Finite Group Databasebeta

← #27143626

Subgroups — #27143626

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#20261178C211yesmaximal, characteristic
2#20260856C211yesmaximal, characteristic
3#20260841C211yesmaximal, characteristic, complemented
4#20260550C211yesmaximal, characteristic, complemented
5#20260544C211yesmaximal, characteristic
6#20260512C211yesmaximal, characteristic, complemented
7#365×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9C2×Dic388C2211yescharacteristic
10#15690818—21no—
11#15690775—21no—
12Dic776C412yescomplemented
13#15690755C412yescomplemented
14#15690750C2211yescharacteristic
15#15690736—21no—
16#15690733—21no—
17#15690720C2211yescharacteristic
18M32×C97C2211yescharacteristic
19#108×C97C2211yescharacteristic
20#106×C97C2211yescharacteristic
21#13095511—21no—
22C2×Dic194—21no—
23Dic388—21no—
24Dic388C2×C412yes—
25Dic388C2×C411yescharacteristic
26#13095500—21no—
27D776C2×C411yescharacteristic
28#13095494—41no—
29C2×#11715895D411yescharacteristic
30C8×D97—41no—
31#13095490—41no—
32C4×Dic97—41no—
33C4×Dic97D411yescharacteristic
34#13095486—21no—
35M16×C97—21no—
36C2×C8×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C16×C97C2×C412yes—
39C2×C4×C97—41no—
40C2×C4×C97C2×D411yescharacteristic
41C22⋊D97—41no—
42C2×Dic97—21no—
43C2×Dic97—41no—
44D388—21no—
45C4×D97—41no—
46Dic194—21no—
47Dic194—22no—
48Dic194—41no—
49C8×C97—41no—
50C8×C97C22×C411yescharacteristic
51C8×C97#4111yescharacteristic
52#11715895—21no—
53C22×C97AGL(1,ℤ/8)11yescharacteristic
54C2×D97—41no—
55C4×C97—81no—
56C4×C97C4×D411yescharacteristic
57C4×C97AGL(1,ℤ/8)11yescharacteristic
58Dic97—41no—
59Dic97—42no—
60Dic97—41no—
61C2×C97—21no—
62C2×C97#44211yescharacteristic
63D97—81no—
64#2180—971nomaximal, Sylow
65C97#218011yescharacteristic, complemented, Sylow
66#510—971no—
67#444—971no—
68#441—971no—
69#365D9711yescharacteristic, complemented
70#362—971no—
71#360—971no—
72#347—971no—
73#135—971no—
74C2×Q16—971no—
75#131—1941no—
76C4×Q8—1941no—
77Q32—972no—
78SD32—972no—
79M32—971no—
80M32C2×D9711yescharacteristic
81#108C2×D9711yescharacteristic
82#106C2×D9711yescharacteristic
83#104—1941no—
84#103—1941no—
85#97—971no—
86#41—1941no—
87C2×Q8—1941no—
88Q16—972no—
89Q16—971no—
90Q16—1941no—
91SD16—1941no—
92D8—971no—
93M16—21no—
94M16—3881no—
95C2×C8—971no—
96C2×C8—3881no—
97C2×C8C22×D9711yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—3881no—
100C42—971no—
101C42—3881no—
102C16—1941no—
103C16C4×D9712yes—
104Q8—3881no—
105Q8—1941no—
106Q8—1942no—
107D4—3881no—
108D4—1941no—
109C2×C4—41no—
110C2×C4—3881no—
111C2×C4—3881no—
112C2×C4—1941no—
113C2×C4D4×D9711yescharacteristic
114C8—41no—
115C8—1941no—
116C8C2×C4×D9711yescharacteristic
117C8#1309551211yescharacteristic
118C22—3881no—
119C22#1569083711yescharacteristic
120C4—81no—
121C4—3881no—
122C4—3881no—
123C4—3882no—
124C4#1569077211yescharacteristic
125C4#1569083711yescharacteristic
126C2—21no—
127C2—7761no—
128C2#2026084611yescharacteristic
subgroupG/H
centerC2#20260846
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#365×C97C2
socleC2×C97#442
prime radical#15690832C22
cosocleC2×C8×C97C23
hypercenter#365D97
hypocenterC97#2180
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
3. #20260841C2semidirect
4. #20260550C2semidirect
6. #20260512C2semidirect
7. #365×C97C2semidirect
12. Dic776C4semidirect
13. #15690755C4semidirect
65. C97#2180semidirect
69. #365D97semidirect

Maximal subgroups

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

subgroupindex
#202611782
#202608562
#202608412
#202605502
#202605442
#202605122
#365×C972
#218097