Finite Group Databasebeta

← #26113908

Subgroups — #26113908

Order 12032.

576 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 103 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#20246500C211yesmaximal, characteristic, complemented
2#20242012C211yesmaximal, characteristic, complemented
3#20241896C212yesmaximal, complemented
4#20241282C212yesmaximal, complemented
5C47×#3153C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689030—21no—
8#15689016—22no—
9#15689003—21no—
10C22⋊C4×Dic47—22no—
11#15688969C2212yes—
12#15688863—22no—
13#15688857—22no—
14#15688857—22no—
15C2×#137×C47—21no—
16C2×AGL(1,ℤ/8)×C47—21no—
17C47×#532C2211yescharacteristic
18C2×#124×C47C2211yescharacteristic
19C47×#497C412yescomplemented
20C47×#496C412yescomplemented
21C47×#495—22no—
22C4×M16×C47C2211yescharacteristic
23C23×Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—21no—
27C2×C4×Dic47—22no—
28C2×C4×Dic47—21no—
29C4⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95D412yes—
35C47⋊#95D412yes—
36C2×#41×C47—21no—
37C22×Q8×C47D411yescharacteristic
38C22×D4×C47D411yescharacteristic
39#137×C47—42no—
40#137×C47—42no—
41AGL(1,ℤ/8)×C47—42no—
42AGL(1,ℤ/8)×C47—42no—
43C2×Q16×C47—22no—
44C2×SD16×C47—22no—
45C2×SD16×C47—22no—
46C2×D8×C47—22no—
47C2×M16×C47—21no—
48C4⋊Q8×C47C2×C411yescharacteristic, complemented
49#127×C47C2×C411yescharacteristic, complemented
50#124×C47—21no—
51#124×C47—22no—
52#124×C47C2×C412yes—
53#124×C47D412yes—
54#121×C47—22no—
55C4×D4×C47—22no—
56C2×C22⋊C4×C47—22no—
57C2×C42×C47C2311yescharacteristic
58#97×C47C2×C412yescomplemented
59C4×C8×C47—21no—
60C24×C47—21no—
61#41×C47—41no—
62#41×C47—41no—
63#41×C47—41no—
64#41×C47—41no—
65C2×Q8×C47—21no—
66C2×Q8×C47—21no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47C22⋊C412yes—
70C2×D4×C47—21no—
71C2×D4×C47—22no—
72C2×D4×C47—21no—
73C2×D4×C47—41no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47C22⋊C412yes—
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D412yes—
81C22×Dic47—22no—
82C22×Dic47—22no—
83C22×Dic47—44no—
84C22×Dic47—81no—
85Q16×C47—44no—
86SD16×C47—44no—
87SD16×C47—44no—
88D8×C47—44no—
89M16×C47—22no—
90M16×C47—22no—
91C2×C8×C47—22no—
92C4⋊C4×C47—22no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—42no—
96C42×C47C22×C411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47#4112yes—
99C22⋊Dic47—44no—
100C22⋊Dic47—81no—
101C4⋊Dic47—81no—
102C4×Dic47—44no—
103C4×Dic47—81no—
104C2×C47⋊C8—42no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47#12011yescharacteristic
112Q8×C47—42no—
113Q8×C47—41no—
114Q8×C47—41no—
115Q8×C47—81no—
116D4×C47—42no—
117D4×C47—41no—
118D4×C47—41no—
119D4×C47—81no—
120D4×C47—82no—
121D4×C47—81no—
122D4×C47—81no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—22no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47C2×C22⋊C411yescharacteristic
133C2×C4×C47C4×D412yes—
134C2×C4×C47#12112yes—
135C2×C4×C47#12311yescharacteristic
136C2×Dic47—24no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—84no—
143C8×C47—24no—
144C47⋊C8—82no—
145#9441—471nomaximal, Sylow
146C22×C47—22no—
147C22×C47—42no—
148C22×C47—44no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47—82no—
152C22×C47—81no—
153C22×C47—82no—
154C22×C47#38611yescharacteristic
155C22×C47#45311yescharacteristic
156C22×C47#45311yescharacteristic
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—22no—
160C4×C47—22no—
161C4×C47—42no—
162C4×C47—81no—
163Dic47—44no—
164Dic47—82no—
165#3153D4711yescharacteristic
166#2372—471no—
167#1533—471no—
168#1505—472no—
169#1300—472no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—82no—
173C2×C47—161no—
174C2×C47#188711yescharacteristic
175C2×C47#190612yes—
176C2×#137—21no—
177C2×AGL(1,ℤ/8)—21no—
178#532C2×D4711yescharacteristic
179C2×#124C2×D4711yescharacteristic
180#497Dic4712yescomplemented
181#496Dic4712yescomplemented
182#495—22no—
183#432—472no—
184C4×M16C2×D4711yescharacteristic
185#391—941no—
186#388—942no—
187#386—941no—
188C4×C22⋊C4—942no—
189#376—472no—
190#329—942no—
191#327—942no—
192#327—942no—
193C47#944111yescharacteristic, complemented, Sylow
194C2×#41—21no—
195C22×Q8C22⋊D4711yescharacteristic
196C22×D4C22⋊D4711yescharacteristic
197C23×C4—941no—
198#137—42no—
199#137—42no—
200AGL(1,ℤ/8)—42no—
201AGL(1,ℤ/8)—42no—
202C2×Q16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×D8—22no—
206C2×M16—21no—
207C4⋊Q8C2×Dic4711yescharacteristic, complemented
208#127C2×Dic4711yescharacteristic, complemented
209#124—22no—
210#124—21no—
211#124C2×Dic4712yes—
212#124C22⋊D4712yes—
213#121—22no—
214C4×D4—22no—
215C2×C4⋊C4—941no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4—942no—
219C2×C42—941no—
220C2×C42—942no—
221C2×C42C22×D4711yescharacteristic
222C4⋊C8—942no—
223C22⋊C8—942no—
224#97—942no—
225#97C2×Dic4712yescomplemented
226C4×C8—21no—
227#95—472no—
228#95—472no—
229#95—942no—
230#95—942no—
231C24—21no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8C22⋊Dic4712yes—
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4C22⋊Dic4712yes—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—942no—
251C22×C4—942no—
252C22×C4—3761no—
253C22×C4—1884no—
254C22×C4D4×D4712yes—
255C22×C4C2×C22⋊D4711yescharacteristic
256Q16—44no—
257SD16—44no—
258SD16—44no—
259D8—44no—
260M16—22no—
261M16—22no—
262C2×C8—22no—
263C2×C8—1882no—
264C4⋊C4—22no—
265C4⋊C4—3761no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—42no—
269C22⋊C4—3761no—
270C22⋊C4—1884no—
271C42—3761no—
272C42—1884no—
273C42#1062535212yes—
274C42C22×Dic4711yescharacteristic
275C42C2×C22⋊D4711yescharacteristic
276C23—22no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1309399211yescharacteristic
283Q8—42no—
284Q8—41no—
285Q8—41no—
286Q8—81no—
287D4—42no—
288D4—41no—
289D4—41no—
290D4—81no—
291D4—82no—
292D4—81no—
293D4—81no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—944no—
304C2×C4—3764no—
305C2×C4—1882no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4C2×C22⋊Dic4711yescharacteristic
311C2×C4D4×Dic4712yes—
312C2×C4#1309399612yes—
313C2×C4#1309400511yescharacteristic
314C8—24no—
315C8—3762no—
316C22—22no—
317C22—42no—
318C22—44no—
319C22—81no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22—82no—
324C22#1568900311yescharacteristic
325C22#1568929011yescharacteristic
326C22#1568929011yescharacteristic
327C4—21no—
328C4—21no—
329C4—22no—
330C4—22no—
331C4—42no—
332C4—81no—
333C4—3762no—
334C4—1884no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2—161no—
339C2#2024378711yescharacteristic
340C2#2024389112yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3153C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3153D47
hypocenterC47#9441
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
1. #20246500C2semidirect
2. #20242012C2semidirect
3. #20241896C2semidirect
4. #20241282C2semidirect
19. C47×#497C4semidirect
20. C47×#496C4semidirect
48. C4⋊Q8×C47C2×C4semidirect
49. #127×C47C2×C4semidirect
58. #97×C47C2×C4semidirect
180. #497Dic47semidirect
181. #496Dic47semidirect
193. C47#9441semidirect
207. C4⋊Q8C2×Dic47semidirect
208. #127C2×Dic47semidirect
225. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465002
#202420122
#202418962
#202412822
C47×#31532
#944147