Finite Group Databasebeta

← #26130562

Subgroups — #26130562

Order 12032.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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#20245644C211yesmaximal, characteristic
2#20242758C211yesmaximal, characteristic, complemented
3#20242748C211yesmaximal, characteristic
4#20241460C211yesmaximal, characteristic, complemented
5#20241438C211yesmaximal, characteristic, complemented
6#20241435C211yesmaximal, characteristic
7C47×#2183C211yesmaximal, characteristic, complemented
8C2×#13093969—21no—
9C2×#13093969—21no—
10#15689503C2211yescharacteristic
11#15689500C2211yescharacteristic
12#15689408—21no—
13#15689323—21no—
14#15688876C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688870—21no—
17#15688866—21no—
18C8×Q8×C47C2211yescharacteristic
19C47×C4⋊C16—21no—
20C47×C4⋊C16C2211yescharacteristic
21C47×#346—21no—
22C4×C16×C47C2211yescharacteristic
23C2×Dic188—21no—
24C2×Dic188—21no—
25C2×#10625336—41no—
26C2×D376—21no—
27C2×D376—21no—
28#13094024—21no—
29#13094021—21no—
30#13093969—42no—
31#13093969—42no—
32#13093955—41no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35#13093948—41no—
36#13093944—41no—
37C4×Q8×C47D411yescharacteristic
38C2×C16×C47—21no—
39C2×C16×C47D411yescharacteristic
40C2×C16×C47D411yescharacteristic
41C4⋊C8×C47—21no—
42C4⋊C8×C47D411yescharacteristic
43C4×C8×C47—21no—
44C4×C8×C47C2311yescharacteristic
45C2×Q8×C47D811yescharacteristic
46C2×D188—41no—
47C2×D188—41no—
48C2×D188—41no—
49C2×Dic94—41no—
50C2×C8×C47—21no—
51C2×C8×C47C2×D411yescharacteristic
52C2×C8×C47#4111yescharacteristic
53C4⋊C4×C47—21no—
54C4⋊C4×C47D811yescharacteristic
55C42×C47—21no—
56C42×C47C2×D411yescharacteristic
57#10625336—82no—
58C4⋊Dic47—41no—
59C4⋊Dic47—41no—
60C2×C47⋊C8—41no—
61C2×C47⋊C8—41no—
62Dic188—41no—
63Dic188—41no—
64Dic188—81no—
65D376—41no—
66D376—41no—
67D376—81no—
68C16×C47—21no—
69C16×C47—41no—
70C16×C47SD1612yescomplemented
71C22×D47—81no—
72Q8×C47—21no—
73Q8×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47#12111yescharacteristic
77C2×C4×C47C2×D811yescharacteristic
78C2×C4×C47#13511yescharacteristic
79C2×Dic47—81no—
80D188—81no—
81D188—81no—
82D188—81no—
83D188—81no—
84D188—161no—
85Dic94—81no—
86Dic94—81no—
87C8×C47—21no—
88C8×C47—41no—
89C8×C47C2×SD1611yescharacteristic
90C8×C47AGL(1,ℤ/8)11yescharacteristic
91C47⋊C8—81no—
92C47⋊C8—81no—
93#12444—471nomaximal, Sylow
94C22×C47#46611yescharacteristic
95C2×D47—161no—
96C2×D47—161no—
97C2×D47—161no—
98C4×C47—21no—
99C4×C47—41no—
100C4×C47—41no—
101C4×C47#46011yescharacteristic
102C4×C47#47211yescharacteristic
103C4×C47#50911yescharacteristic
104C4×C47#51011yescharacteristic
105Dic47—161no—
106#2248—471no—
107#2183D4711yescharacteristic, complemented
108#1670—471no—
109#1668—471no—
110#1371—471no—
111#1362—471no—
112#1361—471no—
113C2×C47#166811yescharacteristic
114C2×C47#222011yescharacteristic
115C2×C47#222311yescharacteristic
116D47—321no—
117C2×SD32—941no—
118C2×SD32—941no—
119#494—471no—
120#493—471no—
121#475—941no—
122#460—941no—
123C8×Q8C2×D4711yescharacteristic
124C4⋊C16—21no—
125C4⋊C16C2×D4711yescharacteristic
126#346—21no—
127C4×C16C2×D4711yescharacteristic
128#335—471no—
129#335—471no—
130#332—941no—
131#331—941no—
132C47#1244411yescharacteristic, complemented, Sylow
133C2×Q16—941no—
134C2×Q16—941no—
135C2×SD16—1881no—
136C2×D8—941no—
137C2×D8—941no—
138C4⋊Q8—941no—
139#127—941no—
140C4×Q8D18811yescharacteristic
141SD32—1882no—
142SD32—1882no—
143C2×C16—21no—
144C2×C16C22⋊D4711yescharacteristic
145C2×C16C22⋊D4711yescharacteristic
146#106—1881no—
147C4⋊C8—21no—
148C4⋊C8—941no—
149C4⋊C8—941no—
150C4⋊C8D18811yescharacteristic
151#103—1881no—
152#102—1881no—
153C4×C8—21no—
154C4×C8C22×D4711yescharacteristic
155C2×Q8—1881no—
156C2×Q8D37611yescharacteristic
157C2×D4—1881no—
158C2×D4—1881no—
159C2×D4—1881no—
160Q16—3761no—
161Q16—1881no—
162Q16—1881no—
163SD16—3762no—
164D8—3761no—
165D8—1881no—
166D8—1881no—
167C2×C8—21no—
168C2×C8—1881no—
169C2×C8—1881no—
170C2×C8#1062535011yescharacteristic
171C2×C8C2×C22⋊D4711yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—1881no—
174C4⋊C4—1881no—
175C4⋊C4D37611yescharacteristic
176C42—21no—
177C42C2×D18811yescharacteristic
178C16—21no—
179C16—41no—
180C16#1062533612yescomplemented
181C23—3761no—
182Q8—21no—
183Q8—21no—
184Q8—3761no—
185Q8—3761no—
186D4—3761no—
187D4—3761no—
188D4—3761no—
189D4—3761no—
190D4—7521no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—3761no—
194C2×C4#1309399811yescharacteristic
195C2×C4C2×D37611yescharacteristic
196C2×C4#1309404611yescharacteristic
197C8—21no—
198C8—41no—
199C8—3761no—
200C8—3761no—
201C8C2×#1062533611yescharacteristic
202C8#1309404911yescharacteristic
203C22—7521no—
204C22—7521no—
205C22—7521no—
206C22#1568935911yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—7521no—
211C4#1568932311yescharacteristic
212C4#1568939311yescharacteristic
213C4#1568956111yescharacteristic
214C4#1568956611yescharacteristic
215C2—15041no—
216C2#2024274811yescharacteristic
217C2#2024550111yescharacteristic
218C2#2024551811yescharacteristic
subgroupG/H
centerC22#15689359
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#2183C2
socleC22×C47#466
prime radical#15689500C22
hypercenter#2183D47
hypocenterC47#12444
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #2183

Complements

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

NH ≅ G/Ntype
2. #20242758C2semidirect
4. #20241460C2semidirect
5. #20241438C2semidirect
7. C47×#2183C2semidirect
70. C16×C47SD16semidirect
107. #2183D47semidirect
132. C47#12444semidirect
180. C16#10625336semidirect

Maximal subgroups

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

subgroupindex
#202456442
#202427582
#202427482
#202414602
#202414382
#202414352
C47×#21832
#1244447