Finite Group Databasebeta

← #27143632

Subgroups — #27143632

Order 12416.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261171C211yesmaximal, characteristic
2#20260858C211yesmaximal, characteristic, complemented
3#20260557C211yesmaximal, characteristic, complemented
4#20260548C211yesmaximal, characteristic, complemented
5#20260544C211yesmaximal, characteristic, complemented
6#20260513C211yesmaximal, characteristic, complemented
7#442×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9#15690816—21no—
10#15690754C412yescomplemented
11#15690751C412yescomplemented
12#15690750C2211yescharacteristic
13#15690750C2211yescharacteristic
14#15690746C2211yescharacteristic
15#15690738—21no—
16C2×D8×C97C2211yescharacteristic
17C4×D4×C97—21no—
18#106×C97C2211yescharacteristic
19#102×C97—21no—
20#97×C97C2211yescharacteristic
21#13095511—21no—
22Dic388C2×C411yescharacteristic, complemented
23#13095500—21no—
24D776C2×C411yescharacteristic, complemented
25#13095495—21no—
26#13095494—41no—
27C8×D97—41no—
28#13095486—21no—
29#13095486C2×C412yescomplemented
30C2×D4×C97—21no—
31C22×C4×C97—41no—
32D8×C97—21no—
33D8×C97C2×C412yes—
34C2×C8×C97D411yescharacteristic
35C2×C8×C97C2311yescharacteristic
36C4⋊C4×C97—21no—
37C22⋊C4×C97—41no—
38C42×C97D411yescharacteristic
39C23×C97—41no—
40D4×C97—22no—
41D4×C97—41no—
42C2×C4×C97—21no—
43C2×C4×C97—41no—
44C2×C4×C97—82no—
45C2×C4×C97C2×D411yescharacteristic
46C22⋊D97—41no—
47D388—21no—
48C4×D97—41no—
49Dic194—21no—
50C8×C97—21no—
51C8×C97C22×C411yescharacteristic
52C8×C97#4111yescharacteristic
53#11715895—41no—
54C22×C97—42no—
55C22×C97—81no—
56C22×C97—81no—
57C22×C97AGL(1,ℤ/8)11yescharacteristic
58C2×D97—41no—
59C4×C97—41no—
60C4×C97—81no—
61C4×C97C4×D411yescharacteristic
62C4×C97AGL(1,ℤ/8)11yescharacteristic
63Dic97—41no—
64C2×C97—21no—
65C2×C97—82no—
66C2×C97#44211yescharacteristic
67D97—81no—
68AGL(1,ℤ/16)—971nomaximal, Sylow
69C97AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—971no—
71#444—971no—
72#442D9711yescharacteristic, complemented
73#365—971no—
74#361—971no—
75#360—971no—
76#347—971no—
77#135—971no—
78C2×D8C2×D9711yescharacteristic
79#131—1941no—
80C4×D4—21no—
81SD32—972no—
82D16—972no—
83M32—971no—
84M32—971no—
85#108—971no—
86#106C2×D9711yescharacteristic
87#104—1941no—
88#102—21no—
89#97C2×D9711yescharacteristic
90#41—1941no—
91C2×D4—21no—
92C22×C4—41no—
93Q16—971no—
94SD16—1941no—
95D8—21no—
96D8—971no—
97D8C4×D9712yes—
98M16—3881no—
99M16—1941no—
100C2×C8—3881no—
101C2×C8C22⋊D9711yescharacteristic
102C2×C8C22×D9711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42C22⋊D9711yescharacteristic
106C16—972no—
107C16—1941no—
108C23—41no—
109Q8—1941no—
110D4—22no—
111D4—41no—
112D4—3881no—
113D4—1941no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—3881no—
118C2×C4C2×C22⋊D9711yescharacteristic
119C8—21no—
120C8—3881no—
121C8C2×C4×D9711yescharacteristic
122C8#1309551111yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—3881no—
127C22#1569083611yescharacteristic
128C4—41no—
129C4—81no—
130C4—3881no—
131C4C4×C22⋊D9711yescharacteristic
132C4#1569083511yescharacteristic
133C2—21no—
134C2—82no—
135C2—7761no—
136C2#2026084511yescharacteristic
subgroupG/H
centerC2#20260845
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#442×C97C2
socleC2×C97#442
prime radical#20261171C2
cosocleC2×C8×C97C23
hypercenter#442D97
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #20260858C2semidirect
3. #20260557C2semidirect
4. #20260548C2semidirect
5. #20260544C2semidirect
6. #20260513C2semidirect
7. #442×C97C2semidirect
10. #15690754C4semidirect
11. #15690751C4semidirect
22. Dic388C2×C4semidirect
24. D776C2×C4semidirect
29. #13095486C2×C4semidirect
69. C97AGL(1,ℤ/16)semidirect
72. #442D97semidirect

Maximal subgroups

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

subgroupindex
#202611712
#202608582
#202605572
#202605482
#202605442
#202605132
#442×C972
AGL(1,ℤ/16)97