Finite Group Databasebeta

← #26109891

Subgroups — #26109891

Order 12032.

384 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 95 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#20246353C211yesmaximal, characteristic
2C4×#13093943C211yesmaximal, characteristic
3#20241280C212yesmaximal
4#20241254C212yesmaximal
5C47×#1467C211yesmaximal, characteristic
6C2×#13093943—21no—
7C4×#10625329C2212yes—
8#15689030—21no—
9#15689009—21no—
10#15689009—21no—
11#15688991C411yescharacteristic
12#15688991C2211yescharacteristic
13#15688989C411yescharacteristic
14C4⋊C4×Dic47—21no—
15C4×C4⋊Dic47C411yescharacteristic
16#15688969C411yescharacteristic
17#15688969C2211yescharacteristic
18#15688892—21no—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#432C2212yes—
21C47×#324—21no—
22C47×#324—21no—
23C2×#10625329—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4×Dic47—21no—
27C2×C4×Dic47—21no—
28C2×C4×Dic47—21no—
29#13093943—24no—
30#13093935—22no—
31C4×C47⋊C8—22no—
32C47⋊#95—21no—
33C47⋊#95—21no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95C2×C411yescharacteristic
40C47⋊#95C2×C411yescharacteristic
41C47⋊#95C2×C411yescharacteristic
42C47⋊#95C2×C411yescharacteristic
43C22×Q8×C47D411yescharacteristic
44C4×Q8×C47—22no—
45C4×Q8×C47—22no—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54#97×C47—22no—
55C2×Q8×C47—22no—
56C2×Q8×C47—22no—
57C2×Q8×C47—41no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—21no—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C22×Dic47—21no—
69C2×C8×C47—42no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—22no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47#4112yes—
78C42×C47#4112yes—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81C4×Dic47—42no—
82C4×Dic47—42no—
83C4×Dic47—42no—
84#10625329—24no—
85#10625329—42no—
86C2×C47⋊C8—42no—
87C23×C47C2×C22⋊C411yescharacteristic
88Q8×C47—44no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47#10412yes—
103C2×C4×C47#11711yescharacteristic
104C2×C4×C47#11711yescharacteristic
105C2×C4×C47#12312yes—
106C2×C4×C47#12312yes—
107C2×Dic47—22no—
108C2×Dic47—41no—
109C2×Dic47—41no—
110C2×Dic47—41no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C8×C47—82no—
120C47⋊C8—44no—
121#8774—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47#38111yescharacteristic
125C22×C47C2×#10411yescharacteristic
126C22×C47#42111yescharacteristic
127C4×C47—22no—
128C4×C47—22no—
129C4×C47—22no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—41no—
133C4×C47—41no—
134C4×C47—41no—
135Dic47—42no—
136Dic47—81no—
137Dic47—81no—
138Dic47—81no—
139#2356—471no—
140C4×#101—471no—
141#1467D4711yescharacteristic
142#1300—472no—
143#1288—472no—
144C2×C47—21no—
145C2×C47—21no—
146C2×C47#182611yescharacteristic
147C2×C47#182711yescharacteristic
148C2×C47#183011yescharacteristic
149C2×C4×Q8C2×D4711yescharacteristic
150#432C2×D4712yes—
151C2×#101—941no—
152C4×M16—472no—
153#391—941no—
154#387—941no—
155#387—941no—
156#383—471no—
157#383—471no—
158#382—471no—
159C4×C4⋊C4—471no—
160C4×C4⋊C4—941no—
161#376—471no—
162#376—471no—
163#343—941no—
164#324—21no—
165#324—21no—
166C47#877411yescharacteristic, complemented, Sylow
167C22×Q8D18811yescharacteristic
168C2×M16—942no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—941no—
172C2×C4⋊C4—941no—
173C2×C4⋊C4D18811yescharacteristic
174C2×C4⋊C4C22⋊D4711yescharacteristic
175C2×C4⋊C4C22⋊D4711yescharacteristic
176C2×C42—941no—
177C2×C42—941no—
178C2×C42—941no—
179C2×C42C4×D4711yescharacteristic
180C2×C42C4×D4711yescharacteristic
181C2×C42C22×D4711yescharacteristic
182C4⋊C8—22no—
183#101—944no—
184C22⋊C8—22no—
185#97—22no—
186#97—942no—
187C4×C8—942no—
188#95—471no—
189#95—471no—
190#95—471no—
191#95—471no—
192#95—941no—
193#95—941no—
194#95—941no—
195#95—941no—
196#95—941no—
197#95—941no—
198#95—941no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C22×C4—941no—
203C22×C4—941no—
204C22×C4—941no—
205C22×C4—941no—
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4(C2×D47)⋊C411yescharacteristic
210C22×C4C2×C4×D4711yescharacteristic
211C22×C4C2×D18811yescharacteristic
212C22×C4C2×C22⋊D4711yescharacteristic
213M16—944no—
214M16—1882no—
215C2×C8—42no—
216C2×C8—1882no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—1882no—
221C4⋊C4—1882no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—1882no—
227C42—1882no—
228C42—1882no—
229C42#1062535212yes—
230C42#1062535212yes—
231C23C2×(C2×D47)⋊C411yescharacteristic
232Q8—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—942no—
247C2×C4—1881no—
248C2×C4—1881no—
249C2×C4—1881no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1881no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4—1881no—
256C2×C4—1881no—
257C2×C4—1881no—
258C2×C4#1309395012yes—
259C2×C4#1309398011yescharacteristic
260C2×C4#1309398011yescharacteristic
261C2×C4#1309400512yes—
262C2×C4#1309400812yes—
263C8—82no—
264C8—1884no—
265C22—21no—
266C22—21no—
267C22#1568898411yescharacteristic
268C22C2×#1309395011yescharacteristic
269C22#1568913911yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—3761no—
279C4—3761no—
280C4—3761no—
281C4—1882no—
282C2—21no—
283C2—21no—
284C2#2024348511yescharacteristic
285C2#2024348911yescharacteristic
286C2#2024350011yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1467C2
socleC22×C47#381
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1467D47
hypocenterC47#8774
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
166. C47#8774semidirect

Maximal subgroups

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

subgroupindex
#202463532
C4×#130939432
#202412802
#202412542
C47×#14672
#877447