Finite Group Databasebeta

← #26131423

Subgroups — #26131423

Order 12032.

548 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 123 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#20246053C211yesmaximal, characteristic
2C2×#15688914C212yesmaximal
3C4×#13093939C211yesmaximal, characteristic
4#20241507C212yesmaximal
5C4×#100×C47C211yesmaximal, characteristic
6C2×#13094005C2211yescharacteristic
7C22×Dic47⋊C4C411yescharacteristic, complemented
8C2×#13093939C2212yes—
9#15689018—21no—
10#15689017—21no—
11#15689007—21no—
12#15689003C2211yescharacteristic
13#15688996—21no—
14#15688995C411yescharacteristic, complemented
15#15688984C411yescharacteristic, complemented
16C4×C22⋊Dic47—21no—
17C4×C22⋊Dic47C411yescharacteristic, complemented
18#15688914C418yescomplemented
19#15688890—21no—
20C2×C4×D4×C47C2211yescharacteristic
21C2×#100×C47C2212yes—
22C4×M16×C47—21no—
23C47×#343—21no—
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26#13094005—22no—
27#13094005C2×C414yes—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—41no—
33C2×Dic47⋊C4—41no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47C2×C411yescharacteristic
38C2×C22⋊Dic47C2×C411yescharacteristic
39C2×C4×Dic47—21no—
40C2×C4×Dic47—41no—
41#13093939—22no—
42#13093939C2×C414yes—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—41no—
49C22×D4×C47C2311yescharacteristic
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C2×M16×C47—22no—
53C4×D4×C47—41no—
54C4×D4×C47—41no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47C2×C411yescharacteristic
57C2×C22⋊C4×C47D411yescharacteristic
58C2×C42×C47D411yescharacteristic
59#100×C47—22no—
60#100×C47C2×C414yes—
61#97×C47—41no—
62C4×C8×C47—41no—
63C24×C47C22×C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—41no—
66C2×D4×C47—41no—
67C2×D4×C47C22×C412yes—
68C2×D4×C47#4112yes—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—81no—
74C22×C4×C47—81no—
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—41no—
86C22×Dic47—41no—
87C22×Dic47—42no—
88C22×Dic47—41no—
89C22×Dic47—81no—
90C22×Dic47C4212yescomplemented
91M16×C47—24no—
92M16×C47—42no—
93C2×C8×C47—42no—
94C4⋊C4×C47—41no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C42×C47—21no—
98C42×C47—21no—
99C22⋊Dic47—21no—
100C22⋊Dic47—24no—
101C22⋊Dic47—24no—
102C22⋊Dic47—42no—
103C22⋊Dic47—41no—
104C22⋊Dic47—81no—
105C22⋊Dic47C4212yescomplemented
106Dic47⋊C4—24no—
107Dic47⋊C4—42no—
108Dic47⋊C4—41no—
109Dic47⋊C4—41no—
110Dic47⋊C4—81no—
111C4×Dic47—81no—
112C4×Dic47—82no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47C2×C4211yescharacteristic
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47#11711yescharacteristic
124C23×C47C4×D412yes—
125D4×C47—42no—
126D4×C47—42no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47#9911yescharacteristic
141C2×C4×C47#9911yescharacteristic
142C2×C4×C47C4×D412yes—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—42no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—44no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—82no—
159C2×Dic47—82no—
160C2×Dic47—82no—
161C8×C47—44no—
162#12621—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—22no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—42no—
171C22×C47—42no—
172C22×C47—41no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—81no—
177C22×C47C4×C22⋊C411yescharacteristic
178C22×C47C2×#9911yescharacteristic
179C22×C47#41011yescharacteristic
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—22no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—41no—
186Dic47—42no—
187Dic47—42no—
188Dic47—81no—
189Dic47—84no—
190Dic47—81no—
191#2322—471no—
192C2×#352—472no—
193C4×#100D4711yescharacteristic
194C4×#99—471no—
195#1390—472no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—42no—
199C2×C47—41no—
200C2×C47—41no—
201C2×C47C4×#9911yescharacteristic
202C2×C47#212111yescharacteristic
203C2×C47#212311yescharacteristic
204C2×#123—471no—
205C2×C4×D4C2×D4711yescharacteristic
206C22×C4⋊C4—471no—
207C2×#100C2×D4712yes—
208C2×#99—472no—
209C4×M16—21no—
210#389—941no—
211#388—941no—
212#387—941no—
213#386—471no—
214#385—471no—
215#385—941no—
216#381—471no—
217C4×C22⋊C4—471no—
218C4×C22⋊C4—941no—
219#352—478no—
220#343—21no—
221#342—941no—
222C47#1262111yescharacteristic, complemented, Sylow
223C22×D4C22×D4711yescharacteristic
224C23×C4—471no—
225C23×C4—471no—
226C23×C4C4×D4711yescharacteristic
227C23×C4D18811yescharacteristic
228C2×M16—22no—
229#123—474no—
230#123—942no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4—1881no—
238C2×C4⋊C4—1881no—
239C2×C4⋊C4C22⋊D4711yescharacteristic
240C2×C22⋊C4—471no—
241C2×C22⋊C4—471no—
242C2×C22⋊C4—941no—
243C2×C22⋊C4—942no—
244C2×C22⋊C4—941no—
245C2×C22⋊C4C4×D4711yescharacteristic
246C2×C22⋊C4D18811yescharacteristic
247C2×C42—941no—
248C2×C42—1881no—
249C2×C42C22⋊D4711yescharacteristic
250#100—22no—
251#100C4×D4714yes—
252#99—474no—
253#99—942no—
254#97—41no—
255C4×C8—41no—
256#95—941no—
257#95—941no—
258#95—941no—
259#95—941no—
260#95—941no—
261#95—1881no—
262C24C2×C4×D4711yescharacteristic
263C24C2×D18811yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C2×C4×D4712yes—
267C2×D4#1062535012yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—472no—
275C22×C4—941no—
276C22×C4—941no—
277C22×C4—941no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—3761no—
281C22×C4—1882no—
282C22×C4—1881no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4(C2×D47)⋊C411yescharacteristic
286C22×C4(C2×D47)⋊C411yescharacteristic
287C22×C4(C2×D47)⋊C411yescharacteristic
288C22×C4(C2×D47)⋊C411yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290M16—24no—
291M16—42no—
292C2×C8—42no—
293C4⋊C4—41no—
294C4⋊C4—944no—
295C4⋊C4—3761no—
296C4⋊C4—1882no—
297C4⋊C4—1881no—
298C4⋊C4—1881no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—472no—
302C22⋊C4—941no—
303C22⋊C4—944no—
304C22⋊C4—944no—
305C22⋊C4—3761no—
306C22⋊C4—1882no—
307C22⋊C4—1881no—
308C42—21no—
309C42—21no—
310C42—3762no—
311C42—3761no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C42×D4711yescharacteristic
321C23C2×(C2×D47)⋊C411yescharacteristic
322C23#1309398111yescharacteristic
323C23C4×D18812yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—942no—
340C2×C4—942no—
341C2×C4—3762no—
342C2×C4—3762no—
343C2×C4—3762no—
344C2×C4—1881no—
345C2×C4—1884no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4—1881no—
352C2×C4—1881no—
353C2×C4—1882no—
354C2×C4—1881no—
355C2×C4—1881no—
356C2×C4—1882no—
357C2×C4#1309393811yescharacteristic
358C2×C4#1309393811yescharacteristic
359C2×C4C4×C22⋊D4712yes—
360C8—44no—
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×(C2×D47)⋊C411yescharacteristic
376C22C2×#1309393811yescharacteristic
377C22#1568909611yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3761no—
385C4—3764no—
386C4—3761no—
387C4—1882no—
388C4—1882no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#1309393811yescharacteristic
395C2#2024503511yescharacteristic
396C2#2024504311yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC23×C47C2×C42
FrattiniC22×D4C22×D47
FittingC4×#100×C47C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenterC4×#100D47
hypocenterC47#12621
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
7. C22×Dic47⋊C4C4semidirect
14. #15688995C4semidirect
15. #15688984C4semidirect
17. C4×C22⋊Dic47C4semidirect
18. #15688914C4semidirect
90. C22×Dic47C42semidirect
105. C22⋊Dic47C42semidirect
222. C47#12621semidirect

Maximal subgroups

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

subgroupindex
#202460532
C2×#156889142
C4×#130939392
#202415072
C4×#100×C472
#1262147