Finite Group Databasebeta

← #26109849

Subgroups — #26109849

Order 12032.

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

normal subgroups only

This group has 498 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#20243380C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241291C212yesmaximal, complemented
4#20241246C212yesmaximal, complemented
5C47×#2349C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688847—22no—
11C2×C4×D4×C47—21no—
12C2×C4×D4×C47C2211yescharacteristic
13C47×#389—21no—
14C47×#385—22no—
15C47×#382C2211yescharacteristic
16C47×#380C412yes—
17C4×C4⋊C4×C47C2211yescharacteristic
18C2×#95×C47C412yes—
19C2×#10625329—22no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)—22no—
22C4⋊(C47⋊C8)—22no—
23#13093941—42no—
24C22⋊(C47⋊C8)—22no—
25C4×C47⋊C8—22no—
26C22×D4×C47D411yescharacteristic
27C23×C4×C47—22no—
28C23×C4×C47C2×C412yes—
29C4×D4×C47—22no—
30C4×D4×C47—22no—
31C4×D4×C47—44no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47D411yescharacteristic
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47D412yes—
38C2×C42×C47—21no—
39C2×C42×C47C2311yescharacteristic
40#95×C47—22no—
41#95×C47—21no—
42#95×C47C2×C411yescharacteristic
43#95×C47C2×C412yes—
44#95×C47C2×C411yescharacteristic
45C24×C47C22⋊C412yes—
46C2×D4×C47—22no—
47C2×D4×C47—22no—
48C2×D4×C47SD1612yes—
49C2×D4×C47Q1612yes—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—44no—
56C22×C4×C47—44no—
57C22×C4×C47—82no—
58C22×C4×C47—82no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C4⋊C4×C47—22no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—42no—
66C4⋊C4×C47—82no—
67C22⋊C4×C47—24no—
68C22⋊C4×C47—44no—
69C42×C47—42no—
70C42×C47#4112yes—
71C42×C47#4112yes—
72#10625329—42no—
73#10625329—42no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C23×C47—22no—
77C23×C47—22no—
78C23×C47—42no—
79C23×C47—42no—
80C23×C47#10314yes—
81C23×C47C2×C22⋊C411yescharacteristic
82D4×C47—24no—
83D4×C47—24no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47—82no—
98C2×C4×C47—82no—
99C2×C4×C47—88no—
100C2×C4×C47#11711yescharacteristic
101C2×C4×C47#11711yescharacteristic
102C2×C4×C47#12312yes—
103C2×C4×C47#12312yes—
104C2×C4×C47C2×SD1611yescharacteristic
105C2×C4×C47C2×Q1611yescharacteristic
106C2×C4×C47AGL(1,ℤ/8)12yes—
107C47⋊C8—82no—
108C47⋊C8—82no—
109#8765—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47—24no—
113C22×C47—44no—
114C22×C47—44no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47#38111yescharacteristic
118C22×C47C2×#10311yescharacteristic
119C22×C47#41811yescharacteristic
120C4×C47—22no—
121C4×C47—21no—
122C4×C47—21no—
123C4×C47—42no—
124C4×C47—42no—
125C4×C47—81no—
126C4×C47—82no—
127C4×C47#48112yes—
128C4×C47#48412yes—
129#2349D4711yescharacteristic
130#1800—471no—
131#1460—471no—
132#1303—472no—
133#1286—472no—
134C2×C47—21no—
135C2×C47—21no—
136C2×C47—44no—
137C2×C47#182411yescharacteristic
138C2×C47#182711yescharacteristic
139C2×C47#182911yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2×D4711yescharacteristic
142#431—472no—
143#423—472no—
144C2×#100—941no—
145#389—21no—
146#385—22no—
147#382C2×D4711yescharacteristic
148#380Dic4712yes—
149C4×C4⋊C4C2×D4711yescharacteristic
150C2×#95Dic4712yes—
151#343—941no—
152#323—942no—
153C47#876511yescharacteristic, complemented, Sylow
154C22×D4C22⋊D4711yescharacteristic
155C23×C4—22no—
156C23×C4C2×Dic4712yes—
157C2×M16—942no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D4712yes—
167C2×C42—21no—
168C2×C42C22×D4711yescharacteristic
169C4⋊C8—942no—
170C4⋊C8—942no—
171C4⋊C8—942no—
172#100—1882no—
173C22⋊C8—942no—
174C4×C8—942no—
175#95—22no—
176#95—21no—
177#95C2×Dic4711yescharacteristic
178#95C2×Dic4712yes—
179#95C2×Dic4711yescharacteristic
180C24C22⋊Dic4712yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#1062533612yes—
184C2×D4#1062533712yes—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C22⋊Dic4712yes—
195C22×C4C22×Dic4711yescharacteristic
196C22×C4C2×C22⋊D4711yescharacteristic
197C22×C4C2×C22⋊D4711yescharacteristic
198M16—1882no—
199M16—1882no—
200C2×C8—1882no—
201C2×C8—1882no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—24no—
207C22⋊C4—44no—
208C42—42no—
209C42#1062535012yes—
210C42#1062535012yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#1309394814yes—
216C23C2×C22⋊Dic4711yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#1309398311yescharacteristic
236C2×C4#1309398311yescharacteristic
237C2×C4#1309400712yes—
238C2×C4#1309400712yes—
239C2×C4C2×#1062533611yescharacteristic
240C2×C4C2×#1062533711yescharacteristic
241C2×C4#1309405012yes—
242C8—3762no—
243C8—3762no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1568898611yescharacteristic
252C22C2×#1309394811yescharacteristic
253C22#1568912811yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—81no—
261C4#1568944112yes—
262C4#1568945812yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2024347811yescharacteristic
267C2#2024349211yescharacteristic
268C2#2024349911yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2349C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2349D47
hypocenterC47#8765
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20241291C2semidirect
4. #20241246C2semidirect
153. C47#8765semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202417092
#202412912
#202412462
C47×#23492
#876547