Finite Group Databasebeta

← #26129625

Subgroups — #26129625

Order 12032.

310 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 59 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#20245605C212yesmaximal, complemented
2#20244941C211yesmaximal, characteristic
3#20241376C212yesmaximal, complemented
4#20241326C211yesmaximal, characteristic
5C47×#3202C211yesmaximal, characteristic
6#15689172C2211yescharacteristic
7#15689149—21no—
8#15689140—21no—
9#15688945—22no—
10#15688939—22no—
11#15688905C2212yes—
12#15688880C2211yescharacteristic
13C47×#536—21no—
14C47×#485—22no—
15C47×#470C2212yes—
16C47×#467—21no—
17C47×#451—22no—
18C47×#408C2211yescharacteristic
19C2×#10625329—21no—
20#13093983—21no—
21#13093964—22no—
22#13093960—22no—
23Dic47⋊C8—42no—
24#13093951—42no—
25C2×#41×C47D411yescharacteristic
26C22×Q8×C47—21no—
27C2×Q16×C47—21no—
28C2×Q16×C47D412yes—
29C2×M16×C47C2311yescharacteristic
30C22×C8×C47D411yescharacteristic
31C4⋊Q8×C47—41no—
32#124×C47—41no—
33#123×C47—41no—
34#123×C47—41no—
35C22⋊Q8×C47—22no—
36C22⋊Q8×C47—22no—
37#117×C47—21no—
38#107×C47—21no—
39#106×C47D412yes—
40#103×C47—22no—
41#103×C47—22no—
42C22⋊C8×C47—22no—
43#41×C47—41no—
44#41×C47—41no—
45C2×Q8×C47—22no—
46C2×Q8×C47—41no—
47C2×Q8×C47—81no—
48C2×Q8×C47—81no—
49C2×Q8×C47SD1611yescharacteristic
50C2×D4×C47—21no—
51C2×D4×C47SD1611yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—41no—
54C22×C4×C47C2×D411yescharacteristic
55Q16×C47—42no—
56Q16×C47—81no—
57M16×C47—41no—
58C2×C8×C47—21no—
59C2×C8×C47—41no—
60C2×C8×C47C2×D412yes—
61C4⋊C4×C47—22no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—42no—
64C22⋊C4×C47—41no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C42×C47—41no—
68C22⋊Dic47—41no—
69C4⋊Dic47—41no—
70C4×Dic47—22no—
71#10625329—41no—
72#10625329—41no—
73C2×C47⋊C8—41no—
74C47⋊C16—42no—
75C23×C47—21no—
76C23×C47AGL(1,ℤ/8)11yescharacteristic
77Q8×C47—41no—
78Q8×C47—42no—
79Q8×C47—42no—
80Q8×C47—82no—
81D4×C47—41no—
82D4×C47—81no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—82no—
91C2×C4×C47#12011yescharacteristic
92C2×C4×C47C2×SD1611yescharacteristic
93C2×Dic47—42no—
94C8×C47—22no—
95C8×C47—41no—
96C47⋊C8—81no—
97#12302—471nomaximal, Sylow
98C22×C47—21no—
99C22×C47—41no—
100C22×C47—42no—
101C22×C47—41no—
102C22×C47#45011yescharacteristic
103C4×C47—21no—
104C4×C47—41no—
105C4×C47—82no—
106C4×C47—82no—
107C4×C47#45311yescharacteristic
108C4×C47#45811yescharacteristic
109Dic47—82no—
110#3202D4711yescharacteristic
111#2239—472no—
112#2098—471no—
113#1340—472no—
114#1321—471no—
115C2×C47—21no—
116C2×C47—41no—
117C2×C47—81no—
118C2×C47#161311yescharacteristic
119#536—21no—
120#485—22no—
121#470C2×D4712yes—
122#467—21no—
123#451—22no—
124#430—471no—
125#424—941no—
126#421—941no—
127#408C2×D4711yescharacteristic
128#365—942no—
129#362—942no—
130#349—472no—
131H(3,ℤ/4)—471no—
132C47#1230211yescharacteristic, complemented, Sylow
133C2×#41C22⋊D4711yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16C22⋊D4712yes—
137C2×M16—941no—
138C2×M16C22×D4711yescharacteristic
139C22×C8C22⋊D4711yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—941no—
148M32—942no—
149#108—942no—
150#107—21no—
151#106C22⋊D4712yes—
152C4⋊C8—1882no—
153#104—1882no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#1062533511yescharacteristic
164C2×D4—21no—
165C2×D4#1062533511yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D4711yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—1881no—
173M16—1881no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—1881no—
177C2×C8C2×C22⋊D4712yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—1881no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—1881no—
186C42—41no—
187C42—942no—
188C16—1882no—
189C23—21no—
190C23#1309405011yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—1882no—
206C2×C4#1309399311yescharacteristic
207C2×C4C2×#1062533511yescharacteristic
208C8—22no—
209C8—41no—
210C8—3761no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1568927411yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—3762no—
221C4#1568929111yescharacteristic
222C4#1568931311yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#2024240811yescharacteristic
subgroupG/H
centerC2#20242408
commutator, cosocleC2×M16×C47C23
FrattiniC2×M16C22×D47
FittingC47×#3202C2
socleC2×C47#1613
prime radicalC2×#41×C47D4
hypercenter#3202D47
hypocenterC47#12302
LCSG ▷ C2×M16×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #20245605C2semidirect
3. #20241376C2semidirect
132. C47#12302semidirect

Maximal subgroups

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

subgroupindex
#202456052
#202449412
#202413762
#202413262
C47×#32022
#1230247