Finite Group Databasebeta

← #26114137

Subgroups — #26114137

Order 12032.

518 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 115 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#20248742C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20241818C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#2975C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689147C2212yes—
8C2×#13093941—21no—
9#15689059C412yes—
10#15689035C412yes—
11#15689026—21no—
12#15689019—21no—
13#15688996—22no—
14#15688989C2211yescharacteristic
15#15688892—21no—
16#15688855—24no—
17C2×C4×D4×C47C2211yescharacteristic
18C47×#436—24no—
19C47×#431C2212yes—
20C47×#407—22no—
21C22×Dic94C2×C411yescharacteristic
22C2×#10625329—22no—
23C4×Dic94—22no—
24C4×Dic94—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C4×Dic47—21no—
32C4⋊(C47⋊C8)—22no—
33C4⋊(C47⋊C8)—22no—
34#13093941—42no—
35C47⋊#95—22no—
36C47⋊#95—21no—
37C47⋊#95C2×C412yes—
38C22×D4×C47D411yescharacteristic
39C23×C4×C47D412yes—
40C2×M16×C47—24no—
41C4×D4×C47—22no—
42C4×D4×C47—22no—
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47C2×C412yes—
45C2×C42×C47C2311yescharacteristic
46C4⋊C8×C47—22no—
47C22⋊C8×C47—24no—
48C22⋊C8×C47—22no—
49C4×C8×C47D414yes—
50C24×C47C22⋊C412yes—
51C2×D4×C47—22no—
52C2×D4×C47—21no—
53C2×D4×C47—22no—
54C2×D4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—82no—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—22no—
64C22×Dic47—21no—
65C2×Dic94—22no—
66C2×Dic94—42no—
67M16×C47—84no—
68C2×C8×C47—24no—
69C4⋊C4×C47—22no—
70C22⋊C4×C47—24no—
71C42×C47C2×D412yes—
72C42×C47#4112yes—
73C22⋊Dic47—44no—
74C22⋊Dic47—82no—
75C4⋊Dic47—22no—
76C4⋊Dic47—41no—
77C4⋊Dic47—41no—
78Dic47⋊C4—41no—
79Dic47⋊C4—41no—
80Dic47⋊C4—82no—
81C4×Dic47—42no—
82#10625329—42no—
83#10625329—42no—
84C2×C47⋊C8—42no—
85C23×C47—22no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—42no—
89C23×C47—42no—
90C23×C47C2×C22⋊C411yescharacteristic
91D4×C47—42no—
92D4×C47—42no—
93C2×C4×C47—22no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—44no—
97C2×C4×C47—42no—
98C2×C4×C47—82no—
99C2×C4×C47C4×D412yes—
100C2×C4×C47#12011yescharacteristic
101C2×C4×C47#12112yes—
102C2×C4×C47#12311yescharacteristic
103C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C47#13712yes—
105C2×C4×C47#13711yescharacteristic
106C2×Dic47—22no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—42no—
115Dic94—24no—
116Dic94—42no—
117C8×C47—44no—
118C47⋊C8—82no—
119#9487—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—44no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47—42no—
126C22×C47—82no—
127C22×C47#38611yescharacteristic
128C22×C47#41711yescharacteristic
129C22×C47#41911yescharacteristic
130C4×C47—22no—
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—42no—
134C4×C47#46812yes—
135C4×C47#47012yes—
136Dic47—42no—
137Dic47—82no—
138Dic47—81no—
139#2975D4711yescharacteristic
140#2580—471no—
141#1800—471no—
142#1487—472no—
143#1286—472no—
144C2×C47—21no—
145C2×C47—21no—
146C2×C47—82no—
147C2×C47#189111yescharacteristic
148C2×C47#189411yescharacteristic
149C2×C47#191111yescharacteristic
150C2×C4×Q8—471no—
151C2×C4×D4C2×D4711yescharacteristic
152#436—24no—
153#431C2×D4712yes—
154#423—472no—
155C2×#100—941no—
156#407—22no—
157#399—472no—
158#393—472no—
159#390—941no—
160#389—941no—
161#385—942no—
162#382—471no—
163#343—941no—
164Q8⋊C8—944no—
165C47#948711yescharacteristic, complemented, Sylow
166C22×Q8—471no—
167C22×D4D18811yescharacteristic
168C23×C4C22⋊D4712yes—
169C2×M16—24no—
170C2×M16—942no—
171C4×Q8—942no—
172C4×Q8—942no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C4⋊C4—471no—
176C2×C4⋊C4—942no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4D18811yescharacteristic
180C2×C22⋊C4—942no—
181C2×C22⋊C4—942no—
182C2×C22⋊C4C4×D4712yes—
183C2×C42—941no—
184C2×C42C22×D4711yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—942no—
187C4⋊C8—942no—
188#100—1882no—
189C22⋊C8—24no—
190C22⋊C8—22no—
191C4×C8C22⋊D4714yes—
192#95—472no—
193#95—942no—
194#95—941no—
195C24(C2×D47)⋊C412yes—
196C2×Q8—942no—
197C2×Q8—1882no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—82no—
205C22×C4—941no—
206C22×C4—942no—
207C22×C4—941no—
208C22×C4(C2×D47)⋊C412yes—
209C22×C4C2×C4×D4711yescharacteristic
210C22×C4C2×D18811yescharacteristic
211C22×C4C2×C22⋊D4711yescharacteristic
212M16—84no—
213M16—1882no—
214M16—1882no—
215C2×C8—24no—
216C2×C8—1882no—
217C4⋊C4—22no—
218C4⋊C4—942no—
219C4⋊C4—3762no—
220C4⋊C4—1881no—
221C4⋊C4—1881no—
222C4⋊C4—1881no—
223C4⋊C4—1881no—
224C22⋊C4—24no—
225C22⋊C4—3762no—
226C22⋊C4—1884no—
227C42—1882no—
228C42#1062535012yes—
229C42C2×C22⋊D4712yes—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23C2×(C2×D47)⋊C411yescharacteristic
236Q8—944no—
237Q8—1882no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—942no—
247C2×C4—1881no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1882no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4C4×C22⋊D4712yes—
256C2×C4#1309399311yescharacteristic
257C2×C4#1309399812yes—
258C2×C4#1309400711yescharacteristic
259C2×C4#1309405211yescharacteristic
260C2×C4#1309405712yes—
261C2×C4#1309405711yescharacteristic
262C8—44no—
263C8—3762no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#1568900211yescharacteristic
272C22#1568912511yescharacteristic
273C22#1568913311yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—3762no—
279C4—3761no—
280C4—1882no—
281C4#1568936912yes—
282C4#1568938112yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#2024380611yescharacteristic
287C2#2024382111yescharacteristic
288C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2975C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2975D47
hypocenterC47#9487
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #20241818C2semidirect
4. #20241247C2semidirect
165. C47#9487semidirect

Maximal subgroups

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

subgroupindex
#202487422
#202433802
#202418182
#202412472
C47×#29752
#948747