Finite Group Databasebeta

← #26109687

Subgroups — #26109687

Order 12032.

526 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 107 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#20245794C211yesmaximal, characteristic
2#20243151C212yesmaximal
3#20241270C212yesmaximal, complemented
4#20241269C211yesmaximal, characteristic, complemented
5C47×#1295C211yesmaximal, characteristic, complemented
6C2×#13093982C411yescharacteristic
7C2×#13093982C411yescharacteristic
8C2×C42×D47C2211yescharacteristic
9C2×(C2×D47)⋊C8—22no—
10C2×#13093935C2212yes—
11#15688992C411yescharacteristic
12#15688992C411yescharacteristic
13#15688989C2211yescharacteristic
14#15688982—22no—
15#15688974—22no—
16#15688877—22no—
17C2×#97×C47C2212yes—
18C47×#382C2211yescharacteristic
19C22×C4×D47C2×C411yescharacteristic
20C22×C4×D47D412yes—
21C22×C47⋊C8—22no—
22#13093982—42no—
23#13093982—42no—
24C2×C4⋊Dic47C2×C411yescharacteristic
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—22no—
28C2×(C2×D47)⋊C4—21no—
29C2×(C2×D47)⋊C4—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34C42×D47—22no—
35C42×D47—21no—
36C42×D47—21no—
37(C2×D47)⋊C8—28no—
38#13093935—22no—
39#13093935—22no—
40C47⋊#95—22no—
41C22×C8×C47—22no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45#97×C47—22no—
46#97×C47—22no—
47#95×C47—22no—
48C23×D47C22⋊C411yescharacteristic
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C2×D412yes—
53C22×Dic47—21no—
54C22×Dic47—21no—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C412yes—
57C2×C4×D47—22no—
58C2×C4×D47—21no—
59C2×C4×D47—21no—
60C2×C4×D47—22no—
61C2×C4×D47—22no—
62C2×C4×D47—22no—
63C2×C4×D47—24no—
64C2×C4×D47—42no—
65C2×C8×C47—24no—
66C2×C8×C47—42no—
67C2×C8×C47—42no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—41no—
70C42×C47#4112yes—
71C42×C47#4111yescharacteristic
72C42×C47#4111yescharacteristic
73(C2×D47)⋊C4—42no—
74(C2×D47)⋊C4—42no—
75C4⋊Dic47—41no—
76C4⋊Dic47—41no—
77Dic47⋊C4—44no—
78Dic47⋊C4—44no—
79C4×Dic47—21no—
80C4×Dic47—21no—
81C4×Dic47—22no—
82C4×Dic47—22no—
83C4×Dic47—42no—
84C4×Dic47—42no—
85C2×C47⋊C8—24no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—42no—
88C23×C47C2×C22⋊C411yescharacteristic
89C22×D47—21no—
90C22×D47—21no—
91C22×D47—22no—
92C22×D47—22no—
93C22×D47#10112yes—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47#11711yescharacteristic
103C2×C4×C47#11711yescharacteristic
104C2×C4×C47#12312yes—
105C2×C4×C47#12312yes—
106C2×Dic47—21no—
107C2×Dic47—22no—
108C2×Dic47—21no—
109C2×Dic47—22no—
110C2×Dic47—41no—
111C2×Dic47—41no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47#10012yes—
115C2×Dic47#10414yes—
116C4×D47—24no—
117C4×D47—24no—
118C4×D47—44no—
119C4×D47—44no—
120C8×C47—44no—
121C47⋊C8—44no—
122#8721—471nomaximal, Sylow
123C22×C47#38111yescharacteristic
124C22×C47C2×#10011yescharacteristic
125C22×C47C2×#10111yescharacteristic
126C22×C47C2×#10412yes—
127C22×C47#42112yes—
128C2×D47—22no—
129C2×D47—24no—
130C2×D47—41no—
131C2×D47—42no—
132C2×D47—41no—
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—22no—
136C4×C47—22no—
137C4×C47—81no—
138C4×C47—81no—
139Dic47—22no—
140Dic47—24no—
141Dic47—81no—
142Dic47—81no—
143#2285—471no—
144#1749—472no—
145#1295—472no—
146#1295—471no—
147#1295D4711yescharacteristic, complemented
148C2×C47#152511yescharacteristic
149C2×C47#153112yes—
150C2×C47#153611yescharacteristic
151C2×C47#182111yescharacteristic
152C2×C47#182612yes—
153D47—42no—
154C2×#117—471no—
155C2×#117—471no—
156C22×C42—471no—
157C2×C22⋊C8—942no—
158C2×#97—472no—
159C2×#97C2×D4712yes—
160#384—471no—
161#384—471no—
162#382—471no—
163#382C2×D4711yescharacteristic
164#380—942no—
165C4×C4⋊C4—942no—
166#336—942no—
167C47#872111yescharacteristic, complemented, Sylow
168C23×C4—471no—
169C23×C4—472no—
170C22×C8—22no—
171C22×C8—942no—
172#117—1882no—
173#117—1882no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—471no—
176C2×C4⋊C4—942no—
177C2×C4⋊C4—942no—
178C2×C4⋊C4C4×D4711yescharacteristic
179C2×C4⋊C4C4×D4711yescharacteristic
180C2×C22⋊C4—941no—
181C2×C22⋊C4—941no—
182C2×C42—471no—
183C2×C42—472no—
184C2×C42—942no—
185C2×C42—941no—
186C2×C42—941no—
187C2×C42—941no—
188C2×C42—941no—
189C2×C42C22×D4711yescharacteristic
190C22⋊C8—948no—
191#97—22no—
192#97—22no—
193#97—942no—
194#97—942no—
195#95—22no—
196#95—942no—
197C24—471no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—471no—
201C22×C4—472no—
202C22×C4—942no—
203C22×C4—944no—
204C22×C4—942no—
205C22×C4—941no—
206C22×C4—941no—
207C22×C4—942no—
208C22×C4—942no—
209C22×C4—941no—
210C22×C4—941no—
211C22×C4—1882no—
212C22×C4C2×C4×D4711yescharacteristic
213C22×C4D4×D4712yes—
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—944no—
218C2×C8—1882no—
219C2×C8—1882no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—1881no—
223C4⋊C4—1881no—
224C4⋊C4—1884no—
225C4⋊C4—1884no—
226C22⋊C4—1882no—
227C22⋊C4—1882no—
228C42—941no—
229C42—941no—
230C42—942no—
231C42—942no—
232C42—1882no—
233C42—1882no—
234C42#1062535012yes—
235C42#1062535211yescharacteristic
236C42#1062535411yescharacteristic
237C23—472no—
238C23—941no—
239C23—941no—
240C23—942no—
241C23—942no—
242C23C22⋊C4×D4711yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—472no—
252C2×C4—474no—
253C2×C4—944no—
254C2×C4—944no—
255C2×C4—941no—
256C2×C4—942no—
257C2×C4—941no—
258C2×C4—942no—
259C2×C4—1884no—
260C2×C4—1884no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4#1309398111yescharacteristic
266C2×C4#1309398211yescharacteristic
267C2×C4#1309400612yes—
268C2×C4#1309400912yes—
269C8—44no—
270C8—1884no—
271C22—942no—
272C22—944no—
273C22—1882no—
274C22—1881no—
275C22—1881no—
276C22#1568898511yescharacteristic
277C22#100×D4711yescharacteristic
278C22#101×D4711yescharacteristic
279C22#104×D4712yes—
280C22#1568913812yes—
281C4—21no—
282C4—21no—
283C4—22no—
284C4—22no—
285C4—81no—
286C4—81no—
287C4—942no—
288C4—944no—
289C4—3761no—
290C4—3761no—
291C2—1882no—
292C2#2024197811yescharacteristic
293C2#2024200112yes—
294C2#2024202111yescharacteristic
295C2#2024346711yescharacteristic
296C2#2024348612yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1295C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1295D47
hypocenterC47#8721
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #20241270C2semidirect
4. #20241269C2semidirect
5. C47×#1295C2semidirect
147. #1295D47semidirect
167. C47#8721semidirect

Maximal subgroups

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

subgroupindex
#202457942
#202431512
#202412702
#202412692
C47×#12952
#872147