Finite Group Databasebeta

← #27357156

Subgroups — #27357156

Order 13568.

974 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288518C211yesmaximal, characteristic
2#20284754C212yesmaximal, complemented
3#20273739C214yesmaximal, complemented
4#20273671C214yesmaximal, complemented
5#20273278C212yesmaximal, complemented
6#20273036C211yesmaximal, characteristic
7C53×#2580C211yesmaximal, characteristic
8#16811332—24no—
9#16811330—22no—
10D4×Dic106—22no—
11#16811254—21no—
12#16811249C2214yes—
13#16811180C2212yes—
14C2×#13133196—22no—
15#16811098—21no—
16#16810958C2214yes—
17#16810918—22no—
18#16810833—22no—
19#16810779C2214yes—
20C4×#11719925—24no—
21#16810676C2212yes—
22#16810359C2218yes—
23#16810347C2214yes—
24C2×C4×Q8×C53C2211yescharacteristic
25C2×C4×D4×C53C2211yescharacteristic
26C53×#399C2212yes—
27C53×#393C2212yes—
28C53×#390—21no—
29C53×#389—21no—
30C53×#385—22no—
31C53×#382C2211yescharacteristic
32C22×Dic106—22no—
33C2×#11719925—24no—
34#13133217D412yes—
35#13133217D412yes—
36#13133216—24no—
37#13133209—22no—
38#13133208—24no—
39#13133196—24no—
40#13133196—24no—
41#13133196—22no—
42#13133196—24no—
43#13133196C2314yes—
44#13133194—28no—
45#13133194—28no—
46C4×Dic106—22no—
47C4×Dic106D414yes—
48D4×Dic53—24no—
49D4×Dic53—24no—
50#13133176—22no—
51C2×C4⋊Dic53—22no—
52C2×#11719919—24no—
53C2×C22⋊Dic53—24no—
54#13133150—24no—
55C4⋊#88322—22no—
56#13133141—24no—
57#13133140—24no—
58C22⋊#88322C2314yes—
59C4×#88322—22no—
60C22×Q8×C53C2311yescharacteristic
61C22×D4×C53D411yescharacteristic
62C23×C4×C53D412yes—
63C4×Q8×C53—22no—
64C4×Q8×C53—22no—
65C4×D4×C53—22no—
66C4×D4×C53—22no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C4⋊C4×C53C2311yescharacteristic
72C2×C22⋊C4×C53—22no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53C2312yes—
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77#95×C53—21no—
78#95×C53—22no—
79#95×C53C2312yes—
80C24×C53C2×D412yes—
81C2×Q8×C53—22no—
82C2×Q8×C53—42no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53Q1614yescomplemented
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—82no—
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2411yescharacteristic
96C22×Dic53—24no—
97C22×Dic53—24no—
98C2×Dic106—22no—
99C2×Dic106—24no—
100C2×Dic106—84no—
101C2×Dic106C2×D414yes—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—82no—
108C22⋊C4×C53—24no—
109C22⋊C4×C53—44no—
110C22⋊C4×C53—82no—
111C42×C53—42no—
112C42×C53C2×D412yes—
113C42×C53C2×D412yes—
114C22⋊Dic53—28no—
115C22⋊Dic53—24no—
116#11719925—48no—
117C4⋊Dic53—24no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53C2×D414yes—
122#11719919—28no—
123#11719919—24no—
124#11719919—24no—
125C4×Dic53—24no—
126C2×#88322—24no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53C2×Q1614yes—
132C23×C53C22×D411yescharacteristic
133Q8×C53—24no—
134Q8×C53—42no—
135D4×C53—28no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—44no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12012yes—
154C2×C4×C53C2×Q1612yes—
155C2×C4×C53#13712yes—
156C2×C4×C53C22×D412yes—
157C2×Dic53—24no—
158C2×Dic53—24no—
159C2×Dic53—84no—
160C2×Dic53—84no—
161Dic106—44no—
162Dic106—44no—
163Dic106—84no—
164#88322—44no—
165#18213—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—82no—
173C22×C53C2×#12011yescharacteristic
174C22×C53C22×Q1611yescharacteristic
175C22×C53C2×#13711yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—42no—
180C4×C53—42no—
181C4×C53—82no—
182C4×C53—81no—
183C4×C53#45114yes—
184Dic53—44no—
185Dic53—44no—
186#3501—531no—
187#3104—532no—
188#2580D5311yescharacteristic
189#1614—534no—
190#1602—534no—
191#1518—532no—
192#1460—531no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—44no—
196C2×C53C2×#45111yescharacteristic
197C2×C53#301911yescharacteristic
198C2×C53#302511yescharacteristic
199#554—1064no—
200#554—1062no—
201D4×Q8—1062no—
202#544—1061no—
203#543—534no—
204#533—532no—
205C2×C22⋊Q8—1062no—
206#518—1061no—
207C2×C4×Q8C2×D5311yescharacteristic
208C2×C4×D4C2×D5311yescharacteristic
209#484—534no—
210#477—1062no—
211#462—1062no—
212#451—534no—
213C4×Q16—1064no—
214#431—532no—
215#399C2×D5312yes—
216#393C2×D5312yes—
217#390—21no—
218#389—21no—
219#385—22no—
220#382C2×D5311yescharacteristic
221#328—538no—
222#323—534no—
223C53#1821311yescharacteristic, complemented, Sylow
224C22×Q8—1062no—
225C22×Q8C22×D5311yescharacteristic
226C22×D4C22⋊D5311yescharacteristic
227C23×C4C22⋊D5312yes—
228C2×Q16—1064no—
229C4⋊Q8—532no—
230C4⋊Q8—532no—
231C4⋊Q8—1064no—
232#125—1064no—
233#125—1062no—
234C22⋊Q8—534no—
235C22⋊Q8—1068no—
236C22⋊Q8—1068no—
237C22⋊Q8—1064no—
238C22⋊Q8—1064no—
239C22⋊Q8—1062no—
240C22⋊Q8—1064no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—534no—
244C4×Q8—1062no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—1064no—
248C4×D4—1064no—
249#117—1062no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—1064no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4C22⋊D5311yescharacteristic
256C2×C4⋊C4C22×D5311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—1064no—
260C2×C22⋊C4C22×D5312yes—
261C2×C42—21no—
262C2×C42C22×D5311yescharacteristic
263#107—1064no—
264C4⋊C8—1062no—
265#103—1064no—
266#103—1064no—
267C22⋊C8—534no—
268C4×C8—1062no—
269#95—21no—
270#95—22no—
271#95C22×D5312yes—
272C24C2×C22⋊D5312yes—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—534no—
276C2×Q8—1064no—
277C2×Q8—1062no—
278C2×Q8—4244no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4#1171992514yescomplemented
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—82no—
288C22×C4—1064no—
289C22×C4—1064no—
290C22×C4C2×C22⋊D5312yes—
291C22×C4C2×C22⋊D5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293C22×C4C23×D5311yescharacteristic
294Q16—2128no—
295C2×C8—1064no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—534no—
303C4⋊C4—1064no—
304C4⋊C4—1068no—
305C4⋊C4—1064no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—1064no—
310C22⋊C4—24no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—1068no—
314C22⋊C4—1064no—
315C42—42no—
316C42—1064no—
317C42D4×D5312yes—
318C42D4×D5312yes—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23C2×#1171992514yes—
324C23C22×C22⋊D5311yescharacteristic
325Q8—24no—
326Q8—42no—
327Q8—4244no—
328Q8—2124no—
329Q8—2124no—
330D4—28no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—1064no—
347C2×C4—1064no—
348C2×C4—4244no—
349C2×C4—4244no—
350C2×C4#1313318511yescharacteristic
351C2×C4#1313318511yescharacteristic
352C2×C4#1313318512yes—
353C2×C4C2×#1171992512yes—
354C2×C4#1313324712yes—
355C2×C4C2×D4×D5312yes—
356C8—2124no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—44no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22C2×#1313318511yescharacteristic
365C22C22×#1171992511yescharacteristic
366C22C2×#1313324711yescharacteristic
367C4—21no—
368C4—21no—
369C4—22no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4—2124no—
375C4—2124no—
376C4#1681077814yes—
377C2—21no—
378C2—21no—
379C2—44no—
380C2C2×#1681077811yescharacteristic
381C2#2028407111yescharacteristic
382C2#2028410611yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC22×C53C2×#120
prime radicalC22×D4×C53D4
hypercenter#2580D53
hypocenterC53#18213
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #20284754C2semidirect
3. #20273739C2semidirect
4. #20273671C2semidirect
5. #20273278C2semidirect
85. C2×D4×C53Q16semidirect
223. C53#18213semidirect
281. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202885182
#202847542
#202737392
#202736712
#202732782
#202730362
C53×#25802
#1821353