Finite Group Databasebeta

← #26127129

Subgroups — #26127129

Order 12032.

280 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 71 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#20254133C211yesmaximal, characteristic
2#20245200C211yesmaximal, characteristic
3#20241381C212yesmaximal
4#20241375C212yesmaximal
5C47×#1841C211yesmaximal, characteristic
6#15689680—21no—
7#15689521C411yescharacteristic, complemented
8#15689503C411yescharacteristic, complemented
9#15689474—21no—
10#15689381—21no—
11#15689381C2212yes—
12#15689231—21no—
13#15689231C412yescomplemented
14#15688944—21no—
15#15688905C2212yes—
16#15688901—21no—
17C4×M16×C47C2211yescharacteristic
18C47×#344—21no—
19C47×#339—21no—
20C47×#338C2211yescharacteristic
21C47×H(3,ℤ/4)C2211yescharacteristic
22C2×Dic188—21no—
23C2×Dic188C2×C412yes—
24#13094024—21no—
25#13094024—21no—
26#13094016—41no—
27#13094003—22no—
28#13094003—41no—
29C4×Dic94—22no—
30C4×Dic94—41no—
31#13093983—41no—
32#13093964—22no—
33#13093956—21no—
34#13093956C2×C412yes—
35#13093949—22no—
36#13093949—22no—
37C2×M16×C47—21no—
38C2×M16×C47D411yescharacteristic
39C2×M16×C47C2311yescharacteristic
40#117×C47—21no—
41C2×C42×C47D411yescharacteristic
42#97×C47C2×C411yescharacteristic
43#97×C47D411yescharacteristic
44C4×C8×C47C2×C411yescharacteristic
45C4×C8×C47D411yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47C2×D411yescharacteristic
48C2×Dic94—22no—
49C2×Dic94—41no—
50M16×C47—41no—
51M16×C47—41no—
52M16×C47—41no—
53M16×C47—41no—
54C2×C8×C47—41no—
55C2×C8×C47C22⋊C411yescharacteristic
56C2×C8×C47C22⋊C411yescharacteristic
57C2×C8×C47C22×C411yescharacteristic
58C2×C8×C47C2×D411yescharacteristic
59C4⋊C4×C47—41no—
60C22⋊C4×C47—41no—
61C42×C47—21no—
62C42×C47—41no—
63C42×C47C22⋊C411yescharacteristic
64C42×C47C22⋊C411yescharacteristic
65C22⋊Dic47—42no—
66C4⋊Dic47—22no—
67C4⋊Dic47—41no—
68C4⋊Dic47—41no—
69C4⋊Dic47—41no—
70Dic47⋊C4—42no—
71Dic47⋊C4—42no—
72C4×Dic47—42no—
73Dic188—42no—
74Dic188—42no—
75C47⋊C16—42no—
76C23×C47AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—42no—
82C2×C4×C47—81no—
83C2×C4×C47C2×C22⋊C411yescharacteristic
84C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
85C2×Dic47—42no—
86C2×Dic47—42no—
87Dic94—42no—
88Dic94—42no—
89Dic94—82no—
90C8×C47—21no—
91C8×C47—21no—
92C8×C47—21no—
93C8×C47—21no—
94C8×C47—81no—
95#11838—471nomaximal, Sylow
96C22×C47—21no—
97C22×C47—41no—
98C22×C47#41811yescharacteristic
99C4×C47—22no—
100C4×C47—21no—
101C4×C47—41no—
102C4×C47—161no—
103C4×C47#41011yescharacteristic
104C4×C47#42111yescharacteristic
105Dic47—82no—
106Dic47—82no—
107#3185—471no—
108#2165—471no—
109#1841D4711yescharacteristic
110#1342—472no—
111#1340—472no—
112C2×C47—21no—
113C2×C47—41no—
114C2×C47#151411yescharacteristic
115#533—941no—
116#498—471no—
117#494—471no—
118#487—941no—
119#470—472no—
120#470—941no—
121#441—472no—
122#441—941no—
123C4×M16C2×D4711yescharacteristic
124#364—941no—
125#349—472no—
126#347—941no—
127#344—21no—
128#339—21no—
129#338C2×D4711yescharacteristic
130H(3,ℤ/4)C2×D4711yescharacteristic
131C47#1183811yescharacteristic, complemented, Sylow
132C2×Q16—472no—
133C2×Q16—941no—
134C2×M16—21no—
135C2×M16C22⋊D4711yescharacteristic
136C2×M16C22×D4711yescharacteristic
137C4⋊Q8—941no—
138C4⋊Q8—941no—
139#125—1881no—
140C22⋊Q8—942no—
141C22⋊Q8—1881no—
142C4×Q8—942no—
143C4×Q8—1881no—
144#117—21no—
145#117—1881no—
146C2×C42C22⋊D4711yescharacteristic
147M32—942no—
148#106—472no—
149#106—941no—
150#103—942no—
151#103—942no—
152#97C4×D4711yescharacteristic
153#97D18811yescharacteristic
154C4×C8C4×D4711yescharacteristic
155C4×C8D18811yescharacteristic
156C2×Q8—942no—
157C2×Q8—1881no—
158C22×C4—21no—
159C22×C4C2×C22⋊D4711yescharacteristic
160Q16—1882no—
161Q16—1882no—
162M16—41no—
163M16—41no—
164M16—41no—
165M16—41no—
166C2×C8—41no—
167C2×C8(C2×D47)⋊C411yescharacteristic
168C2×C8(C2×D47)⋊C411yescharacteristic
169C2×C8C2×C4×D4711yescharacteristic
170C2×C8C2×D18811yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—942no—
173C4⋊C4—1882no—
174C4⋊C4—1881no—
175C4⋊C4—1881no—
176C4⋊C4—1881no—
177C4⋊C4—1882no—
178C22⋊C4—41no—
179C22⋊C4—1882no—
180C42—21no—
181C42—41no—
182C42—1882no—
183C42(C2×D47)⋊C411yescharacteristic
184C42(C2×D47)⋊C411yescharacteristic
185C16—1882no—
186C23#1309404911yescharacteristic
187Q8—3762no—
188Q8—1882no—
189Q8—1882no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—1882no—
197C2×C4—1882no—
198C2×C4C2×(C2×D47)⋊C411yescharacteristic
199C2×C4#1309404911yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—81no—
205C22—21no—
206C22—41no—
207C22#1568912711yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—161no—
212C4—3762no—
213C4—3762no—
214C4#1568909611yescharacteristic
215C4#1568914111yescharacteristic
216C2—21no—
217C2—41no—
218C2#2024193411yescharacteristic
subgroupG/H
centerC2#20241934
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1841C2
socleC2×C47#1514
prime radicalC23×C47AGL(1,ℤ/8)
cosocleC2×M16×C47C23
hypercenter#1841D47
hypocenterC47#11838
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
7. #15689521C4semidirect
8. #15689503C4semidirect
13. #15689231C4semidirect
131. C47#11838semidirect

Maximal subgroups

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

subgroupindex
#202541332
#202452002
#202413812
#202413752
C47×#18412
#1183847