Finite Group Databasebeta

← #26112023

Subgroups — #26112023

Order 12032.

756 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 139 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#20251719C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic, complemented
3C2×#15688856C212yesmaximal
4#20241267C212yesmaximal, complemented
5C2×C47×#376C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689025C412yescomplemented
10#15689021C412yescomplemented
11#15689014—22no—
12#15689002—21no—
13#15688983C2211yescharacteristic
14#15688878—22no—
15#15688856—24no—
16C22×C42×C47—21no—
17C22×C42×C47C2211yescharacteristic
18C47×#376—24no—
19C2×#95×C47—21no—
20C2×#95×C47C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C22×C47⋊C8—22no—
24#13094007—44no—
25#13093998—22no—
26#13093998—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—21no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—21no—
34C22⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C23×C4×C47—22no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C2×C42×C47—28no—
43C2×C42×C47—24no—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47C2×C412yes—
47C2×C42×C47D412yes—
48#95×C47—24no—
49#95×C47—22no—
50#95×C47D414yes—
51C24×C47C2×D411yescharacteristic
52C23×D47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—24no—
56C22×C4×C47—28no—
57C22×C4×C47—24no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—42no—
63C22×C4×C47—44no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47#4112yes—
72C2×C22⋊D47—41no—
73C2×C22⋊D47—41no—
74C2×C22⋊D47—42no—
75C2×C22⋊D47—42no—
76C22×Dic47—22no—
77C22×Dic47—21no—
78C2×D188—22no—
79C2×D188—42no—
80C42×C47—216no—
81C42×C47—24no—
82C42×C47—24no—
83C22⋊Dic47—42no—
84(C2×D47)⋊C4—44no—
85(C2×D47)⋊C4—44no—
86(C2×D47)⋊C4—42no—
87C4⋊Dic47—22no—
88C4⋊Dic47—42no—
89Dic47⋊C4—44no—
90C2×C47⋊C8—44no—
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47C2×C22⋊C411yescharacteristic
95C23×C47#12011yescharacteristic
96C23×C47#12311yescharacteristic
97C23×C47#13512yes—
98C23×C47AGL(1,ℤ/8)12yes—
99C22×D47—41no—
100C22×D47—42no—
101C22×D47—41no—
102C22×D47—41no—
103C22×D47—42no—
104C2×C4×C47—24no—
105C2×C4×C47—24no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—28no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—44no—
117C2×C4×C47—48no—
118C2×C4×C47—48no—
119C2×C4×C47—42no—
120C2×C4×C47—44no—
121C2×C4×C47—42no—
122C2×C4×C47#10414yes—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12112yes—
125C22⋊D47—82no—
126C22⋊D47—82no—
127C2×Dic47—41no—
128C2×Dic47—42no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133D188—82no—
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#38611yescharacteristic
143C22×C47#41012yes—
144C22×C47#41611yescharacteristic
145C22×C47#41811yescharacteristic
146C22×C47C2×#10412yes—
147C22×C47#44912yes—
148C22×C47#48212yes—
149C2×D47—81no—
150C2×D47—81no—
151C2×D47—81no—
152C2×D47—82no—
153C2×D47—82no—
154C4×C47—24no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—48no—
158C4×C47—42no—
159Dic47—81no—
160Dic47—82no—
161#2854—471no—
162C2×#376D4711yescharacteristic, complemented
163#1817—471no—
164C2×#327—472no—
165#1294—472no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47#150012yes—
169C2×C47#150512yes—
170C2×C47#187511yescharacteristic
171C2×C47#189611yescharacteristic
172C2×C47#189911yescharacteristic
173D47—161no—
174C2×#123—941no—
175C2×#121—471no—
176C22×C42—21no—
177C22×C42C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#390—472no—
180#389—472no—
181#388—942no—
182#386—941no—
183#380—471no—
184#376—24no—
185C2×#95—21no—
186C2×#95C2×D4712yes—
187#336—942no—
188#327—944no—
189C47#914911yescharacteristic, complemented, Sylow
190C22×D4—471no—
191C22×D4—941no—
192C23×C4—22no—
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22⋊D4711yescharacteristic
195C23×C4C22×D4711yescharacteristic
196C22×C8—942no—
197#123—1884no—
198#121—942no—
199#121—942no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—472no—
202C2×C4⋊C4—942no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—941no—
207C2×C42—24no—
208C2×C42—28no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C4×D4712yes—
212C2×C42D18812yes—
213C22⋊C8—942no—
214C22⋊C8—942no—
215#95—24no—
216#95—22no—
217#95—942no—
218#95—941no—
219#95C22⋊D4714yes—
220C24—941no—
221C24C2×C22⋊D4711yescharacteristic
222C2×D4—942no—
223C2×D4—1882no—
224C2×D4—1881no—
225C2×D4—1881no—
226C2×D4—1882no—
227C2×D4—1882no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—942no—
242C22×C4—941no—
243C22×C4(C2×D47)⋊C412yes—
244C22×C4(C2×D47)⋊C412yes—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4C2×D18811yescharacteristic
247C22×C4#1062535012yes—
248C22×C4C2×C22⋊D4712yes—
249C2×C8—1884no—
250C2×C8—1882no—
251C4⋊C4—942no—
252C4⋊C4—1882no—
253C4⋊C4—1884no—
254C22⋊C4—1884no—
255C22⋊C4—1882no—
256C22⋊C4—1884no—
257C22⋊C4—1882no—
258C42—216no—
259C42—24no—
260C42—24no—
261C23—22no—
262C23—22no—
263C23—1881no—
264C23—1882no—
265C23—1881no—
266C23—1881no—
267C23—1882no—
268C23C2×(C2×D47)⋊C411yescharacteristic
269C23#1309399311yescharacteristic
270C23#1309400711yescharacteristic
271C23#1309404412yes—
272C23#1309404912yes—
273D4—3762no—
274D4—3762no—
275D4—3762no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—1881no—
295C2×C4—1882no—
296C2×C4—1881no—
297C2×C4—1882no—
298C2×C4—1881no—
299C2×C4—1882no—
300C2×C4#1309395214yes—
301C2×C4C4×C22⋊D4712yes—
302C2×C4#1309399812yes—
303C8—3762no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—3761no—
311C22—3761no—
312C22—3762no—
313C22—3762no—
314C22—3761no—
315C22#1568900211yescharacteristic
316C22#1568909612yes—
317C22#1568911711yescharacteristic
318C22#1568912711yescharacteristic
319C22C2×#1309395212yes—
320C22#1568926812yes—
321C22#1568944712yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—3762no—
328C4—3761no—
329C2—22no—
330C2—22no—
331C2—7521no—
332C2#2024187212yes—
333C2#2024189512yes—
334C2#2024371011yescharacteristic
335C2#2024383111yescharacteristic
336C2#2024384611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#376C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenterC2×#376D47
hypocenterC47#9149
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
2. #20243456C2semidirect
4. #20241267C2semidirect
5. C2×C47×#376C2semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
162. C2×#376D47semidirect
189. C47#9149semidirect

Maximal subgroups

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

subgroupindex
#202517192
#202434562
C2×#156888562
#202412672
C2×C47×#3762
#914947