Finite Group Databasebeta

← #26129478

Subgroups — #26129478

Order 12032.

396 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 67 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#20254139C211yesmaximal, characteristic
2#20245568C212yesmaximal, complemented
3#20241396C212yesmaximal, complemented
4#20241312C211yesmaximal, characteristic, complemented
5C47×#2075C211yesmaximal, characteristic, complemented
6C2×#13093998—21no—
7#15689437—22no—
8#15689359—21no—
9#15689359C2212yes—
10#15689259—22no—
11#15688948—22no—
12#15688923—22no—
13C22⋊(C47⋊C16)C2212yescomplemented
14C2×#107×C47C2211yescharacteristic, complemented
15C2×C22⋊C8×C47C2211yescharacteristic
16C47×#380—21no—
17C47×#340—21no—
18C47×#340C2211yescharacteristic, complemented
19C22×D188—21no—
20C22×C22⋊D47—41no—
21C2×D376—21no—
22C2×D376D412yes—
23#13093998—22no—
24#13093998—22no—
25#13093998—41no—
26#13093998—41no—
27C2×C4⋊Dic47—21no—
28C2×(C2×D47)⋊C4—41no—
29C2×C47⋊C16—22no—
30#13093958—21no—
31#13093958D412yes—
32#13093946—22no—
33#13093946—22no—
34C23×C4×C47D411yescharacteristic
35C22×C8×C47D411yescharacteristic
36C22×C8×C47C2311yescharacteristic
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C22⋊C4×C47—41no—
40#107×C47—22no—
41C22⋊C8×C47—22no—
42#95×C47—41no—
43C24×C47D811yescharacteristic
44C23×D47—41no—
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47—41no—
50C22×C4×C47—41no—
51C22×C4×C47D811yescharacteristic
52C22×C4×C47C2×D411yescharacteristic
53C2×C22⋊D47—42no—
54C2×C22⋊D47—42no—
55C2×C22⋊D47—82no—
56C2×C22⋊D47—82no—
57C22×Dic47—41no—
58C2×D188—22no—
59C2×D188—41no—
60C2×D188—81no—
61C2×D188—81no—
62C2×C8×C47—21no—
63C2×C8×C47—41no—
64C2×C8×C47—41no—
65C2×C8×C47C2×D412yes—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—41no—
70C22⋊C4×C47—82no—
71(C2×D47)⋊C4—42no—
72(C2×D47)⋊C4—42no—
73C4⋊Dic47—22no—
74C4⋊Dic47—41no—
75D376—42no—
76D376—81no—
77C47⋊C16—42no—
78C23×C47—21no—
79C23×C47—21no—
80C23×C47—41no—
81C23×C47—41no—
82C23×C47D1612yes—
83C23×C47C2×D811yescharacteristic
84C22×D47—42no—
85C22×D47—82no—
86C22×D47—82no—
87C22×D47—82no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—81no—
95C2×C4×C47—81no—
96C2×C4×C47—81no—
97C2×C4×C47—81no—
98C2×C4×C47—81no—
99C2×C4×C47—81no—
100C2×C4×C47—81no—
101C2×C4×C47#12011yescharacteristic
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C22⋊D47—84no—
104C22⋊D47—82no—
105C22⋊D47—82no—
106C2×Dic47—42no—
107C2×Dic47—82no—
108D188—42no—
109D188—42no—
110D188—82no—
111C8×C47—22no—
112C8×C47—41no—
113#12278—471nomaximal, Sylow
114C22×C47—21no—
115C22×C47—21no—
116C22×C47—22no—
117C22×C47—42no—
118C22×C47—41no—
119C22×C47—41no—
120C22×C47—81no—
121C22×C47#44711yescharacteristic
122C22×C47C2×D1611yescharacteristic
123C22×C47#50911yescharacteristic
124C2×D47—82no—
125C2×D47—82no—
126C2×D47—82no—
127C2×D47—164no—
128C4×C47—21no—
129C4×C47—41no—
130C4×C47—161no—
131C4×C47—161no—
132C4×C47#45511yescharacteristic
133C4×C47#45811yescharacteristic
134Dic47—82no—
135#3186—471no—
136#2232—472no—
137#2075D4711yescharacteristic, complemented
138#1347—472no—
139#1314—471no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47—42no—
143C2×C47#160711yescharacteristic
144C2×C47#218411yescharacteristic
145C2×C47#219111yescharacteristic
146D47—162no—
147C2×#121—941no—
148#480—942no—
149#466—472no—
150#466—941no—
151#447—942no—
152C2×#107C2×D4711yescharacteristic, complemented
153C2×C22⋊C8C2×D4711yescharacteristic
154#380—21no—
155#366—942no—
156#357—942no—
157C22⋊C16—472no—
158#340—21no—
159#340C2×D4711yescharacteristic, complemented
160C47#1227811yescharacteristic, complemented, Sylow
161C22×D4—941no—
162C22×D4—1881no—
163C23×C4C22⋊D4711yescharacteristic
164C2×D8—472no—
165C2×D8—941no—
166C22×C8C22⋊D4711yescharacteristic
167C22×C8C22×D4711yescharacteristic
168#121—942no—
169#121—942no—
170#121—1881no—
171#121—1881no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—941no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—1881no—
177C2×C16—942no—
178#107—22no—
179#107—472no—
180#107—941no—
181#102—942no—
182#102—942no—
183C22⋊C8—22no—
184#95—41no—
185C24—1881no—
186C24#1062533411yescharacteristic
187C2×D4—942no—
188C2×D4—3761no—
189C2×D4—3761no—
190C2×D4—3762no—
191C2×D4—3762no—
192C2×D4—1881no—
193C2×D4—1882no—
194C2×D4—1882no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—1881no—
202C22×C4#1062533411yescharacteristic
203C22×C4C2×C22⋊D4711yescharacteristic
204D8—3761no—
205D8—1882no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D4712yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—942no—
215C4⋊C4—1881no—
216C22⋊C4—82no—
217C22⋊C4—1882no—
218C22⋊C4—1882no—
219C16—1882no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—3762no—
225C23—3762no—
226C23—3762no—
227C23—1882no—
228C23#1309396512yes—
229C23C2×#1062533411yescharacteristic
230D4—3762no—
231D4—3764no—
232D4—3762no—
233D4—3762no—
234D4—1882no—
235D4—1882no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—3762no—
250C2×C4—1882no—
251C2×C4#1309399211yescharacteristic
252C2×C4#1309404911yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—3762no—
263C22—3762no—
264C22—3762no—
265C22—7524no—
266C22#1568925611yescharacteristic
267C22C2×#1309396511yescharacteristic
268C22#1568955811yescharacteristic
269C4—21no—
270C4—41no—
271C4—161no—
272C4—161no—
273C4—3762no—
274C4#1568930011yescharacteristic
275C4#1568931211yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—7522no—
280C2#2024237311yescharacteristic
281C2#2024529911yescharacteristic
282C2#2024533811yescharacteristic
subgroupG/H
centerC22#15689256
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#2075C2
socleC22×C47#447
prime radical#20254139C2
hypercenter#2075D47
hypocenterC47#12278
LCSG ▷ C22×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2075

Complements

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

NH ≅ G/Ntype
2. #20245568C2semidirect
3. #20241396C2semidirect
4. #20241312C2semidirect
5. C47×#2075C2semidirect
13. C22⋊(C47⋊C16)C22semidirect
14. C2×#107×C47C22semidirect
18. C47×#340C22semidirect
137. #2075D47semidirect
152. C2×#107C2×D47semidirect
159. #340C2×D47semidirect
160. C47#12278semidirect

Maximal subgroups

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

subgroupindex
#202541392
#202455682
#202413962
#202413122
C47×#20752
#1227847