Finite Group Databasebeta

← #26104592

Subgroups — #26104592

Order 12032.

468 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 99 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#20243530C212yesmaximal, complemented
3#20241214C211yesmaximal, characteristic, complemented
4#20241208C212yesmaximal
5C47×#1272C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093986C411yescharacteristic
9#15689025C2211yescharacteristic
10#15689015—22no—
11#15688988C411yescharacteristic
12#15688970—22no—
13C42×Dic47C2211yescharacteristic
14#15688878C2212yes—
15#15688877—22no—
16#15688877—22no—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×D188C2×C411yescharacteristic
20C22×C4×D47—21no—
21C22×C47⋊C8—22no—
22#13094013—22no—
23#13094013—22no—
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—
36C47⋊#95—22no—
37C22×Q8×C47C2×C411yescharacteristic
38C22×C8×C47—22no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C42×C47C2311yescharacteristic
41#95×C47—22no—
42C23×D47—21no—
43C2×Q8×C47—21no—
44C2×Q8×C47—41no—
45C2×Q8×C47—41no—
46C2×Q8×C47M1612yes—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47C22×C411yescharacteristic
50C22×C4×C47C2×D412yes—
51C22×Dic47—21no—
52C22×Dic47C22⋊C412yes—
53C22×Dic47C22⋊C411yescharacteristic
54C22×Dic47C22⋊C411yescharacteristic
55C2×D188—21no—
56C2×D188—41no—
57C2×D188—41no—
58C2×D188M1612yes—
59C2×C4×D47—44no—
60C2×C4×D47—42no—
61C2×C8×C47—42no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C4⋊C4×C47—41no—
65C42×C47—41no—
66(C2×D47)⋊C4—28no—
67(C2×D47)⋊C4—42no—
68(C2×D47)⋊C4—42no—
69(C2×D47)⋊C4—42no—
70C4⋊Dic47—42no—
71C4⋊Dic47—41no—
72C4×Dic47—21no—
73C4×Dic47—21no—
74C4×Dic47—22no—
75C4×Dic47—22no—
76C4×Dic47—22no—
77C4×Dic47—42no—
78C4×Dic47—42no—
79C2×C47⋊C8—42no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C23×C47C2×C22⋊C411yescharacteristic
83C22×D47—22no—
84C22×D47—41no—
85C22×D47—41no—
86C22×D47—42no—
87C22×D47—42no—
88Q8×C47—42no—
89Q8×C47—42no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—42no—
99C2×C4×C47C2×M1611yescharacteristic
100C2×C4×C47C2×M1611yescharacteristic
101C2×C4×C47#13512yes—
102C2×C4×C47#13512yes—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—21no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47#10012yes—
112D188—42no—
113D188—42no—
114C4×D47—84no—
115C8×C47—82no—
116C47⋊C8—82no—
117#7649—471nomaximal, Sylow
118C22×C47#40711yescharacteristic
119C22×C47#41011yescharacteristic
120C22×C47C2×#10011yescharacteristic
121C22×C47#41612yes—
122C22×C47#42112yes—
123C2×D47—42no—
124C2×D47—44no—
125C2×D47—81no—
126C2×D47—82no—
127C2×D47—81no—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—22no—
131C4×C47—22no—
132C4×C47—42no—
133C4×C47—81no—
134Dic47—22no—
135Dic47—41no—
136Dic47—42no—
137Dic47—81no—
138C4×#124—471no—
139#1839—472no—
140#1272—471no—
141#1272D4711yescharacteristic, complemented
142#1270—472no—
143C2×C47#146611yescharacteristic
144C2×C47#148812yes—
145C2×C47#152112yes—
146C2×C47#153311yescharacteristic
147C2×C47#154311yescharacteristic
148D47—82no—
149C2×#124—471no—
150C2×C4×Q8—471no—
151C2×C4×D4—471no—
152#391C2×D4711yescharacteristic
153#390—471no—
154#388—942no—
155#382—471no—
156C4×C22⋊C4—942no—
157C43—471no—
158#336—472no—
159#336—942no—
160#336—942no—
161#336C2×D4712yes—
162C47#764911yescharacteristic, complemented, Sylow
163C22×Q8C4×D4711yescharacteristic
164C22×D4—471no—
165C23×C4—941no—
166C22×C8—22no—
167C22×C8—942no—
168#124—942no—
169#124—942no—
170C4×Q8—1882no—
171C4×D4—1882no—
172C2×C4⋊C4—471no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4C4×D4711yescharacteristic
175C2×C22⋊C4—942no—
176C2×C22⋊C4—941no—
177C2×C22⋊C4—942no—
178C2×C42—471no—
179C2×C42—471no—
180C2×C42—472no—
181C2×C42—472no—
182C2×C42—941no—
183C2×C42C22×D4711yescharacteristic
184#95—22no—
185#95—942no—
186C24—941no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1062532412yes—
191C2×D4—472no—
192C2×D4—941no—
193C2×D4—1881no—
194C2×D4—1881no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—471no—
198C22×C4—472no—
199C22×C4—471no—
200C22×C4—941no—
201C22×C4—1882no—
202C22×C4—1884no—
203C22×C4C2×C4×D4711yescharacteristic
204C22×C4D4×D4712yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—1882no—
209C2×C8—1882no—
210C2×C8—1882no—
211C4⋊C4—41no—
212C4⋊C4—1881no—
213C4⋊C4—1882no—
214C22⋊C4—948no—
215C22⋊C4—1882no—
216C22⋊C4—1882no—
217C22⋊C4—1882no—
218C42—41no—
219C42—942no—
220C42—941no—
221C42—941no—
222C42—942no—
223C42—942no—
224C42—1882no—
225C42—1882no—
226C23—942no—
227C23—1881no—
228C23—1881no—
229C23—1882no—
230C23—1882no—
231C23C22⋊C4×D4711yescharacteristic
232Q8—42no—
233Q8—42no—
234D4—1882no—
235D4—1882no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—472no—
246C2×C4—942no—
247C2×C4—942no—
248C2×C4—941no—
249C2×C4—942no—
250C2×C4—942no—
251C2×C4—942no—
252C2×C4—3764no—
253C2×C4—1881no—
254C2×C4—1882no—
255C2×C4M16×D4711yescharacteristic
256C2×C4C2×#1062532411yescharacteristic
257C2×C4#1309404512yes—
258C2×C4#1309404712yes—
259C8—82no—
260C8—3762no—
261C22—3762no—
262C22—3761no—
263C22—3761no—
264C22—1882no—
265C22—1884no—
266C22#1568908511yescharacteristic
267C22#1568909511yescharacteristic
268C22#100×D4711yescharacteristic
269C22#1568911812yes—
270C22#1568913812yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—942no—
278C4—3761no—
279C4—1881no—
280C4—1882no—
281C2—3762no—
282C2#2024172711yescharacteristic
283C2#2024182112yes—
284C2#2024196412yes—
285C2#2024201011yescharacteristic
286C2#2024204911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7649
LCSG ▷ C22×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. #20243530C2semidirect
3. #20241214C2semidirect
5. C47×#1272C2semidirect
141. #1272D47semidirect
162. C47#7649semidirect

Maximal subgroups

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

subgroupindex
#202459312
#202435302
#202412142
#202412082
C47×#12722
#764947