Finite Group Databasebeta

← #27357939

Subgroups — #27357939

Order 13568.

1142 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1142 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×#16811249C211yesmaximal, characteristic, complemented
2#20285472C212yesmaximal, complemented
3#20273705C214yesmaximal, complemented
4#20273637C214yesmaximal, complemented
5#20273286C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic, complemented
7C53×#2617C211yesmaximal, characteristic
8#16811249—24no—
9#16811249—22no—
10#16811249C2214yes—
11C2×#13133196—21no—
12C2×#13133196C2212yes—
13C2×#13133194—22no—
14C2×#13133194—22no—
15C22×C22⋊Dic53—21no—
16#16810953C2214yescomplemented
17#16810939—22no—
18#16810850—24no—
19#16810774C2214yescomplemented
20#16810761—22no—
21C2×C22⋊#88322C2212yes—
22#16810359C2218yescomplemented
23#16810347C2214yes—
24C2×#121×C53C2211yescharacteristic
25C2×#120×C53—21no—
26C22×C22⋊C4×C53C2211yescharacteristic
27C53×#397C2212yes—
28C53×#394—22no—
29C53×#392C2212yes—
30C53×#386—21no—
31C53×#380C2211yescharacteristic
32C22×Dic106—21no—
33C22×Dic106D412yescomplemented
34C23×Dic53—21no—
35C23×Dic53—21no—
36C2×#11719923—24no—
37C22×#88322—22no—
38#13133196—24no—
39#13133196—24no—
40#13133196—22no—
41#13133196—22no—
42#13133196D414yescomplemented
43#13133196C2314yes—
44#13133194—28no—
45#13133194—28no—
46#13133194—28no—
47#13133194—28no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53D412yescomplemented
50C2×#11719919—22no—
51C2×#11719919—22no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—24no—
56#13133148—24no—
57#13133140—24no—
58#13133138—24no—
59C22⋊#88322—22no—
60C22⋊#88322C2314yes—
61C25×C53D411yescharacteristic
62C22×D4×C53—21no—
63C22×D4×C53—22no—
64C22×D4×C53C2311yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67#121×C53—22no—
68#121×C53—22no—
69#120×C53—44no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53C2311yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—21no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53D412yes—
79C2×C22⋊C4×C53C2312yes—
80#95×C53—22no—
81#95×C53C2312yes—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—22no—
85C24×C53—21no—
86C24×C53—81no—
87C24×C53—81no—
88C24×C53SD1614yescomplemented
89C24×C53C2×D412yes—
90C24×C53C2×D411yescharacteristic
91C2×D4×C53—22no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—44no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—81no—
98C2×D4×C53—82no—
99C2×D4×C53—82no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—81no—
105C22×C4×C53—81no—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2×D412yes—
110C22×C4×C53C2411yescharacteristic
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—24no—
114C22×Dic53—24no—
115C22×Dic53—82no—
116C22×Dic53—82no—
117C2×Dic106—24no—
118C2×Dic106—22no—
119C2×Dic106—82no—
120C2×Dic106—82no—
121C2×Dic106C2×D414yes—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—81no—
124C22⋊C4×C53—24no—
125C22⋊C4×C53—24no—
126C22⋊C4×C53—44no—
127C22⋊C4×C53—42no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—82no—
130C22⋊Dic53—28no—
131C22⋊Dic53—24no—
132C22⋊Dic53—24no—
133#11719923—84no—
134C4⋊Dic53—24no—
135C4⋊Dic53—22no—
136C4⋊Dic53C2×D414yes—
137#11719919—28no—
138#11719919—24no—
139#11719919—24no—
140C2×#88322—24no—
141C2×#88322—82no—
142C23×C53—22no—
143C23×C53—24no—
144C23×C53—22no—
145C23×C53—24no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53—84no—
160C23×C53—84no—
161C23×C53#12011yescharacteristic
162C23×C53#12011yescharacteristic
163C23×C53C2×SD1614yes—
164C23×C53C2×SD1612yes—
165C23×C53#13712yes—
166C23×C53C22×D411yescharacteristic
167D4×C53—44no—
168D4×C53—88no—
169D4×C53—82no—
170D4×C53—82no—
171C2×C4×C53—22no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—81no—
185C2×C4×C53#12012yes—
186C2×C4×C53C22×D412yes—
187C2×Dic53—24no—
188C2×Dic53—24no—
189C2×Dic53—44no—
190C2×Dic53—44no—
191C2×Dic53—84no—
192C2×Dic53—84no—
193Dic106—44no—
194Dic106—82no—
195Dic106—84no—
196#88322—44no—
197#18286—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—24no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—44no—
208C22×C53—48no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—82no—
212C22×C53—84no—
213C22×C53—82no—
214C22×C53—82no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53#44812yes—
219C22×C53#44812yes—
220C22×C53C2×#12011yescharacteristic
221C22×C53C22×SD1611yescharacteristic
222C22×C53C2×#13711yescharacteristic
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—82no—
227C4×C53—81no—
228Dic53—44no—
229Dic53—44no—
230C2×#543—531no—
231#3187—532no—
232#2617D5311yescharacteristic
233#1608—534no—
234#1596—534no—
235#1520—532no—
236C2×#323—531no—
237C2×C53—21no—
238C2×C53—22no—
239C2×C53—21no—
240C2×C53—22no—
241C2×C53—44no—
242C2×C53—161no—
243C2×C53C2×#44811yescharacteristic
244C2×C53#301911yescharacteristic
245C2×C53#302611yescharacteristic
246#543—534no—
247#543—1064no—
248#543—1062no—
249C2×C22⋊Q8—532no—
250C2×C22⋊Q8—1062no—
251C2×C22⋊Q8—1062no—
252C2×C22⋊Q8—1061no—
253C2×#121C2×D5311yescharacteristic
254C2×#120—21no—
255C22×C22⋊C4—1061no—
256C22×C22⋊C4C2×D5311yescharacteristic
257#483—534no—
258#481—1062no—
259#465—1064no—
260#450—534no—
261#448—1062no—
262C2×C22⋊C8—532no—
263#397C2×D5312yes—
264#394—22no—
265#392C2×D5312yes—
266#386—21no—
267#380C2×D5311yescharacteristic
268#328—538no—
269#323—534no—
270C53#1828611yescharacteristic, complemented, Sylow
271C25C22⋊D5311yescharacteristic
272C22×Q8—532no—
273C22×Q8—1061no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4C22×D5311yescharacteristic
277C23×C4—1061no—
278C23×C4—1061no—
279C23×C4C22⋊D5311yescharacteristic
280C23×C4C22×D5311yescharacteristic
281C2×SD16—1064no—
282C22×C8—1062no—
283C22⋊Q8—534no—
284C22⋊Q8—534no—
285C22⋊Q8—1068no—
286C22⋊Q8—1068no—
287C22⋊Q8—1068no—
288C22⋊Q8—1068no—
289C22⋊Q8—1064no—
290C22⋊Q8—1062no—
291C22⋊Q8—1062no—
292C22⋊Q8—1064no—
293#121—22no—
294#121—22no—
295#120—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—532no—
298C2×C4⋊C4—1062no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1062no—
301C2×C4⋊C4C22×D5311yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—1062no—
309C2×C22⋊C4—1064no—
310C2×C22⋊C4—1062no—
311C2×C22⋊C4—1064no—
312C2×C22⋊C4C22⋊D5312yes—
313C2×C22⋊C4C22×D5312yes—
314#106—1064no—
315#103—1064no—
316#102—1064no—
317C22⋊C8—534no—
318C22⋊C8—1062no—
319#95—22no—
320#95C22×D5312yes—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—81no—
326C24—81no—
327C24#1171992314yescomplemented
328C24C2×C22⋊D5312yes—
329C24C2×C22⋊D5311yescharacteristic
330C2×Q8—534no—
331C2×Q8—1064no—
332C2×Q8—1062no—
333C2×Q8—4242no—
334C2×Q8—4242no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—82no—
343C2×D4—82no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—1064no—
351C22×C4—1064no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—4242no—
355C22×C4—4242no—
356C22×C4D4×D5312yes—
357C22×C4D4×D5312yes—
358C22×C4C2×C22⋊D5312yes—
359C22×C4C2×C22⋊D5311yescharacteristic
360C22×C4C23×D5311yescharacteristic
361SD16—4244no—
362C2×C8—1064no—
363C2×C8—4242no—
364C4⋊C4—22no—
365C4⋊C4—81no—
366C4⋊C4—534no—
367C4⋊C4—1068no—
368C4⋊C4—1064no—
369C4⋊C4—1062no—
370C4⋊C4—1064no—
371C4⋊C4—1064no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—82no—
378C22⋊C4—1068no—
379C22⋊C4—1064no—
380C22⋊C4—1064no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—81no—
397C23—81no—
398C23—84no—
399C23—84no—
400C23#1313318511yescharacteristic
401C23#1313318511yescharacteristic
402C23C2×#1171992314yes—
403C23C2×#1171992312yes—
404C23#1313324712yes—
405C23C22×C22⋊D5311yescharacteristic
406Q8—4242no—
407Q8—4244no—
408Q8—2124no—
409D4—44no—
410D4—88no—
411D4—82no—
412D4—82no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—1064no—
428C2×C4—1064no—
429C2×C4—4244no—
430C2×C4—4244no—
431C2×C4—2124no—
432C2×C4—2124no—
433C2×C4#1313318512yes—
434C2×C4C2×D4×D5312yes—
435C8—2124no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#1681076112yes—
457C22#1681076112yes—
458C22C2×#1313318511yescharacteristic
459C22C22×#1171992311yescharacteristic
460C22C2×#1313324711yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—2124no—
467C4—2124no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—161no—
474C2C2×#1681076111yescharacteristic
475C2#2028407111yescharacteristic
476C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#2617C2
socleC22×C53C2×#120
hypercenter#2617D53
hypocenterC53#18286
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. C2×#16811249C2semidirect
2. #20285472C2semidirect
3. #20273705C2semidirect
4. #20273637C2semidirect
5. #20273286C2semidirect
6. C2×#16810347C2semidirect
16. #16810953C22semidirect
19. #16810774C22semidirect
22. #16810359C22semidirect
33. C22×Dic106D4semidirect
42. #13133196D4semidirect
49. C2×C4⋊Dic53D4semidirect
88. C24×C53SD16semidirect
270. C53#18286semidirect
327. C24#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168112492
#202854722
#202737052
#202736372
#202732862
C2×#168103472
C53×#26172
#1828653