Finite Group Databasebeta

← #26129295

Subgroups — #26129295

Order 12032.

454 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244738C211yesmaximal, characteristic, complemented
2#20242327C212yesmaximal, complemented
3#20241391C212yesmaximal, complemented
4#20241332C211yesmaximal, characteristic, complemented
5C47×#2819C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689274—22no—
8#15689163—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C2×#124×C47—21no—
14C47×#394—21no—
15C47×#394C2212yes—
16C47×#393—22no—
17C47×#392—22no—
18C47×#382C2211yescharacteristic
19C2×#10625335—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093957—41no—
26#13093955—41no—
27#13093945—42no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47—21no—
30C22×D4×C47—21no—
31C22×D4×C47—21no—
32C22×D4×C47D412yes—
33#124×C47—42no—
34#124×C47—42no—
35C2×C4⋊C4×C47D411yescharacteristic
36C2×C4⋊C4×C47C2311yescharacteristic
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C42×C47D411yescharacteristic
42#95×C47—21no—
43#95×C47—22no—
44#95×C47D412yes—
45C24×C47—22no—
46C2×Q8×C47—42no—
47C2×Q8×C47—81no—
48C2×Q8×C47—81no—
49C2×D4×C47—42no—
50C2×D4×C47—42no—
51C2×D4×C47—82no—
52C2×D4×C47—81no—
53C2×D4×C47—81no—
54C2×D4×C47—82no—
55C2×D4×C47—81no—
56C22×C4×C47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D412yes—
61C2×Dic94—22no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—41no—
64C22⋊C4×C47—44no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—82no—
68C22⋊C4×C47—82no—
69C42×C47—21no—
70C42×C47SD1612yescomplemented
71C22⋊Dic47—41no—
72#10625335—84no—
73C4⋊Dic47—22no—
74C4⋊Dic47—21no—
75C4⋊Dic47—21no—
76Dic47⋊C4—42no—
77C4×Dic47—41no—
78#10625329—22no—
79#10625329—41no—
80C2×C47⋊C8—41no—
81C2×C47⋊C8—42no—
82C23×C47—42no—
83C23×C47—42no—
84C23×C47—42no—
85C23×C47—42no—
86C23×C47—42no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47#12011yescharacteristic
90Q8×C47—84no—
91D4×C47—44no—
92D4×C47—82no—
93D4×C47—84no—
94D4×C47—84no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47C2×SD1611yescharacteristic
108C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
109C2×Dic47—41no—
110C2×Dic47—41no—
111Dic94—82no—
112C47⋊C8—42no—
113C47⋊C8—82no—
114#12250—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—82no—
118C22×C47—82no—
119C22×C47—82no—
120C22×C47—82no—
121C22×C47—82no—
122C22×C47—82no—
123C22×C47—82no—
124C22×C47#45011yescharacteristic
125C22×C47#45611yescharacteristic
126C22×C47B(4,𝔽2)11yescharacteristic
127C4×C47—21no—
128C4×C47—21no—
129C4×C47—42no—
130C4×C47—42no—
131C4×C47—82no—
132C4×C47—81no—
133Dic47—81no—
134Dic47—81no—
135#2819D4711yescharacteristic
136#2061—471no—
137#1599—472no—
138#1345—472no—
139#1324—471no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47—162no—
143C2×C47#160411yescharacteristic
144C2×C47#219911yescharacteristic
145C2×C47#220211yescharacteristic
146#544—471no—
147C2×#124—21no—
148#450—942no—
149#428—941no—
150#394—21no—
151#394C2×D4712yes—
152#393—22no—
153#392—22no—
154#382C2×D4711yescharacteristic
155#343—471no—
156#339—941no—
157#328—942no—
158#324—472no—
159C47#1225011yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊D4712yes—
164C2×SD16—1882no—
165C2×M16—941no—
166C4⋊Q8—1881no—
167#125—1881no—
168#124—42no—
169#124—42no—
170C22⋊Q8—942no—
171#117—941no—
172C2×C4⋊C4C22⋊D4711yescharacteristic
173C2×C4⋊C4C22×D4711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42C22⋊D4711yescharacteristic
179#107—1881no—
180#106—1881no—
181#102—1882no—
182C22⋊C8—942no—
183#95—21no—
184#95—22no—
185#95C22⋊D4712yes—
186C24—22no—
187C2×Q8—42no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8—942no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×C22⋊D4711yescharacteristic
202C22×C4C2×C22⋊D4712yes—
203SD16—3764no—
204M16—942no—
205M16—1881no—
206C2×C8—1882no—
207C2×C8—1881no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—942no—
211C4⋊C4—941no—
212C4⋊C4—941no—
213C4⋊C4—1882no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—1881no—
220C42—21no—
221C42—1881no—
222C42#1062533512yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23#1309399311yescharacteristic
231Q8—84no—
232Q8—3762no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—1881no—
250C2×C4—1881no—
251C2×C4C2×#1062533511yescharacteristic
252C2×C4#1309405011yescharacteristic
253C8—3762no—
254C8—1882no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#1568927411yescharacteristic
265C22#1568930611yescharacteristic
266C22#1568931011yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—3761no—
274C4—3761no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2#2024235711yescharacteristic
279C2#2024538911yescharacteristic
280C2#2024540511yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
Fitting, prime radicalC47×#2819C2
socleC22×C47B(4,𝔽2)
hypercenter#2819D47
hypocenterC47#12250
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. #20244738C2semidirect
2. #20242327C2semidirect
3. #20241391C2semidirect
4. #20241332C2semidirect
70. C42×C47SD16semidirect
159. C47#12250semidirect
222. C42#10625335semidirect

Maximal subgroups

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

subgroupindex
#202447382
#202423272
#202413912
#202413322
C47×#28192
#1225047