Finite Group Databasebeta

← #26112024

Subgroups — #26112024

Order 12032.

756 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15688969C211yesmaximal, characteristic, complemented
2#20243456C211yesmaximal, characteristic, complemented
3C2×#15688857C212yesmaximal
4#20241266C212yesmaximal, complemented
5C47×#2854C211yesmaximal, characteristic
6C22×C4×Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688969—24no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—22no—
12#15688857—24no—
13C2×#123×C47—21no—
14C2×#121×C47C2211yescharacteristic
15C22×C42×C47C2211yescharacteristic
16C47×#390C412yescomplemented
17C47×#389C412yescomplemented
18C47×#388—22no—
19C47×#386—21no—
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C4×Dic47—24no—
24C2×C4×Dic47—28no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—24no—
28C47⋊#95—22no—
29C47⋊#95D414yes—
30C22×D4×C47—21no—
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35#123×C47—44no—
36#121×C47—22no—
37#121×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C412yes—
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—21no—
45C2×C42×C47—22no—
46C2×C42×C47—22no—
47C2×C42×C47C2×C412yes—
48C2×C42×C47D412yes—
49#95×C47—22no—
50#95×C47—21no—
51C24×C47—21no—
52C24×C47C2×D411yescharacteristic
53C2×D4×C47—22no—
54C2×D4×C47—42no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47—42no—
58C2×D4×C47—42no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—21no—
68C22×C4×C47—42no—
69C22×C4×C47C22⋊C412yes—
70C22×C4×C47C22⋊C412yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D412yes—
74C22×C4×C47#4112yes—
75C22×Dic47—28no—
76C22×Dic47—24no—
77C22×Dic47—44no—
78C22×Dic47—42no—
79C22×Dic47—42no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—44no—
82C4⋊C4×C47—42no—
83C22⋊C4×C47—44no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—44no—
86C22⋊C4×C47—42no—
87C42×C47—24no—
88C42×C47—24no—
89C4×Dic47—216no—
90C2×C47⋊C8—44no—
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—41no—
95C23×C47—42no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—42no—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47#12011yescharacteristic
101C23×C47#12311yescharacteristic
102C23×C47#13512yes—
103C23×C47AGL(1,ℤ/8)12yes—
104D4×C47—82no—
105D4×C47—82no—
106D4×C47—82no—
107C2×C4×C47—24no—
108C2×C4×C47—24no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47#10414yes—
127C2×C4×C47C4×D412yes—
128C2×C4×C47#12112yes—
129C2×Dic47—28no—
130C2×Dic47—44no—
131C2×Dic47—48no—
132C2×Dic47—48no—
133C2×Dic47—44no—
134C47⋊C8—82no—
135#9149—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—24no—
141C22×C47—24no—
142C22×C47—81no—
143C22×C47—81no—
144C22×C47—81no—
145C22×C47—82no—
146C22×C47—82no—
147C22×C47#38611yescharacteristic
148C22×C47#41012yes—
149C22×C47#41611yescharacteristic
150C22×C47#41811yescharacteristic
151C22×C47C2×#10412yes—
152C22×C47#44912yes—
153C22×C47#48212yes—
154C4×C47—24no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—81no—
159C4×C47—82no—
160Dic47—48no—
161#2854D4711yescharacteristic
162C2×#376—471no—
163#1817—471no—
164C2×#327—472no—
165#1294—472no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47—161no—
169C2×C47#150012yes—
170C2×C47#150512yes—
171C2×C47#187511yescharacteristic
172C2×C47#189611yescharacteristic
173C2×C47#189911yescharacteristic
174C2×#123—21no—
175C2×#121C2×D4711yescharacteristic
176C22×C42—941no—
177C22×C42C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#390Dic4712yescomplemented
180#389Dic4712yescomplemented
181#388—22no—
182#386—21no—
183#380C2×D4711yescharacteristic
184#376—944no—
185C2×#95—472no—
186C2×#95—941no—
187#336—942no—
188#327—944no—
189C47#914911yescharacteristic, complemented, Sylow
190C22×D4—21no—
191C22×D4C2×Dic4711yescharacteristic
192C23×C4—942no—
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22⋊D4711yescharacteristic
195C23×C4C22×D4711yescharacteristic
196C22×C8—942no—
197#123—44no—
198#121—22no—
199#121—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C2×Dic4711yescharacteristic
202C2×C4⋊C4C2×Dic4712yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—944no—
210C2×C42—948no—
211C2×C42C2×Dic4712yes—
212C2×C42C22⋊D4712yes—
213C22⋊C8—942no—
214C22⋊C8—942no—
215#95—22no—
216#95—21no—
217#95—474no—
218#95—944no—
219#95—942no—
220C24—21no—
221C24C2×C22⋊D4711yescharacteristic
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—948no—
239C22×C4—944no—
240C22×C4—1884no—
241C22×C4—1882no—
242C22×C4—1882no—
243C22×C4C22⋊Dic4712yes—
244C22×C4C22⋊Dic4712yes—
245C22×C4D4×D4712yes—
246C22×C4#1062535212yes—
247C22×C4C22×Dic4711yescharacteristic
248C22×C4C2×C22⋊D4711yescharacteristic
249C2×C8—1884no—
250C2×C8—1882no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C42—24no—
259C42—24no—
260C42—9416no—
261C23—22no—
262C23—22no—
263C23—41no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23C2×C22⋊Dic4711yescharacteristic
269C23#1309399211yescharacteristic
270C23#1309400511yescharacteristic
271C23#1309404412yes—
272C23#1309405012yes—
273D4—82no—
274D4—82no—
275D4—82no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—948no—
296C2×C4—1884no—
297C2×C4—1888no—
298C2×C4—1888no—
299C2×C4—1884no—
300C2×C4#1309395114yes—
301C2×C4D4×Dic4712yes—
302C2×C4#1309399612yes—
303C8—3762no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—81no—
311C22—81no—
312C22—82no—
313C22—82no—
314C22—81no—
315C22#1568900311yescharacteristic
316C22#1568909712yes—
317C22#1568911911yescharacteristic
318C22#1568912811yescharacteristic
319C22C2×#1309395112yes—
320C22#1568926612yes—
321C22#1568944412yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—1888no—
329C2—22no—
330C2—22no—
331C2—161no—
332C2#2024187412yes—
333C2#2024189612yes—
334C2#2024371111yescharacteristic
335C2#2024383211yescharacteristic
336C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2854C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2854D47
hypocenterC47#9149
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#15688969C2semidirect
2. #20243456C2semidirect
4. #20241266C2semidirect
16. C47×#390C4semidirect
17. C47×#389C4semidirect
179. #390Dic47semidirect
180. #389Dic47semidirect
189. C47#9149semidirect

Maximal subgroups

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

subgroupindex
C2×#156889692
#202434562
C2×#156888572
#202412662
C47×#28542
#914947