Finite Group Databasebeta

← #26104392

Subgroups — #26104392

Order 12032.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#20245931C211yesmaximal, characteristic
2#20243503C212yesmaximal, complemented
3#20241214C211yesmaximal, characteristic, complemented
4#20241208C212yesmaximal
5C47×#1272C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093986C411yescharacteristic
9C2×Dic47⋊C8—22no—
10#15689025C2211yescharacteristic
11#15689015—22no—
12#15688988C411yescharacteristic
13#15688970—22no—
14C42×Dic47C2211yescharacteristic
15#15688878C2212yes—
16#15688877—22no—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×D188C2×C411yescharacteristic
20C22×C4×D47—21no—
21C22×C47⋊C8—22no—
22#13094013—22no—
23#13094013C814yescomplemented
24Q8×Dic47—42no—
25#13093986—42no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35C2×C4×Dic47D412yes—
36Dic47⋊C8—44no—
37C47⋊#95—22no—
38C22×Q8×C47C2×C411yescharacteristic
39C22×C8×C47—22no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C42×C47C2311yescharacteristic
42#95×C47—22no—
43C23×D47—21no—
44C2×Q8×C47—21no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47C2×C812yes—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22×C411yescharacteristic
51C22×C4×C47C2×D412yes—
52C22×Dic47—21no—
53C22×Dic47C22⋊C411yescharacteristic
54C22×Dic47C22⋊C412yes—
55C22×Dic47C22⋊C411yescharacteristic
56C2×D188—21no—
57C2×D188—41no—
58C2×D188—41no—
59C2×D188C2×C812yes—
60C2×C4×D47—44no—
61C2×C4×D47—42no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C2×C8×C47—42no—
65C4⋊C4×C47—41no—
66C42×C47—41no—
67(C2×D47)⋊C4—28no—
68(C2×D47)⋊C4—42no—
69(C2×D47)⋊C4—42no—
70(C2×D47)⋊C4—42no—
71C4⋊Dic47—42no—
72C4⋊Dic47—41no—
73C4×Dic47—22no—
74C4×Dic47—22no—
75C4×Dic47—22no—
76C4×Dic47—42no—
77C4×Dic47—42no—
78C4×Dic47C2×C812yes—
79C4×Dic47M1612yes—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C23×C47C2×C22⋊C411yescharacteristic
84C22×D47—22no—
85C22×D47—41no—
86C22×D47—41no—
87C22×D47—42no—
88C22×D47—42no—
89Q8×C47—42no—
90Q8×C47—42no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47C22×C811yescharacteristic
101C2×C4×C47C2×M1611yescharacteristic
102C2×C4×C47AGL(1,ℤ/8)12yes—
103C2×C4×C47#13712yes—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—21no—
108C2×Dic47—22no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47C22⋊C814yes—
112C2×Dic47#10012yes—
113D188—42no—
114D188—42no—
115C4×D47—84no—
116C8×C47—82no—
117C47⋊C8—82no—
118#7601—471nomaximal, Sylow
119C22×C47C2×C22⋊C811yescharacteristic
120C22×C47#41011yescharacteristic
121C22×C47C2×#10011yescharacteristic
122C22×C47#41712yes—
123C22×C47#42112yes—
124C2×D47—42no—
125C2×D47—44no—
126C2×D47—81no—
127C2×D47—82no—
128C2×D47—81no—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—42no—
134C4×C47—81no—
135Dic47—24no—
136Dic47—22no—
137Dic47—41no—
138Dic47—81no—
139C4×#124—471no—
140#1831—472no—
141#1272—471no—
142#1272D4711yescharacteristic, complemented
143#1270—472no—
144C2×C47#146011yescharacteristic
145C2×C47#147712yes—
146C2×C47#151212yes—
147C2×C47#153311yescharacteristic
148C2×C47#156011yescharacteristic
149D47—82no—
150C2×#124—471no—
151C2×C4×Q8—471no—
152C2×C4×D4—471no—
153C2×C4⋊C8—942no—
154#391C2×D4711yescharacteristic
155#390—471no—
156#388—942no—
157#382—471no—
158C4×C22⋊C4—942no—
159C43—471no—
160#336—472no—
161#336—942no—
162#336C2×D4712yes—
163C47#760111yescharacteristic, complemented, Sylow
164C22×Q8C4×D4711yescharacteristic
165C22×D4—471no—
166C23×C4—941no—
167C22×C8—22no—
168C22×C8—942no—
169#124—474no—
170#124—942no—
171C4×Q8—1882no—
172C4×D4—1882no—
173C2×C4⋊C4—471no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4C4×D4711yescharacteristic
176C2×C22⋊C4—942no—
177C2×C22⋊C4—941no—
178C2×C22⋊C4—942no—
179C2×C42—471no—
180C2×C42—471no—
181C2×C42—472no—
182C2×C42—472no—
183C2×C42—941no—
184C2×C42C22×D4711yescharacteristic
185C4⋊C8—1884no—
186#95—22no—
187#95—942no—
188C24—941no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D4712yes—
193C2×D4—472no—
194C2×D4—941no—
195C2×D4—1881no—
196C2×D4—1881no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—472no—
200C22×C4—471no—
201C22×C4—471no—
202C22×C4—941no—
203C22×C4—1882no—
204C22×C4—1884no—
205C22×C4C2×C4×D4711yescharacteristic
206C22×C4D4×D4712yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—1882no—
211C2×C8—1882no—
212C2×C8—1882no—
213C4⋊C4—41no—
214C4⋊C4—1881no—
215C4⋊C4—1882no—
216C22⋊C4—948no—
217C22⋊C4—1882no—
218C22⋊C4—1882no—
219C22⋊C4—1882no—
220C42—41no—
221C42—472no—
222C42—472no—
223C42—942no—
224C42—942no—
225C42—942no—
226C42—1882no—
227C42—1882no—
228C23—942no—
229C23—1881no—
230C23—1881no—
231C23—1882no—
232C23—1882no—
233C23C22⋊C4×D4711yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—1882no—
237D4—1882no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—474no—
248C2×C4—472no—
249C2×C4—942no—
250C2×C4—942no—
251C2×C4—942no—
252C2×C4—941no—
253C2×C4—942no—
254C2×C4—3764no—
255C2×C4—1881no—
256C2×C4—1882no—
257C2×C4C2×C8×D4711yescharacteristic
258C2×C4M16×D4711yescharacteristic
259C2×C4#1309405112yes—
260C2×C4#1309405612yes—
261C8—82no—
262C8—3762no—
263C22—3762no—
264C22—3761no—
265C22—3761no—
266C22—1882no—
267C22—1884no—
268C22C22⋊C8×D4711yescharacteristic
269C22#1568909511yescharacteristic
270C22#100×D4711yescharacteristic
271C22#1568912312yes—
272C22#1568913812yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—942no—
280C4—944no—
281C4—3761no—
282C4—1881no—
283C2—3762no—
284C2#2024170711yescharacteristic
285C2#2024177412yes—
286C2#2024192512yes—
287C2#2024201011yescharacteristic
288C2#2024212211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7601
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20243503C2semidirect
3. #20241214C2semidirect
5. C47×#1272C2semidirect
23. #13094013C8semidirect
142. #1272D47semidirect
163. C47#7601semidirect

Maximal subgroups

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

subgroupindex
#202459312
#202435032
#202412142
#202412082
C47×#12722
#760147