220C53#2356611yescharacteristic, complemented, Sylow
221C23×C4C22×D5312yes—
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—22no—
239#117—538no—
240#117—1064no—
241#117—1062no—
242#117—1062no—
243#117—1062no—
244#117—1062no—
245#117Dic10614yescomplemented
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—532no—
253C2×C4⋊C4—532no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1068no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C4⋊C4C22⋊D5311yescharacteristic
258C2×C22⋊C4—534no—
259C2×C22⋊C4—534no—
260C2×C22⋊C4C22×D5312yes—
261C2×C22⋊C4C22×D5312yes—
262C2×C42—21no—
263C2×C42—534no—
264C2×C42—1062no—
265C2×C42C22⋊D5312yes—
266#95—24no—
267#95—1068no—
268C24C23×D5311yescharacteristic
269C2×Q8—1064no—
270C2×Q8—1064no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—82no—
275C22×C4—534no—
276C22×C4—534no—
277C22×C4—1064no—
278C22×C4C2×Dic10612yes—
279C22×C4#1171994512yes—
280C22×C4D4×D5312yes—
281C22×C4D4×D5312yes—
282C22×C4Q8×D5314yes—
283C22×C4C2×C22⋊D5312yes—
284C22×C4C2×C22⋊D5312yes—
285C22×C4C2×C22⋊D5311yescharacteristic
286C22×C4C2×C22⋊D5311yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—84no—
294C4⋊C4—1068no—
295C4⋊C4—1068no—
296C4⋊C4—1064no—
297C4⋊C4—1064no—
298C4⋊C4—1062no—
299C4⋊C4—1062no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—1068no—
303C4⋊C4—4248no—
304C22⋊C4—24no—
305C22⋊C4—22no—
306C22⋊C4—538no—
307C22⋊C4—1064no—
308C22⋊C4—1064no—
309C22⋊C4—1064no—
310C22⋊C4C2×Dic10614yes—
311C42—22no—
312C42—22no—
313C42—24no—
314C42—1064no—
315C42—1064no—
316C42—1068no—
317C23—41no—
318C23—41no—
319C23C2×D4×D5312yes—
320C23C22×C22⋊D5311yescharacteristic
321C23C2×Q8×D5312yes—
322C23C22×Dic10611yescharacteristic
323C23C2×#1171994511yescharacteristic
324Q8—4244no—
325Q8—4244no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—1064no—
337C2×C4—1064no—
338C2×C4—1068no—
339C2×C4—2124no—
340C2×C4—2124no—
341C2×C4—2124no—
342C2×C4—2124no—
343C2×C4#1313318511yescharacteristic
344C2×C4#1313318511yescharacteristic
345C2×C4#1313318512yes—
346C2×C4#1313319514yes—
347C2×C4#1313319514yes—
348C2×C4#1313319612yes—
349C2×C4#1313319612yes—
350C2×C4#1313321512yes—
351C2×C4#1313321512yes—
352C2×C4#1313321614yes—
353C2×C4#1313321614yes—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22C2×#1313318511yescharacteristic
362C22C2×#1313319512yes—
363C22C2×#1313319611yescharacteristic
364C22C2×#1313321511yescharacteristic
365C22C2×#1313321612yes—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—82no—
370C4—2128no—
371C4—2124no—
372C4—2124no—
373C4#1681054312yes—
374C4#1681054312yes—
375C4#1681054314yes—
376C2—21no—
377C2—22no—
378C2—21no—
379C2—22no—
380C2C2×#1681054311yescharacteristic
381C2#2028411611yescharacteristic
382C2#2028412011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, prime radical, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#2317C2
socleC22×C53C2×#120
hypercenter#2317D53
hypocenterC53#23566
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
47. #13133176Q8semidirect
62. #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
#202773392
#202760682
#202744482
#202744472
C53×#23172
#2356653