Finite Group Databasebeta

← #27284284

Subgroups — #27284284

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic
2#20272942C211yesmaximal, characteristic
3#20272593C212yesmaximal
4#20272593C212yesmaximal
5C53×#2851C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×C22⋊#88322C2212yes—
8C2×#13133124—21no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12C2×#125×C53C411yescharacteristic, complemented
13C22×C42×C53C2211yescharacteristic
14C53×#389C411yescharacteristic, complemented
15C53×#389C411yescharacteristic, complemented
16C53×#387—21no—
17C53×#387—21no—
18C53×#385—21no—
19C53×#385—21no—
20C53×#382C411yescharacteristic, complemented
21C53×#380C2211yescharacteristic
22C53×#380C2211yescharacteristic
23C23×Dic53—22no—
24C22×#88322—22no—
25C4⋊#88322—42no—
26C22⋊#88322—24no—
27#13133124—22no—
28#13133124—42no—
29#13133124—42no—
30#13133124Q814yes—
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34#125×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C42×C53—42no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×Q812yes—
77C22×C4×C53#4112yes—
78C22×Dic53—24no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C42×C53—22no—
97C42×C53—22no—
98C42×C53—41no—
99C42×C53—41no—
100C2×#88322—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53C2×C22⋊C411yescharacteristic
108C23×C53C22⋊Q812yes—
109C23×C53#13512yes—
110C23×C53AGL(1,ℤ/8)11yescharacteristic
111C23×C53#13711yescharacteristic
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—44no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53#9912yes—
145C2×C4×C53C4×Q812yes—
146C2×C4×C53#12411yescharacteristic
147C2×C4×C53C4⋊Q811yescharacteristic
148C2×Dic53—44no—
149C2×Dic53—44no—
150C2×Dic53—44no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C2×Dic53—82no—
155#88322—82no—
156#9913—531nomaximal, Sylow
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53#39111yescharacteristic
166C22×C53C2×#9911yescharacteristic
167C22×C53#41011yescharacteristic
168C22×C53#41611yescharacteristic
169C22×C53#41711yescharacteristic
170C22×C53#42111yescharacteristic
171C22×C53#42111yescharacteristic
172C22×C53#47812yes—
173C22×C53#47912yes—
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—21no—
177C4×C53—22no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—81no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53—81no—
185Dic53—84no—
186#2851D5311yescharacteristic
187#1797—531no—
188#1434—531no—
189#1294—532no—
190#1294—532no—
191C2×C53—22no—
192C2×C53—22no—
193C2×C53#151211yescharacteristic
194C2×C53#151311yescharacteristic
195C2×C53#151611yescharacteristic
196C2×C53#152111yescharacteristic
197C2×C53#198911yescharacteristic
198C2×C53#199511yescharacteristic
199C2×C53#199611yescharacteristic
200C2×#125Dic5311yescharacteristic, complemented
201C22×C42C2×D5311yescharacteristic
202C2×C4⋊C8—1061no—
203C2×C22⋊C8—532no—
204#389Dic5311yescharacteristic, complemented
205#389Dic5311yescharacteristic, complemented
206#387—21no—
207#387—21no—
208#385—21no—
209#385—21no—
210#382Dic5311yescharacteristic, complemented
211#380C2×D5311yescharacteristic
212#380C2×D5311yescharacteristic
213C2×#95—532no—
214C2×#95—1061no—
215C2×#95—1061no—
216#336—1061no—
217C53#991311yescharacteristic, complemented, Sylow
218C23×C4—1062no—
219C23×C4C22⋊D5311yescharacteristic
220C23×C4C22⋊D5311yescharacteristic
221C23×C4C22×D5311yescharacteristic
222C22×C8—1062no—
223#125—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic5311yescharacteristic
229C2×C4⋊C4C2×Dic5311yescharacteristic
230C2×C4⋊C4C2×Dic5311yescharacteristic
231C2×C4⋊C4C2×Dic5311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—42no—
235C2×C42C2×Dic5311yescharacteristic
236C2×C42C2×Dic5311yescharacteristic
237C2×C42C22⋊D5311yescharacteristic
238C2×C42C22⋊D5311yescharacteristic
239C4⋊C8—2122no—
240C22⋊C8—1064no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—534no—
248#95—1062no—
249#95—2122no—
250#95—2122no—
251C24C2×C22⋊D5311yescharacteristic
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—1064no—
266C22×C4—2122no—
267C22×C4—2122no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4C22⋊Dic5311yescharacteristic
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22⋊Dic5311yescharacteristic
276C22×C4Q8×D5312yes—
277C22×C4#1171994912yes—
278C22×C4C22×Dic5311yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280C2×C8—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×C22⋊Dic5311yescharacteristic
303C23#1313319312yes—
304C23#1313323712yes—
305C23#1313324211yescharacteristic
306C23#1313324711yescharacteristic
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—4242no—
340C2×C4—4242no—
341C2×C4—4242no—
342C2×C4—4242no—
343C2×C4—2124no—
344C2×C4—2124no—
345C2×C4—2124no—
346C2×C4#1313313212yes—
347C2×C4Q8×Dic5312yes—
348C2×C4#1313320611yescharacteristic
349C2×C4#1313321511yescharacteristic
350C8—4242no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22#1681053011yescharacteristic
360C22C2×#1313313211yescharacteristic
361C22#1681059711yescharacteristic
362C22#1681061911yescharacteristic
363C22#1681062411yescharacteristic
364C22#1681064011yescharacteristic
365C22#1681064011yescharacteristic
366C22#1681092012yes—
367C22#1681092712yes—
368C4—22no—
369C4—21no—
370C4—22no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—4244no—
380C2—22no—
381C2—22no—
382C2#2027325311yescharacteristic
383C2#2027325811yescharacteristic
384C2#2027327111yescharacteristic
385C2#2027329211yescharacteristic
386C2#2027567311yescharacteristic
387C2#2027569711yescharacteristic
388C2#2027570111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2851C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2851D53
hypocenterC53#9913
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C53C4semidirect
14. C53×#389C4semidirect
15. C53×#389C4semidirect
20. C53×#382C4semidirect
200. C2×#125Dic53semidirect
204. #389Dic53semidirect
205. #389Dic53semidirect
210. #382Dic53semidirect
217. C53#9913semidirect

Maximal subgroups

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

subgroupindex
#202746952
#202729422
#202725932
C53×#28512
#991353