Finite Group Databasebeta

← #27143624

Subgroups — #27143624

Order 12416.

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

normal subgroups only

This group has 142 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#20261175C211yesmaximal, characteristic
2#20260857C211yesmaximal, characteristic
3#20260841C211yesmaximal, characteristic, complemented
4#20260557C211yesmaximal, characteristic, complemented
5#20260552C211yesmaximal, characteristic, complemented
6#20260512C211yesmaximal, characteristic, complemented
7#360×C97C211yesmaximal, characteristic
8C2×Dic388C2211yescharacteristic
9#15690817—21no—
10#15690775—21no—
11#15690756C412yescomplemented
12#15690754C412yescomplemented
13#15690750C2211yescharacteristic
14#15690746C2211yescharacteristic
15#15690737—21no—
16#15690733—21no—
17#15690720C2211yescharacteristic
18#135×C97C2211yescharacteristic
19M32×C97C2211yescharacteristic
20#106×C97C2211yescharacteristic
21C2×Dic194—21no—
22Dic388—21no—
23Dic388C2×C412yes—
24#13095495—21no—
25#13095495—41no—
26C2×#11715895—41no—
27C2×#11715895D411yescharacteristic
28#13095490—41no—
29C4×Dic97—41no—
30C4×Dic97D411yescharacteristic
31#13095486C2×C412yes—
32#41×C97—21no—
33Q16×C97C2×C411yescharacteristic
34SD16×C97—21no—
35D8×C97C2×C411yescharacteristic
36C2×C8×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C16×C97—21no—
39Q8×C97—21no—
40D4×C97—21no—
41D4×C97—41no—
42C2×C4×C97—41no—
43C2×C4×C97—41no—
44C2×C4×C97C2×D411yescharacteristic
45C2×Dic97—21no—
46C2×Dic97—41no—
47Dic194—22no—
48Dic194—41no—
49C8×C97C22×C411yescharacteristic
50C8×C97#4111yescharacteristic
51#11715895—21no—
52#11715895—41no—
53C22×C97—41no—
54C22×C97AGL(1,ℤ/8)11yescharacteristic
55C4×C97—41no—
56C4×C97—81no—
57C4×C97C4×D411yescharacteristic
58C4×C97AGL(1,ℤ/8)11yescharacteristic
59Dic97—41no—
60Dic97—42no—
61C2×C97—21no—
62C2×C97—81no—
63C2×C97#44211yescharacteristic
64#2180—971nomaximal, Sylow
65C97#218011yescharacteristic, complemented, Sylow
66#510—971no—
67#444—971no—
68#441—971no—
69#365—971no—
70#362—971no—
71#360D9711yescharacteristic
72#347—971no—
73#135C2×D9711yescharacteristic
74C2×Q16—971no—
75#131—1941no—
76C4×Q8—1941no—
77Q32—972no—
78SD32—972no—
79M32—971no—
80M32C2×D9711yescharacteristic
81#108—971no—
82#106C2×D9711yescharacteristic
83#104—1941no—
84#103—1941no—
85#97—971no—
86#41—21no—
87C2×Q8—1941no—
88Q16—972no—
89Q16—1941no—
90Q16C4×D9711yescharacteristic
91SD16—21no—
92D8C4×D9711yescharacteristic
93M16—3881no—
94M16—1941no—
95C2×C8—971no—
96C2×C8—3881no—
97C2×C8C22×D9711yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—3881no—
100C42—971no—
101C42—3881no—
102C16—21no—
103C16—972no—
104Q8—21no—
105Q8—3881no—
106Q8—1942no—
107D4—21no—
108D4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—3881no—
112C2×C4—1941no—
113C2×C4D4×D9711yescharacteristic
114C8—3881no—
115C8—1941no—
116C8C2×C4×D9711yescharacteristic
117C8#1309551011yescharacteristic
118C22—41no—
119C22#1569083411yescharacteristic
120C4—41no—
121C4—81no—
122C4—3881no—
123C4—3882no—
124C4#1569077011yescharacteristic
125C4#1569083711yescharacteristic
126C2—21no—
127C2—81no—
128C2#2026084411yescharacteristic
subgroupG/H
centerC2#20260844
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#360×C97C2
socleC2×C97#442
prime radical#135×C97C22
cosocleC2×C8×C97C23
hypercenter#360D97
hypocenterC97#2180
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
3. #20260841C2semidirect
4. #20260557C2semidirect
5. #20260552C2semidirect
6. #20260512C2semidirect
11. #15690756C4semidirect
12. #15690754C4semidirect
65. C97#2180semidirect

Maximal subgroups

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

subgroupindex
#202611752
#202608572
#202608412
#202605572
#202605522
#202605122
#360×C972
#218097