Finite Group Databasebeta

← #26104538

Subgroups — #26104538

Order 12032.

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

normal subgroups only

This group has 520 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#20245931C211yesmaximal, characteristic
2#20243240C212yesmaximal, complemented
3#20241214C211yesmaximal, characteristic, complemented
4#20241208C212yesmaximal
5C47×#1272C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093986C411yescharacteristic
9C2×Dic47⋊C8—22no—
10C2×#13093934—22no—
11#15689025C2211yescharacteristic
12#15689015—22no—
13#15688988C411yescharacteristic
14#15688970—22no—
15C42×Dic47C2211yescharacteristic
16#15688878C2212yes—
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—
36Dic47⋊C8—28no—
37#13093934—44no—
38C47⋊#95—22no—
39C22×Q8×C47C2×C411yescharacteristic
40C22×C8×C47—22no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C42×C47C2311yescharacteristic
43#95×C47—22no—
44C23×D47—21no—
45C2×Q8×C47—21no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47M1612yes—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C2×D412yes—
53C22×Dic47—21no—
54C22×Dic47C22⋊C412yes—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C411yescharacteristic
57C2×D188—21no—
58C2×D188—41no—
59C2×D188—41no—
60C2×D188M1612yes—
61C2×C4×D47—42no—
62C2×C4×D47—44no—
63C2×C8×C47—24no—
64C2×C8×C47—42no—
65C2×C8×C47—42no—
66C4⋊C4×C47—41no—
67C42×C47—41no—
68(C2×D47)⋊C4—28no—
69(C2×D47)⋊C4—42no—
70(C2×D47)⋊C4—42no—
71(C2×D47)⋊C4—42no—
72C4⋊Dic47—42no—
73C4⋊Dic47—41no—
74C4×Dic47—21no—
75C4×Dic47—21no—
76C4×Dic47—22no—
77C4×Dic47—22no—
78C4×Dic47—22no—
79C4×Dic47—24no—
80C4×Dic47—42no—
81C2×C47⋊C8—24no—
82C2×C47⋊C8—42no—
83C2×C47⋊C8—42no—
84C23×C47C2×C22⋊C411yescharacteristic
85C22×D47—22no—
86C22×D47—41no—
87C22×D47—41no—
88C22×D47—42no—
89C22×D47—42no—
90Q8×C47—42no—
91Q8×C47—42no—
92C2×C4×C47—22no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47C2×M1611yescharacteristic
102C2×C4×C47C2×M1611yescharacteristic
103C2×C4×C47AGL(1,ℤ/8)12yes—
104C2×C4×C47#13712yes—
105C2×Dic47—22no—
106C2×Dic47—21no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—41no—
111C2×Dic47—42no—
112C2×Dic47#10012yes—
113C2×Dic47#10414yes—
114D188—42no—
115D188—42no—
116C4×D47—84no—
117C8×C47—44no—
118C47⋊C8—44no—
119#7635—471nomaximal, Sylow
120C22×C47#40711yescharacteristic
121C22×C47#41011yescharacteristic
122C22×C47C2×#10011yescharacteristic
123C22×C47#41712yes—
124C22×C47C2×#10412yes—
125C2×D47—42no—
126C2×D47—44no—
127C2×D47—81no—
128C2×D47—82no—
129C2×D47—81no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—42no—
135C4×C47—81no—
136Dic47—41no—
137Dic47—42no—
138Dic47—81no—
139Dic47#32914yes—
140C4×#124—471no—
141#1767—472no—
142#1272—471no—
143#1272D4711yescharacteristic, complemented
144#1270—472no—
145C2×C47#146611yescharacteristic
146C2×C47#148912yes—
147C2×C47#150912yes—
148C2×C47C2×#32911yescharacteristic
149C2×C47#156011yescharacteristic
150D47—82no—
151C2×#124—471no—
152C2×C4×Q8—471no—
153C2×C4×D4—471no—
154C2×C4⋊C8—942no—
155C2×#97—942no—
156#391C2×D4711yescharacteristic
157#390—471no—
158#388—942no—
159#382—471no—
160C4×C22⋊C4—942no—
161C43—471no—
162#336—472no—
163#336C2×D4712yes—
164C47#763511yescharacteristic, complemented, Sylow
165C22×Q8C4×D4711yescharacteristic
166C22×D4—471no—
167C23×C4—941no—
168C22×C8—22no—
169C22×C8—942no—
170#124—942no—
171#124—942no—
172C4×Q8—1882no—
173C4×D4—1882no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—941no—
176C2×C4⋊C4C4×D4711yescharacteristic
177C2×C22⋊C4—942no—
178C2×C22⋊C4—941no—
179C2×C22⋊C4—942no—
180C2×C42—471no—
181C2×C42—471no—
182C2×C42—472no—
183C2×C42—472no—
184C2×C42—941no—
185C2×C42C22×D4711yescharacteristic
186C4⋊C8—948no—
187#97—1884no—
188#95—22no—
189#95—942no—
190C24—941no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8#1062532412yes—
195C2×D4—472no—
196C2×D4—941no—
197C2×D4—1881no—
198C2×D4—1881no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—471no—
202C22×C4—472no—
203C22×C4—471no—
204C22×C4—941no—
205C22×C4—1882no—
206C22×C4—1884no—
207C22×C4C2×C4×D4711yescharacteristic
208C22×C4D4×D4712yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—944no—
213C2×C8—1882no—
214C2×C8—1882no—
215C4⋊C4—41no—
216C4⋊C4—1881no—
217C4⋊C4—1882no—
218C22⋊C4—948no—
219C22⋊C4—1882no—
220C22⋊C4—1882no—
221C22⋊C4—1882no—
222C42—41no—
223C42—944no—
224C42—942no—
225C42—941no—
226C42—941no—
227C42—942no—
228C42—942no—
229C42—1882no—
230C23—942no—
231C23—1881no—
232C23—1881no—
233C23—1882no—
234C23—1882no—
235C23C22⋊C4×D4711yescharacteristic
236Q8—42no—
237Q8—42no—
238D4—1882no—
239D4—1882no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—472no—
250C2×C4—474no—
251C2×C4—942no—
252C2×C4—941no—
253C2×C4—942no—
254C2×C4—942no—
255C2×C4—942no—
256C2×C4—3764no—
257C2×C4—1881no—
258C2×C4—1882no—
259C2×C4M16×D4711yescharacteristic
260C2×C4C2×#1062532411yescharacteristic
261C2×C4#1309405112yes—
262C2×C4#1309405612yes—
263C8—44no—
264C8—1884no—
265C22—3761no—
266C22—3761no—
267C22—3762no—
268C22—1882no—
269C22—1884no—
270C22#1568908511yescharacteristic
271C22#1568909511yescharacteristic
272C22#100×D4711yescharacteristic
273C22#1568912312yes—
274C22#104×D4712yes—
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—81no—
281C4—474no—
282C4—3761no—
283C4—1882no—
284C4—1881no—
285C2—3762no—
286C2#2024172711yescharacteristic
287C2#2024182612yes—
288C2#2024191212yes—
289C2#329×D4711yescharacteristic
290C2#2024212211yescharacteristic
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#7635
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. #20243240C2semidirect
3. #20241214C2semidirect
5. C47×#1272C2semidirect
143. #1272D47semidirect
164. C47#7635semidirect

Maximal subgroups

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

subgroupindex
#202459312
#202432402
#202412142
#202412082
C47×#12722
#763547