Finite Group Databasebeta

← #26109175

Subgroups — #26109175

Order 12032.

550 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×(C22⋊D47)⋊C8C211yesmaximal, characteristic
2#20243415C212yesmaximal, complemented
3#20243373C212yesmaximal
4#20241256C211yesmaximal, characteristic, complemented
5C47×#1289C211yesmaximal, characteristic, complemented
6C23×C47⋊C8C2211yescharacteristic
7C22×C8×D47C411yescharacteristic, complemented
8C2×C4×C22⋊D47C2211yescharacteristic
9(C22⋊D47)⋊C8—22no—
10(C22⋊D47)⋊C8—22no—
11C2×#13093957—22no—
12C2×Dic47⋊C8C411yescharacteristic, complemented
13C2×#13093938—22no—
14C2×(C2×D47)⋊C8C411yescharacteristic, complemented
15C2×C8×Dic47C411yescharacteristic, complemented
16#15688983C2212yes—
17#15688889—22no—
18#15688885—22no—
19C2×C22⋊C8×C47C2211yescharacteristic
20C47×#380C2212yes—
21C22×C22⋊D47D411yescharacteristic
22C22×C4×D47C2×C411yescharacteristic
23C22×C47⋊C8—42no—
24C22×C47⋊C8C2×C411yescharacteristic
25C22×C47⋊C8C2×C411yescharacteristic
26C22×C47⋊C8D412yes—
27C22×C47⋊C8D412yes—
28C2×C8×D47—21no—
29C2×C8×D47—21no—
30C2×C8×D47—42no—
31C4×C22⋊D47—22no—
32C4×C22⋊D47—22no—
33C2×C4⋊Dic47—22no—
34C2×Dic47⋊C4Q811yescharacteristic
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47D411yescharacteristic
37C2×(C2×D47)⋊C4Q811yescharacteristic
38C2×C4×Dic47C2×C411yescharacteristic
39#13093957—44no—
40Dic47⋊C8—22no—
41#13093938—44no—
42(C2×D47)⋊C8—22no—
43C8×Dic47—21no—
44C8×Dic47—21no—
45C47⋊#95—22no—
46C23×C4×C47C2311yescharacteristic
47C22×C8×C47C2×C411yescharacteristic
48C22×C8×C47C2×C411yescharacteristic
49C2×C4⋊C4×C47—22no—
50C2×C22⋊C4×C47—22no—
51C22⋊C8×C47—22no—
52#95×C47—22no—
53C24×C47C2×D411yescharacteristic
54C23×D47C4⋊C411yescharacteristic
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—42no—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D412yes—
60C22×C4×C47C2×Q811yescharacteristic
61C22×C4×C47#4112yes—
62C2×C22⋊D47—22no—
63C2×C22⋊D47—22no—
64C2×C22⋊D47D812yes—
65C2×C22⋊D47Q1612yes—
66C22×Dic47—42no—
67C22×Dic47C4⋊C411yescharacteristic
68C22×Dic47C4⋊C411yescharacteristic
69C22×Dic47C4⋊C411yescharacteristic
70C2×C4×D47—21no—
71C2×C4×D47—21no—
72C2×C4×D47—41no—
73C2×C4×D47—41no—
74C2×C8×C47—21no—
75C2×C8×C47—21no—
76C2×C8×C47—41no—
77C2×C8×C47—41no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—42no—
80C22⋊C4×C47—42no—
81C22⋊C4×C47—42no—
82C22⋊Dic47—22no—
83C22⋊Dic47—42no—
84C22⋊Dic47—42no—
85(C2×D47)⋊C4—22no—
86C4⋊Dic47—42no—
87C4⋊Dic47—42no—
88Dic47⋊C4—22no—
89C4×Dic47—21no—
90C4×Dic47—21no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—22no—
93C2×C47⋊C8—41no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8C22⋊C411yescharacteristic
96C2×C47⋊C8C22⋊C412yes—
97C2×C47⋊C8C22⋊C411yescharacteristic
98C8×D47—42no—
99C8×D47—42no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47C2×C4⋊C411yescharacteristic
103C23×C47#12112yes—
104C23×C47C2×D811yescharacteristic
105C23×C47C2×Q1611yescharacteristic
106C23×C47#13512yes—
107C22×D47—21no—
108C22×D47—21no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47#10712yescomplemented
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—82no—
119C2×C4×C47—82no—
120C2×C4×C47—82no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47#11711yescharacteristic
123C2×C4×C47C22⋊Q812yes—
124C22⋊D47—24no—
125C22⋊D47—24no—
126C2×Dic47—21no—
127C2×Dic47—21no—
128C2×Dic47—21no—
129C2×Dic47—21no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—82no—
133C2×Dic47—82no—
134C2×Dic47—82no—
135C2×Dic47#10712yescomplemented
136C4×D47—42no—
137C4×D47—41no—
138C4×D47—41no—
139C8×C47—41no—
140C8×C47—41no—
141C47⋊C8—21no—
142C47⋊C8—22no—
143C47⋊C8#9912yescomplemented
144#8609—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—22no—
147C22×C47—21no—
148C22×C47—22no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—42no—
152C22×C47#38011yescharacteristic
153C22×C47C2×#10711yescharacteristic
154C22×C47#42711yescharacteristic
155C22×C47#46612yes—
156C22×C47#46712yes—
157C2×D47—22no—
158C2×D47—42no—
159C2×D47—41no—
160C2×D47—41no—
161C2×D47—42no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—162no—
165C4×C47C2×#9911yescharacteristic
166C4×C47#41011yescharacteristic
167Dic47—22no—
168Dic47—41no—
169Dic47—41no—
170Dic47—162no—
171C2×C8×D4—471no—
172#1808—472no—
173#1799—472no—
174#1289—471no—
175#1289D4711yescharacteristic, complemented
176C2×C47—22no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—22no—
180C2×C47#179911yescharacteristic
181C2×C47#180811yescharacteristic
182C2×C47#181211yescharacteristic
183D47—42no—
184C23×C8—471no—
185C23×C8—471no—
186C2×C4×D4—471no—
187C8×D4—942no—
188C8×D4—942no—
189C2×#107—942no—
190C2×C4⋊C8—471no—
191C2×#99—942no—
192C2×C22⋊C8—471no—
193C2×C22⋊C8C2×D4711yescharacteristic
194C2×C4×C8—471no—
195#380—472no—
196#380C2×D4712yes—
197#342—942no—
198#340—942no—
199C47#860911yescharacteristic, complemented, Sylow
200C22×D4—471no—
201C23×C4—471no—
202C23×C4C22×D4711yescharacteristic
203C22×C8—471no—
204C22×C8—472no—
205C22×C8—471no—
206C22×C8—472no—
207C22×C8—941no—
208C22×C8—941no—
209C22×C8—1882no—
210C22×C8—1882no—
211C22×C8C4×D4711yescharacteristic
212C22×C8C4×D4711yescharacteristic
213C4×D4—942no—
214C4×D4—942no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4—942no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—471no—
220C2×C22⋊C4—471no—
221C2×C22⋊C4—942no—
222C2×C42—471no—
223#107—1884no—
224C4⋊C8—942no—
225#99—1884no—
226C22⋊C8—22no—
227C22⋊C8—942no—
228C4×C8—941no—
229C4×C8—941no—
230#95—22no—
231#95—942no—
232C24—471no—
233C24D4×D4711yescharacteristic
234C2×D4—472no—
235C2×D4—472no—
236C2×D4—942no—
237C2×D4—942no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—471no—
242C22×C4—471no—
243C22×C4—471no—
244C22×C4—941no—
245C22×C4—941no—
246C22×C4—1881no—
247C22×C4—1881no—
248C22×C4—1882no—
249C22×C4C2×C4×D4711yescharacteristic
250C22×C4#1062535012yes—
251C22×C4D4×D4712yes—
252C22×C4Q8×D4711yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—471no—
258C2×C8—471no—
259C2×C8—472no—
260C2×C8—944no—
261C2×C8—942no—
262C2×C8—1882no—
263C2×C8—1882no—
264C2×C8—1881no—
265C2×C8—1882no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—942no—
269C4⋊C4—1882no—
270C4⋊C4—1882no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—942no—
274C22⋊C4—942no—
275C22⋊C4—1882no—
276C22⋊C4—1882no—
277C42—941no—
278C42—941no—
279C23—41no—
280C23—41no—
281C23—472no—
282C23—941no—
283C23—941no—
284C23—1881no—
285C23—1881no—
286C23C4⋊C4×D4711yescharacteristic
287C23#1309399712yes—
288C23D8×D4711yescharacteristic
289C23Q16×D4711yescharacteristic
290C23#1309404512yes—
291D4—944no—
292D4—944no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—472no—
303C2×C4—941no—
304C2×C4—941no—
305C2×C4—941no—
306C2×C4—941no—
307C2×C4—3762no—
308C2×C4—3762no—
309C2×C4—3762no—
310C2×C4—1882no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4C22⋊C4×D4711yescharacteristic
316C2×C4#1309398111yescharacteristic
317C2×C4#1309400212yes—
318C8—41no—
319C8—41no—
320C8—472no—
321C8—942no—
322C8—941no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—942no—
331C22—1882no—
332C22—1882no—
333C22—1881no—
334C22—1881no—
335C22#1568898211yescharacteristic
336C22#107×D4711yescharacteristic
337C22#1568915811yescharacteristic
338C22#1568935812yes—
339C22#1568936312yes—
340C4—21no—
341C4—22no—
342C4—162no—
343C4—942no—
344C4—1881no—
345C4—1881no—
346C4—7522no—
347C4#99×D4711yescharacteristic
348C4#1568909511yescharacteristic
349C2—22no—
350C2—21no—
351C2—21no—
352C2—22no—
353C2—1882no—
354C2#2024337511yescharacteristic
355C2#2024341611yescharacteristic
356C2#2024343311yescharacteristic
subgroupG/H
centerC22#15688982
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1289C2
socleC22×C47#380
prime radicalC22×C22⋊D47D4
cosocleC23×C4×C47C23
hypercenter#1289D47
hypocenterC47#8609
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #20243415C2semidirect
4. #20241256C2semidirect
5. C47×#1289C2semidirect
7. C22×C8×D47C4semidirect
12. C2×Dic47⋊C8C4semidirect
14. C2×(C2×D47)⋊C8C4semidirect
15. C2×C8×Dic47C4semidirect
111. C22×D47#107semidirect
135. C2×Dic47#107semidirect
143. C47⋊C8#99semidirect
175. #1289D47semidirect
199. C47#8609semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D47)⋊C82
#202434152
#202433732
#202412562
C47×#12892
#860947