Finite Group Databasebeta

← #27284167

Subgroups — #27284167

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊#88322C211yesmaximal, characteristic
2#20272942C211yesmaximal, characteristic
3#20272593C212yesmaximal
4#20272593C212yesmaximal
5C53×#2851C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7C2×C4×#88322—21no—
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—
25C22⋊#88322—22no—
26C22⋊#88322Q814yes—
27C4×#88322—42no—
28#13133124—24no—
29#13133124—24no—
30#13133124—42no—
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—22no—
59C22×C4×C53—22no—
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×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×Q812yes—
76C22×C4×C53#4112yes—
77C22×Dic53—24no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C42×C53—22no—
95C42×C53—22no—
96C42×C53—41no—
97C42×C53—41no—
98C2×#88322—24no—
99C2×#88322—42no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53#12411yescharacteristic
107C23×C53C4⋊Q811yescharacteristic
108C23×C53#13512yes—
109C23×C53#13512yes—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
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—42no—
125C2×C4×C53—42no—
126C2×C4×C53—44no—
127C2×C4×C53—44no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
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#9912yes—
141C2×C4×C53#10412yes—
142C2×C4×C53C4×Q812yes—
143C2×C4×C53C22⋊Q812yes—
144C2×Dic53—44no—
145C2×Dic53—44no—
146C2×Dic53—44no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149C2×Dic53—82no—
150C2×Dic53—82no—
151#88322—44no—
152#9886—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53#39111yescharacteristic
162C22×C53C2×#9911yescharacteristic
163C22×C53#41011yescharacteristic
164C22×C53#41611yescharacteristic
165C22×C53#41611yescharacteristic
166C22×C53C2×#10411yescharacteristic
167C22×C53#42111yescharacteristic
168C22×C53#48812yes—
169C22×C53#49912yes—
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53—81no—
180Dic53—84no—
181#2851D5311yescharacteristic
182C4×C22⋊C8—531no—
183#1434—531no—
184#1294—532no—
185#1294—532no—
186C2×C53—22no—
187C2×C53—22no—
188C2×C53#150011yescharacteristic
189C2×C53#150111yescharacteristic
190C2×C53#151611yescharacteristic
191C2×C53#152111yescharacteristic
192C2×C53#198611yescharacteristic
193C2×C53#199111yescharacteristic
194C2×C53#199611yescharacteristic
195C2×#125Dic5311yescharacteristic, complemented
196C22×C42C2×D5311yescharacteristic
197C2×C22⋊C8—532no—
198C2×C4×C8—1061no—
199#389Dic5311yescharacteristic, complemented
200#389Dic5311yescharacteristic, complemented
201#387—21no—
202#387—21no—
203#385—21no—
204#385—21no—
205#382Dic5311yescharacteristic, complemented
206#380C2×D5311yescharacteristic
207#380C2×D5311yescharacteristic
208C2×#95—532no—
209C2×#95—1061no—
210C2×#95—1061no—
211#336—1061no—
212C53#988611yescharacteristic, complemented, Sylow
213C23×C4—1062no—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C22×C8—1062no—
218#125—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×Dic5311yescharacteristic
224C2×C4⋊C4C2×Dic5311yescharacteristic
225C2×C4⋊C4C2×Dic5311yescharacteristic
226C2×C4⋊C4C2×Dic5311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—42no—
230C2×C42C2×Dic5311yescharacteristic
231C2×C42C2×Dic5311yescharacteristic
232C2×C42C22⋊D5311yescharacteristic
233C2×C42C22⋊D5311yescharacteristic
234C22⋊C8—534no—
235C22⋊C8—1062no—
236C4×C8—2122no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—1064no—
244#95—1064no—
245#95—2122no—
246C24C2×C22⋊D5311yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—1064no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4—2122no—
263C22×C4—2122no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C22⋊Dic5311yescharacteristic
269C22×C4C22⋊Dic5311yescharacteristic
270C22×C4Q8×D5312yes—
271C22×C4#1171994912yes—
272C22×C4C22×Dic5311yescharacteristic
273C22×C4C2×C22⋊D5311yescharacteristic
274C2×C8—1064no—
275C2×C8—2122no—
276C2×C8—2122no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—22no—
288C42—22no—
289C42—41no—
290C42—41no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23C2×C22⋊Dic5311yescharacteristic
296C23#1313320611yescharacteristic
297C23#1313321511yescharacteristic
298C23#1313323712yes—
299C23#1313323712yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—4242no—
331C2×C4—4242no—
332C2×C4—4242no—
333C2×C4—4242no—
334C2×C4—2124no—
335C2×C4—2124no—
336C2×C4—2124no—
337C2×C4#1313313212yes—
338C2×C4#1313314412yes—
339C2×C4Q8×Dic5312yes—
340C2×C4#1313319312yes—
341C8—2124no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22#1681053011yescharacteristic
351C22C2×#1313313211yescharacteristic
352C22#1681059711yescharacteristic
353C22#1681061911yescharacteristic
354C22#1681061911yescharacteristic
355C22C2×#1313314411yescharacteristic
356C22#1681064011yescharacteristic
357C22#1681098012yes—
358C22#1681102212yes—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—4244no—
370C2—22no—
371C2—22no—
372C2#2027320111yescharacteristic
373C2#2027320611yescharacteristic
374C2#2027327111yescharacteristic
375C2#2027329211yescharacteristic
376C2#2027566011yescharacteristic
377C2#2027568111yescharacteristic
378C2#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#9886
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
195. C2×#125Dic53semidirect
199. #389Dic53semidirect
200. #389Dic53semidirect
205. #382Dic53semidirect
212. C53#9886semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#883222
#202729422
#202725932
C53×#28512
#988653