Finite Group Databasebeta

← #26111826

Subgroups — #26111826

Order 12032.

560 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252417C211yesmaximal, characteristic
2#20246053C211yesmaximal, characteristic
3C2×#15688848C212yesmaximal, complemented
4#20241235C212yesmaximal, complemented
5C47×#1459C211yesmaximal, characteristic
6C2×#13094005C411yescharacteristic
7C22×Dic47⋊C4C2211yescharacteristic
8#15689195—24no—
9#15689180—21no—
10#15689087—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15689003C411yescharacteristic
16#15688996—21no—
17#15688995C411yescharacteristic
18#15688984C411yescharacteristic
19C4×C22⋊Dic47C2211yescharacteristic
20#15688848—24no—
21C2×C4×D4×C47C2211yescharacteristic
22C2×C22⋊C8×C47C2212yes—
23C47×#324—21no—
24C47×#323—21no—
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C2×#10625329—22no—
28C22×C47⋊C8—22no—
29#13094005—22no—
30#13094005—22no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—22no—
35C2×Dic47⋊C4—22no—
36C2×Dic47⋊C4—21no—
37C2×C22⋊Dic47—21no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47C2×C411yescharacteristic
40C2×C22⋊Dic47C2×C411yescharacteristic
41C2×C4×Dic47—21no—
42C4⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—22no—
44C22⋊(C47⋊C8)—22no—
45C22⋊(C47⋊C8)D414yes—
46#13093935—22no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C22×D4×C47C2×C411yescharacteristic
53C23×C4×C47D411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55C22×C8×C47—22no—
56C4×D4×C47—22no—
57C4×D4×C47—22no—
58C2×C4⋊C4×C47D411yescharacteristic
59C2×C22⋊C4×C47C2×C411yescharacteristic
60C2×C22⋊C4×C47D411yescharacteristic
61C2×C42×C47D411yescharacteristic
62C22⋊C8×C47—22no—
63C22⋊C8×C47—22no—
64C24×C47C22⋊C411yescharacteristic
65C24×C47C22×C411yescharacteristic
66C2×D4×C47—22no—
67C2×D4×C47—21no—
68C2×D4×C47—21no—
69C2×D4×C47—22no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—81no—
73C22×C4×C47—81no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4112yes—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—22no—
84C22×Dic47—21no—
85C22×Dic47—22no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—22no—
89C22×Dic47—81no—
90C22×Dic47M1612yescomplemented
91C2×C8×C47—42no—
92C2×C8×C47—82no—
93C4⋊C4×C47—22no—
94C22⋊C4×C47—22no—
95C22⋊C4×C47—22no—
96C42×C47—21no—
97C42×C47—21no—
98C22⋊Dic47—24no—
99C22⋊Dic47—21no—
100C22⋊Dic47—22no—
101C22⋊Dic47—81no—
102C22⋊Dic47M1612yescomplemented
103Dic47⋊C4—24no—
104Dic47⋊C4—41no—
105Dic47⋊C4—42no—
106Dic47⋊C4—41no—
107Dic47⋊C4—81no—
108C4×Dic47—81no—
109#10625329—82no—
110C2×C47⋊C8—24no—
111C2×C47⋊C8—82no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47C4×D412yes—
121C23×C47C2×M1611yescharacteristic
122C23×C47C2×M1611yescharacteristic
123C23×C47#13112yes—
124D4×C47—42no—
125D4×C47—42no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—42no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—81no—
139C2×C4×C47#12011yescharacteristic
140C2×C4×C47#12112yes—
141C2×C4×C47#12311yescharacteristic
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—24no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—42no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—82no—
157C8×C47—82no—
158C47⋊C8—44no—
159#9110—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—22no—
162C22×C47—21no—
163C22×C47—22no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—42no—
167C22×C47—42no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—81no—
172C22×C47#38611yescharacteristic
173C22×C47#40711yescharacteristic
174C22×C47#40811yescharacteristic
175C22×C47#43512yes—
176C22×C47#43612yes—
177C4×C47—21no—
178C4×C47—22no—
179C4×C47—21no—
180C4×C47—41no—
181C4×C47—41no—
182C4×C47—41no—
183Dic47—42no—
184Dic47—42no—
185Dic47—81no—
186Dic47—81no—
187#2973—471no—
188#2322—471no—
189#1459D4711yescharacteristic
190C2×#324—472no—
191#1282—472no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—22no—
195C2×C47—22no—
196C2×C47—81no—
197C2×C47#187011yescharacteristic
198C2×C47#191311yescharacteristic
199C2×C47#191511yescharacteristic
200C2×#123—471no—
201C2×C4×D4C2×D4711yescharacteristic
202C22×C4⋊C4—471no—
203#435—944no—
204#432—941no—
205#407—941no—
206C2×C22⋊C8—472no—
207C2×C22⋊C8C2×D4712yes—
208#389—941no—
209#388—941no—
210#387—941no—
211#386—471no—
212#385—471no—
213#385—941no—
214#381—471no—
215C4×C22⋊C4—471no—
216#324—21no—
217#324—944no—
218#323—21no—
219C47#911011yescharacteristic, complemented, Sylow
220C22×D4C4×D4711yescharacteristic
221C23×C4—471no—
222C23×C4—471no—
223C23×C4D18811yescharacteristic
224C23×C4C22×D4711yescharacteristic
225C2×M16—942no—
226C22×C8—22no—
227C22×C8—942no—
228#123—942no—
229#123—942no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—941no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—942no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4C22⋊D4711yescharacteristic
239C2×C22⋊C4—471no—
240C2×C22⋊C4—471no—
241C2×C22⋊C4—941no—
242C2×C22⋊C4—941no—
243C2×C22⋊C4C4×D4711yescharacteristic
244C2×C22⋊C4C22⋊D4711yescharacteristic
245C2×C42—941no—
246C2×C42D18811yescharacteristic
247C4⋊C8—942no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—474no—
251C22⋊C8—942no—
252C22⋊C8—942no—
253#97—942no—
254#95—941no—
255#95—941no—
256#95—941no—
257#95—941no—
258#95—941no—
259C24(C2×D47)⋊C411yescharacteristic
260C24C2×C4×D4711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—472no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—942no—
273C22×C4—941no—
274C22×C4—942no—
275C22×C4—942no—
276C22×C4—941no—
277C22×C4—941no—
278C22×C4—3761no—
279C22×C4(C2×D47)⋊C411yescharacteristic
280C22×C4(C2×D47)⋊C411yescharacteristic
281C22×C4(C2×D47)⋊C411yescharacteristic
282C22×C4C2×D18811yescharacteristic
283C22×C4D4×D4712yes—
284C22×C4#1062535212yes—
285C22×C4C2×C22⋊D4711yescharacteristic
286M16—3762no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—944no—
290C2×C8—3762no—
291C4⋊C4—22no—
292C4⋊C4—944no—
293C4⋊C4—3761no—
294C4⋊C4—1881no—
295C4⋊C4—1882no—
296C4⋊C4—1881no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—472no—
300C22⋊C4—944no—
301C22⋊C4—941no—
302C22⋊C4—942no—
303C22⋊C4—3761no—
304C42—21no—
305C42—21no—
306C42—3761no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×(C2×D47)⋊C411yescharacteristic
315C23(C22⋊D47)⋊C412yes—
316C23M16×D4711yescharacteristic
317C23M16×D4711yescharacteristic
318C23#1309403012yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—944no—
337C2×C4—3762no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4—1882no—
349C2×C4#1309399411yescharacteristic
350C2×C4#1309399612yes—
351C2×C4#1309400811yescharacteristic
352C8—82no—
353C8—1884no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1568900011yescharacteristic
367C22#1568908611yescharacteristic
368C22#1568908911yescharacteristic
369C22#1568919712yes—
370C22#1568919812yes—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3761no—
378C4—3761no—
379C4—1882no—
380C4—1882no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2024368311yescharacteristic
387C2#2024392811yescharacteristic
388C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1459C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#1459D47
hypocenterC47#9110
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. C2×#15688848C2semidirect
4. #20241235C2semidirect
90. C22×Dic47M16semidirect
102. C22⋊Dic47M16semidirect
219. C47#9110semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202460532
C2×#156888482
#202412352
C47×#14592
#911047