Finite Group Databasebeta

← #27276480

Subgroups — #27276480

Order 13568.

820 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#20275540C211yesmaximal, characteristic, complemented
2#20273429C211yesmaximal, characteristic, complemented
3#20273335C212yesmaximal, complemented
4#20273253C212yesmaximal, complemented
5C53×#3543C211yesmaximal, characteristic
6#16810680C2212yescomplemented
7C2×#13133138—22no—
8C2×#13133128C2211yescharacteristic, complemented
9#16810489C2211yescharacteristic, complemented
10#16810367—22no—
11#16810365—22no—
12#16810363—22no—
13#16810360—22no—
14#16810357—22no—
15C53×#560—22no—
16C53×#559C411yescharacteristic, complemented
17C53×#550C412yescomplemented
18C53×#546—22no—
19D42×C53—22no—
20D42×C53C411yescharacteristic, complemented
21C53×#540—22no—
22C53×#535—22no—
23C53×#534—22no—
24C2×#127×C53C2211yescharacteristic
25C53×#529C2212yes—
26C22×#88322—21no—
27C2×C4⋊Dic53—21no—
28C4⋊#88322—22no—
29#13133138—44no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33#13133124—22no—
34C2×#41×C53—22no—
35C22×D4×C53—22no—
36C22×D4×C53—22no—
37C22×D4×C53—22no—
38C22×D4×C53D412yes—
39#127×C53—22no—
40#127×C53—22no—
41#127×C53C2×C411yescharacteristic
42#127×C53C2×C411yescharacteristic
43#127×C53D412yes—
44#126×C53—22no—
45#125×C53D412yes—
46#124×C53—22no—
47#124×C53D412yes—
48#123×C53—22no—
49#123×C53—24no—
50#123×C53—22no—
51#123×C53—22no—
52C22⋊Q8×C53D412yes—
53#121×C53—24no—
54#121×C53—24no—
55#121×C53—24no—
56#121×C53—22no—
57#121×C53—22no—
58#121×C53—24no—
59#121×C53D412yes—
60#120×C53—22no—
61#120×C53—24no—
62#120×C53—24no—
63#120×C53—22no—
64C4×Q8×C53C2×C412yes—
65C4×D4×C53—22no—
66C4×D4×C53—22no—
67C4×D4×C53—22no—
68C4×D4×C53C2×C412yes—
69#117×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C24×C53—22no—
74#41×C53—84no—
75C2×Q8×C53C22⋊C412yes—
76C2×D4×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—24no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—84no—
87C2×D4×C53—84no—
88C2×D4×C53—82no—
89C2×D4×C53—82no—
90C2×D4×C53C22⋊C412yes—
91C2×D4×C53C22⋊C412yes—
92C2×D4×C53C22⋊C412yes—
93C22×C4×C53—24no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D412yes—
98C22×Dic53—41no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53C22⋊C412yes—
104C4⋊C4×C53C22⋊C412yes—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—24no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—22no—
114C42×C53—22no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D412yes—
117C42×C53C2×D411yescharacteristic
118C4⋊Dic53—42no—
119C2×#88322—22no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—42no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—84no—
129C23×C53—82no—
130C23×C53—84no—
131C23×C53#12011yescharacteristic
132Q8×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—44no—
137D4×C53—44no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—82no—
141D4×C53—84no—
142D4×C53—84no—
143D4×C53—84no—
144D4×C53—84no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—42no—
151C2×C4×C53—44no—
152C2×C4×C53—82no—
153C2×C4×C53—84no—
154C2×C4×C53C2×C22⋊C412yes—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12012yes—
158C2×Dic53—82no—
159C2×Dic53—81no—
160#88322—42no—
161#88322—82no—
162#8497—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—162no—
172C22×C53—162no—
173C22×C53—164no—
174C22×C53#37911yescharacteristic
175C22×C53#45311yescharacteristic
176C22×C53#45511yescharacteristic
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53—42no—
184Dic53—161no—
185#3543D5311yescharacteristic
186#1961—531no—
187#1555—531no—
188#1532—532no—
189#1512—532no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—82no—
193C2×C53—82no—
194C2×C53—82no—
195C2×C53#179211yescharacteristic
196C2×C53#179512yes—
197#560—22no—
198#559Dic5311yescharacteristic, complemented
199#550Dic5312yescomplemented
200#546—22no—
201D42—22no—
202D42Dic5311yescharacteristic, complemented
203#540—22no—
204#535—22no—
205#534—22no—
206C2×#127C2×D5311yescharacteristic
207#529C2×D5312yes—
208#432—532no—
209C2×#102—1062no—
210C2×#97—531no—
211#382—531no—
212#331—1062no—
213#330—1062no—
214#329—1062no—
215#328—1062no—
216#327—1062no—
217C53#849711yescharacteristic, complemented, Sylow
218C2×#41—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4C22⋊D5312yes—
223C22×C8—1061no—
224#127—22no—
225#127—22no—
226#127C2×Dic5311yescharacteristic
227#127C2×Dic5311yescharacteristic
228#127C22⋊D5312yes—
229#126—22no—
230#125C22⋊D5312yes—
231#124—22no—
232#124C22⋊D5312yes—
233#123—22no—
234#123—24no—
235#123—22no—
236#123—22no—
237C22⋊Q8C22⋊D5312yes—
238#121—24no—
239#121—24no—
240#121—24no—
241#121—24no—
242#121—22no—
243#121—22no—
244#121C22⋊D5312yes—
245#120—22no—
246#120—24no—
247#120—24no—
248#120—22no—
249C4×Q8C2×Dic5312yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C2×Dic5312yes—
254#117—22no—
255C2×C4⋊C4—1061no—
256C2×C22⋊C4—22no—
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—1062no—
259#102—2124no—
260C22⋊C8—1062no—
261#97—1062no—
262#97—1062no—
263#95—1062no—
264C24—22no—
265C24—22no—
266#41—84no—
267C2×Q8C22⋊Dic5312yes—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C22⋊Dic5312yes—
283C2×D4C22⋊Dic5312yes—
284C2×D4C22⋊Dic5312yes—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—2121no—
289C22×C4C2×C22⋊D5311yescharacteristic
290C22×C4C2×C22⋊D5312yes—
291C2×C8—1062no—
292C2×C8—2122no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—2122no—
300C4⋊C4C22⋊Dic5312yes—
301C4⋊C4C22⋊Dic5312yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42—22no—
312C42C22×Dic5311yescharacteristic
313C42C2×C22⋊D5312yes—
314C42C2×C22⋊D5311yescharacteristic
315C23—22no—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—84no—
321C23—82no—
322C23—84no—
323C23#1313318611yescharacteristic
324Q8—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—44no—
329D4—44no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—4242no—
347C2×C4—4241no—
348C2×C4C2×C22⋊Dic5312yes—
349C2×C4C2×C22⋊Dic5311yescharacteristic
350C2×C4#1313318611yescharacteristic
351C2×C4#1313318612yes—
352C8—4242no—
353C8—2122no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—162no—
363C22—162no—
364C22—164no—
365C22#1681047911yescharacteristic
366C22#1681079111yescharacteristic
367C22#1681080111yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—8481no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—82no—
380C2—82no—
381C2#2027466911yescharacteristic
382C2#2027468512yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3543C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3543D53
hypocenterC53#8497
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3543

Complements

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

NH ≅ G/Ntype
1. #20275540C2semidirect
2. #20273429C2semidirect
3. #20273335C2semidirect
4. #20273253C2semidirect
6. #16810680C22semidirect
8. C2×#13133128C22semidirect
9. #16810489C22semidirect
16. C53×#559C4semidirect
17. C53×#550C4semidirect
20. D42×C53C4semidirect
198. #559Dic53semidirect
199. #550Dic53semidirect
202. D42Dic53semidirect
217. C53#8497semidirect

Maximal subgroups

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

subgroupindex
#202755402
#202734292
#202733352
#202732532
C53×#35432
#849753