Finite Group Databasebeta

← #26130678

Subgroups — #26130678

Order 12032.

196 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 55 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#20245561C211yesmaximal, characteristic, complemented
2#20245461C211yesmaximal, characteristic
3#20245231C211yesmaximal, characteristic, complemented
4#20241456C211yesmaximal, characteristic, complemented
5#20241449C211yesmaximal, characteristic, complemented
6#20241425C211yesmaximal, characteristic, complemented
7C47×#2244C211yesmaximal, characteristic, complemented
8#15689559—21no—
9#15689539—21no—
10#15689241C2211yescharacteristic
11#15688945—21no—
12#15688944C2211yescharacteristic
13#15688944C2211yescharacteristic
14#15688944C2211yescharacteristic
15#15688936—21no—
16#15688932—21no—
17#15688906—21no—
18C47×#499C2211yescharacteristic
19C47×#493C2211yescharacteristic
20C47×#357—21no—
21C47×#357—21no—
22C47×#346C2211yescharacteristic
23#13094046D411yescharacteristic
24#13094030—21no—
25#13093968—41no—
26#13093965—41no—
27#13093964—21no—
28#13093964—21no—
29#13093964—21no—
30#13093963—41no—
31C16×D47—41no—
32#13093960D411yescharacteristic
33#13093952—21no—
34C2×D8×C47—21no—
35C2×D8×C47—21no—
36#127×C47—21no—
37#125×C47—21no—
38C2×C16×C47D411yescharacteristic
39C2×C16×C47D411yescharacteristic
40#107×C47—21no—
41#106×C47—21no—
42C4×C8×C47C2311yescharacteristic
43C2×D4×C47—41no—
44C2×D4×C47—81no—
45#10625350—21no—
46D8×C47—41no—
47D8×C47—41no—
48D8×C47—41no—
49D8×C47—41no—
50C2×C8×C47C2×D411yescharacteristic
51C2×C8×C47C2×D411yescharacteristic
52C4⋊C4×C47—41no—
53C4⋊C4×C47—81no—
54C42×C47#4111yescharacteristic
55#10625329—21no—
56Dic188SD1611yescharacteristic
57D376SD1611yescharacteristic
58#10625325—21no—
59#10625324—41no—
60C8×D47—41no—
61C16×C47—21no—
62C16×C47—21no—
63C47⋊C16—41no—
64C47⋊C16—41no—
65C47⋊C16—41no—
66C23×C47—81no—
67D4×C47—81no—
68D4×C47—81no—
69D4×C47—162no—
70C2×C4×C47—21no—
71C2×C4×C47—81no—
72C2×C4×C47#12111yescharacteristic
73C22⋊D47—41no—
74D188—21no—
75C4×D47—41no—
76Dic94—21no—
77C8×C47C2×SD1611yescharacteristic
78C8×C47#13511yescharacteristic
79C8×C47#13511yescharacteristic
80C8×C47AGL(1,ℤ/8)11yescharacteristic
81C47⋊C8—41no—
82#12465—471nomaximal, Sylow
83C22×C47—162no—
84C22×C47—161no—
85C22×C47#47211yescharacteristic
86C2×D47—41no—
87C4×C47—41no—
88C4×C47—161no—
89C4×C47#46311yescharacteristic
90C4×C47#46511yescharacteristic
91Dic47—41no—
92#2244D4711yescharacteristic, complemented
93#2231—471no—
94#2213—471no—
95#2171—471no—
96#1369—471no—
97#1366—471no—
98#1357—471no—
99C2×C47—21no—
100C2×C47—321no—
101C2×C47#167711yescharacteristic
102D47—81no—
103#509—941no—
104#504—941no—
105#499C2×D4711yescharacteristic
106#493C2×D4711yescharacteristic
107#443—471no—
108#365—941no—
109#364—471no—
110#364—471no—
111#364—471no—
112#361—941no—
113#360—941no—
114#357—21no—
115#357—21no—
116#350—941no—
117#346C2×D4711yescharacteristic
118C47#1246511yescharacteristic, complemented, Sylow
119#135—471no—
120C2×D8—21no—
121C2×D8—21no—
122#131—941no—
123#127—21no—
124#125—21no—
125SD32—1881no—
126D16—1881no—
127M32—941no—
128M32—941no—
129M32—941no—
130M32—1881no—
131C2×C16—1881no—
132C2×C16C22⋊D4711yescharacteristic
133C2×C16C22⋊D4711yescharacteristic
134#108—471no—
135#107—21no—
136#106—21no—
137#104—941no—
138C4×C8C22×D4711yescharacteristic
139#41—941no—
140C2×D4—41no—
141C2×D4—81no—
142Q16—471no—
143SD16—941no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8—471no—
149M16—941no—
150M16—1881no—
151C2×C8—1881no—
152C2×C8D4×D4711yescharacteristic
153C2×C8C2×C22⋊D4711yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1062535211yescharacteristic
157C16—21no—
158C16—21no—
159C16—1881no—
160C16—1881no—
161C16—1881no—
162C23—81no—
163Q8—941no—
164D4—81no—
165D4—81no—
166D4—162no—
167D4—941no—
168D4—1881no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—1881no—
172C2×C4#1309399511yescharacteristic
173C8—1881no—
174C8SD16×D4711yescharacteristic
175C8#1309404411yescharacteristic
176C8#1309404511yescharacteristic
177C8#1309404911yescharacteristic
178C22—162no—
179C22—161no—
180C22—1881no—
181C22#1568939011yescharacteristic
182C4—41no—
183C4—161no—
184C4—1881no—
185C4#1568933511yescharacteristic
186C4#1568934911yescharacteristic
187C2—21no—
188C2—321no—
189C2—3761no—
190C2#2024279411yescharacteristic
subgroupG/H
centerC2#20242794
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#2244C2
socleC2×C47#1677
prime radical#20245461C2
hypercenter#2244D47
hypocenterC47#12465
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #20245561C2semidirect
3. #20245231C2semidirect
4. #20241456C2semidirect
5. #20241449C2semidirect
6. #20241425C2semidirect
7. C47×#2244C2semidirect
92. #2244D47semidirect
118. C47#12465semidirect

Maximal subgroups

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

subgroupindex
#202455612
#202454612
#202452312
#202414562
#202414492
#202414252
C47×#22442
#1246547