Finite Group Databasebeta

← #26126827

Subgroups — #26126827

Order 12032.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#20254047C211yesmaximal, characteristic
2#20243143C211yesmaximal, characteristic
3#20241383C212yesmaximal
4#20241377C212yesmaximal
5C47×#2166C211yesmaximal, characteristic
6#15689735—21no—
7#15689525C411yescharacteristic, complemented
8#15689503C411yescharacteristic, complemented
9#15689492—21no—
10#15689381C2212yes—
11#15689364—21no—
12#15689231C412yescomplemented
13C4×Dic188—21no—
14#15689078—21no—
15C47⋊#338C2211yescharacteristic
16#15688880C2211yescharacteristic
17#15688879—21no—
18C47×#405C2211yescharacteristic
19C47×#364—21no—
20C47×#349C2212yes—
21C47×#347—21no—
22C2×Dic188—21no—
23C2×Dic188C2×C412yes—
24C2×#10625329—21no—
25#13094024—21no—
26#13094017—42no—
27#13094016—21no—
28#13094014—42no—
29#13094003—22no—
30C4×Dic94—22no—
31#13093983—21no—
32#13093958—21no—
33#13093956C2×C412yes—
34#13093949—22no—
35#13093949—22no—
36#13093934—22no—
37C8×Dic47—22no—
38C2×M16×C47C2311yescharacteristic
39C22×C8×C47D411yescharacteristic
40#117×C47D411yescharacteristic
41M32×C47—22no—
42#97×C47C2×C411yescharacteristic
43#97×C47D411yescharacteristic
44C4×C8×C47C2×C411yescharacteristic
45C4×C8×C47D411yescharacteristic
46C22×C4×C47C2×D411yescharacteristic
47C2×Dic94—22no—
48M16×C47—41no—
49C2×C8×C47—21no—
50C2×C8×C47—21no—
51C2×C8×C47C22⋊C411yescharacteristic
52C2×C8×C47C22⋊C411yescharacteristic
53C2×C8×C47C22×C411yescharacteristic
54C2×C8×C47C2×D411yescharacteristic
55C4⋊C4×C47—21no—
56C22⋊C4×C47—21no—
57C42×C47C22⋊C411yescharacteristic
58C42×C47C22⋊C411yescharacteristic
59C22⋊Dic47—41no—
60C22⋊Dic47—42no—
61C4⋊Dic47—22no—
62C4⋊Dic47—41no—
63C4⋊Dic47—42no—
64Dic47⋊C4—42no—
65Dic47⋊C4—42no—
66Dic47⋊C4—42no—
67C4×Dic47—22no—
68C4×Dic47—42no—
69#10625329—41no—
70#10625329—41no—
71C2×C47⋊C8—22no—
72Dic188—42no—
73Dic188—41no—
74Dic188—41no—
75C16×C47—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×C47C2×C22⋊C411yescharacteristic
82C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
83C2×Dic47—42no—
84C2×Dic47—42no—
85C2×Dic47—42no—
86Dic94—42no—
87Dic94—42no—
88C8×C47—21no—
89C8×C47—21no—
90C8×C47—21no—
91C8×C47#10411yescharacteristic
92C8×C47#10411yescharacteristic
93C47⋊C8—42no—
94#11771—471nomaximal, Sylow
95C22×C47—21no—
96C22×C47—41no—
97C22×C47#41811yescharacteristic
98C4×C47—21no—
99C4×C47—41no—
100C4×C47—41no—
101C4×C47#41011yescharacteristic
102C4×C47C2×#10411yescharacteristic
103Dic47—82no—
104Dic47—82no—
105Dic47—82no—
106#3177—471no—
107#2166D4711yescharacteristic
108#1747—471no—
109#1342—472no—
110#1340—472no—
111C2×C47—21no—
112C2×C47—41no—
113C2×C47#150511yescharacteristic
114#541—941no—
115#499—471no—
116#494—471no—
117#491—941no—
118#470—472no—
119#467—941no—
120#441—472no—
121C4×Q16—941no—
122#405—941no—
123#405C2×D4711yescharacteristic
124#364—21no—
125#349C2×D4712yes—
126#347—21no—
127#338—471no—
128H(3,ℤ/4)—471no—
129H(3,ℤ/4)—941no—
130C47#1177111yescharacteristic, complemented, Sylow
131C2×Q16—472no—
132C2×Q16—941no—
133C2×M16—941no—
134C2×M16C22×D4711yescharacteristic
135C22×C8D18811yescharacteristic
136C4⋊Q8—941no—
137#126—1882no—
138#125—941no—
139#125—1882no—
140C22⋊Q8—942no—
141C4×Q8—942no—
142#117—941no—
143#117D18811yescharacteristic
144M32—22no—
145#107—941no—
146#106—472no—
147#103—942no—
148#103—942no—
149#97—942no—
150#97C4×D4711yescharacteristic
151#97C22⋊D4711yescharacteristic
152C4×C8—942no—
153C4×C8C4×D4711yescharacteristic
154C4×C8C22⋊D4711yescharacteristic
155C2×Q8—942no—
156C22×C4C2×D18811yescharacteristic
157Q16—1882no—
158Q16—1881no—
159Q16—1881no—
160M16—41no—
161M16—1881no—
162M16—1881no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—942no—
166C2×C8(C2×D47)⋊C411yescharacteristic
167C2×C8(C2×D47)⋊C411yescharacteristic
168C2×C8C2×C4×D4711yescharacteristic
169C2×C8C2×C22⋊D4711yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—942no—
172C4⋊C4—1882no—
173C4⋊C4—1882no—
174C4⋊C4—1881no—
175C4⋊C4—1882no—
176C4⋊C4—1882no—
177C22⋊C4—21no—
178C22⋊C4—1881no—
179C22⋊C4—1882no—
180C42—942no—
181C42—1882no—
182C42(C2×D47)⋊C411yescharacteristic
183C42(C2×D47)⋊C411yescharacteristic
184C16—42no—
185C23#1309405211yescharacteristic
186Q8—1882no—
187Q8—1882no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—1882no—
193C2×C4—1882no—
194C2×C4—1882no—
195C2×C4C2×(C2×D47)⋊C411yescharacteristic
196C2×C4#1309405211yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—1882no—
201C8#1309395011yescharacteristic
202C8#1309395011yescharacteristic
203C22—21no—
204C22—41no—
205C22#1568912911yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—3762no—
210C4—3762no—
211C4—3762no—
212C4#1568909611yescharacteristic
213C4C2×#1309395011yescharacteristic
214C2—21no—
215C2—41no—
216C2#2024189711yescharacteristic
subgroupG/H
centerC2#20241897
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2166C2
socleC2×C47#1505
prime radicalC23×C47AGL(1,ℤ/8)
cosocleC2×M16×C47C23
hypercenter#2166D47
hypocenterC47#11771
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
7. #15689525C4semidirect
8. #15689503C4semidirect
12. #15689231C4semidirect
130. C47#11771semidirect

Maximal subgroups

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

subgroupindex
#202540472
#202431432
#202413832
#202413772
C47×#21662
#1177147