Finite Group Databasebeta

← #26119383

Subgroups — #26119383

Order 12032.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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#20244216C211yesmaximal, characteristic
2#20241947C212yesmaximal
3#20241246C212yesmaximal, complemented
4#20241221C211yesmaximal, characteristic, complemented
5C47×#2864C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689176C2212yes—
8#15689147C2211yescharacteristic
9C2×#13093952—22no—
10#15688856—24no—
11C47×#389—22no—
12C47×#387—24no—
13C47×#382C2212yes—
14C43×C47C2211yescharacteristic
15C2×#10625350—21no—
16C2×#10625329—21no—
17#13094024Q811yescharacteristic
18#13094021Q811yescharacteristic
19#13094012—21no—
20#13093998—22no—
21C4×D188—22no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24#13093952—44no—
25C22⋊(C47⋊C8)—22no—
26C4×C47⋊C8—22no—
27C2×C4⋊C4×C47—24no—
28C2×C4⋊C4×C47—22no—
29C2×C42×C47—22no—
30C2×C42×C47D412yes—
31C2×C42×C47C2311yescharacteristic
32#95×C47—24no—
33#95×C47—22no—
34#95×C47D414yes—
35C22×C4×C47—22no—
36C22×C4×C47—24no—
37C22×C4×C47C2×D412yes—
38C22×C4×C47C2×D411yescharacteristic
39C2×C22⋊D47—41no—
40#10625350—42no—
41#10625350—42no—
42C2×D188—21no—
43C2×D188—22no—
44C2×C4×D47—41no—
45C2×Dic94—21no—
46C4⋊C4×C47—48no—
47C4⋊C4×C47—42no—
48C4⋊C4×C47—42no—
49C42×C47—22no—
50C42×C47—22no—
51C42×C47—44no—
52C42×C47C2×Q811yescharacteristic
53C42×C47#4111yescharacteristic
54C42×C47#4112yes—
55(C2×D47)⋊C4—42no—
56C4⋊Dic47—22no—
57#10625329—42no—
58C2×C47⋊C8—42no—
59C2×C47⋊C8—41no—
60C23×C47#12011yescharacteristic
61C22×D47—41no—
62C2×C4×C47—24no—
63C2×C4×C47—42no—
64C2×C4×C47—42no—
65C2×C4×C47—42no—
66C2×C4×C47—48no—
67C2×C4×C47—44no—
68C2×C4×C47C22⋊Q812yes—
69C2×C4×C47C22⋊Q811yescharacteristic
70C2×C4×C47#12412yes—
71C2×C4×C47#12411yescharacteristic
72C2×C4×C47#13512yes—
73C2×C4×C47AGL(1,ℤ/8)12yes—
74C22⋊D47—82no—
75C2×Dic47—41no—
76D188—42no—
77D188—82no—
78C4×D47—81no—
79C4×D47—81no—
80Dic94—42no—
81C47⋊C8—82no—
82C47⋊C8—81no—
83#10366—471nomaximal, Sylow
84C22×C47—22no—
85C22×C47#39311yescharacteristic
86C22×C47#44912yes—
87C2×D47—81no—
88C2×D47—82no—
89C4×C47—22no—
90C4×C47—22no—
91C4×C47—44no—
92C4×C47—84no—
93C4×C47#47812yes—
94C4×C47#48811yescharacteristic
95C4×C47#49011yescharacteristic
96Dic47—81no—
97#2864D4711yescharacteristic, complemented
98#1962—471no—
99#1517—472no—
100#1286—472no—
101#1276—471no—
102C2×C47—22no—
103C2×C47#202511yescharacteristic
104C2×C47#203112yes—
105D47—161no—
106#532—471no—
107#431—472no—
108#423—471no—
109C2×#104—942no—
110#389—22no—
111#387—24no—
112#382C2×D4712yes—
113C43C2×D4711yescharacteristic
114#327—944no—
115C47#1036611yescharacteristic, complemented, Sylow
116C2×#41—941no—
117C2×M16—941no—
118C4⋊Q8—471no—
119#127—471no—
120#124—941no—
121#121—942no—
122C4×D4—942no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C22⋊D4712yes—
127C2×C42C22×D4711yescharacteristic
128C4⋊C8—942no—
129C4⋊C8—942no—
130#104—1884no—
131C22⋊C8—942no—
132C4×C8—942no—
133#95—24no—
134#95—22no—
135#95C22⋊D4714yes—
136#41—1882no—
137#41—1882no—
138C2×Q8—941no—
139C2×D4—942no—
140C2×D4—941no—
141C2×D4—1881no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—1881no—
145C22×C4C2×C22⋊D4712yes—
146C22×C4C2×C22⋊D4711yescharacteristic
147M16—1882no—
148C2×C8—1882no—
149C2×C8—1881no—
150C4⋊C4—48no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—942no—
154C22⋊C4—1882no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42Q8×D4711yescharacteristic
159C42#1062535411yescharacteristic
160C42#1062535412yes—
161C23—1881no—
162C23#1309399311yescharacteristic
163Q8—1882no—
164D4—3762no—
165D4—3762no—
166D4—1882no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—3761no—
174C2×C4—3761no—
175C2×C4—1881no—
176C2×C4#1309400012yes—
177C2×C4#1309400011yescharacteristic
178C2×C4#1309401312yes—
179C2×C4#1309401311yescharacteristic
180C2×C4#1309404412yes—
181C2×C4#1309404912yes—
182C8—3762no—
183C8—3761no—
184C22—22no—
185C22—3761no—
186C22—3762no—
187C22#1568903411yescharacteristic
188C22#1568926812yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—3761no—
194C4#1568942012yes—
195C4#1568948011yescharacteristic
196C4#1568948911yescharacteristic
197C2—22no—
198C2—7521no—
199C2#2024453111yescharacteristic
200C2#2024455712yes—
subgroupG/H
centerC22#15689034
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2864C2
socleC22×C47#393
prime radical#15689673C22
hypercenter#2864D47
hypocenterC47#10366
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
3. #20241246C2semidirect
4. #20241221C2semidirect
5. C47×#2864C2semidirect
97. #2864D47semidirect
115. C47#10366semidirect

Maximal subgroups

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

subgroupindex
#202442162
#202419472
#202412462
#202412212
C47×#28642
#1036647