Finite Group Databasebeta

← #27281877

Subgroups — #27281877

Order 13568.

912 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 123 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#20283744C211yesmaximal, characteristic, complemented
2#20274703C211yesmaximal, characteristic, complemented
3#20274646C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#2408C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133138—24no—
9C2×#13133132—21no—
10#16810587—21no—
11C2×C22⋊#88322C2212yes—
12#16810483C2212yes—
13#16810390—21no—
14C23×D4×C53C2211yescharacteristic
15C2×#121×C53—21no—
16C2×#120×C53—21no—
17C2×C4×D4×C53C2211yescharacteristic
18C53×#394C412yescomplemented
19C53×#392C412yescomplemented
20C53×#390—21no—
21C53×#380C2211yescharacteristic
22C53×#379—21no—
23C2×#11719917—22no—
24C22×#88322D414yescomplemented
25C2×C4⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C4⋊#88322—22no—
28#13133138—48no—
29#13133132—42no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34#13133124—22no—
35C25×C53—21no—
36C22×D4×C53—22no—
37C22×D4×C53—22no—
38C22×D4×C53—21no—
39C22×D4×C53—22no—
40C22×D4×C53—44no—
41C22×D4×C53C2×C412yes—
42C22×D4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45#121×C53—44no—
46#120×C53—44no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58#95×C53C2×C412yes—
59C24×C53—22no—
60C24×C53—44no—
61C24×C53—42no—
62C24×C53—42no—
63C24×C53—41no—
64C24×C53—41no—
65C24×C53—82no—
66C24×C53C22⋊C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C2×D4×C53—28no—
69C2×D4×C53—24no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—24no—
75C2×D4×C53—42no—
76C2×D4×C53—44no—
77C2×D4×C53—44no—
78C2×D4×C53—48no—
79C2×D4×C53—44no—
80C2×D4×C53—44no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—82no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4112yes—
96C22×Dic53—42no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—42no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—44no—
102C22⋊C4×C53—44no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—82no—
105C42×C53—21no—
106C42×C53—21no—
107C22⋊Dic53—42no—
108C22⋊Dic53—42no—
109C4⋊Dic53—44no—
110#11719917—82no—
111C2×#88322—24no—
112C2×#88322—44no—
113C23×C53—24no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—48no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—84no—
129C23×C53—88no—
130C23×C53—84no—
131C23×C53—84no—
132C23×C53—82no—
133C23×C53—82no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12112yes—
136C23×C53AGL(1,ℤ/8)11yescharacteristic
137C23×C53AGL(1,ℤ/8)12yes—
138C23×C53#13711yescharacteristic
139D4×C53—28no—
140D4×C53—44no—
141D4×C53—44no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—44no—
145D4×C53—88no—
146C2×C4×C53—22no—
147C2×C4×C53—24no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—81no—
162C2×C4×C53C4×D412yes—
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×Dic53—82no—
166C2×Dic53—82no—
167C2×Dic53—82no—
168#88322—44no—
169#9472—531nomaximal, Sylow
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—41no—
175C22×C53—42no—
176C22×C53—41no—
177C22×C53—42no—
178C22×C53—44no—
179C22×C53—44no—
180C22×C53—41no—
181C22×C53—44no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—81no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53—88no—
188C22×C53—88no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53#38611yescharacteristic
192C22×C53#41711yescharacteristic
193C22×C53#41811yescharacteristic
194C22×C53#46812yes—
195C22×C53#46912yes—
196C4×C53—21no—
197C4×C53—22no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—82no—
202C4×C53B(4,𝔽2)12yes—
203Dic53—162no—
204#2973—531no—
205#2408D5311yescharacteristic
206#1799—531no—
207#1786—532no—
208#1289—532no—
209C2×C53—22no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—81no—
214C2×C53—84no—
215C2×C53#188911yescharacteristic
216C2×C53#189511yescharacteristic
217C2×C53#191011yescharacteristic
218C23×D4C2×D5311yescharacteristic
219C2×#121—21no—
220C2×#120—21no—
221C2×C4×D4C2×D5311yescharacteristic
222#435—1064no—
223#432—1061no—
224C2×#102—1064no—
225C2×#99—1061no—
226#407—1061no—
227C2×C22⋊C8—532no—
228#394Dic5312yescomplemented
229#392Dic5312yescomplemented
230#390—21no—
231#380—532no—
232#380C2×D5311yescharacteristic
233#379—21no—
234#342—1061no—
235C53#947211yescharacteristic, complemented, Sylow
236C25—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—44no—
242C22×D4C2×Dic5312yes—
243C22×D4C22⋊D5311yescharacteristic
244C23×C4C22⋊D5311yescharacteristic
245C23×C4C22×D5311yescharacteristic
246C2×M16—1062no—
247C22×C8—534no—
248#121—44no—
249#120—44no—
250C4×D4—22no—
251C4×D4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4C2×Dic5311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—1062no—
260C2×C22⋊C4C2×Dic5311yescharacteristic
261C2×C22⋊C4C22⋊D5311yescharacteristic
262C2×C42C22⋊D5311yescharacteristic
263C4⋊C8—1062no—
264#102—2128no—
265#99—2122no—
266C22⋊C8—1062no—
267C22⋊C8—1062no—
268C22⋊C8—1062no—
269#97—1062no—
270#95—1062no—
271#95C2×Dic5312yes—
272C24—22no—
273C24—44no—
274C24—42no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—82no—
279C24C22⋊Dic5311yescharacteristic
280C24C2×C22⋊D5311yescharacteristic
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—24no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—2122no—
303C22×C4C22⋊Dic5311yescharacteristic
304C22×C4C22⋊Dic5311yescharacteristic
305C22×C4C22⋊Dic5311yescharacteristic
306C22×C4D4×D5312yes—
307C22×C4#1171994712yes—
308C22×C4C22×Dic5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310M16—4242no—
311C2×C8—1064no—
312C2×C8—2124no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—2124no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—2122no—
323C22⋊C4—2122no—
324C42—21no—
325C42—21no—
326C23—24no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—48no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—84no—
342C23—84no—
343C23—88no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23C2×C22⋊Dic5311yescharacteristic
348C23#1313318912yes—
349C23#1313324211yescharacteristic
350C23#1313324312yes—
351C23#1313324711yescharacteristic
352D4—28no—
353D4—44no—
354D4—44no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—88no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—4242no—
376C2×C4—4242no—
377C2×C4—4242no—
378C2×C4D4×Dic5312yes—
379C2×C4#1313318511yescharacteristic
380C2×C4#1313319811yescharacteristic
381C8—2124no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—41no—
393C22—41no—
394C22—44no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—88no—
400C22—88no—
401C22—82no—
402C22—82no—
403C22#1681050311yescharacteristic
404C22#1681062411yescharacteristic
405C22#1681062811yescharacteristic
406C22#1681086712yes—
407C22#1681087412yes—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4—8482no—
415C4#1681080912yes—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—84no—
422C2#2027512411yescharacteristic
423C2#2027515411yescharacteristic
424C2#2027524311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2408C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2408D53
hypocenterC53#9472
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. #20274703C2semidirect
3. #20274646C2semidirect
4. #20272582C2semidirect
18. C53×#394C4semidirect
19. C53×#392C4semidirect
24. C22×#88322D4semidirect
228. #394Dic53semidirect
229. #392Dic53semidirect
235. C53#9472semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202747032
#202746462
#202725822
C53×#24082
#947253