Finite Group Databasebeta

← #26109879

Subgroups — #26109879

Order 12032.

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

normal subgroups only

This group has 426 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#20243380C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241291C212yesmaximal, complemented
4#20241246C212yesmaximal, complemented
5C47×#2357C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688847—22no—
11C2×C4×D4×C47C2211yescharacteristic
12C47×#390—21no—
13C47×#388—22no—
14C47×#382C2211yescharacteristic
15C47×#381C412yes—
16C4×C4⋊C4×C47—21no—
17C4×C4⋊C4×C47C2211yescharacteristic
18C4×C22⋊C4×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×C47C2×C412yes—
28C4×D4×C47—22no—
29C4×D4×C47—22no—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47D411yescharacteristic
33C2×C22⋊C4×C47—22no—
34C2×C22⋊C4×C47—22no—
35C2×C22⋊C4×C47D412yes—
36C2×C42×C47—21no—
37C2×C42×C47—22no—
38C2×C42×C47C2311yescharacteristic
39#95×C47—22no—
40#95×C47—21no—
41#95×C47C2×C411yescharacteristic
42#95×C47C2×C412yes—
43#95×C47C2×C411yescharacteristic
44C24×C47C22⋊C412yes—
45C2×D4×C47—22no—
46C2×D4×C47—22no—
47C2×D4×C47—21no—
48C2×D4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—82no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C4⋊C4×C47—22no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C22⋊C4×C47—24no—
64C22⋊C4×C47—44no—
65C22⋊C4×C47—82no—
66C42×C47—42no—
67C42×C47—82no—
68C42×C47#4112yes—
69C42×C47#4112yes—
70#10625329—42no—
71#10625329—42no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—42no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47C2×C22⋊C411yescharacteristic
80D4×C47—42no—
81D4×C47—42no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—41no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—82no—
96C2×C4×C47—82no—
97C2×C4×C47#11711yescharacteristic
98C2×C4×C47#11711yescharacteristic
99C2×C4×C47#12312yes—
100C2×C4×C47#12312yes—
101C2×C4×C47#13512yes—
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C47#13711yescharacteristic
104C47⋊C8—82no—
105C47⋊C8—82no—
106#8771—471nomaximal, Sylow
107C22×C47—21no—
108C22×C47—21no—
109C22×C47—42no—
110C22×C47—42no—
111C22×C47—42no—
112C22×C47—82no—
113C22×C47—82no—
114C22×C47#38111yescharacteristic
115C22×C47#41611yescharacteristic
116C22×C47#41711yescharacteristic
117C4×C47—22no—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—42no—
121C4×C47—42no—
122C4×C47—81no—
123C4×C47—82no—
124C4×C47#48212yes—
125C4×C47#48512yes—
126#2357D4711yescharacteristic
127#1800—471no—
128#1460—471no—
129#1303—472no—
130#1286—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—82no—
134C2×C47#182511yescharacteristic
135C2×C47#182711yescharacteristic
136C2×C47#182911yescharacteristic
137C2×C4×D4C2×D4711yescharacteristic
138#431—472no—
139#423—472no—
140C2×#100—941no—
141#390—21no—
142#388—22no—
143#382C2×D4711yescharacteristic
144#381Dic4712yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×D4711yescharacteristic
147C4×C22⋊C4Dic4712yes—
148#343—941no—
149#323—942no—
150C47#877111yescharacteristic, complemented, Sylow
151C22×D4C22⋊D4711yescharacteristic
152C23×C4C2×Dic4712yes—
153C2×M16—942no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C22⋊D4711yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊D4712yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C22×D4711yescharacteristic
165C4⋊C8—942no—
166C4⋊C8—942no—
167C4⋊C8—942no—
168#100—1882no—
169C22⋊C8—942no—
170C4×C8—942no—
171#95—22no—
172#95—21no—
173#95C2×Dic4711yescharacteristic
174#95C2×Dic4712yes—
175#95C2×Dic4711yescharacteristic
176C24C22⋊Dic4712yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊Dic4712yes—
188C22×C4C22×Dic4711yescharacteristic
189C22×C4C2×C22⋊D4711yescharacteristic
190C22×C4C2×C22⋊D4711yescharacteristic
191M16—1882no—
192M16—1882no—
193C2×C8—1882no—
194C2×C8—1882no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—24no—
200C22⋊C4—44no—
201C22⋊C4—82no—
202C42—42no—
203C42—82no—
204C42#1062535012yes—
205C42#1062535012yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊Dic4711yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#1309398311yescharacteristic
230C2×C4#1309398311yescharacteristic
231C2×C4#1309400712yes—
232C2×C4#1309400712yes—
233C2×C4#1309404412yes—
234C2×C4#1309404911yescharacteristic
235C2×C4#1309405411yescharacteristic
236C8—3762no—
237C8—3762no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1568898611yescharacteristic
246C22#1568911911yescharacteristic
247C22#1568912411yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4#1568944712yes—
256C4#1568946412yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2024348311yescharacteristic
261C2#2024349211yescharacteristic
262C2#2024349911yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2357C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2357D47
hypocenterC47#8771
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #20241291C2semidirect
4. #20241246C2semidirect
150. C47#8771semidirect

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×#23572
#877147