Finite Group Databasebeta

← #27279997

Subgroups — #27279997

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284512C211yesmaximal, characteristic
2#20277625C211yesmaximal, characteristic
3#20273280C211yesmaximal, characteristic, complemented
4#20273215C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#1478C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×D4×Dic53—21no—
10#16810923—22no—
11#16810913C412yescomplemented
12#16810903C412yescomplemented
13#16810663—21no—
14#16810647C2211yescharacteristic
15#16810628—21no—
16#16810522—21no—
17#16810498—22no—
18#16810489C2211yescharacteristic
19#16810483—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21C2×#13133124—21no—
22#16810361—21no—
23#16810361—21no—
24#16810361—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C53×D4⋊C8—22no—
29C23×Dic53—22no—
30C2×#11719917—21no—
31#13133217C2×C411yescharacteristic, complemented
32#13133217C2×C411yescharacteristic, complemented
33#13133209—21no—
34#13133196—21no—
35#13133196—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38D4×Dic53—42no—
39D4×Dic53—42no—
40#13133176—21no—
41C2×C4⋊Dic53—22no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C22⋊Dic53—22no—
45C2×C4×Dic53—21no—
46#13133150—22no—
47#13133148—22no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yescomplemented
50#13133138—22no—
51#13133138—22no—
52#13133124—22no—
53#13133124—21no—
54#13133124D412yes—
55#13133124D411yescharacteristic
56#13133124D411yescharacteristic
57C22×D4×C53D411yescharacteristic
58C23×C4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53C2×C412yes—
62C4×D4×C53D412yes—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C24×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53C22⋊C412yes—
77C2×D4×C53SD1612yes—
78C2×D4×C53Q1612yes—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—82no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—21no—
89C22×Dic53—44no—
90C22×Dic53—44no—
91C22×Dic53—82no—
92C2×Dic106—21no—
93C2×Dic106—21no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53C22⋊C412yes—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C42×C53C22×C411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104C22⋊Dic53—44no—
105C22⋊Dic53—41no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—42no—
111C4⋊Dic53—41no—
112C4⋊Dic53—41no—
113C4⋊Dic53—82no—
114#11719919—41no—
115#11719919—41no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C4×Dic53—41no—
119#11719917—41no—
120#11719917—41no—
121C2×#88322—22no—
122C23×C53—22no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53#12011yescharacteristic
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—22no—
131D4×C53#10314yes—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—82no—
137C2×C4×C53—82no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53C2×Q1611yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—88no—
160Dic106—81no—
161Dic106—81no—
162C8×C53—81no—
163C8×C53—81no—
164#88322—42no—
165#9175—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53#38611yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45111yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53C2×#10311yescharacteristic
182C4×C53#41711yescharacteristic
183C4×C53#48311yescharacteristic
184C4×C53#48411yescharacteristic
185Dic53—42no—
186Dic53—81no—
187Dic53—82no—
188Dic53—81no—
189Dic53—81no—
190#3077—531no—
191#2349—531no—
192#1518—531no—
193#1503—531no—
194#1478D5311yescharacteristic
195#1303—531no—
196#1286—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—44no—
200C2×C53#187211yescharacteristic
201C2×C53#190411yescharacteristic
202C2×C53#190611yescharacteristic
203#533—531no—
204C2×C4×D4—1061no—
205C2×C4×D4C2×D5311yescharacteristic
206#478—1062no—
207#476—532no—
208#474—532no—
209#431C2×D5311yescharacteristic
210#431C2×D5311yescharacteristic
211#428—1061no—
212#423—531no—
213#418—1061no—
214#389—1061no—
215#385—1062no—
216#382—531no—
217#380—1061no—
218C4×C4⋊C4—531no—
219C2×#95—1061no—
220#328—1061no—
221#328—1061no—
222#328—1062no—
223D4⋊C8—22no—
224C53#917511yescharacteristic, complemented, Sylow
225C22×D4D21211yescharacteristic
226C23×C4—21no—
227C23×C4—1062no—
228C2×M16—1061no—
229C4⋊Q8—531no—
230C4⋊Q8—531no—
231#125—1061no—
232C22⋊Q8—1061no—
233C22⋊Q8—1061no—
234C4×Q8—1061no—
235C4×Q8—1061no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—2122no—
239C4×D4—2122no—
240C4×D4C4×D5312yes—
241C4×D4C22⋊D5312yes—
242#117—1061no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4D21211yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1062no—
249C2×C42—1061no—
250C2×C42C22×D5311yescharacteristic
251#107—1062no—
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—532no—
256C4⋊C8—1061no—
257#102—1062no—
258#102—1062no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263#95—531no—
264#95—531no—
265#95—532no—
266#95—1062no—
267#95—1061no—
268C24—21no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1171991312yes—
275C2×D4Dic21212yes—
276C2×D4#1171992112yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1061no—
284C22×C4—4242no—
285C22×C4—2124no—
286C22×C4—2124no—
287C22×C4C2×D21211yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290M16—2121no—
291M16—2121no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—1062no—
295C4⋊C4—21no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—4242no—
301C4⋊C4—2122no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C4⋊C4#1171992112yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—2124no—
310C22⋊C4—2121no—
311C42—2121no—
312C42—2121no—
313C42—2121no—
314C42C2×C4×D5311yescharacteristic
315C42#1171994711yescharacteristic
316C42#1171994711yescharacteristic
317C42C2×C22⋊D5311yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1313318711yescharacteristic
324Q8—4241no—
325Q8—4241no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1313314214yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—1062no—
337C2×C4—4248no—
338C2×C4—2121no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2122no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C2×#1171992111yescharacteristic
349C2×C4#1313317711yescharacteristic
350C2×C4#1313317711yescharacteristic
351C2×C4#1313318911yescharacteristic
352C2×C4#1313318911yescharacteristic
353C2×C4#1313320111yescharacteristic
354C2×C4C2×#1171991311yescharacteristic
355C2×C4C2×Dic21211yescharacteristic
356C2×C4#1313324511yescharacteristic
357C2×C4#1313325011yescharacteristic
358C8—81no—
359C8—81no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1681050011yescharacteristic
369C22#1681077611yescharacteristic
370C22#1681078111yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—4242no—
377C4—4241no—
378C4—4241no—
379C4—4241no—
380C4—2122no—
381C4C2×#1313314211yescharacteristic
382C4#1681062511yescharacteristic
383C4#1681095411yescharacteristic
384C4#1681095911yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2027502011yescharacteristic
389C2#2027520011yescharacteristic
390C2#2027521411yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1478C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1478D53
hypocenterC53#9175
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #20273280C2semidirect
4. #20273215C2semidirect
5. #20272617C2semidirect
6. #20272574C2semidirect
11. #16810913C4semidirect
12. #16810903C4semidirect
31. #13133217C2×C4semidirect
32. #13133217C2×C4semidirect
49. C4⋊#88322C2×C4semidirect
224. C53#9175semidirect

Maximal subgroups

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

subgroupindex
#202845122
#202776252
#202732802
#202732152
#202726172
#202725742
C53×#14782
#917553