Finite Group Databasebeta

← #26129681

Subgroups — #26129681

Order 12032.

372 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 67 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#20245469C212yesmaximal, complemented
2#20242993C211yesmaximal, characteristic, complemented
3#20241425C211yesmaximal, characteristic, complemented
4#20241361C212yesmaximal, complemented
5C47×#3292C211yesmaximal, characteristic
6#15689563—22no—
7#15689503C2211yescharacteristic, complemented
8#15689469—22no—
9#15688944C2212yescomplemented
10#15688938—22no—
11C8×C47⋊C8C2211yescharacteristic, complemented
12C47×#576—22no—
13C47×#493C2211yescharacteristic
14C47×#472—22no—
15C47×#453—22no—
16C47×#443C2212yes—
17C2×Dic188—22no—
18#13094024—21no—
19#13093970—42no—
20#13093968—42no—
21#13093964—22no—
22#13093945—44no—
23C4×C47⋊C8—21no—
24H(5,𝔽2)×C47—22no—
25AGL(1,ℤ/8)×C47—24no—
26AGL(1,ℤ/8)×C47—44no—
27#135×C47—42no—
28#135×C47D412yes—
29C2×D8×C47—22no—
30C2×D8×C47—42no—
31#131×C47—22no—
32#127×C47D412yes—
33#108×C47D412yes—
34#104×C47—22no—
35#100×C47—22no—
36C4×C8×C47C2311yescharacteristic
37#41×C47—22no—
38#41×C47—44no—
39C2×D4×C47—22no—
40C2×D4×C47—42no—
41C2×D4×C47—44no—
42C2×D4×C47—82no—
43C2×Dic94—41no—
44Q16×C47D812yes—
45SD16×C47—22no—
46SD16×C47—44no—
47D8×C47—44no—
48D8×C47—42no—
49D8×C47—42no—
50D8×C47D812yes—
51M16×C47—22no—
52M16×C47—42no—
53C2×C8×C47—42no—
54C2×C8×C47C2×D412yes—
55C42×C47C2×D411yescharacteristic
56C4⋊Dic47—42no—
57C2×C47⋊C8—22no—
58Dic188—42no—
59Dic188—42no—
60C47⋊C16—42no—
61C23×C47—42no—
62C23×C47—82no—
63Q8×C47—22no—
64D4×C47—22no—
65D4×C47—42no—
66D4×C47—44no—
67D4×C47—84no—
68D4×C47—84no—
69C2×C4×C47—21no—
70C2×C4×C47—42no—
71C2×C4×C47—82no—
72C2×C4×C47#12011yescharacteristic
73C2×Dic47—81no—
74Dic94—82no—
75C8×C47—42no—
76C8×C47C2×D812yes—
77C8×C47AGL(1,ℤ/8)12yes—
78C47⋊C8—42no—
79#12312—471nomaximal, Sylow
80C22×C47—42no—
81C22×C47—82no—
82C22×C47—162no—
83C22×C47—322no—
84C22×C47#45311yescharacteristic
85C4×C47—22no—
86C4×C47—42no—
87C4×C47#44712yes—
88Dic47—161no—
89#3292D4711yescharacteristic
90#2214—472no—
91#1709—471no—
92#1357—471no—
93#1335—472no—
94C2×C47—21no—
95C2×C47—82no—
96C2×C47—162no—
97C2×C47#161911yescharacteristic
98#576—22no—
99#510—942no—
100#494—471no—
101#493C2×D4711yescharacteristic
102#486—942no—
103#472—22no—
104#453—22no—
105#443C2×D4712yes—
106#364—472no—
107#362—942no—
108C82—471no—
109C47#1231211yescharacteristic, complemented, Sylow
110H(5,𝔽2)—22no—
111AGL(1,ℤ/8)—24no—
112AGL(1,ℤ/8)—44no—
113#135—42no—
114#135C22⋊D4712yes—
115C2×Q16—942no—
116C2×D8—22no—
117C2×D8—42no—
118#131—22no—
119C4⋊Q8—941no—
120#127C22⋊D4712yes—
121Q32—1882no—
122SD32—1882no—
123M32—942no—
124#108C22⋊D4712yes—
125#104—22no—
126#102—1884no—
127#100—22no—
128C4×C8—941no—
129C4×C8C22×D4711yescharacteristic
130#41—22no—
131#41—44no—
132C2×Q8—1881no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—82no—
137Q16—1882no—
138Q16—1882no—
139Q16#1062533412yes—
140SD16—22no—
141SD16—44no—
142D8—44no—
143D8—42no—
144D8—42no—
145D8#1062533412yes—
146M16—22no—
147M16—42no—
148C2×C8—42no—
149C2×C8—942no—
150C2×C8C2×C22⋊D4712yes—
151C4⋊C4—1882no—
152C42C2×C22⋊D4711yescharacteristic
153C16—1882no—
154C23—42no—
155C23—82no—
156Q8—22no—
157Q8—3762no—
158D4—22no—
159D4—42no—
160D4—44no—
161D4—84no—
162D4—84no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—3761no—
167C2×C4#1309399311yescharacteristic
168C8—42no—
169C8—1882no—
170C8C2×#1062533412yes—
171C8#1309405012yes—
172C22—42no—
173C22—82no—
174C22—162no—
175C22—322no—
176C22#1568929111yescharacteristic
177C4—22no—
178C4—42no—
179C4—7521no—
180C4#1568925712yes—
181C2—21no—
182C2—82no—
183C2—162no—
184C2#2024244211yescharacteristic
subgroupG/H
centerC2#20242442
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3292C2
socleC2×C47#1619
hypercenter#3292D47
hypocenterC47#12312
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3292

Complements

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

NH ≅ G/Ntype
1. #20245469C2semidirect
2. #20242993C2semidirect
3. #20241425C2semidirect
4. #20241361C2semidirect
7. #15689503C22semidirect
9. #15688944C22semidirect
11. C8×C47⋊C8C22semidirect
109. C47#12312semidirect

Maximal subgroups

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

subgroupindex
#202454692
#202429932
#202414252
#202413612
C47×#32922
#1231247