Finite Group Databasebeta

← #26132801

Subgroups — #26132801

Order 12032.

424 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 135 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#20251978C211yesmaximal, characteristic
2#20245176C212yesmaximal, complemented
3#20243421C211yesmaximal, characteristic
4#20241494C212yesmaximal, complemented
5C2×C47×#349C211yesmaximal, characteristic
6C2×#13093983—21no—
7#15689165—22no—
8#15689165—22no—
9#15689161—21no—
10#15689161C2212yes—
11C2×#13093958C2211yescharacteristic
12C2×#13093956C2211yescharacteristic
13#15688983—21no—
14#15688951—22no—
15#15688949—22no—
16#15688886—21no—
17C22×M16×C47C2211yescharacteristic
18C2×M32×C47C2212yes—
19C47×#349C414yescomplemented
20#13093983—22no—
21#13093983—22no—
22#13093983—41no—
23#13093983—41no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C22⋊Dic47—41no—
29C2×C22⋊Dic47—41no—
30C2×C4×Dic47—41no—
31#13093958—21no—
32#13093958—21no—
33#13093958—22no—
34#13093958D412yes—
35#13093958D412yes—
36#13093956—22no—
37#13093956—22no—
38#13093956—22no—
39C47⋊#95—41no—
40C23×C4×C47D411yescharacteristic
41C2×M16×C47—22no—
42C2×M16×C47—22no—
43C2×M16×C47C2×C412yes—
44C2×M16×C47Q812yes—
45C22×C8×C47D411yescharacteristic
46C22×C8×C47C2311yescharacteristic
47M32×C47—22no—
48M32×C47C2×C414yescomplemented
49C2×C16×C47—22no—
50C24×C47D811yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47C4⋊C412yes—
56C22×C4×C47D811yescharacteristic
57C22×C4×C47Q1612yes—
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—41no—
60C22×Dic47—41no—
61C22×Dic47—41no—
62C22×Dic47—41no—
63M16×C47—24no—
64M16×C47—24no—
65C2×C8×C47—21no—
66C2×C8×C47—21no—
67C2×C8×C47C4⋊C412yes—
68C2×C8×C47C22×C411yescharacteristic
69C2×C8×C47C2×D412yes—
70C2×C8×C47C2×Q811yescharacteristic
71C2×C8×C47#4112yes—
72C22⋊Dic47—42no—
73C22⋊Dic47—42no—
74C22⋊Dic47—82no—
75C4⋊Dic47—22no—
76C4⋊Dic47—22no—
77C4⋊Dic47—41no—
78C4⋊Dic47—41no—
79C4⋊Dic47—41no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4⋊Dic47—41no—
83C4×Dic47—42no—
84C4×Dic47—42no—
85C16×C47—24no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47#10712yes—
91C23×C47C2×D811yescharacteristic
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47#10712yes—
99C2×C4×C47C2×C4⋊C411yescharacteristic
100C2×C4×C47#12011yescharacteristic
101C2×C4×C47#12311yescharacteristic
102C2×C4×C47C2×Q1611yescharacteristic
103C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C47#13711yescharacteristic
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—82no—
108C2×Dic47—82no—
109C2×Dic47—81no—
110C2×Dic47—81no—
111C2×Dic47—81no—
112C2×Dic47—81no—
113C2×Dic47—81no—
114C2×Dic47—81no—
115C8×C47—22no—
116C8×C47C4×D412yes—
117C8×C47C22⋊Q812yes—
118#12923—471nomaximal, Sylow
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—22no—
122C22×C47—42no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47—81no—
126C22×C47C2×#10711yescharacteristic
127C22×C47#44711yescharacteristic
128C22×C47#48011yescharacteristic
129C4×C47—22no—
130C4×C47#38511yescharacteristic
131C4×C47#42811yescharacteristic
132C4×C47#45111yescharacteristic
133C4×C47#48411yescharacteristic
134Dic47—82no—
135Dic47—82no—
136Dic47—161no—
137Dic47—161no—
138#2911—471no—
139#2159—472no—
140C2×#349D4711yescharacteristic
141#1809—471no—
142#1386—472no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—42no—
146C2×C47#185211yescharacteristic
147C2×C47#219011yescharacteristic
148C2×C47#219111yescharacteristic
149C22×M16C2×D4711yescharacteristic
150C2×#117—941no—
151C2×M32C2×D4712yes—
152#428—942no—
153#428—942no—
154#427—472no—
155#427—941no—
156C2×#107—471no—
157C2×#106—471no—
158#380—941no—
159#367—942no—
160#366—942no—
161#349Dic4714yescomplemented
162#340—941no—
163C47#1292311yescharacteristic, complemented, Sylow
164C23×C4D18811yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16Dic9412yes—
168C2×M16C2×Dic4712yes—
169C22×C8D18811yescharacteristic
170C22×C8C22×D4711yescharacteristic
171#117—942no—
172#117—942no—
173#117—1881no—
174#117—1881no—
175C2×C4⋊C4—941no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—941no—
179C2×C22⋊C4—1881no—
180C2×C22⋊C4—1881no—
181C2×C42—1881no—
182M32—22no—
183M32C2×Dic4714yescomplemented
184C2×C16—22no—
185#107—472no—
186#107—472no—
187#107—941no—
188#107—941no—
189#107—942no—
190#106—942no—
191#106—942no—
192#106—942no—
193#95—1881no—
194C24D37611yescharacteristic
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—1881no—
200C22×C4—1881no—
201C22×C4—1881no—
202C22×C4—1881no—
203C22×C4D37611yescharacteristic
204C22×C4Dic18812yes—
205C22×C4C4⋊Dic4712yes—
206C22×C4C2×D18811yescharacteristic
207M16—24no—
208M16—24no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8C4⋊Dic4712yes—
212C2×C8C2×Dic9411yescharacteristic
213C2×C8D4×D4712yes—
214C2×C8#1062535212yes—
215C2×C8C22×Dic4711yescharacteristic
216C4⋊C4—942no—
217C4⋊C4—942no—
218C4⋊C4—1881no—
219C4⋊C4—1881no—
220C4⋊C4—1881no—
221C4⋊C4—1881no—
222C4⋊C4—1881no—
223C4⋊C4—1881no—
224C22⋊C4—3762no—
225C22⋊C4—1882no—
226C22⋊C4—1882no—
227C42—1882no—
228C42—1882no—
229C16—24no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23#1309395812yes—
235C23C2×D37611yescharacteristic
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—3762no—
243C2×C4—3762no—
244C2×C4—3761no—
245C2×C4—3761no—
246C2×C4—3761no—
247C2×C4—3761no—
248C2×C4—3761no—
249C2×C4—3761no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4#1309395812yes—
253C2×C4C2×C4⋊Dic4711yescharacteristic
254C2×C4#1309399411yescharacteristic
255C2×C4#1309400811yescharacteristic
256C2×C4C2×Dic18811yescharacteristic
257C2×C4#1309405211yescharacteristic
258C2×C4#1309405711yescharacteristic
259C8—22no—
260C8D4×Dic4712yes—
261C8#1309400112yes—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22C2×#1309395811yescharacteristic
270C22#1568925911yescharacteristic
271C22#1568943711yescharacteristic
272C4—22no—
273C4—3762no—
274C4—3762no—
275C4—7521no—
276C4—7521no—
277C4#1568899811yescharacteristic
278C4#1568916511yescharacteristic
279C4#1568928111yescharacteristic
280C4#1568945911yescharacteristic
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2024357811yescharacteristic
285C2#2024533611yescharacteristic
286C2#2024534111yescharacteristic
subgroupG/H
centerC22#15689259
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#349C2
socleC22×C47#447
prime radicalC24×C47D8
cosocleC22×C8×C47C23
hypercenterC2×#349D47
hypocenterC47#12923
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
2. #20245176C2semidirect
4. #20241494C2semidirect
19. C47×#349C4semidirect
48. M32×C47C2×C4semidirect
161. #349Dic47semidirect
163. C47#12923semidirect
183. M32C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202519782
#202451762
#202434212
#202414942
C2×C47×#3492
#1292347