Finite Group Databasebeta

← #26104244

Subgroups — #26104244

Order 12032.

528 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 119 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#20245935C211yesmaximal, characteristic
2#20243132C212yesmaximal, complemented
3#20241213C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#1270C211yesmaximal, characteristic
6C2×#13094011C411yescharacteristic
7C2×#13093989C411yescharacteristic
8C2×D4×Dic47C411yescharacteristic
9C2×C8×Dic47—22no—
10#15689029C2211yescharacteristic
11#15689011—22no—
12#15688988C411yescharacteristic
13C22⋊C4×Dic47—22no—
14C42×Dic47C2211yescharacteristic
15#15688878C2212yes—
16#15688877—22no—
17C47×#390C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94C2×C411yescharacteristic
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22#13094011—22no—
23#13094011C814yescomplemented
24#13093989—42no—
25D4×Dic47—42no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34C2×C4×Dic47D412yes—
35C8×Dic47—28no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C22×D4×C47C2×C411yescharacteristic
39C22×C8×C47—22no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C22⋊C4×C47—22no—
42C2×C42×C47C2311yescharacteristic
43C24×C47—21no—
44C2×D4×C47—21no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47C2×C812yes—
48C22×C4×C47—21no—
49C22×C4×C47C22×C411yescharacteristic
50C22×C4×C47C2×D412yes—
51C22×Dic47—21no—
52C22×Dic47—21no—
53C22×Dic47—42no—
54C22×Dic47—44no—
55C22×Dic47C22⋊C411yescharacteristic
56C22×Dic47C22⋊C412yes—
57C22×Dic47C22⋊C411yescharacteristic
58C2×Dic94—21no—
59C2×Dic94—41no—
60C2×Dic94—41no—
61C2×Dic94C2×C812yes—
62C2×C8×C47—24no—
63C2×C8×C47—42no—
64C2×C8×C47—42no—
65C4⋊C4×C47—41no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—42no—
68C42×C47—41no—
69C22⋊Dic47—28no—
70C22⋊Dic47—42no—
71C4⋊Dic47—41no—
72Dic47⋊C4—42no—
73C4×Dic47—22no—
74C4×Dic47—22no—
75C4×Dic47—22no—
76C4×Dic47—24no—
77C4×Dic47—42no—
78C4×Dic47C2×C812yes—
79C4×Dic47M1612yes—
80C2×C47⋊C8—24no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C23×C47—22no—
84C23×C47—41no—
85C23×C47—41no—
86C23×C47—42no—
87C23×C47—42no—
88C23×C47C2×C22⋊C411yescharacteristic
89D4×C47—42no—
90D4×C47—42no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47C22×C811yescharacteristic
98C2×C4×C47C2×M1611yescharacteristic
99C2×C4×C47#13512yes—
100C2×C4×C47#13512yes—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—21no—
105C2×Dic47—22no—
106C2×Dic47—41no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109C2×Dic47—42no—
110C2×Dic47—84no—
111C2×Dic47C22⋊C814yes—
112C2×Dic47#10012yes—
113C2×Dic47#10414yes—
114Dic94—42no—
115Dic94—42no—
116C8×C47—44no—
117C47⋊C8—44no—
118#7567—471nomaximal, Sylow
119C22×C47—42no—
120C22×C47—44no—
121C22×C47—81no—
122C22×C47—82no—
123C22×C47—81no—
124C22×C47C2×C22⋊C811yescharacteristic
125C22×C47#41011yescharacteristic
126C22×C47C2×#10011yescharacteristic
127C22×C47#41612yes—
128C22×C47C2×#10412yes—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—81no—
134Dic47—24no—
135Dic47—41no—
136Dic47—42no—
137Dic47—81no—
138Dic47#32914yes—
139C4×#124—471no—
140#1744—472no—
141#1272—472no—
142#1270—471no—
143#1270D4711yescharacteristic
144C2×C47—82no—
145C2×C47#146011yescharacteristic
146C2×C47#147612yes—
147C2×C47#150012yes—
148C2×C47C2×#32911yescharacteristic
149C2×C47#154311yescharacteristic
150C2×#124—471no—
151C2×C4×Q8—471no—
152C2×C4×D4—471no—
153C2×C4×C8—942no—
154#391—471no—
155#390C2×D4711yescharacteristic
156#388—942no—
157#382—471no—
158C4×C22⋊C4—942no—
159C43—471no—
160#336—472no—
161#336—942no—
162#336C2×D4712yes—
163C47#756711yescharacteristic, complemented, Sylow
164C22×Q8—471no—
165C22×D4C4×D4711yescharacteristic
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—22no—
177C2×C22⋊C4—942no—
178C2×C22⋊C4—941no—
179C2×C42—471no—
180C2×C42—471no—
181C2×C42—472no—
182C2×C42—472no—
183C2×C42—941no—
184C2×C42C22×D4711yescharacteristic
185C4×C8—948no—
186#95—942no—
187#95—942no—
188C24—21no—
189C2×Q8—472no—
190C2×Q8—941no—
191C2×Q8—1881no—
192C2×Q8—1881no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C8×D4712yes—
197C22×C4—21no—
198C22×C4—471no—
199C22×C4—472no—
200C22×C4—471no—
201C22×C4—941no—
202C22×C4—941no—
203C22×C4—1882no—
204C22×C4—1884no—
205C22×C4C2×C4×D4711yescharacteristic
206C22×C4D4×D4712yes—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—944no—
211C2×C8—1882no—
212C2×C8—1882no—
213C4⋊C4—41no—
214C4⋊C4—1881no—
215C4⋊C4—1882no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—948no—
219C22⋊C4—1882no—
220C42—41no—
221C42—472no—
222C42—472no—
223C42—944no—
224C42—942no—
225C42—942no—
226C42—942no—
227C42—1882no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23C22⋊C4×D4711yescharacteristic
234Q8—1882no—
235Q8—1882no—
236D4—42no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—472no—
245C2×C4—474no—
246C2×C4—474no—
247C2×C4—942no—
248C2×C4—942no—
249C2×C4—942no—
250C2×C4—941no—
251C2×C4—942no—
252C2×C4—3764no—
253C2×C4—1881no—
254C2×C4—1882no—
255C2×C4—1881no—
256C2×C4—1881no—
257C2×C4C2×C8×D4711yescharacteristic
258C2×C4M16×D4711yescharacteristic
259C2×C4#1309404312yes—
260C2×C4#1309404512yes—
261C8—44no—
262C8—1884no—
263C22—42no—
264C22—44no—
265C22—81no—
266C22—81no—
267C22—82no—
268C22C22⋊C8×D4711yescharacteristic
269C22#1568909511yescharacteristic
270C22#100×D4711yescharacteristic
271C22#1568911612yes—
272C22#104×D4712yes—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—81no—
278C4—474no—
279C4—944no—
280C4—3761no—
281C4—1882no—
282C4—1881no—
283C2—82no—
284C2#2024170711yescharacteristic
285C2#2024176712yes—
286C2#2024187112yes—
287C2#329×D4711yescharacteristic
288C2#2024204711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1270C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1270D47
hypocenterC47#7567
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20243132C2semidirect
3. #20241213C2semidirect
23. #13094011C8semidirect
163. C47#7567semidirect

Maximal subgroups

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

subgroupindex
#202459352
#202431322
#202412132
#202412092
C47×#12702
#756747