Finite Group Databasebeta

← #28310586

Subgroups — #28310586

Order 13568.

548 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811525C212yesmaximal
2C4×#13133277C211yesmaximal, characteristic
3C4×#13133275C211yesmaximal, characteristic
4#20289799C212yesmaximal
5#20277384C211yesmaximal, characteristic
6C2×#13133277C2212yes—
7C2×#13133275C2212yes—
8C4×#11719940—21no—
9C4×C22⋊#6157—21no—
10#16811525C418yescomplemented
11#16811503—21no—
12#16811501—21no—
13C2×#13133200C2211yescharacteristic
14C2×C4×C22⋊D53C2211yescharacteristic
15#16810518—21no—
16#16810513—21no—
17#16810504—21no—
18#16810502C2211yescharacteristic
19#16810496C411yescharacteristic, complemented
20#16810494—21no—
21#16810486C411yescharacteristic, complemented
22C4×C22⋊Dic53C411yescharacteristic, complemented
23C22×C4⋊C4×C53C411yescharacteristic, complemented
24C2×#11719940—22no—
25C2×C22⋊#6157—22no—
26C2×C4×#6157—41no—
27#13133277—22no—
28#13133277C2×C414yes—
29#13133275—22no—
30#13133275C2×C414yes—
31#13133272—41no—
32C4×#88324—41no—
33(C2×#6157)⋊C4—41no—
34C22×C22⋊D53C2311yescharacteristic
35C22×C4×D53D411yescharacteristic
36#13133200—22no—
37#13133200C2×C414yes—
38C4×C22⋊D53—41no—
39C4×C22⋊D53—41no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919D411yescharacteristic
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C411yescharacteristic
45C2×C22⋊Dic53C2×C411yescharacteristic
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×#11719921D411yescharacteristic
49C2×C4×Dic53—21no—
50C2×C4×Dic53D411yescharacteristic
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—41no—
62C2×C4⋊C4×C53—41no—
63C24×C53C22×C411yescharacteristic
64C23×D53C2×D411yescharacteristic
65C22×#6157—42no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C4212yescomplemented
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53C22×C412yes—
82C2×C22⋊D53#4112yes—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C2×D411yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—81no—
91C22⋊#6157—24no—
92C22⋊#6157—42no—
93#11719940—24no—
94#11719940—42no—
95C2×#88324—42no—
96C4×#6157—82no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C22⋊Dic53—21no—
101C22⋊Dic53—41no—
102C22⋊Dic53—41no—
103C22⋊Dic53C4212yescomplemented
104#11719921—24no—
105#11719921—41no—
106#11719921—81no—
107#11719919—24no—
108#11719919—41no—
109#11719919—81no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—81no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53C2×C4211yescharacteristic
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53#11711yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53C4×D412yes—
125C2×#6157—44no—
126C2×#6157—82no—
127C2×#6157—82no—
128C2×C4×C53—22no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—82no—
142C2×C4×C53#9911yescharacteristic
143C2×C4×C53#9911yescharacteristic
144C22⋊D53—42no—
145C22⋊D53—42no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53C4×D412yes—
159C4×D53—42no—
160C4×D53—81no—
161#88324—44no—
162#12621—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53C4×C22⋊C411yescharacteristic
173C22×C53C2×#9911yescharacteristic
174C22×C53#41011yescharacteristic
175C2×D53—22no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—42no—
179C2×D53—81no—
180#6157—84no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—42no—
185C4×C53—81no—
186Dic53—22no—
187Dic53—42no—
188Dic53—41no—
189Dic53—41no—
190Dic53—81no—
191#2322—531no—
192C2×#352—532no—
193C4×#100—531no—
194C4×#99—531no—
195#1390—532no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—41no—
199C2×C53—41no—
200C2×C53C4×#9911yescharacteristic
201C2×C53#212111yescharacteristic
202C2×C53#212311yescharacteristic
203D53—42no—
204C2×#123—531no—
205C2×C4×D4—531no—
206C22×C4⋊C4#615711yescharacteristic, complemented
207C2×#100—532no—
208C2×#99—532no—
209C4×M16—1061no—
210#389—1061no—
211#388—1061no—
212#387—1061no—
213#386—531no—
214#385—531no—
215#385—1061no—
216#381—531no—
217C4×C22⋊C4—531no—
218C4×C22⋊C4—1061no—
219#352—538no—
220#343—1061no—
221#342—1061no—
222C53#1262111yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C23×C4—531no—
225C23×C4C2×#615711yescharacteristic
226C23×C4C2×#615711yescharacteristic
227C23×C4C2×#615711yescharacteristic
228C2×M16—1062no—
229#123—534no—
230#123—1062no—
231C4×D4—2121no—
232C4×D4—2121no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1061no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—1061no—
247C2×C42—531no—
248C2×C42—1061no—
249C2×C42—2121no—
250#100—534no—
251#100—1062no—
252#99—534no—
253#99—1062no—
254#97—2121no—
255C4×C8—2121no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—2121no—
262C24—531no—
263C24C22×#615711yescharacteristic
264C2×D4—532no—
265C2×D4—532no—
266C2×D4—2121no—
267C2×D4—2121no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—4241no—
286C22×C4—2122no—
287C22×C4C4×#615712yescomplemented
288C22×C4C22⋊#615711yescharacteristic
289C22×C4C22⋊#615711yescharacteristic
290M16—1064no—
291M16—2122no—
292C2×C8—2122no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—1064no—
297C4⋊C4—4241no—
298C4⋊C4—2121no—
299C22⋊C4—532no—
300C22⋊C4—1064no—
301C22⋊C4—1061no—
302C22⋊C4—1064no—
303C22⋊C4—4241no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C22⋊C4—2122no—
307C22⋊C4—2121no—
308C42—1061no—
309C42—1061no—
310C42—4241no—
311C42—4242no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—532no—
317C23—1061no—
318C23—1061no—
319C23—2121no—
320C23—2121no—
321C23C2×C4×#615711yescharacteristic
322C23C2×C22⋊#615711yescharacteristic
323C23#1313329611yescharacteristic
324D4—2122no—
325D4—2122no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—532no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1062no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—4241no—
347C2×C4—4242no—
348C2×C4—4242no—
349C2×C4—2122no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2124no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4#1313327511yescharacteristic
359C2×C4#1313327511yescharacteristic
360C8—2124no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—1062no—
371C22—4241no—
372C22—2122no—
373C22—2121no—
374C22—2121no—
375C22C4×C22⋊#615711yescharacteristic
376C22C2×#1313327511yescharacteristic
377C22#1681160211yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—1062no—
384C4—4241no—
385C4—4244no—
386C4—2122no—
387C4—2121no—
388C4—2121no—
389C2—21no—
390C2—21no—
391C2—41no—
392C2—41no—
393C2—2122no—
394C2C4×#1313327511yescharacteristic
395C2#2029097011yescharacteristic
396C2#2029097711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radical, cosocleC22×C22⋊D53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#12621
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #16811525C4semidirect
19. #16810496C4semidirect
21. #16810486C4semidirect
22. C4×C22⋊Dic53C4semidirect
23. C22×C4⋊C4×C53C4semidirect
76. C22×C4×C53C42semidirect
103. C22⋊Dic53C42semidirect
206. C22×C4⋊C4#6157semidirect
222. C53#12621semidirect
287. C22×C4C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115252
C4×#131332772
C4×#131332752
#202897992
#202773842
#1262153