Finite Group Databasebeta

← #26127125

Subgroups — #26127125

Order 12032.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#20245200C211yesmaximal, characteristic, complemented
2#20243537C211yesmaximal, characteristic, complemented
3#20241379C212yesmaximal, complemented
4#20241373C212yesmaximal, complemented
5C47×#3184C211yesmaximal, characteristic
6#15688944—21no—
7#15688905C2212yes—
8#15688901—21no—
9#15688894—21no—
10#15688883—21no—
11C47⋊#338C2211yescharacteristic
12#15688880C2211yescharacteristic
13C47×#532—21no—
14C47×#498C411yescharacteristic, complemented
15C47×#493C411yescharacteristic, complemented
16C47×#486—21no—
17C47×#469—21no—
18C47×#469C2212yes—
19C47×#442—21no—
20C47×#442C412yescomplemented
21C4×M16×C47C2211yescharacteristic
22C2×#10625329—21no—
23#13093983—21no—
24#13093964—22no—
25C2×#41×C47—41no—
26C2×D8×C47—21no—
27C2×D8×C47C2×C412yes—
28C2×M16×C47D411yescharacteristic
29C2×M16×C47C2311yescharacteristic
30C4⋊Q8×C47—21no—
31#127×C47—21no—
32#124×C47—41no—
33#121×C47—22no—
34#121×C47—41no—
35C4×D4×C47—22no—
36C4×D4×C47—41no—
37C2×C42×C47D411yescharacteristic
38#106×C47—21no—
39#106×C47C2×C412yes—
40#102×C47—22no—
41#102×C47—22no—
42#97×C47C2×C411yescharacteristic
43#97×C47D411yescharacteristic
44C4×C8×C47C2×C411yescharacteristic
45C4×C8×C47D411yescharacteristic
46#41×C47—84no—
47C2×Q8×C47—41no—
48C2×D4×C47—22no—
49C2×D4×C47—41no—
50C2×D4×C47—41no—
51C2×D4×C47—42no—
52C22×C4×C47—21no—
53C22×C4×C47—42no—
54C22×C4×C47C2×D411yescharacteristic
55D8×C47—42no—
56D8×C47—42no—
57M16×C47—41no—
58M16×C47—41no—
59C2×C8×C47C22⋊C411yescharacteristic
60C2×C8×C47C22⋊C411yescharacteristic
61C2×C8×C47C22×C411yescharacteristic
62C2×C8×C47C2×D411yescharacteristic
63C4⋊C4×C47—22no—
64C4⋊C4×C47—41no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C42×C47—21no—
68C42×C47C22⋊C411yescharacteristic
69C42×C47C22⋊C411yescharacteristic
70C22⋊Dic47—41no—
71C4⋊Dic47—41no—
72C4×Dic47—41no—
73#10625329—41no—
74#10625329—41no—
75C2×C47⋊C8—41no—
76C47⋊C16—42no—
77C23×C47—42no—
78C23×C47AGL(1,ℤ/8)11yescharacteristic
79Q8×C47—82no—
80D4×C47—42no—
81D4×C47—42no—
82D4×C47—82no—
83D4×C47—82no—
84D4×C47—84no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—84no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
94C2×Dic47—81no—
95C8×C47—21no—
96C8×C47—21no—
97C8×C47—21no—
98C8×C47—21no—
99C47⋊C8—81no—
100#11837—471nomaximal, Sylow
101C22×C47—21no—
102C22×C47—41no—
103C22×C47—82no—
104C22×C47—82no—
105C22×C47—82no—
106C22×C47#41811yescharacteristic
107C4×C47—22no—
108C4×C47—21no—
109C4×C47—41no—
110C4×C47—82no—
111C4×C47#41011yescharacteristic
112C4×C47#42111yescharacteristic
113Dic47—161no—
114#3184D4711yescharacteristic
115#2165—471no—
116#1841—471no—
117#1341—472no—
118#1339—472no—
119C2×C47—21no—
120C2×C47—41no—
121C2×C47—162no—
122C2×C47#151411yescharacteristic
123#532—21no—
124#498Dic4711yescharacteristic, complemented
125#493Dic4711yescharacteristic, complemented
126#486—21no—
127#469—21no—
128#469C2×D4712yes—
129#442—21no—
130#442Dic4712yescomplemented
131C4×M16C2×D4711yescharacteristic
132#364—941no—
133#349—472no—
134#347—941no—
135#344—941no—
136#339—941no—
137#338—471no—
138H(3,ℤ/4)—471no—
139C47#1183711yescharacteristic, complemented, Sylow
140C2×#41—41no—
141C2×D8—21no—
142C2×D8C2×Dic4712yes—
143C2×M16—941no—
144C2×M16C22⋊D4711yescharacteristic
145C2×M16C22×D4711yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#124—41no—
149#121—22no—
150#121—41no—
151C4×D4—22no—
152C4×D4—41no—
153#117—941no—
154C2×C42C22⋊D4711yescharacteristic
155M32—942no—
156#106—21no—
157#106C2×Dic4712yes—
158#102—22no—
159#102—22no—
160#97C2×Dic4711yescharacteristic
161#97C22⋊D4711yescharacteristic
162C4×C8C2×Dic4711yescharacteristic
163C4×C8C22⋊D4711yescharacteristic
164#41—84no—
165C2×Q8—41no—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4C2×C22⋊D4711yescharacteristic
173D8—42no—
174D8—42no—
175M16—41no—
176M16—41no—
177M16—1881no—
178M16—1881no—
179C2×C8—1881no—
180C2×C8C22⋊Dic4711yescharacteristic
181C2×C8C22⋊Dic4711yescharacteristic
182C2×C8C22×Dic4711yescharacteristic
183C2×C8C2×C22⋊D4711yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—1881no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—1881no—
190C42—21no—
191C42—1881no—
192C42C22⋊Dic4711yescharacteristic
193C42C22⋊Dic4711yescharacteristic
194C16—1882no—
195C23—42no—
196C23#1309405011yescharacteristic
197Q8—82no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—84no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4—3761no—
211C2×C4C2×C22⋊Dic4711yescharacteristic
212C2×C4#1309405011yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—3761no—
218C22—21no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22#1568912811yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—82no—
228C4—7521no—
229C4#1568909711yescharacteristic
230C4#1568914011yescharacteristic
231C2—21no—
232C2—41no—
233C2—162no—
234C2#2024193511yescharacteristic
subgroupG/H
centerC2#20241935
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
Fitting, prime radicalC47×#3184C2
socleC2×C47#1514
cosocleC2×M16×C47C23
hypercenter#3184D47
hypocenterC47#11837
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #20245200C2semidirect
2. #20243537C2semidirect
3. #20241379C2semidirect
4. #20241373C2semidirect
14. C47×#498C4semidirect
15. C47×#493C4semidirect
20. C47×#442C4semidirect
124. #498Dic47semidirect
125. #493Dic47semidirect
130. #442Dic47semidirect
139. C47#11837semidirect

Maximal subgroups

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

subgroupindex
#202452002
#202435372
#202413792
#202413732
C47×#31842
#1183747