Finite Group Databasebeta

← #27299679

Subgroups — #27299679

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276400C211yesmaximal, characteristic, complemented
2#20276380C211yesmaximal, characteristic, complemented
3#20276356C212yesmaximal, complemented
4#20274819C212yesmaximal, complemented
5C53×#3444C211yesmaximal, characteristic
6#16810607C2212yes—
7#16810597C2212yes—
8#16810422—21no—
9#16810420—21no—
10#16810418—21no—
11#16810416—21no—
12#16810413—22no—
13#16810410—22no—
14#16810383—22no—
15#16810379—22no—
16C53×#533—21no—
17C53×#533C411yescharacteristic, complemented
18C53×#532—21no—
19C53×#532C2211yescharacteristic
20C53×#529—21no—
21C53×#529—21no—
22C53×#529—22no—
23C53×#529—22no—
24C53×#525—22no—
25C53×#525C411yescharacteristic, complemented
26C53×#525C2211yescharacteristic
27C4×#41×C53—22no—
28C4×#41×C53C411yescharacteristic, complemented
29C4×#41×C53C2211yescharacteristic
30C22×C42×C53C411yescharacteristic, complemented
31C2×#11719917—22no—
32#13133176—22no—
33#13133136—22no—
34#13133134—22no—
35#13133132—22no—
36#13133132—22no—
37C2×#41×C53—22no—
38C2×#41×C53C2×C411yescharacteristic
39C2×#41×C53C2311yescharacteristic
40C23×C4×C53—21no—
41C23×C4×C53C2×C411yescharacteristic
42C4⋊Q8×C53—21no—
43C4⋊Q8×C53D411yescharacteristic
44#127×C53D411yescharacteristic
45#126×C53—22no—
46#126×C53—22no—
47#125×C53—22no—
48#125×C53—21no—
49#124×C53—22no—
50#124×C53D411yescharacteristic
51#124×C53D411yescharacteristic
52#123×C53—22no—
53#123×C53—22no—
54#123×C53—21no—
55#123×C53D412yes—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60#121×C53—22no—
61#121×C53—21no—
62#121×C53—22no—
63#121×C53—21no—
64#120×C53—21no—
65#120×C53D412yes—
66C4×Q8×C53—22no—
67C4×Q8×C53—22no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×D4×C53—22no—
71C4×D4×C53—22no—
72C4×D4×C53—22no—
73C4×D4×C53—22no—
74C4×D4×C53—22no—
75C4×D4×C53—21no—
76C4×D4×C53—22no—
77C4×D4×C53—21no—
78C4×D4×C53—22no—
79C4×D4×C53—22no—
80#117×C53—22no—
81#117×C53—22no—
82#117×C53—22no—
83#117×C53—21no—
84#117×C53C2×C411yescharacteristic
85#117×C53C2×C411yescharacteristic
86#117×C53D412yes—
87C2×C42×C53—21no—
88C2×C42×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53C2×C411yescharacteristic
94C2×C42×C53C2×C411yescharacteristic
95C2×C42×C53D412yes—
96C24×C53C22⋊C411yescharacteristic
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—48no—
100#41×C53—41no—
101#41×C53—41no—
102C2×Q8×C53—22no—
103C2×Q8×C53C22⋊C411yescharacteristic
104C2×Q8×C53C2×D411yescharacteristic
105C2×D4×C53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53C22⋊C411yescharacteristic
110C2×D4×C53C2×D411yescharacteristic
111C2×D4×C53C2×D412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—44no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53C22⋊C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D412yes—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—21no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—22no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53C22⋊C412yes—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—22no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—22no—
158C42×C53C22⋊C411yescharacteristic
159C42×C53C22⋊C411yescharacteristic
160C42×C53C22⋊C411yescharacteristic
161C42×C53C22⋊C412yes—
162C42×C53C22⋊C411yescharacteristic
163C22⋊Dic53—42no—
164C4⋊Dic53—42no—
165C4×Dic53—42no—
166#11719917—42no—
167#11719917—42no—
168C2×#88322—42no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—42no—
172C23×C53—41no—
173C23×C53C2×C22⋊C411yescharacteristic
174C23×C53#12012yes—
175Q8×C53—41no—
176Q8×C53—44no—
177Q8×C53—41no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—44no—
182D4×C53—44no—
183D4×C53—44no—
184D4×C53—42no—
185C2×C4×C53—24no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53#12012yes—
214C2×Dic53—82no—
215#88322—82no—
216#12719—531nomaximal, Sylow
217C22×C53—22no—
218C22×C53—22no—
219C22×C53—21no—
220C22×C53—41no—
221C22×C53—42no—
222C22×C53—41no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53#37911yescharacteristic
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—41no—
234C4×C53—41no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53#41011yescharacteristic
240C4×C53#41011yescharacteristic
241Dic53—162no—
242#3444D5311yescharacteristic
243#2132—531no—
244#2126—531no—
245#2120—532no—
246#1827—532no—
247C2×C53—21no—
248C2×C53—42no—
249C2×C53—42no—
250C2×C53—41no—
251C2×C53—81no—
252C2×C53#178211yescharacteristic
253#533—21no—
254#533Dic5311yescharacteristic, complemented
255#532—21no—
256#532C2×D5311yescharacteristic
257#529—21no—
258#529—21no—
259#529—22no—
260#529—22no—
261#525—22no—
262#525Dic5311yescharacteristic, complemented
263#525C2×D5311yescharacteristic
264C4×#41—22no—
265C4×#41Dic5311yescharacteristic, complemented
266C4×#41C2×D5311yescharacteristic
267C22×C42Dic5311yescharacteristic, complemented
268#413—532no—
269#410—532no—
270#356—1061no—
271#355—1061no—
272#354—1061no—
273#353—1061no—
274#352—1062no—
275AΣL(1,𝔽16)—1062no—
276#339—1062no—
277H(3,ℤ/4)—1062no—
278C53#1271911yescharacteristic, complemented, Sylow
279C2×#41—22no—
280C2×#41C2×Dic5311yescharacteristic
281C2×#41C22×D5311yescharacteristic
282C23×C4—21no—
283C23×C4C2×Dic5311yescharacteristic
284C2×M16—1062no—
285C4⋊Q8—21no—
286C4⋊Q8C22⋊D5311yescharacteristic
287#127C22⋊D5311yescharacteristic
288#126—22no—
289#126—22no—
290#125—22no—
291#125—21no—
292#124—22no—
293#124C22⋊D5311yescharacteristic
294#124C22⋊D5311yescharacteristic
295#123—22no—
296#123—22no—
297#123—21no—
298#123C22⋊D5312yes—
299C22⋊Q8—22no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302C22⋊Q8—22no—
303#121—22no—
304#121—22no—
305#121—21no—
306#121—21no—
307#120—21no—
308#120C22⋊D5312yes—
309C4×Q8—22no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—21no—
327#117—1062no—
328#117C2×Dic5311yescharacteristic
329#117C2×Dic5311yescharacteristic
330#117C22⋊D5312yes—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C2×Dic5311yescharacteristic
338C2×C42C2×Dic5311yescharacteristic
339C2×C42C22⋊D5312yes—
340#101—1062no—
341#100—1062no—
342#99—1062no—
343#99—1062no—
344C24C22⋊Dic5311yescharacteristic
345#41—41no—
346#41—41no—
347#41—48no—
348#41—41no—
349#41—41no—
350C2×Q8—22no—
351C2×Q8C22⋊Dic5311yescharacteristic
352C2×Q8C2×C22⋊D5311yescharacteristic
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4C22⋊Dic5311yescharacteristic
358C2×D4C2×C22⋊D5311yescharacteristic
359C2×D4C2×C22⋊D5312yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4C22⋊Dic5311yescharacteristic
375C22×C4C22×Dic5311yescharacteristic
376C22×C4C2×C22⋊D5312yes—
377M16—2122no—
378M16—2122no—
379C2×C8—2122no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—2122no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—21no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—2122no—
401C22⋊C4C22⋊Dic5312yes—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—22no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42—2122no—
412C42C22⋊Dic5311yescharacteristic
413C42C22⋊Dic5311yescharacteristic
414C42C22⋊Dic5311yescharacteristic
415C42C22⋊Dic5312yes—
416C42C22⋊Dic5311yescharacteristic
417C23—21no—
418C23—21no—
419C23—42no—
420C23—41no—
421C23C2×C22⋊Dic5311yescharacteristic
422C23#1313318612yes—
423Q8—41no—
424Q8—41no—
425Q8—44no—
426D4—42no—
427D4—41no—
428D4—41no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—42no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—4242no—
460C2×C4C2×C22⋊Dic5311yescharacteristic
461C2×C4C2×C22⋊Dic5311yescharacteristic
462C2×C4#1313318612yes—
463C8—4242no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—41no—
468C22—42no—
469C22—41no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22#1681047911yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—41no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—8482no—
487C4#1681059711yescharacteristic
488C4#1681059711yescharacteristic
489C2—21no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—81no—
494C2#2027463311yescharacteristic
subgroupG/H
centerC4#16810597
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3444C2
socleC2×C53#1782
cosocleC2×#41×C53C23
hypercenter#3444D53
hypocenterC53#12719
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ #3444

Complements

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

NH ≅ G/Ntype
1. #20276400C2semidirect
2. #20276380C2semidirect
3. #20276356C2semidirect
4. #20274819C2semidirect
17. C53×#533C4semidirect
25. C53×#525C4semidirect
28. C4×#41×C53C4semidirect
30. C22×C42×C53C4semidirect
254. #533Dic53semidirect
262. #525Dic53semidirect
265. C4×#41Dic53semidirect
267. C22×C42Dic53semidirect
278. C53#12719semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202763802
#202763562
#202748192
C53×#34442
#1271953