Finite Group Databasebeta

← #27299234

Subgroups — #27299234

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810413C212yesmaximal
2C4×#13133134C211yesmaximal, characteristic
3C4×#13133132C211yesmaximal, characteristic
4#20272833C212yesmaximal
5C53×#2322C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×#11719917—21no—
9C4×C22⋊Dic53—21no—
10#16810413C418yescomplemented
11#16810392—21no—
12#16810390—21no—
13C2×#123×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C411yescharacteristic, complemented
16C53×#389—21no—
17C53×#388—21no—
18C53×#387—21no—
19C53×#386C2211yescharacteristic
20C53×#385—21no—
21C53×#385C411yescharacteristic, complemented
22C53×#381C411yescharacteristic, complemented
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—41no—
27#13133134—22no—
28#13133134C2×C414yes—
29#13133132—22no—
30#13133132C2×C414yes—
31#13133128—41no—
32C4×#88322—41no—
33#13133124—41no—
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#123×C53—22no—
40#123×C53C2×C414yes—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53—21no—
57C2×C42×C53D411yescharacteristic
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53C22×C412yes—
68C2×D4×C53#4112yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53—81no—
84C22×C4×C53C4212yescomplemented
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×Dic53—42no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—81no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—24no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—81no—
103C22⋊C4×C53C4212yescomplemented
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—81no—
107C22⋊Dic53—24no—
108C22⋊Dic53—42no—
109C4×Dic53—82no—
110#11719917—24no—
111#11719917—42no—
112C2×#88322—42no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C4211yescharacteristic
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#11711yescharacteristic
124C23×C53C4×D412yes—
125D4×C53—42no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53—82no—
155C2×C4×C53#9911yescharacteristic
156C2×C4×C53#9911yescharacteristic
157C2×C4×C53C4×D412yes—
158C2×Dic53—44no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161#88322—44no—
162#12621—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—81no—
177C22×C53C4×C22⋊C411yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53#41011yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—42no—
188C4×C53—81no—
189C4×C53—81no—
190Dic53—84no—
191#2322D5311yescharacteristic
192C2×#352—532no—
193C4×#100—531no—
194C4×#99—531no—
195#1390—532no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—42no—
199C2×C53—41no—
200C2×C53—41no—
201C2×C53C4×#9911yescharacteristic
202C2×C53#212111yescharacteristic
203C2×C53#212311yescharacteristic
204C2×#123C2×D5311yescharacteristic
205C2×C4×D4C2×D5311yescharacteristic
206C22×C4⋊C4Dic5311yescharacteristic, complemented
207C2×#100—532no—
208C2×#99—532no—
209C4×M16—1061no—
210#389—21no—
211#388—21no—
212#387—21no—
213#386C2×D5311yescharacteristic
214#385—21no—
215#385Dic5311yescharacteristic, complemented
216#381Dic5311yescharacteristic, complemented
217C4×C22⋊C4—1061no—
218C4×C22⋊C4Dic5311yescharacteristic, complemented
219#352—538no—
220#343—1061no—
221#342—1061no—
222C53#1262111yescharacteristic, complemented, Sylow
223C22×D4C22×D5311yescharacteristic
224C23×C4C2×Dic5311yescharacteristic
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C2×Dic5311yescharacteristic
227C23×C4C22⋊D5311yescharacteristic
228C2×M16—1062no—
229#123—22no—
230#123C4×D5314yes—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊D5311yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4C2×Dic5311yescharacteristic
244C2×C22⋊C4C2×Dic5311yescharacteristic
245C2×C22⋊C4C2×Dic5311yescharacteristic
246C2×C22⋊C4C22⋊D5311yescharacteristic
247C2×C42—21no—
248C2×C42—2121no—
249C2×C42C22⋊D5311yescharacteristic
250#100—534no—
251#100—1062no—
252#99—534no—
253#99—1062no—
254#97—2121no—
255C4×C8—2121no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—2121no—
262C24C22×Dic5311yescharacteristic
263C24C2×C22⋊D5311yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C2×C4×D5312yes—
267C2×D4#1171994512yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—2122no—
284C22×C4C4×Dic5312yescomplemented
285C22×C4C22⋊Dic5311yescharacteristic
286C22×C4C22⋊Dic5311yescharacteristic
287C22×C4C22⋊Dic5311yescharacteristic
288C22×C4C22⋊Dic5311yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—1064no—
291M16—2122no—
292C2×C8—2122no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C22⋊C4—21no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—1064no—
306C22⋊C4—2122no—
307C22⋊C4C4×Dic5312yescomplemented
308C42—21no—
309C42—21no—
310C42—81no—
311C42—4242no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C2×C4×Dic5311yescharacteristic
321C23C2×C22⋊Dic5311yescharacteristic
322C23#1313317611yescharacteristic
323C23C4×C22⋊D5312yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2124no—
357C2×C4#1313313211yescharacteristic
358C2×C4#1313313211yescharacteristic
359C2×C4C4×C22⋊D5312yes—
360C8—2124no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C4×C22⋊Dic5311yescharacteristic
376C22C2×#1313313211yescharacteristic
377C22#1681059711yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—4244no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#1313313211yescharacteristic
395C2#2027636111yescharacteristic
396C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#2322C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2322D53
hypocenterC53#12621
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2322

Complements

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

NH ≅ G/Ntype
10. #16810413C4semidirect
15. C22×C4⋊C4×C53C4semidirect
21. C53×#385C4semidirect
22. C53×#381C4semidirect
23. C4×C22⋊C4×C53C4semidirect
84. C22×C4×C53C42semidirect
103. C22⋊C4×C53C42semidirect
206. C22×C4⋊C4Dic53semidirect
215. #385Dic53semidirect
216. #381Dic53semidirect
218. C4×C22⋊C4Dic53semidirect
222. C53#12621semidirect
284. C22×C4C4×Dic53semidirect
307. C22⋊C4C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
C4×#131331342
C4×#131331322
#202728332
C53×#23222
#1262153