Finite Group Databasebeta

← #26130565

Subgroups — #26130565

Order 12032.

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

normal subgroups only

This group has 228 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#20245645C211yesmaximal, characteristic
2#20245293C211yesmaximal, characteristic, complemented
3#20242755C211yesmaximal, characteristic, complemented
4#20242749C211yesmaximal, characteristic
5#20241459C211yesmaximal, characteristic, complemented
6#20241435C211yesmaximal, characteristic
7C47×#1362C211yesmaximal, characteristic, complemented
8C2×#13093967—21no—
9C2×#13093967—21no—
10#15689500C2211yescharacteristic
11#15689405—21no—
12#15689324—21no—
13C8×Dic94C2211yescharacteristic
14Dic47⋊C16—21no—
15#15688898—21no—
16C4×C47⋊C16C2211yescharacteristic
17#15688876C2211yescharacteristic
18#15688869—21no—
19#15688866—21no—
20C47×#494C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×#10625325—41no—
24C2×D376—21no—
25C2×D376—21no—
26#13094021—21no—
27C4×Dic94D411yescharacteristic
28#13093967—42no—
29#13093967—42no—
30C2×C47⋊C16D411yescharacteristic
31C2×C47⋊C16D411yescharacteristic
32#13093955—41no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)D411yescharacteristic
35Dic47⋊C8—21no—
36#13093947—41no—
37#13093946—41no—
38C8×Dic47—21no—
39C2×Q16×C47—21no—
40C2×Q16×C47—21no—
41C4⋊Q8×C47—21no—
42C2×C16×C47—21no—
43C4⋊C8×C47—21no—
44C4×C8×C47C2311yescharacteristic
45C2×Q8×C47—41no—
46C2×D188—41no—
47C2×D188—41no—
48C2×D188—41no—
49C2×Dic94D811yescharacteristic
50Q16×C47—41no—
51Q16×C47—41no—
52Q16×C47—81no—
53C2×C8×C47—41no—
54C2×C8×C47C2×D411yescharacteristic
55C2×C8×C47#4111yescharacteristic
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C42×C47C2×D411yescharacteristic
59C4⋊Dic47D811yescharacteristic
60Dic47⋊C4—21no—
61C4×Dic47—21no—
62C2×C47⋊C8—21no—
63C2×C47⋊C8—41no—
64D376—41no—
65D376—41no—
66D376—81no—
67#10625325—82no—
68C16×C47—41no—
69C47⋊C16—21no—
70C47⋊C16SD1612yescomplemented
71C22×D47—81no—
72Q8×C47—81no—
73Q8×C47—81no—
74C2×C4×C47—81no—
75C2×C4×C47#12111yescharacteristic
76C2×C4×C47C2×D811yescharacteristic
77C2×C4×C47#13511yescharacteristic
78C2×Dic47—21no—
79C2×Dic47—21no—
80D188—81no—
81D188—81no—
82D188—81no—
83D188—81no—
84D188—161no—
85Dic94—21no—
86Dic94—21no—
87C8×C47—21no—
88C8×C47—81no—
89C8×C47C2×SD1611yescharacteristic
90C8×C47AGL(1,ℤ/8)11yescharacteristic
91C47⋊C8—41no—
92C47⋊C8—81no—
93#12444—471nomaximal, Sylow
94C22×C47#46611yescharacteristic
95C2×D47—161no—
96C2×D47—161no—
97C2×D47—161no—
98C4×C47—21no—
99C4×C47—161no—
100C4×C47#46011yescharacteristic
101C4×C47#47211yescharacteristic
102C4×C47#50911yescharacteristic
103C4×C47#51011yescharacteristic
104Dic47—41no—
105Dic47—41no—
106#2248—471no—
107#2183—471no—
108#1670—471no—
109#1668—471no—
110#1371—471no—
111#1362D4711yescharacteristic, complemented
112#1361—471no—
113C2×C47#166811yescharacteristic
114C2×C47#222011yescharacteristic
115C2×C47#222311yescharacteristic
116D47—321no—
117C2×SD32—941no—
118C2×SD32—941no—
119#494C2×D4711yescharacteristic
120#493—471no—
121#475—941no—
122#460—941no—
123C8×Q8—471no—
124C4⋊C16—941no—
125C4⋊C16C2×D4711yescharacteristic
126#346—941no—
127C4×C16—471no—
128#335—471no—
129#335C2×D4711yescharacteristic
130#332—941no—
131#331—941no—
132C47#1244411yescharacteristic, complemented, Sylow
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—1881no—
136C2×D8—941no—
137C2×D8—941no—
138C4⋊Q8—21no—
139#127—941no—
140C4×Q8—471no—
141SD32—1882no—
142SD32—1882no—
143C2×C16—21no—
144C2×C16—471no—
145C2×C16—471no—
146#106—1881no—
147C4⋊C8—21no—
148C4⋊C8—471no—
149C4⋊C8—941no—
150C4⋊C8—941no—
151#103—1881no—
152#102—1881no—
153C4×C8—941no—
154C4×C8C22×D4711yescharacteristic
155C2×Q8—41no—
156C2×Q8—471no—
157C2×D4—1881no—
158C2×D4—1881no—
159C2×D4—1881no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—3762no—
164D8—3761no—
165D8—1881no—
166D8—1881no—
167C2×C8—41no—
168C2×C8—941no—
169C2×C8—1881no—
170C2×C8#1062535011yescharacteristic
171C2×C8D4×D4711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—471no—
175C4⋊C4—941no—
176C42—941no—
177C42D4×D4711yescharacteristic
178C16—41no—
179C16—472no—
180C16—941no—
181C23—3761no—
182Q8—81no—
183Q8—81no—
184Q8—941no—
185Q8—941no—
186D4—3761no—
187D4—3761no—
188D4—3761no—
189D4—3761no—
190D4—7521no—
191C2×C4—81no—
192C2×C4—941no—
193C2×C4—941no—
194C2×C4#1309399711yescharacteristic
195C2×C4D8×D4711yescharacteristic
196C2×C4#1309404511yescharacteristic
197C8—21no—
198C8—81no—
199C8—3761no—
200C8—1881no—
201C8SD16×D4711yescharacteristic
202C8#1309404811yescharacteristic
203C22—7521no—
204C22—7521no—
205C22—7521no—
206C22#1568935811yescharacteristic
207C4—21no—
208C4—161no—
209C4—1881no—
210C4—1881no—
211C4#1568932211yescharacteristic
212C4#1568939211yescharacteristic
213C4#1568956011yescharacteristic
214C4#1568956511yescharacteristic
215C2—15041no—
216C2#2024274711yescharacteristic
217C2#2024550411yescharacteristic
218C2#2024552111yescharacteristic
subgroupG/H
centerC22#15689358
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1362C2
socleC22×C47#466
prime radical#15689500C22
hypercenter#1362D47
hypocenterC47#12444
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
2. #20245293C2semidirect
3. #20242755C2semidirect
5. #20241459C2semidirect
7. C47×#1362C2semidirect
70. C47⋊C16SD16semidirect
111. #1362D47semidirect
132. C47#12444semidirect

Maximal subgroups

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

subgroupindex
#202456452
#202452932
#202427552
#202427492
#202414592
#202414352
C47×#13622
#1244447