Finite Group Databasebeta

← #26131042

Subgroups — #26131042

Order 12032.

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

normal subgroups only

This group has 306 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#20245645C211yesmaximal, characteristic, complemented
2#20245637C211yesmaximal, characteristic
3#20245615C211yesmaximal, characteristic, complemented
4#20245611C211yesmaximal, characteristic, complemented
5#20242980C211yesmaximal, characteristic, complemented
6#20241306C211yesmaximal, characteristic, complemented
7C47×#1709C211yesmaximal, characteristic, complemented
8C2×#13093967—22no—
9C2×#13093965—22no—
10#15689528C2211yescharacteristic
11#15689500C2211yescharacteristic
12#15689496—21no—
13#15689470—21no—
14#15688927—22no—
15#15688924—22no—
16C4×C47⋊C16C2211yescharacteristic
17C4×C47⋊C16C2211yescharacteristic
18C47×#494C2211yescharacteristic
19C47×#493C2211yescharacteristic
20C47×#486—21no—
21C47×#486—21no—
22C82×C47C2211yescharacteristic
23C2×#10625325—42no—
24C2×D376—21no—
25C2×D376—21no—
26#13094024—21no—
27#13094021—21no—
28#13093967—44no—
29#13093965—44no—
30C2×C47⋊C16D412yes—
31C2×C47⋊C16D412yes—
32#13093958—21no—
33#13093958—21no—
34#13093946—42no—
35C2×Q16×C47—21no—
36C2×Q16×C47—21no—
37C2×D8×C47—21no—
38C2×D8×C47—21no—
39C4⋊Q8×C47—21no—
40#127×C47—21no—
41#102×C47—42no—
42#102×C47—42no—
43C4×C8×C47D411yescharacteristic
44C4×C8×C47D411yescharacteristic
45C4×C8×C47C2311yescharacteristic
46C2×Q8×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47—41no—
49C2×D4×C47—41no—
50C2×D188—41no—
51C2×D188—41no—
52C2×D188—41no—
53C2×Dic94—41no—
54Q16×C47—42no—
55Q16×C47—42no—
56D8×C47—42no—
57D8×C47—42no—
58C2×C8×C47D812yes—
59C2×C8×C47D812yes—
60C2×C8×C47C2×D411yescharacteristic
61C2×C8×C47C2×D411yescharacteristic
62C4⋊C4×C47—41no—
63C4⋊C4×C47—41no—
64C42×C47C2×D411yescharacteristic
65C4⋊Dic47—41no—
66C4⋊Dic47—41no—
67D376—42no—
68D376—42no—
69#10625325—84no—
70C47⋊C16—22no—
71C47⋊C16—22no—
72C23×C47—81no—
73C22×D47—81no—
74Q8×C47—81no—
75Q8×C47—81no—
76D4×C47—81no—
77D4×C47—81no—
78D4×C47—82no—
79D4×C47—82no—
80C2×C4×C47—81no—
81C2×C4×C47#12711yescharacteristic
82C2×C4×C47C2×D811yescharacteristic
83C2×C4×C47C2×D811yescharacteristic
84C2×Dic47—81no—
85D188—81no—
86D188—81no—
87D188—82no—
88D188—82no—
89Dic94—82no—
90C8×C47—22no—
91C8×C47—22no—
92C8×C47AGL(1,ℤ/8)12yes—
93C8×C47AGL(1,ℤ/8)12yes—
94#12527—471nomaximal, Sylow
95C22×C47—161no—
96C22×C47—161no—
97C22×C47—161no—
98C22×C47#49311yescharacteristic
99C2×D47—161no—
100C2×D47—161no—
101C2×D47—161no—
102C4×C47—161no—
103C4×C47#49611yescharacteristic
104C4×C47#49611yescharacteristic
105C4×C47#50812yes—
106C4×C47#50912yes—
107Dic47—161no—
108#2248—471no—
109#2246—471no—
110#2241—471no—
111#2240—471no—
112#1709D4711yescharacteristic, complemented
113#1706—471no—
114#1312—471no—
115C2×C47—321no—
116C2×C47#172111yescharacteristic
117C2×C47#224911yescharacteristic
118C2×C47#225011yescharacteristic
119D47—321no—
120C2×SD32—942no—
121C2×D16—942no—
122#500—471no—
123#494C2×D4711yescharacteristic
124#493—471no—
125#493C2×D4711yescharacteristic
126#492—941no—
127#486—21no—
128#486—21no—
129#486—941no—
130#358—942no—
131#357—942no—
132C4×C16—471no—
133C4×C16—471no—
134C82C2×D4711yescharacteristic
135C47#1252711yescharacteristic, complemented, Sylow
136C2×Q16—21no—
137C2×Q16—21no—
138C2×SD16—1882no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—941no—
142C2×D8—941no—
143C4⋊Q8—21no—
144C4⋊Q8—941no—
145#127—21no—
146#127—941no—
147SD32—1884no—
148D16—1884no—
149C2×C16—472no—
150C2×C16—472no—
151#107—941no—
152#107—941no—
153#102—42no—
154#102—42no—
155#102—1882no—
156C4×C8C22⋊D4711yescharacteristic
157C4×C8C22⋊D4711yescharacteristic
158C4×C8C22×D4711yescharacteristic
159C2×Q8—41no—
160C2×Q8—1881no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—1881no—
165C2×D4—1881no—
166C2×D4—1881no—
167Q16—42no—
168Q16—42no—
169SD16—3764no—
170D8—42no—
171D8—42no—
172D8—1882no—
173D8—1882no—
174C2×C8#1062533412yes—
175C2×C8#1062533412yes—
176C2×C8D4×D4711yescharacteristic
177C2×C8D4×D4711yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—1881no—
181C4⋊C4—1881no—
182C42C2×C22⋊D4711yescharacteristic
183C16—942no—
184C16—942no—
185C23—81no—
186C23—3761no—
187Q8—81no—
188Q8—81no—
189Q8—3762no—
190D4—81no—
191D4—81no—
192D4—82no—
193D4—82no—
194D4—3761no—
195D4—3761no—
196D4—3762no—
197D4—3762no—
198C2×C4—81no—
199C2×C4—3761no—
200C2×C4#1309402011yescharacteristic
201C2×C4C2×#1062533411yescharacteristic
202C2×C4C2×#1062533411yescharacteristic
203C8—22no—
204C8—22no—
205C8#1309404812yes—
206C8#1309404812yes—
207C22—161no—
208C22—161no—
209C22—161no—
210C22—7521no—
211C22—7521no—
212C22—7521no—
213C22#1568949911yescharacteristic
214C4—161no—
215C4—7521no—
216C4#1568951111yescharacteristic
217C4#1568951111yescharacteristic
218C4#1568955312yes—
219C4#1568955812yes—
220C2—321no—
221C2—15041no—
222C2#2024305511yescharacteristic
223C2#2024564711yescharacteristic
224C2#2024565511yescharacteristic
subgroupG/H
centerC22#15689499
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1709C2
socleC22×C47#493
prime radical#20245637C2
hypercenter#1709D47
hypocenterC47#12527
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1709

Complements

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

NH ≅ G/Ntype
1. #20245645C2semidirect
3. #20245615C2semidirect
4. #20245611C2semidirect
5. #20242980C2semidirect
6. #20241306C2semidirect
7. C47×#1709C2semidirect
112. #1709D47semidirect
135. C47#12527semidirect

Maximal subgroups

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

subgroupindex
#202456452
#202456372
#202456152
#202456112
#202429802
#202413062
C47×#17092
#1252747