Finite Group Databasebeta

← #26114117

Subgroups — #26114117

Order 12032.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2972C211yesmaximal, 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
18C8×D4×C47—24no—
19C47×#431C2212yes—
20C2×C22⋊C8×C47—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—
40C22×C8×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—22no—
53C2×D4×C47SD1612yes—
54C2×D4×C47Q1612yes—
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—
67C2×C8×C47—24no—
68C2×C8×C47—84no—
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—42no—
77Dic47⋊C4—41no—
78Dic47⋊C4—41no—
79Dic47⋊C4—82no—
80C4×Dic47—42no—
81#10625329—42no—
82#10625329—42no—
83C2×C47⋊C8—42no—
84C23×C47—22no—
85C23×C47—22no—
86C23×C47—42no—
87C23×C47—42no—
88C23×C47#10314yes—
89C23×C47C2×C22⋊C411yescharacteristic
90D4×C47—24no—
91D4×C47—24no—
92C2×C4×C47—22no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—44no—
96C2×C4×C47—42no—
97C2×C4×C47—82no—
98C2×C4×C47C4×D412yes—
99C2×C4×C47#12011yescharacteristic
100C2×C4×C47#12112yes—
101C2×C4×C47#12311yescharacteristic
102C2×C4×C47C2×SD1611yescharacteristic
103C2×C4×C47C2×Q1611yescharacteristic
104C2×C4×C47#13512yes—
105C2×Dic47—22no—
106C2×Dic47—41no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—42no—
114Dic94—24no—
115Dic94—42no—
116C8×C47—44no—
117C47⋊C8—82no—
118#9483—471nomaximal, Sylow
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—24no—
122C22×C47—44no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47—82no—
126C22×C47#38611yescharacteristic
127C22×C47C2×#10311yescharacteristic
128C22×C47#41611yescharacteristic
129C4×C47—22no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—42no—
133C4×C47#46512yes—
134C4×C47#46712yes—
135Dic47—42no—
136Dic47—82no—
137Dic47—81no—
138#2972D4711yescharacteristic
139#2580—471no—
140#1800—471no—
141#1487—472no—
142#1286—472no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—44no—
146C2×C47#189111yescharacteristic
147C2×C47#189211yescharacteristic
148C2×C47#191111yescharacteristic
149C2×C4×Q8—471no—
150C2×C4×D4C2×D4711yescharacteristic
151C8×D4—24no—
152#431C2×D4712yes—
153#423—472no—
154C2×#100—941no—
155C2×C22⋊C8—22no—
156#399—472no—
157#393—472no—
158#390—941no—
159#389—941no—
160#385—942no—
161#382—471no—
162#343—941no—
163Q8⋊C8—944no—
164C47#948311yescharacteristic, complemented, Sylow
165C22×Q8—471no—
166C22×D4D18811yescharacteristic
167C23×C4C22⋊D4712yes—
168C2×M16—942no—
169C22×C8—24no—
170C4×Q8—942no—
171C4×Q8—942no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—942no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4D18811yescharacteristic
179C2×C22⋊C4—942no—
180C2×C22⋊C4—942no—
181C2×C22⋊C4C4×D4712yes—
182C2×C42—941no—
183C2×C42C22×D4711yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—942no—
186C4⋊C8—942no—
187#100—1882no—
188C22⋊C8—24no—
189C22⋊C8—22no—
190C4×C8C22⋊D4714yes—
191#95—472no—
192#95—942no—
193#95—941no—
194C24(C2×D47)⋊C412yes—
195C2×Q8—942no—
196C2×Q8—1882no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1062532512yes—
200C2×D4Dic18812yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—82no—
204C22×C4—941no—
205C22×C4—942no—
206C22×C4—941no—
207C22×C4(C2×D47)⋊C412yes—
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4C2×D18811yescharacteristic
210C22×C4C2×C22⋊D4711yescharacteristic
211M16—1882no—
212M16—1882no—
213C2×C8—24no—
214C2×C8—84no—
215C2×C8—1882no—
216C4⋊C4—22no—
217C4⋊C4—942no—
218C4⋊C4—3762no—
219C4⋊C4—1881no—
220C4⋊C4—1881no—
221C4⋊C4—1882no—
222C22⋊C4—24no—
223C22⋊C4—3762no—
224C22⋊C4—1884no—
225C42—1882no—
226C42#1062535012yes—
227C42C2×C22⋊D4712yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1309394914yes—
233C23C2×(C2×D47)⋊C411yescharacteristic
234Q8—944no—
235Q8—1882no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—942no—
245C2×C4—1881no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1881no—
249C2×C4—1881no—
250C2×C4—1882no—
251C2×C4—1881no—
252C2×C4—1881no—
253C2×C4C4×C22⋊D4712yes—
254C2×C4#1309399311yescharacteristic
255C2×C4#1309399812yes—
256C2×C4#1309400711yescharacteristic
257C2×C4C2×#1062532511yescharacteristic
258C2×C4C2×Dic18811yescharacteristic
259C2×C4#1309404612yes—
260C8—44no—
261C8—3762no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1568900211yescharacteristic
270C22C2×#1309394911yescharacteristic
271C22#1568912011yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—3762no—
277C4—3761no—
278C4—1882no—
279C4#1568935212yes—
280C4#1568936412yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2024380611yescharacteristic
285C2#2024381111yescharacteristic
286C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9483
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #20241818C2semidirect
4. #20241247C2semidirect
164. C47#9483semidirect

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×#29722
#948347