Finite Group Databasebeta

← #27276637

Subgroups — #27276637

Order 13568.

728 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 131 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#20275527C211yesmaximal, characteristic, complemented
2#20273339C211yesmaximal, characteristic, complemented
3#20273253C212yesmaximal, complemented
4#20273175C212yesmaximal, complemented
5C53×#3545C211yesmaximal, characteristic
6#16810680C2212yescomplemented
7C2×#13133141—21no—
8C2×#13133138—21no—
9C2×C4×#88322C2211yescharacteristic, complemented
10#16810489C2211yescharacteristic, complemented
11#16810363—21no—
12#16810363—21no—
13#16810360—22no—
14#16810357—22no—
15Q8⋊#88322—22no—
16D4⋊#88322—22no—
17C53×#563C411yescharacteristic, complemented
18C53×#562C412yescomplemented
19C53×H(3,𝔽4)C411yescharacteristic, complemented
20C53×#555—21no—
21C53×#551—21no—
22C53×#551—22no—
23C53×#546—22no—
24C53×#542—22no—
25C53×#538—22no—
26C53×#536—21no—
27C53×#534—21no—
28C53×#529C2212yes—
29C2×#124×C53C2211yescharacteristic
30C22×#88322—21no—
31C2×C4⋊Dic53—21no—
32C4⋊#88322—22no—
33#13133141—42no—
34#13133138—42no—
35C22⋊#88322—22no—
36#13133128—22no—
37C4×#88322—22no—
38#13133124—22no—
39C2×#41×C53—21no—
40C22×Q8×C53D411yescharacteristic
41C22×D4×C53—21no—
42C22×D4×C53D411yescharacteristic
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45#126×C53—22no—
46#126×C53—22no—
47#126×C53—22no—
48#125×C53C2×C412yes—
49#124×C53—21no—
50#124×C53—22no—
51#124×C53—22no—
52#124×C53—21no—
53#124×C53—22no—
54#124×C53C2×C411yescharacteristic
55#124×C53C2×C411yescharacteristic
56#124×C53C2×C412yes—
57#124×C53D411yescharacteristic
58#124×C53D411yescharacteristic
59#123×C53—21no—
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—22no—
64#123×C53—21no—
65#123×C53—22no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53D412yes—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53—22no—
74#121×C53—22no—
75#121×C53D412yes—
76#120×C53—21no—
77#120×C53—22no—
78#120×C53—21no—
79#120×C53—22no—
80C4×Q8×C53D412yes—
81C4×D4×C53—22no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53D412yes—
85#117×C53—21no—
86#117×C53—22no—
87#117×C53—22no—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C42×C53C2311yescharacteristic
92C24×C53—21no—
93C24×C53—21no—
94#41×C53—81no—
95#41×C53—81no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53—41no—
99C2×Q8×C53—41no—
100C2×Q8×C53C22⋊C412yes—
101C2×Q8×C53C22⋊C411yescharacteristic
102C2×Q8×C53C22⋊C411yescharacteristic
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—81no—
112C2×D4×C53—81no—
113C2×D4×C53C22⋊C412yes—
114C2×D4×C53C22⋊C411yescharacteristic
115C2×D4×C53C22⋊C411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×Dic53—41no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53C22⋊C412yes—
132C4⋊C4×C53C22⋊C412yes—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—22no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—22no—
151C42×C53C22×C411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D412yes—
154C4⋊Dic53—41no—
155C4⋊Dic53—41no—
156C2×#88322—22no—
157C2×#88322—41no—
158C2×#88322—41no—
159C2×#88322—42no—
160C23×C53—22no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—82no—
166C23×C53—81no—
167C23×C53—82no—
168C23×C53#12011yescharacteristic
169Q8×C53—41no—
170Q8×C53—41no—
171Q8×C53—42no—
172Q8×C53—81no—
173Q8×C53—81no—
174D4×C53—41no—
175D4×C53—41no—
176D4×C53—42no—
177D4×C53—42no—
178D4×C53—42no—
179D4×C53—81no—
180D4×C53—82no—
181D4×C53—82no—
182D4×C53—82no—
183D4×C53—81no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—82no—
195C2×C4×C53—82no—
196C2×C4×C53—81no—
197C2×C4×C53—82no—
198C2×C4×C53C2×C22⋊C411yescharacteristic
199C2×C4×C53C2×C22⋊C412yes—
200C2×C4×C53#12012yes—
201C2×C4×C53#12011yescharacteristic
202C2×Dic53—82no—
203C2×Dic53—81no—
204#88322—42no—
205#88322—82no—
206#8528—531nomaximal, Sylow
207C22×C53—22no—
208C22×C53—42no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—161no—
216C22×C53—161no—
217C22×C53—162no—
218C22×C53#37911yescharacteristic
219C22×C53#45311yescharacteristic
220C22×C53#45511yescharacteristic
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—41no—
226C4×C53—41no—
227C4×C53—42no—
228C4×C53—42no—
229C4×C53—42no—
230Dic53—161no—
231#3545D5311yescharacteristic
232#1958—531no—
233#1533—531no—
234#1512—532no—
235#1494—532no—
236C2×C53—21no—
237C2×C53—21no—
238C2×C53—82no—
239C2×C53—81no—
240C2×C53—81no—
241C2×C53#179211yescharacteristic
242C2×C53#179412yes—
243#563Dic5311yescharacteristic, complemented
244#562Dic5312yescomplemented
245H(3,𝔽4)Dic5311yescharacteristic, complemented
246#555—21no—
247#551—21no—
248#551—22no—
249#546—22no—
250#542—22no—
251#538—22no—
252#536—21no—
253#534—21no—
254#529C2×D5312yes—
255C2×#124C2×D5311yescharacteristic
256#432—532no—
257C2×#103—1061no—
258C2×#102—1061no—
259C2×C4×C8—531no—
260#382—531no—
261#329—1061no—
262#329—1061no—
263#328—1062no—
264#327—1062no—
265Q8⋊C8—1062no—
266D4⋊C8—1062no—
267C53#852811yescharacteristic, complemented, Sylow
268C2×#41—21no—
269C22×Q8C22⋊D5311yescharacteristic
270C22×D4—21no—
271C22×D4C22⋊D5311yescharacteristic
272C22×C8—1061no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275#126—22no—
276#126—22no—
277#126—22no—
278#125C2×Dic5312yes—
279#124—21no—
280#124—22no—
281#124—21no—
282#124—22no—
283#124—22no—
284#124C2×Dic5311yescharacteristic
285#124C2×Dic5311yescharacteristic
286#124C2×Dic5312yes—
287#124C22⋊D5311yescharacteristic
288#124C22⋊D5311yescharacteristic
289#123—21no—
290#123—22no—
291#123—22no—
292#123—21no—
293#123—22no—
294#123—22no—
295#123—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8C22⋊D5312yes—
301#121—22no—
302#121—22no—
303#121—22no—
304#121—22no—
305#121C22⋊D5312yes—
306#120—21no—
307#120—22no—
308#120—21no—
309#120—22no—
310C4×Q8C22⋊D5312yes—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4C22⋊D5312yes—
315#117—21no—
316#117—22no—
317#117—22no—
318C2×C4⋊C4—1061no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42C22×D5311yescharacteristic
323C4⋊C8—1062no—
324#103—2122no—
325#102—2122no—
326C22⋊C8—1062no—
327#97—1062no—
328C4×C8—1062no—
329#95—1062no—
330C24—21no—
331C24—21no—
332#41—81no—
333#41—81no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8C22⋊Dic5312yes—
339C2×Q8C22⋊Dic5311yescharacteristic
340C2×Q8C22⋊Dic5311yescharacteristic
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—81no—
350C2×D4—81no—
351C2×D4C22⋊Dic5312yes—
352C2×D4C22⋊Dic5311yescharacteristic
353C2×D4C22⋊Dic5311yescharacteristic
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—2121no—
360C22×C4C2×C22⋊D5311yescharacteristic
361C22×C4C2×C22⋊D5312yes—
362C2×C8—1062no—
363C2×C8—2122no—
364C2×C8—2121no—
365C2×C8—2121no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4C22⋊Dic5312yes—
376C4⋊C4C22⋊Dic5312yes—
377C22⋊C4—22no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C42—21no—
393C42—21no—
394C42—22no—
395C42C22×Dic5311yescharacteristic
396C42C2×C22⋊D5312yes—
397C42C2×C22⋊D5311yescharacteristic
398C23—21no—
399C23—22no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—82no—
404C23—81no—
405C23—82no—
406C23#1313318611yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—42no—
410Q8—81no—
411Q8—81no—
412D4—41no—
413D4—41no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—4242no—
437C2×C4—4241no—
438C2×C4C2×C22⋊Dic5312yes—
439C2×C4C2×C22⋊Dic5311yescharacteristic
440C2×C4#1313318612yes—
441C2×C4#1313318611yescharacteristic
442C8—4242no—
443C8—2122no—
444C22—22no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—161no—
453C22—161no—
454C22—162no—
455C22#1681047911yescharacteristic
456C22#1681079111yescharacteristic
457C22#1681080111yescharacteristic
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—8481no—
468C2—21no—
469C2—21no—
470C2—81no—
471C2—82no—
472C2—81no—
473C2#2027466911yescharacteristic
474C2#2027467912yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3545C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3545D53
hypocenterC53#8528
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3545

Complements

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

NH ≅ G/Ntype
1. #20275527C2semidirect
2. #20273339C2semidirect
3. #20273253C2semidirect
4. #20273175C2semidirect
6. #16810680C22semidirect
9. C2×C4×#88322C22semidirect
10. #16810489C22semidirect
17. C53×#563C4semidirect
18. C53×#562C4semidirect
19. C53×H(3,𝔽4)C4semidirect
243. #563Dic53semidirect
244. #562Dic53semidirect
245. H(3,𝔽4)Dic53semidirect
267. C53#8528semidirect

Maximal subgroups

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

subgroupindex
#202755272
#202733392
#202732532
#202731752
C53×#35452
#852853