Finite Group Databasebeta

← #27357150

Subgroups — #27357150

Order 13568.

1058 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 203 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#20284719C212yesmaximal, complemented
2#20279612C211yesmaximal, characteristic
3#20273706C214yesmaximal, complemented
4#20273638C214yesmaximal, complemented
5#20273279C212yesmaximal, complemented
6#20273036C211yesmaximal, characteristic, complemented
7C53×#3501C211yesmaximal, characteristic, complemented
8C2×#13133188—21no—
9C2×C4×D212C2211yescharacteristic
10#16810951C2214yes—
11#16810910—22no—
12#16810825—22no—
13#16810773C2214yes—
14C4×#11719924—24no—
15#16810676C2212yes—
16#16810548C2212yes—
17#16810532C2212yes—
18#16810502—22no—
19#16810489C2211yescharacteristic
20#16810480—21no—
21#16810360C2218yes—
22#16810347C2214yes—
23C53×#554—22no—
24C53×#554—24no—
25D4×Q8×C53—22no—
26C53×#544—21no—
27C53×#543C2214yes—
28C53×#533C2212yes—
29C2×C22⋊Q8×C53—22no—
30C53×#518—21no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—22no—
34C22×C4×D53—21no—
35C2×#11719924—24no—
36#13133188—44no—
37C4×D212—22no—
38C4×D212—22no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×C22⋊Dic53—22no—
42C2×#11719921—21no—
43C2×#11719921—22no—
44#13133148—24no—
45C4⋊#88322—22no—
46#13133141—24no—
47#13133137—24no—
48C22⋊#88322C2314yes—
49C4×#88322—22no—
50#13133124—21no—
51#13133124—22no—
52#13133124C2312yes—
53C22×Q8×C53—22no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53D412yes—
56C4⋊Q8×C53—24no—
57C4⋊Q8×C53D412yescomplemented
58C4⋊Q8×C53D412yescomplemented
59#125×C53—24no—
60#125×C53—22no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53C2314yes—
68C4×Q8×C53—22no—
69C4×Q8×C53D414yescomplemented
70C4×D4×C53—24no—
71C4×D4×C53—24no—
72C4×D4×C53—22no—
73C4×D4×C53—22no—
74#117×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53—24no—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—24no—
79C2×C22⋊C4×C53C2312yes—
80C2×C42×C53C2311yescharacteristic
81C24×C53C2×D412yes—
82C23×D53—21no—
83C2×Q8×C53—24no—
84C2×Q8×C53—22no—
85C2×Q8×C53—84no—
86C2×Q8×C53C2×D414yes—
87C2×D4×C53—22no—
88C2×D4×C53—22no—
89C2×D4×C53SD1614yescomplemented
90C22×C4×C53—24no—
91C22×C4×C53—24no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—82no—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2411yescharacteristic
99C2×C22⋊D53—44no—
100C2×C22⋊D53—44no—
101C2×C22⋊D53—82no—
102C22×Dic53—22no—
103C22×Dic53—21no—
104C2×D212—22no—
105C2×D212—42no—
106C2×C4×D53—42no—
107C2×C4×D53—42no—
108C2×C4×D53—42no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—28no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53C2×D414yes—
118C22⋊C4×C53—28no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C42×C53—24no—
122C42×C53C2×D412yes—
123C42×C53C2×D412yes—
124C22⋊Dic53—44no—
125#11719924—48no—
126#11719921—41no—
127#11719921—41no—
128#11719921—82no—
129C4⋊Dic53—22no—
130C4⋊Dic53—42no—
131C2×#88322—24no—
132C23×C53—22no—
133C23×C53—22no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53C2×SD1614yes—
137C23×C53C22×D411yescharacteristic
138C22×D53—22no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145Q8×C53—44no—
146Q8×C53—44no—
147Q8×C53—84no—
148D4×C53—28no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—24no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—44no—
155C2×C4×C53—42no—
156C2×C4×C53—82no—
157C2×C4×C53—84no—
158C2×C4×C53—84no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12012yes—
162C2×C4×C53C2×SD1612yes—
163C2×C4×C53#13712yes—
164C2×C4×C53C22×D412yes—
165C22⋊D53—88no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—41no—
172D212—24no—
173D212—42no—
174C4×D53—84no—
175#88322—44no—
176#18212—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—24no—
180C22×C53—44no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—82no—
184C22×C53C2×#12011yescharacteristic
185C22×C53C22×SD1611yescharacteristic
186C22×C53C2×#13711yescharacteristic
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—42no—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C2×D53—81no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—22no—
197C4×C53—42no—
198C4×C53—44no—
199C4×C53—44no—
200C4×C53#44814yes—
201Dic53—82no—
202Dic53—81no—
203#3501D5311yescharacteristic, complemented
204#3101—532no—
205#2535—531no—
206#1608—534no—
207#1596—534no—
208#1518—532no—
209#1460—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—44no—
213C2×C53C2×#44811yescharacteristic
214C2×C53#301911yescharacteristic
215C2×C53#302511yescharacteristic
216D53—82no—
217#554—22no—
218#554—24no—
219D4×Q8—22no—
220#544—21no—
221#543C2×D5314yes—
222#533C2×D5312yes—
223C2×C22⋊Q8—22no—
224C2×#121—1061no—
225#518—21no—
226C2×C4×D4—531no—
227C2×C4×D4C2×D5311yescharacteristic
228#483—534no—
229#476—1062no—
230#461—1062no—
231#450—534no—
232C4×SD16—1064no—
233#431—532no—
234#397—532no—
235#392—532no—
236#386—1062no—
237#382—531no—
238#380—1061no—
239#328—538no—
240#323—534no—
241C53#1821211yescharacteristic, complemented, Sylow
242C22×Q8—22no—
243C22×D4—531no—
244C22×D4—1062no—
245C22×D4C22⋊D5311yescharacteristic
246C23×C4—1061no—
247C23×C4C22⋊D5312yes—
248C2×SD16—1064no—
249C4⋊Q8—24no—
250C4⋊Q8C22⋊D5312yescomplemented
251C4⋊Q8C22⋊D5312yescomplemented
252#125—24no—
253#125—22no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—28no—
257C22⋊Q8—22no—
258C22⋊Q8—24no—
259C22⋊Q8—28no—
260C22⋊Q8C22×D5314yes—
261#121—2124no—
262C4×Q8—22no—
263C4×Q8C22⋊D5314yescomplemented
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—1062no—
269C4×D4—1062no—
270#117—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4C22⋊D5311yescharacteristic
276C2×C22⋊C4—24no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1062no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4C22×D5312yes—
281C2×C42C22×D5311yescharacteristic
282#106—1064no—
283C4⋊C8—1062no—
284#103—1064no—
285#102—1064no—
286C22⋊C8—534no—
287C4×C8—1062no—
288#95—532no—
289#95—1062no—
290#95—1061no—
291C24—1061no—
292C24C2×C22⋊D5312yes—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—84no—
296C2×Q8C2×C22⋊D5314yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—1062no—
300C2×D4—4242no—
301C2×D4—2122no—
302C2×D4—2124no—
303C2×D4—2124no—
304C2×D4#1171992414yescomplemented
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4—1062no—
311C22×C4—1061no—
312C22×C4—2122no—
313C22×C4—2122no—
314C22×C4—2122no—
315C22×C4C2×C22⋊D5312yes—
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318C22×C4C23×D5311yescharacteristic
319SD16—2128no—
320C2×C8—1064no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—1062no—
330C4⋊C4—2122no—
331C4⋊C4C2×C22⋊D5314yes—
332C22⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—4242no—
336C22⋊C4—2124no—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C42—24no—
340C42C2×C22⋊D5312yes—
341C42C2×C22⋊D5312yes—
342C23—22no—
343C23—22no—
344C23—42no—
345C23—42no—
346C23—1062no—
347C23—2121no—
348C23—2121no—
349C23—2121no—
350C23—2121no—
351C23—2121no—
352C23—2121no—
353C23C2×#1171992414yes—
354C23C22×C22⋊D5311yescharacteristic
355Q8—44no—
356Q8—44no—
357Q8—84no—
358D4—28no—
359D4—1064no—
360D4—4248no—
361D4—2122no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—4244no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2122no—
378C2×C4—2121no—
379C2×C4#1313318611yescharacteristic
380C2×C4#1313318611yescharacteristic
381C2×C4#1313318612yes—
382C2×C4C2×#1171992412yes—
383C2×C4#1313324812yes—
384C2×C4C22×C22⋊D5312yes—
385C8—2124no—
386C22—21no—
387C22—21no—
388C22—24no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—82no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22—4241no—
397C22—2122no—
398C22—2122no—
399C22—2122no—
400C22C2×#1313318611yescharacteristic
401C22C22×#1171992411yescharacteristic
402C22C2×#1313324811yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—42no—
407C4—44no—
408C4—44no—
409C4—4242no—
410C4—4241no—
411C4#1681076214yes—
412C2—21no—
413C2—21no—
414C2—44no—
415C2—4242no—
416C2C2×#1681076211yescharacteristic
417C2#2028407211yescharacteristic
418C2#2028410711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3501C2
socleC22×C53C2×#120
prime radical#20279612C2
hypercenter#3501D53
hypocenterC53#18212
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #20284719C2semidirect
3. #20273706C2semidirect
4. #20273638C2semidirect
5. #20273279C2semidirect
6. #20273036C2semidirect
7. C53×#3501C2semidirect
57. C4⋊Q8×C53D4semidirect
58. C4⋊Q8×C53D4semidirect
69. C4×Q8×C53D4semidirect
89. C2×D4×C53SD16semidirect
203. #3501D53semidirect
241. C53#18212semidirect
250. C4⋊Q8C22⋊D53semidirect
251. C4⋊Q8C22⋊D53semidirect
263. C4×Q8C22⋊D53semidirect
304. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202796122
#202737062
#202736382
#202732792
#202730362
C53×#35012
#1821253