Finite Group Databasebeta

← #27357993

Subgroups — #27357993

Order 13568.

1282 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1282 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
1C2×#16811204C211yesmaximal, characteristic, complemented
2#20285463C212yesmaximal, complemented
3#20273717C214yesmaximal, complemented
4#20273649C214yesmaximal, complemented
5#20273282C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C53×#2618C211yesmaximal, characteristic, complemented
8C22×#11719945—21no—
9#16811204—24no—
10#16811204—22no—
11#16811204C2214yes—
12C2×#13133197—22no—
13C2×#13133192—22no—
14C2×#13133191—21no—
15C2×#13133191C2212yes—
16#16810936C2214yescomplemented
17#16810934—22no—
18#16810857—24no—
19#16810757C2214yescomplemented
20#16810756—22no—
21C2×C22⋊#88322C2212yes—
22#16810356C2218yescomplemented
23#16810349C2214yescomplemented
24C2×#123×C53—21no—
25C2×#121×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic
27C53×#398C2212yescomplemented
28C53×#396—22no—
29C53×#394C2212yescomplemented
30C53×#386—21no—
31C53×#380C2211yescharacteristic
32C2×#11719945—24no—
33C2×#11719945—24no—
34C2×#11719945—44no—
35C2×#11719945—44no—
36C22×Dic106—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212D412yes—
40C22×C22⋊D53—22no—
41C22×C4×D53—22no—
42C2×#11719922—24no—
43C22×#88322—22no—
44#13133197—28no—
45#13133197—28no—
46#13133192—28no—
47#13133192—28no—
48#13133191—24no—
49#13133191—24no—
50#13133191—22no—
51#13133191—22no—
52#13133191D414yes—
53#13133191C2314yes—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53D412yes—
56C2×#11719921—22no—
57C2×#11719921—22no—
58#13133150—24no—
59#13133138—24no—
60#13133137—24no—
61C22⋊#88322—22no—
62C22⋊#88322C2314yes—
63C22×D4×C53—21no—
64C22×D4×C53C2311yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53D411yescharacteristic
67C23×C4×C53C2311yescharacteristic
68#123×C53—44no—
69#121×C53—22no—
70#121×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53D412yes—
76C2×C4⋊C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53C2312yes—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53—22no—
82#95×C53—22no—
83#95×C53C2312yes—
84C24×C53—21no—
85C24×C53C2×D411yescharacteristic
86C23×D53—22no—
87C2×D4×C53—22no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—81no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—81no—
101C22×C4×C53—81no—
102C22×C4×C53D814yescomplemented
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2411yescharacteristic
109C2×C22⋊D53—24no—
110C2×C22⋊D53—24no—
111C2×C22⋊D53—44no—
112C2×C22⋊D53—44no—
113C22×Dic53—22no—
114#11719945—48no—
115#11719945—416no—
116#11719945—48no—
117C2×D212—24no—
118C2×D212—24no—
119C2×D212—22no—
120C2×D212—44no—
121C2×D212—82no—
122C2×D212—82no—
123C2×D212C2×D414yes—
124C2×C4×D53—28no—
125C2×C4×D53—48no—
126C2×Dic106—24no—
127C2×Dic106—44no—
128C4⋊C4×C53—24no—
129C4⋊C4×C53—24no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—44no—
132C4⋊C4×C53—81no—
133C22⋊C4×C53—42no—
134C22⋊C4×C53—42no—
135C22⋊C4×C53—44no—
136C22⋊C4×C53—82no—
137#11719922—84no—
138#11719921—28no—
139#11719921—24no—
140#11719921—24no—
141C4⋊Dic53—24no—
142C4⋊Dic53—22no—
143C4⋊Dic53C2×D414yes—
144C2×#88322—24no—
145C2×#88322—82no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53#12011yescharacteristic
154C23×C53#12011yescharacteristic
155C23×C53C2×D812yes—
156C23×C53AGL(1,ℤ/8)12yes—
157C23×C53C22×D411yescharacteristic
158C22×D53—24no—
159C22×D53—44no—
160C22×D53—44no—
161C22×D53—44no—
162D4×C53—44no—
163D4×C53—82no—
164D4×C53—82no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—24no—
168C2×C4×C53—24no—
169C2×C4×C53—22no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—81no—
183C2×C4×C53#12012yes—
184C2×C4×C53C2×D814yes—
185C2×C4×C53C22×D412yes—
186C22⋊D53—44no—
187C22⋊D53—44no—
188C22⋊D53—48no—
189C2×Dic53—24no—
190C2×Dic53—44no—
191D212—44no—
192D212—48no—
193D212—82no—
194D212—84no—
195C4×D53—416no—
196Dic106—48no—
197#88322—44no—
198#18292—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—82no—
208C22×C53—82no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53#44712yes—
213C22×C53#44712yes—
214C22×C53C2×#12011yescharacteristic
215C22×C53C22×D811yescharacteristic
216C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
217C2×D53—44no—
218C2×D53—44no—
219C2×D53—44no—
220C2×D53—88no—
221C4×C53—24no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—42no—
225C4×C53—82no—
226C4×C53—81no—
227Dic53—44no—
228C2×#537—531no—
229#3186—532no—
230#2618D5311yescharacteristic, complemented
231#1610—534no—
232#1598—534no—
233#1519—532no—
234C2×#324—531no—
235C2×C53—21no—
236C2×C53—22no—
237C2×C53—21no—
238C2×C53—22no—
239C2×C53—161no—
240C2×C53C2×#44711yescharacteristic
241C2×C53#302011yescharacteristic
242C2×C53#302611yescharacteristic
243D53—84no—
244C22×#41—1061no—
245#537—534no—
246#537—1064no—
247#537—1062no—
248C2×#123—21no—
249C2×C22⋊Q8—1062no—
250C2×#121—532no—
251C2×#121—1062no—
252C2×#121—1061no—
253C2×#121C2×D5311yescharacteristic
254C22×C4⋊C4C2×D5311yescharacteristic
255#480—534no—
256#480—1062no—
257#466—1064no—
258#447—534no—
259#447—1062no—
260C2×C22⋊C8—532no—
261#398C2×D5312yescomplemented
262#396—22no—
263#394C2×D5312yescomplemented
264#386—21no—
265#380C2×D5311yescharacteristic
266#327—538no—
267#324—534no—
268C53#1829211yescharacteristic, complemented, Sylow
269C2×#41—1064no—
270C2×#41—1064no—
271C2×#41—2124no—
272C2×#41—2124no—
273C22×Q8—1061no—
274C22×D4—21no—
275C22×D4—532no—
276C22×D4—1061no—
277C22×D4—1061no—
278C22×D4—1062no—
279C22×D4C22×D5311yescharacteristic
280C23×C4—1062no—
281C23×C4C22⋊D5311yescharacteristic
282C23×C4C22⋊D5311yescharacteristic
283C23×C4C22×D5311yescharacteristic
284C2×D8—1064no—
285C22×C8—1062no—
286#123—44no—
287C22⋊Q8—1068no—
288C22⋊Q8—1068no—
289#121—22no—
290#121—22no—
291#121—534no—
292#121—534no—
293#121—1068no—
294#121—1068no—
295#121—1064no—
296#121—1062no—
297#121—1062no—
298#121—1064no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—532no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4C22⋊D5312yes—
306C2×C4⋊C4C22×D5311yescharacteristic
307C2×C4⋊C4C22×D5312yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—1062no—
313C2×C22⋊C4—1062no—
314#107—1064no—
315#102—1064no—
316#102—1064no—
317C22⋊C8—534no—
318C22⋊C8—1062no—
319#95—22no—
320#95C22×D5312yes—
321C24—21no—
322C24—1062no—
323C24C2×C22⋊D5311yescharacteristic
324#41—21216no—
325#41—2128no—
326#41—2128no—
327C2×Q8—1064no—
328C2×Q8—2124no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—534no—
336C2×D4—1064no—
337C2×D4—1064no—
338C2×D4—1064no—
339C2×D4—1064no—
340C2×D4—1062no—
341C2×D4—4242no—
342C2×D4—4242no—
343C2×D4—2124no—
344C2×D4—2124no—
345C2×D4—2124no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—1068no—
356C22×C4—1062no—
357C22×C4—2128no—
358C22×C4#1171992214yescomplemented
359C22×C4D4×D5312yes—
360C22×C4D4×D5312yes—
361C22×C4C2×C22⋊D5312yes—
362C22×C4C2×C22⋊D5312yes—
363C22×C4C2×C22⋊D5311yescharacteristic
364C22×C4C23×D5311yescharacteristic
365D8—4244no—
366C2×C8—1064no—
367C2×C8—4242no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—81no—
373C4⋊C4—534no—
374C4⋊C4—1064no—
375C4⋊C4—1062no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—1068no—
381C22⋊C4—1064no—
382C22⋊C4—1064no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—1064no—
391C23—2124no—
392C23—2124no—
393C23—2124no—
394C23#1313318511yescharacteristic
395C23#1313318511yescharacteristic
396C23C2×#1171992212yes—
397C23#1313324212yes—
398C23C22×C22⋊D5311yescharacteristic
399Q8—2128no—
400D4—44no—
401D4—82no—
402D4—82no—
403D4—4242no—
404D4—4244no—
405D4—2124no—
406D4—2128no—
407D4—2128no—
408D4—2124no—
409D4—2124no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—81no—
428C2×C4—1064no—
429C2×C4—21216no—
430C2×C4—2124no—
431C2×C4#1313318512yes—
432C2×C4C2×#1171992214yes—
433C2×C4C2×D4×D5312yes—
434C8—2124no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—4248no—
449C22—2124no—
450C22—2124no—
451C22—2124no—
452C22#1681075612yes—
453C22#1681075612yes—
454C22C2×#1313318511yescharacteristic
455C22C22×#1171992211yescharacteristic
456C22C2×#1313324211yescharacteristic
457C4—22no—
458C4—22no—
459C4—24no—
460C4—42no—
461C4—82no—
462C4—81no—
463C4—2124no—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—22no—
468C2—161no—
469C2—4244no—
470C2C2×#1681075611yescharacteristic
471C2#2028407811yescharacteristic
472C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2618C2
socleC22×C53C2×#120
hypercenter#2618D53
hypocenterC53#18292
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#16811204C2semidirect
2. #20285463C2semidirect
3. #20273717C2semidirect
4. #20273649C2semidirect
5. #20273282C2semidirect
6. C2×#16810349C2semidirect
7. C53×#2618C2semidirect
16. #16810936C22semidirect
19. #16810757C22semidirect
22. #16810356C22semidirect
23. #16810349C22semidirect
27. C53×#398C22semidirect
29. C53×#394C22semidirect
102. C22×C4×C53D8semidirect
230. #2618D53semidirect
261. #398C2×D53semidirect
263. #394C2×D53semidirect
268. C53#18292semidirect
358. C22×C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168112042
#202854632
#202737172
#202736492
#202732822
C2×#168103492
C53×#26182
#1829253