Finite Group Databasebeta

← #26112284

Subgroups — #26112284

Order 12032.

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

normal subgroups only

This group has 458 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#20252026C211yesmaximal, characteristic
2#20249209C211yesmaximal, characteristic
3#20242068C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7#15689184—24no—
8#15689147C2212yes—
9#15689087—21no—
10C4×#10625329—21no—
11#15689062—21no—
12#15689059—21no—
13#15689043C412yes—
14#15689035C412yes—
15#15689021—21no—
16#15688989C2211yescharacteristic
17#15688983—21no—
18#15688871—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×Dic94C2×C412yes—
24C2×#10625329—22no—
25#13094024—22no—
26#13094024—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C4⋊(C47⋊C8)—22no—
33C4⋊(C47⋊C8)D414yes—
34C22⋊(C47⋊C8)—24no—
35#13093935—22no—
36C4×C47⋊C8—22no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95C2×C412yes—
41C23×C4×C47D411yescharacteristic
42#117×C47—22no—
43#117×C47—22no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47C2×C411yescharacteristic
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47C2311yescharacteristic
50C4⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C4×C8×C47—22no—
53C24×C47C22⋊C411yescharacteristic
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C2×Dic94—24no—
67C2×Dic94—82no—
68C2×C8×C47—42no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—22no—
71C22⋊C4×C47—22no—
72C22⋊C4×C47—22no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47C2×D412yes—
76C42×C47#4112yes—
77C22⋊Dic47—82no—
78C4⋊Dic47—41no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4⋊Dic47—42no—
82Dic47⋊C4—82no—
83#10625329—42no—
84#10625329—42no—
85C2×C47⋊C8—24no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47C2×C22⋊C411yescharacteristic
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47#12311yescharacteristic
107C2×C4×C47#13512yes—
108C2×C4×C47#13712yes—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115Dic94—84no—
116C8×C47—82no—
117C47⋊C8—44no—
118#9192—471nomaximal, Sylow
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47—81no—
126C22×C47#38611yescharacteristic
127C22×C47#41611yescharacteristic
128C22×C47#41911yescharacteristic
129C4×C47—22no—
130C4×C47—22no—
131C4×C47—41no—
132C4×C47—41no—
133C4×C47—41no—
134C4×C47#45212yes—
135C4×C47#48512yes—
136Dic47—82no—
137Dic47—82no—
138#2921—471no—
139#2628—471no—
140#1548—472no—
141#1458D4711yescharacteristic
142#1286—472no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—81no—
146C2×C47#187311yescharacteristic
147C2×C47#190911yescharacteristic
148C2×C47#191411yescharacteristic
149C2×C4⋊Q8—471no—
150C2×#117C2×D4711yescharacteristic
151#433—944no—
152#431C2×D4712yes—
153#423—472no—
154#407—941no—
155C4×M16—941no—
156#400—941no—
157#399—941no—
158#395—472no—
159#393—472no—
160#389—941no—
161#382—471no—
162#380—941no—
163#333—944no—
164#324—21no—
165#323—21no—
166C47#919211yescharacteristic, complemented, Sylow
167C22×Q8—472no—
168C23×C4D18811yescharacteristic
169C2×M16—942no—
170C4⋊Q8—942no—
171C4⋊Q8—942no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—942no—
175C2×C4⋊C4—941no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4C4×D4711yescharacteristic
179C2×C4⋊C4C22⋊D4711yescharacteristic
180C2×C22⋊C4—942no—
181C2×C22⋊C4C4×D4711yescharacteristic
182C2×C22⋊C4C22⋊D4711yescharacteristic
183C2×C42D18811yescharacteristic
184C2×C42C22×D4711yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—474no—
187C4⋊C8—942no—
188C22⋊C8—22no—
189C22⋊C8—944no—
190#97—942no—
191C4×C8—22no—
192C4×C8—942no—
193#95—472no—
194#95—942no—
195#95—942no—
196#95—941no—
197C24(C2×D47)⋊C411yescharacteristic
198C2×Q8—944no—
199C2×Q8—3762no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—942no—
205C22×C4—942no—
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4C2×C4×D4711yescharacteristic
210C22×C4C2×D18811yescharacteristic
211C22×C4C2×C22⋊D4711yescharacteristic
212M16—1882no—
213M16—1882no—
214C2×C8—42no—
215C2×C8—944no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—3762no—
219C4⋊C4—1881no—
220C4⋊C4—1882no—
221C4⋊C4—1881no—
222C4⋊C4—1882no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—3762no—
226C42—21no—
227C42—21no—
228C42D4×D4712yes—
229C42#1062535212yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×(C2×D47)⋊C411yescharacteristic
236Q8—3764no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4(C22⋊D47)⋊C412yes—
255C2×C4#1309399411yescharacteristic
256C2×C4#1309399612yes—
257C2×C4#1309400811yescharacteristic
258C2×C4#1309404612yes—
259C2×C4#1309405712yes—
260C8—82no—
261C8—1884no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1568900011yescharacteristic
270C22#1568912011yescharacteristic
271C22#1568913311yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—3762no—
278C4—3762no—
279C4#1568928712yes—
280C4#1568946512yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2024369811yescharacteristic
285C2#2024390811yescharacteristic
286C2#2024393311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9192
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20242068C2semidirect
4. #20241247C2semidirect
166. C47#9192semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202492092
#202420682
#202412472
C47×#14582
#919247