Finite Group Databasebeta

← #26131043

Subgroups — #26131043

Order 12032.

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

normal subgroups only

This group has 306 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#20245645C211yesmaximal, characteristic, complemented
2#20245638C211yesmaximal, characteristic
3#20245613C211yesmaximal, characteristic, complemented
4#20242995C211yesmaximal, characteristic, complemented
5#20242981C211yesmaximal, characteristic, complemented
6#20241307C211yesmaximal, characteristic, complemented
7C47×#2241C211yesmaximal, characteristic, complemented
8C2×#13093967—22no—
9C2×D752—22no—
10#15689500C2211yescharacteristic
11#15689500C2211yescharacteristic
12#15689497—21no—
13#15689471—21no—
14#15689471—21no—
15#15689471—21no—
16#15688923—22no—
17C4×C47⋊C16C2211yescharacteristic
18C8×C47⋊C8C2211yescharacteristic
19C47×#500C2211yescharacteristic
20C47×#494C2211yescharacteristic
21C47×#358—22no—
22C4×C16×C47C2211yescharacteristic
23C2×#10625336—42no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376—21no—
28#13094021—21no—
29#13094021—21no—
30#13093967—44no—
31D752—44no—
32C2×C47⋊C16D412yes—
33#13093944—42no—
34#13093944—42no—
35#13093944—42no—
36C4×C47⋊C8D411yescharacteristic
37C4×C47⋊C8D411yescharacteristic
38C2×Q16×C47—21no—
39C2×Q16×C47—21no—
40C4⋊Q8×C47—21no—
41C4⋊Q8×C47—21no—
42C2×C16×C47D412yes—
43#107×C47—21no—
44#107×C47—21no—
45C4×C8×C47C2311yescharacteristic
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×D188—41no—
49C2×D188—41no—
50C2×D188—41no—
51C2×D188—41no—
52C2×D188—41no—
53C2×D188—41no—
54Q16×C47—42no—
55Q16×C47—42no—
56C2×C8×C47C2×D411yescharacteristic
57C2×C8×C47C2×D411yescharacteristic
58C4⋊C4×C47—41no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—41no—
62C42×C47C2×D411yescharacteristic
63#10625336—84no—
64C2×C47⋊C8D812yes—
65C2×C47⋊C8D812yes—
66D376—42no—
67D376—42no—
68D376—42no—
69D376—42no—
70C16×C47—22no—
71C47⋊C16—22no—
72C22×D47—81no—
73C22×D47—81no—
74Q8×C47—81no—
75Q8×C47—81no—
76Q8×C47—82no—
77C2×C4×C47—81no—
78C2×C4×C47—81no—
79C2×C4×C47#12711yescharacteristic
80C2×C4×C47C2×D811yescharacteristic
81C2×C4×C47C2×D811yescharacteristic
82D188—81no—
83D188—81no—
84D188—81no—
85D188—81no—
86D188—82no—
87D188—82no—
88D188—82no—
89D188—82no—
90C8×C47AGL(1,ℤ/8)12yes—
91C8×C47AGL(1,ℤ/8)12yes—
92C47⋊C8—22no—
93C47⋊C8—22no—
94#12527—471nomaximal, Sylow
95C22×C47#49311yescharacteristic
96C2×D47—161no—
97C2×D47—161no—
98C2×D47—161no—
99C2×D47—161no—
100C2×D47—161no—
101C2×D47—161no—
102C4×C47—161no—
103C4×C47—161no—
104C4×C47#49611yescharacteristic
105C4×C47#49611yescharacteristic
106C4×C47#50812yes—
107C4×C47#50912yes—
108#2248—471no—
109#2246—471no—
110#2241D4711yescharacteristic, complemented
111#2240—471no—
112#1709—471no—
113#1706—471no—
114#1312—471no—
115C2×C47#172111yescharacteristic
116C2×C47#224911yescharacteristic
117C2×C47#225011yescharacteristic
118D47—321no—
119D47—321no—
120C2×SD32—942no—
121C2×D16—942no—
122#500C2×D4711yescharacteristic
123#494C2×D4711yescharacteristic
124#493—471no—
125#493—471no—
126#492—941no—
127#486—941no—
128#486—941no—
129#486—941no—
130#358—22no—
131#357—942no—
132C4×C16—471no—
133C4×C16C2×D4711yescharacteristic
134C82—471no—
135C47#1252711yescharacteristic, complemented, Sylow
136C2×Q16—21no—
137C2×Q16—21no—
138C2×SD16—1882no—
139C2×D8—941no—
140C2×D8—941no—
141C2×D8—941no—
142C2×D8—941no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145#127—941no—
146#127—941no—
147SD32—1884no—
148D16—1884no—
149C2×C16—472no—
150C2×C16C22⋊D4712yes—
151#107—21no—
152#107—21no—
153#102—1882no—
154#102—1882no—
155#102—1882no—
156C4×C8—471no—
157C4×C8—471no—
158C4×C8C22×D4711yescharacteristic
159C2×Q8—41no—
160C2×Q8—41no—
161C2×D4—1881no—
162C2×D4—1881no—
163C2×D4—1881no—
164C2×D4—1881no—
165C2×D4—1881no—
166C2×D4—1881no—
167Q16—42no—
168Q16—42no—
169SD16—3764no—
170D8—1882no—
171D8—1882no—
172D8—1882no—
173D8—1882no—
174C2×C8—472no—
175C2×C8—472no—
176C2×C8D4×D4711yescharacteristic
177C2×C8C2×C22⋊D4711yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C42D4×D4711yescharacteristic
183C16—22no—
184C16—942no—
185C23—3761no—
186C23—3761no—
187Q8—81no—
188Q8—81no—
189Q8—82no—
190D4—3761no—
191D4—3761no—
192D4—3761no—
193D4—3761no—
194D4—3762no—
195D4—3762no—
196D4—3762no—
197D4—3762no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4#1309402011yescharacteristic
201C2×C4D8×D4711yescharacteristic
202C2×C4D8×D4711yescharacteristic
203C8—942no—
204C8—942no—
205C8#1309404912yes—
206C8#1309405112yes—
207C22—7521no—
208C22—7521no—
209C22—7521no—
210C22—7521no—
211C22—7521no—
212C22—7521no—
213C22#1568949811yescharacteristic
214C4—161no—
215C4—161no—
216C4#1568950911yescharacteristic
217C4#1568951311yescharacteristic
218C4#1568955612yes—
219C4#1568956012yes—
220C2—15041no—
221C2—15041no—
222C2#2024305611yescharacteristic
223C2#2024565011yescharacteristic
224C2#2024565311yescharacteristic
subgroupG/H
centerC22#15689498
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#2241C2
socleC22×C47#493
prime radical#20245638C2
hypercenter#2241D47
hypocenterC47#12527
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
1. #20245645C2semidirect
3. #20245613C2semidirect
4. #20242995C2semidirect
5. #20242981C2semidirect
6. #20241307C2semidirect
7. C47×#2241C2semidirect
110. #2241D47semidirect
135. C47#12527semidirect

Maximal subgroups

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

subgroupindex
#202456452
#202456382
#202456132
#202429952
#202429812
#202413072
C47×#22412
#1252747