Finite Group Databasebeta

← #27276525

Subgroups — #27276525

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#20275548C211yesmaximal, characteristic, complemented
2#20273377C211yesmaximal, characteristic, complemented
3#20273201C212yesmaximal, complemented
4#20273180C212yesmaximal, complemented
5C53×#3529C211yesmaximal, characteristic
6#16810676C2212yescomplemented
7#16810647C2211yescharacteristic, complemented
8C2×#13133144—21no—
9C2×#13133144—21no—
10#16810469C2211yescharacteristic, complemented
11#16810372—21no—
12#16810367—21no—
13#16810360—22no—
14#16810357—22no—
15Q8⋊#88322—22no—
16D4⋊#88322—22no—
17C53×#563—21no—
18C53×#563C411yescharacteristic, complemented
19C53×#562—22no—
20C53×#562C412yescomplemented
21C53×H(3,𝔽4)C411yescharacteristic, complemented
22C53×#560—21no—
23C53×#560—22no—
24C53×#532—21no—
25C53×#532C2211yescharacteristic
26C53×#529—22no—
27C53×#529C2212yes—
28C53×#525—21no—
29C53×#525—22no—
30C2×#11719917—21no—
31C2×C4×Dic53—21no—
32C4⋊#88322—22no—
33C4⋊#88322—22no—
34#13133144—42no—
35#13133144—42no—
36C22⋊#88322—22no—
37C4×#88322—22no—
38#13133124—22no—
39C2×#41×C53—21no—
40C2×#41×C53D411yescharacteristic
41C2×#41×C53D411yescharacteristic
42C23×C4×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53D411yescharacteristic
45#127×C53—21no—
46#127×C53D411yescharacteristic
47#126×C53—22no—
48#126×C53—22no—
49#126×C53—22no—
50#125×C53—22no—
51#125×C53C2×C412yes—
52#124×C53—21no—
53#124×C53—22no—
54#124×C53—21no—
55#124×C53C2×C412yes—
56#124×C53C2×C411yescharacteristic
57#124×C53C2×C411yescharacteristic
58#123×C53—22no—
59#123×C53—21no—
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—21no—
64#123×C53—22no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53D412yes—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53—22no—
74#121×C53—22no—
75#121×C53—22no—
76#121×C53—22no—
77#121×C53D412yes—
78#120×C53—22no—
79#120×C53—21no—
80#120×C53—21no—
81#120×C53—22no—
82C4×Q8×C53D412yes—
83C4×D4×C53—22no—
84C4×D4×C53—22no—
85C4×D4×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53D412yes—
88#117×C53—21no—
89#117×C53—22no—
90C2×C42×C53—21no—
91C2×C42×C53C2311yescharacteristic
92C24×C53—21no—
93#41×C53—41no—
94#41×C53—41no—
95#41×C53—41no—
96#41×C53—41no—
97#41×C53—81no—
98#41×C53—81no—
99C2×Q8×C53—21no—
100C2×Q8×C53C22⋊C412yes—
101C2×Q8×C53C22⋊C411yescharacteristic
102C2×Q8×C53C22⋊C411yescharacteristic
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—22no—
111C2×D4×C53C22⋊C412yes—
112C2×D4×C53C22⋊C411yescharacteristic
113C2×D4×C53C22⋊C411yescharacteristic
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—81no—
121C22×C4×C53—81no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D412yes—
124C22×Dic53—41no—
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×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53C22⋊C412yes—
134C4⋊C4×C53C22⋊C412yes—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
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×D412yes—
153C42×C53C2×D411yescharacteristic
154C4×Dic53—22no—
155C4×Dic53—41no—
156#11719917—41no—
157#11719917—41no—
158C2×#88322—42no—
159C2×#88322—41no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—22no—
163C23×C53—81no—
164C23×C53—82no—
165C23×C53#12011yescharacteristic
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—41no—
169Q8×C53—81no—
170D4×C53—42no—
171D4×C53—41no—
172D4×C53—41no—
173D4×C53—82no—
174D4×C53—82no—
175D4×C53—81no—
176D4×C53—81no—
177D4×C53—82no—
178D4×C53—82no—
179D4×C53—82no—
180D4×C53—81no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—82no—
196C2×C4×C53—82no—
197C2×C4×C53—81no—
198C2×C4×C53C2×C22⋊C411yescharacteristic
199C2×C4×C53C2×C22⋊C412yes—
200C2×C4×C53#12012yes—
201C2×C4×C53#12011yescharacteristic
202C2×Dic53—42no—
203C2×Dic53—82no—
204#88322—82no—
205#88322—81no—
206#8504—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—22no—
209C22×C53—41no—
210C22×C53—42no—
211C22×C53—41no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—84no—
217C22×C53#37911yescharacteristic
218C4×C53—21no—
219C4×C53—22no—
220C4×C53—22no—
221C4×C53—41no—
222C4×C53—42no—
223C4×C53—41no—
224C4×C53—42no—
225C4×C53—42no—
226C4×C53#45311yescharacteristic
227C4×C53#45511yescharacteristic
228Dic53—82no—
229#3529D5311yescharacteristic
230#1963—531no—
231#1543—531no—
232#1500—532no—
233#1495—532no—
234C2×C53—41no—
235C2×C53—81no—
236C2×C53—81no—
237C2×C53—82no—
238C2×C53#179211yescharacteristic
239C2×C53#179412yes—
240#563—21no—
241#563Dic5311yescharacteristic, complemented
242#562—22no—
243#562Dic5312yescomplemented
244H(3,𝔽4)Dic5311yescharacteristic, complemented
245#560—21no—
246#560—22no—
247#532—21no—
248#532C2×D5311yescharacteristic
249#529—22no—
250#529C2×D5312yes—
251#525—21no—
252#525—22no—
253#431—532no—
254#423—531no—
255C2×#104—1061no—
256C2×#104—1061no—
257#376—531no—
258#333—1061no—
259#331—1061no—
260#328—1062no—
261#327—1062no—
262Q8⋊C8—1062no—
263D4⋊C8—1062no—
264C53#850411yescharacteristic, complemented, Sylow
265C2×#41—21no—
266C2×#41C22⋊D5311yescharacteristic
267C2×#41C22⋊D5311yescharacteristic
268C23×C4—21no—
269C2×M16—1061no—
270C4⋊Q8—21no—
271C4⋊Q8C22⋊D5311yescharacteristic
272#127—21no—
273#127C22⋊D5311yescharacteristic
274#126—22no—
275#126—22no—
276#126—22no—
277#125—22no—
278#125C2×Dic5312yes—
279#124—21no—
280#124—22no—
281#124—21no—
282#124C2×Dic5312yes—
283#124C2×Dic5311yescharacteristic
284#124C2×Dic5311yescharacteristic
285#123—22no—
286#123—21no—
287#123—22no—
288#123—21no—
289#123—22no—
290#123—22no—
291#123—22no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8C22⋊D5312yes—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121C22⋊D5312yes—
305#120—21no—
306#120—22no—
307#120—21no—
308#120—22no—
309C4×Q8C22⋊D5312yes—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4C22⋊D5312yes—
315#117—21no—
316#117—22no—
317C2×C42—21no—
318C2×C42—1061no—
319C2×C42C22×D5311yescharacteristic
320C4⋊C8—1062no—
321C4⋊C8—1062no—
322#104—2122no—
323#104—2122no—
324C22⋊C8—1062no—
325C4×C8—1062no—
326#95—1062no—
327C24—21no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—81no—
333#41—81no—
334C2×Q8—21no—
335C2×Q8C22⋊Dic5312yes—
336C2×Q8C22⋊Dic5311yescharacteristic
337C2×Q8C22⋊Dic5311yescharacteristic
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4C22⋊Dic5312yes—
347C2×D4C22⋊Dic5311yescharacteristic
348C2×D4C22⋊Dic5311yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—2121no—
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C2×C22⋊D5312yes—
360M16—2121no—
361M16—2121no—
362C2×C8—2122no—
363C2×C8—2121no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4C22⋊Dic5312yes—
373C4⋊C4C22⋊Dic5312yes—
374C22⋊C4—21no—
375C22⋊C4—21no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—1062no—
391C42—2121no—
392C42C22×Dic5311yescharacteristic
393C42C2×C22⋊D5311yescharacteristic
394C42C2×C22⋊D5312yes—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—81no—
399C23—82no—
400C23#1313318611yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—41no—
404Q8—81no—
405D4—42no—
406D4—41no—
407D4—41no—
408D4—82no—
409D4—82no—
410D4—81no—
411D4—81no—
412D4—82no—
413D4—82no—
414D4—82no—
415D4—81no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—4242no—
434C2×C4—2122no—
435C2×C4C2×C22⋊Dic5312yes—
436C2×C4C2×C22⋊Dic5311yescharacteristic
437C2×C4#1313318612yes—
438C2×C4#1313318611yescharacteristic
439C8—4242no—
440C8—4241no—
441C22—21no—
442C22—22no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—84no—
451C22#1681047911yescharacteristic
452C4—21no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—42no—
457C4—41no—
458C4—42no—
459C4—42no—
460C4—4242no—
461C4#1681079111yescharacteristic
462C4#1681080111yescharacteristic
463C2—41no—
464C2—81no—
465C2—82no—
466C2—81no—
467C2#2027466911yescharacteristic
468C2#2027467912yes—
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3529C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3529D53
hypocenterC53#8504
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3529

Complements

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

NH ≅ G/Ntype
1. #20275548C2semidirect
2. #20273377C2semidirect
3. #20273201C2semidirect
4. #20273180C2semidirect
6. #16810676C22semidirect
7. #16810647C22semidirect
10. #16810469C22semidirect
18. C53×#563C4semidirect
20. C53×#562C4semidirect
21. C53×H(3,𝔽4)C4semidirect
241. #563Dic53semidirect
243. #562Dic53semidirect
244. H(3,𝔽4)Dic53semidirect
264. C53#8504semidirect

Maximal subgroups

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

subgroupindex
#202755482
#202733772
#202732012
#202731802
C53×#35292
#850453