Finite Group Databasebeta

← #26126617

Subgroups — #26126617

Order 12032.

470 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13093939C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241409C212yesmaximal, complemented
4#20241388C212yesmaximal, complemented
5C47×#2398C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C4×C22⋊Dic47—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×C4×Q8×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#400—21no—
15C47×#397C411yescharacteristic, complemented
16C47×#397C2211yescharacteristic
17C47×#394—21no—
18C47×#393C411yescharacteristic, complemented
19C47×#393C2211yescharacteristic
20C4×C22⋊C4×C47—21no—
21C4×C22⋊C4×C47—21no—
22C47×#376C411yescharacteristic, complemented
23C22×C47⋊C8—41no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—41no—
26#13093939—24no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—41no—
29C47⋊#95—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×D4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C4×Q8×C47—41no—
35C4×Q8×C47—41no—
36C4×D4×C47—41no—
37C4×D4×C47—41no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47C2×C42×C47—21no—
48C2×C42×C47—21no—
49C2×C42×C47C2×C411yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×Q8×C47—42no—
60C2×Q8×C47—41no—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47—41no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—42no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—42no—
82C4⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C42×C47—21no—
90C42×C47—21no—
91C42×C47—42no—
92C42×C47—41no—
93C42×C47—41no—
94C22⋊Dic47—24no—
95C22⋊Dic47—42no—
96C4×Dic47—82no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—82no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47#10412yes—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#13512yes—
110Q8×C47—42no—
111Q8×C47—42no—
112D4×C47—44no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47#9911yescharacteristic
140C2×C4×C47#9911yescharacteristic
141C2×Dic47—44no—
142C2×Dic47—82no—
143C2×Dic47—82no—
144C47⋊C8—82no—
145#11728—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—81no—
157C22×C47—81no—
158C22×C47C2×#9911yescharacteristic
159C22×C47#41611yescharacteristic
160C22×C47C2×#10411yescharacteristic
161C4×C47—21no—
162C4×C47—42no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—41no—
166C4×C47—41no—
167C4×C47—81no—
168C4×C47—81no—
169C4×C47—81no—
170C4×C47#35412yes—
171Dic47—84no—
172#2398D4711yescharacteristic
173C4×#99—471no—
174#1459—471no—
175#1351—472no—
176#1344—472no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—42no—
180C2×C47—42no—
181C2×C47#150111yescharacteristic
182C2×C47#212011yescharacteristic
183C2×C47C2×#35411yescharacteristic
184C2×C4×Q8Dic4711yescharacteristic, complemented
185C2×C4×D4C2×D4711yescharacteristic
186C2×#99—941no—
187C2×C22⋊C8—941no—
188#400—21no—
189#397Dic4711yescharacteristic, complemented
190#397C2×D4711yescharacteristic
191#394—21no—
192#393Dic4711yescharacteristic, complemented
193#393C2×D4711yescharacteristic
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—941no—
197#376Dic4711yescharacteristic, complemented
198#342—472no—
199#324—941no—
200#323—472no—
201C47#1172811yescharacteristic, complemented, Sylow
202C22×Q8C2×Dic4711yescharacteristic
203C22×D4C22⋊D4711yescharacteristic
204C23×C4C22⋊D4711yescharacteristic
205C23×C4C22⋊D4711yescharacteristic
206C22×C8—1881no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic4711yescharacteristic
213C2×C4⋊C4C22⋊D4711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—942no—
219C2×C22⋊C4C2×Dic4711yescharacteristic
220C2×C22⋊C4C22×D4711yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—1881no—
224C2×C42C2×Dic4711yescharacteristic
225#99—944no—
226C22⋊C8—942no—
227C22⋊C8—1881no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—1881no—
234#95C2×Dic4711yescharacteristic
235#95C2×Dic4711yescharacteristic
236C24C22⋊Dic4711yescharacteristic
237C24C2×C22⋊D4711yescharacteristic
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—1882no—
254C22×C4C22⋊Dic4711yescharacteristic
255C22×C4C22⋊Dic4711yescharacteristic
256C22×C4C22⋊Dic4711yescharacteristic
257C22×C4C22×Dic4711yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259C2×C8—3762no—
260C2×C8—1882no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—944no—
271C22⋊C4—1882no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—3762no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1309395112yes—
287C23C2×C22⋊Dic4711yescharacteristic
288C23#1309404412yes—
289Q8—42no—
290Q8—42no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3762no—
319C2×C4—3762no—
320C2×C4—1884no—
321C2×C4#1309393911yescharacteristic
322C2×C4#1309393911yescharacteristic
323C8—3762no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1309393911yescharacteristic
337C22#1568911911yescharacteristic
338C22C2×#1309395111yescharacteristic
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—3764no—
349C4#1568891812yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2024187911yescharacteristic
355C2#2024502911yescharacteristic
356C2C2×#1568891811yescharacteristic
subgroupG/H
centerC2×C4#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2398C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2398D47
hypocenterC47#11728
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #20241409C2semidirect
4. #20241388C2semidirect
12. C2×C4×Q8×C47C4semidirect
15. C47×#397C4semidirect
18. C47×#393C4semidirect
22. C47×#376C4semidirect
184. C2×C4×Q8Dic47semidirect
189. #397Dic47semidirect
192. #393Dic47semidirect
197. #376Dic47semidirect
201. C47#11728semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417062
#202414092
#202413882
C47×#23982
#1172847