Finite Group Databasebeta

← #26130719

Subgroups — #26130719

Order 12032.

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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#20245617C211yesmaximal, characteristic, complemented
2#20245207C211yesmaximal, characteristic
3#20242774C211yesmaximal, characteristic
4#20242754C211yesmaximal, characteristic
5#20241461C211yesmaximal, characteristic, complemented
6#20241438C211yesmaximal, characteristic, complemented
7C47×#1364C211yesmaximal, characteristic, complemented
8C2×C16×D47—21no—
9#15689503C2211yescharacteristic
10#15689404—21no—
11#15689325—21no—
12C8×D188C2211yescharacteristic
13#15688926—21no—
14#15688926—21no—
15(C2×D47)⋊C16—21no—
16C4×C47⋊C16C2211yescharacteristic
17#15688876C2211yescharacteristic
18#15688869—21no—
19#15688868—21no—
20C47×#500C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×Dic188—21no—
24C2×Dic188—21no—
25C2×#10625336—41no—
26C2×C8×D47—21no—
27#13094024—21no—
28C4×D188D411yescharacteristic
29C2×C47⋊C16D411yescharacteristic
30C2×C47⋊C16D411yescharacteristic
31C16×D47—42no—
32#13093958—41no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)D411yescharacteristic
35#13093948—41no—
36#13093946—41no—
37(C2×D47)⋊C8—21no—
38C4⋊Q8×C47—21no—
39C2×C16×C47—21no—
40#107×C47—21no—
41#107×C47—21no—
42C4⋊C8×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×Q8×C47—41no—
45C2×D188D811yescharacteristic
46C2×C4×D47—21no—
47C2×Dic94—41no—
48C2×C8×C47—41no—
49C2×C8×C47C2×D411yescharacteristic
50C2×C8×C47#4111yescharacteristic
51C4⋊C4×C47—41no—
52C4⋊C4×C47—41no—
53C42×C47C2×D411yescharacteristic
54#10625336—82no—
55(C2×D47)⋊C4—21no—
56C4⋊Dic47—41no—
57C4⋊Dic47—41no—
58C4⋊Dic47D811yescharacteristic
59C2×C47⋊C8—21no—
60C2×C47⋊C8—41no—
61Dic188—41no—
62Dic188—41no—
63Dic188—81no—
64C8×D47—41no—
65C8×D47—41no—
66C16×C47—41no—
67C47⋊C16—21no—
68C47⋊C16—21no—
69C22×D47—21no—
70Q8×C47—81no—
71Q8×C47—81no—
72C2×C4×C47—81no—
73C2×C4×C47#12111yescharacteristic
74C2×C4×C47C2×D811yescharacteristic
75C2×C4×C47#13511yescharacteristic
76C2×Dic47—21no—
77C2×Dic47—81no—
78D188—21no—
79D188SD3212yes—
80C4×D47—41no—
81C4×D47—41no—
82Dic94—81no—
83Dic94—81no—
84C8×C47—21no—
85C8×C47—81no—
86C8×C47#13511yescharacteristic
87C8×C47AGL(1,ℤ/8)11yescharacteristic
88C47⋊C8—41no—
89C47⋊C8—81no—
90#12470—471nomaximal, Sylow
91C22×C47#46611yescharacteristic
92C2×D47—22no—
93C2×D47—41no—
94C2×D47—41no—
95C4×C47—21no—
96C4×C47—161no—
97C4×C47#46311yescharacteristic
98C4×C47#47111yescharacteristic
99C4×C47C2×SD3211yescharacteristic
100C4×C47#50811yescharacteristic
101Dic47—41no—
102Dic47—161no—
103#2241—471no—
104D4×C16—471no—
105#1673—471no—
106#1669—471no—
107#1372—471no—
108#1364D4711yescharacteristic, complemented
109#1362—471no—
110C2×C47#168111yescharacteristic
111C2×C47#221711yescharacteristic
112C2×C47#222211yescharacteristic
113D47—42no—
114C22×C16—941no—
115#500C2×D4711yescharacteristic
116#494—471no—
117#474—941no—
118#461—941no—
119C8×D4—471no—
120C4⋊C16C2×D4711yescharacteristic
121#358—941no—
122#358—941no—
123C22⋊C16—941no—
124C4×C16—471no—
125#335—471no—
126#335C2×D4711yescharacteristic
127#332—941no—
128#332—941no—
129C47#1247011yescharacteristic, complemented, Sylow
130C2×Q16—941no—
131C2×Q16—941no—
132C2×SD16—1881no—
133C22×C8—941no—
134C4⋊Q8—21no—
135C4⋊Q8—941no—
136C4×D4—471no—
137C2×C16—21no—
138C2×C16—471no—
139C2×C16—471no—
140C2×C16—1882no—
141#107—21no—
142#107—21no—
143#107—1881no—
144C4⋊C8—21no—
145C4⋊C8—471no—
146C4⋊C8—941no—
147#103—1881no—
148#102—1881no—
149C22⋊C8—941no—
150C4×C8C22×D4711yescharacteristic
151C2×Q8—41no—
152C2×Q8—1881no—
153C2×D4—471no—
154C22×C4—941no—
155Q16—3761no—
156Q16—1881no—
157Q16—1881no—
158SD16—3762no—
159C2×C8—41no—
160C2×C8—941no—
161C2×C8—1881no—
162C2×C8—1881no—
163C2×C8—1881no—
164C2×C8#1062535011yescharacteristic
165C2×C8D4×D4711yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—471no—
169C4⋊C4—1881no—
170C4⋊C4—1881no—
171C22⋊C4—941no—
172C42D4×D4711yescharacteristic
173C16—41no—
174C16—941no—
175C16—941no—
176C23—941no—
177Q8—81no—
178Q8—81no—
179Q8—3761no—
180Q8—3761no—
181D4—472no—
182D4—941no—
183C2×C4—81no—
184C2×C4—941no—
185C2×C4—3761no—
186C2×C4—1881no—
187C2×C4—1881no—
188C2×C4#1309399711yescharacteristic
189C2×C4D8×D4711yescharacteristic
190C2×C4#1309404511yescharacteristic
191C8—21no—
192C8—81no—
193C8—3761no—
194C8—1881no—
195C8#1309404711yescharacteristic
196C8#1309405111yescharacteristic
197C22—942no—
198C22—1881no—
199C22—1881no—
200C22#1568935811yescharacteristic
201C4—21no—
202C4—161no—
203C4—1881no—
204C4—7521no—
205C4#1568934111yescharacteristic
206C4#1568938911yescharacteristic
207C4SD32×D4711yescharacteristic
208C4#1568955211yescharacteristic
209C2—1882no—
210C2#2024282811yescharacteristic
211C2#2024548311yescharacteristic
212C2#2024551211yescharacteristic
subgroupG/H
centerC22#15689358
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1364C2
socleC22×C47#466
prime radicalC2×D188D8
hypercenter#1364D47
hypocenterC47#12470
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #20245617C2semidirect
5. #20241461C2semidirect
6. #20241438C2semidirect
7. C47×#1364C2semidirect
108. #1364D47semidirect
129. C47#12470semidirect

Maximal subgroups

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

subgroupindex
#202456172
#202452072
#202427742
#202427542
#202414612
#202414382
C47×#13642
#1247047