Finite Group Databasebeta

← #27143633

Subgroups — #27143633

Order 12416.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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#20261173C211yesmaximal, characteristic
2#20260856C211yesmaximal, characteristic, complemented
3#20260846C211yesmaximal, characteristic, complemented
4#20260547C211yesmaximal, characteristic, complemented
5#20260544C211yesmaximal, characteristic, complemented
6#20260512C211yesmaximal, characteristic, complemented
7#365×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9C2×D776C2211yescharacteristic
10#15690818—21no—
11#15690772—21no—
12#15690755C412yescomplemented
13D1552C412yescomplemented
14#15690750C2211yescharacteristic
15#15690736—21no—
16#15690730—21no—
17#15690720C2211yescharacteristic
18M32×C97C2211yescharacteristic
19#108×C97C2211yescharacteristic
20#106×C97C2211yescharacteristic
21#13095511—21no—
22C2×D388—21no—
23C2×C4×D97—41no—
24Dic388C2×C411yescharacteristic, complemented
25#13095500—21no—
26D776—21no—
27D776C2×C412yes—
28D776C2×C411yescharacteristic, complemented
29#13095494—41no—
30C2×#11715895D411yescharacteristic
31C8×D97—41no—
32#13095488—41no—
33C4×Dic97D411yescharacteristic
34#13095486—21no—
35M16×C97—21no—
36C2×C8×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C16×C97C2×C412yescomplemented
39C22×D97—41no—
40C2×C4×C97—41no—
41C2×C4×C97C2×D411yescharacteristic
42C22⋊D97—41no—
43C2×Dic97—21no—
44D388—22no—
45D388—21no—
46D388—41no—
47C4×D97—41no—
48C4×D97—82no—
49Dic194—21no—
50C8×C97—41no—
51C8×C97C22×C411yescharacteristic
52C8×C97#4111yescharacteristic
53#11715895—21no—
54C22×C97AGL(1,ℤ/8)11yescharacteristic
55C2×D97—41no—
56C2×D97—42no—
57C2×D97—81no—
58C2×D97—81no—
59C4×C97—81no—
60C4×C97C4×D411yescharacteristic
61C4×C97AGL(1,ℤ/8)11yescharacteristic
62Dic97—41no—
63Dic97—41no—
64C2×C97—21no—
65C2×C97#44211yescharacteristic
66D97—81no—
67D97—82no—
68AGL(1,ℤ/16)—971nomaximal, Sylow
69C97AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—971no—
71#444—971no—
72#442—971no—
73#365D9711yescharacteristic, complemented
74#361—971no—
75#360—971no—
76#347—971no—
77#135—971no—
78C2×D8—971no—
79#131—1941no—
80C4×D4—1941no—
81SD32—972no—
82D16—972no—
83M32—971no—
84M32C2×D9711yescharacteristic
85#108C2×D9711yescharacteristic
86#106C2×D9711yescharacteristic
87#104—1941no—
88#102—1941no—
89#97—971no—
90#41—1941no—
91C2×D4—1941no—
92C22×C4—3881no—
93Q16—971no—
94SD16—1941no—
95D8—972no—
96D8—971no—
97D8—1941no—
98M16—21no—
99M16—3881no—
100C2×C8—971no—
101C2×C8—3881no—
102C2×C8C22×D9711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—3881no—
105C42—971no—
106C16—1941no—
107C16C4×D9712yescomplemented
108C23—3881no—
109Q8—1941no—
110D4—3881no—
111D4—3881no—
112D4—1942no—
113D4—1941no—
114C2×C4—41no—
115C2×C4—7762no—
116C2×C4—3881no—
117C2×C4—1941no—
118C2×C4D4×D9711yescharacteristic
119C8—41no—
120C8—1941no—
121C8C2×C4×D9711yescharacteristic
122C8#1309551211yescharacteristic
123C22—7761no—
124C22—7761no—
125C22—3881no—
126C22—3882no—
127C22#1569083711yescharacteristic
128C4—81no—
129C4—3881no—
130C4—3881no—
131C4#1569077211yescharacteristic
132C4#1569083711yescharacteristic
133C2—21no—
134C2—7762no—
135C2—7761no—
136C2#2026084611yescharacteristic
subgroupG/H
centerC2#20260846
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#365×C97C2
socleC2×C97#442
prime radical#20261173C2
cosocleC2×C8×C97C23
hypercenter#365D97
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #365

Complements

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

NH ≅ G/Ntype
2. #20260856C2semidirect
3. #20260846C2semidirect
4. #20260547C2semidirect
5. #20260544C2semidirect
6. #20260512C2semidirect
7. #365×C97C2semidirect
12. #15690755C4semidirect
13. D1552C4semidirect
24. Dic388C2×C4semidirect
28. D776C2×C4semidirect
38. C16×C97C2×C4semidirect
69. C97AGL(1,ℤ/16)semidirect
73. #365D97semidirect
107. C16C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202611732
#202608562
#202608462
#202605472
#202605442
#202605122
#365×C972
AGL(1,ℤ/16)97